3.307 \(\int \frac {\tan ^6(c+d x)}{(a+b \sec (c+d x))^2} \, dx\)

Optimal. Leaf size=200 \[ -\frac {a \left (4 a^2-5 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{b^5 d}+\frac {2 (a-b)^{3/2} (a+b)^{3/2} \left (4 a^2+b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^2 b^5 d}+\frac {\left (3 a^2-2 b^2\right ) \tan (c+d x)}{b^4 d}+\frac {\left (a^2-b^2\right )^2 \sin (c+d x)}{a b^4 d (a \cos (c+d x)+b)}-\frac {x}{a^2}-\frac {a \tan (c+d x) \sec (c+d x)}{b^3 d}+\frac {\tan ^3(c+d x)}{3 b^2 d} \]

[Out]

-x/a^2-a*(4*a^2-5*b^2)*arctanh(sin(d*x+c))/b^5/d+2*(a-b)^(3/2)*(a+b)^(3/2)*(4*a^2+b^2)*arctanh((a-b)^(1/2)*tan
(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^2/b^5/d+(a^2-b^2)^2*sin(d*x+c)/a/b^4/d/(b+a*cos(d*x+c))+(3*a^2-2*b^2)*tan(d*x+c
)/b^4/d-a*sec(d*x+c)*tan(d*x+c)/b^3/d+1/3*tan(d*x+c)^3/b^2/d

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Rubi [A]  time = 0.43, antiderivative size = 283, normalized size of antiderivative = 1.42, number of steps used = 16, number of rules used = 10, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3898, 2897, 2664, 12, 2659, 208, 3770, 3767, 8, 3768} \[ \frac {3 \left (a^2-b^2\right ) \tan (c+d x)}{b^4 d}-\frac {2 a \left (2 a^2-3 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{b^5 d}+\frac {\left (a^2-b^2\right )^2 \sin (c+d x)}{a b^4 d (a \cos (c+d x)+b)}-\frac {2 (a-b)^{3/2} (a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^2 b^3 d}+\frac {4 (a-b)^{3/2} (a+b)^{3/2} \left (2 a^2+b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^2 b^5 d}-\frac {x}{a^2}-\frac {a \tanh ^{-1}(\sin (c+d x))}{b^3 d}-\frac {a \tan (c+d x) \sec (c+d x)}{b^3 d}+\frac {\tan ^3(c+d x)}{3 b^2 d}+\frac {\tan (c+d x)}{b^2 d} \]

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^6/(a + b*Sec[c + d*x])^2,x]

[Out]

-(x/a^2) - (a*ArcTanh[Sin[c + d*x]])/(b^3*d) - (2*a*(2*a^2 - 3*b^2)*ArcTanh[Sin[c + d*x]])/(b^5*d) - (2*(a - b
)^(3/2)*(a + b)^(3/2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*b^3*d) + (4*(a - b)^(3/2)*(a +
b)^(3/2)*(2*a^2 + b^2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*b^5*d) + ((a^2 - b^2)^2*Sin[
c + d*x])/(a*b^4*d*(b + a*Cos[c + d*x])) + Tan[c + d*x]/(b^2*d) + (3*(a^2 - b^2)*Tan[c + d*x])/(b^4*d) - (a*Se
c[c + d*x]*Tan[c + d*x])/(b^3*d) + Tan[c + d*x]^3/(3*b^2*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 2664

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(a + b*Sin[c + d*x])^(n +
1))/(d*(n + 1)*(a^2 - b^2)), x] + Dist[1/((n + 1)*(a^2 - b^2)), Int[(a + b*Sin[c + d*x])^(n + 1)*Simp[a*(n + 1
) - b*(n + 2)*Sin[c + d*x], x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && Integer
Q[2*n]

Rule 2897

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(
m_), x_Symbol] :> Int[ExpandTrig[(d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m*(1 - sin[e + f*x]^2)^(p/2), x], x]
/; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[m, 2*n, p/2] && (LtQ[m, -1] || (EqQ[m, -1] && G
tQ[p, 0]))

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3898

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Int[(Cos[c + d*x]^
m*(b + a*Sin[c + d*x])^n)/Sin[c + d*x]^(m + n), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[
n] && IntegerQ[m] && (IntegerQ[m/2] || LeQ[m, 1])

Rubi steps

\begin {align*} \int \frac {\tan ^6(c+d x)}{(a+b \sec (c+d x))^2} \, dx &=\int \frac {\sin ^2(c+d x) \tan ^4(c+d x)}{(b+a \cos (c+d x))^2} \, dx\\ &=\int \left (-\frac {1}{a^2}+\frac {\left (a^2-b^2\right )^3}{a^2 b^4 (b+a \cos (c+d x))^2}+\frac {2 \left (2 a^6-3 a^4 b^2+b^6\right )}{a^2 b^5 (b+a \cos (c+d x))}+\frac {2 \left (-2 a^3+3 a b^2\right ) \sec (c+d x)}{b^5}-\frac {3 \left (-a^2+b^2\right ) \sec ^2(c+d x)}{b^4}-\frac {2 a \sec ^3(c+d x)}{b^3}+\frac {\sec ^4(c+d x)}{b^2}\right ) \, dx\\ &=-\frac {x}{a^2}-\frac {(2 a) \int \sec ^3(c+d x) \, dx}{b^3}+\frac {\int \sec ^4(c+d x) \, dx}{b^2}-\frac {\left (2 a \left (2 a^2-3 b^2\right )\right ) \int \sec (c+d x) \, dx}{b^5}+\frac {\left (3 \left (a^2-b^2\right )\right ) \int \sec ^2(c+d x) \, dx}{b^4}+\frac {\left (a^2-b^2\right )^3 \int \frac {1}{(b+a \cos (c+d x))^2} \, dx}{a^2 b^4}+\frac {\left (2 \left (2 a^6-3 a^4 b^2+b^6\right )\right ) \int \frac {1}{b+a \cos (c+d x)} \, dx}{a^2 b^5}\\ &=-\frac {x}{a^2}-\frac {2 a \left (2 a^2-3 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{b^5 d}+\frac {\left (a^2-b^2\right )^2 \sin (c+d x)}{a b^4 d (b+a \cos (c+d x))}-\frac {a \sec (c+d x) \tan (c+d x)}{b^3 d}-\frac {a \int \sec (c+d x) \, dx}{b^3}-\frac {\left (a^2-b^2\right )^2 \int \frac {b}{b+a \cos (c+d x)} \, dx}{a^2 b^4}-\frac {\operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{b^2 d}-\frac {\left (3 \left (a^2-b^2\right )\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{b^4 d}+\frac {\left (4 \left (2 a^6-3 a^4 b^2+b^6\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+b+(-a+b) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^2 b^5 d}\\ &=-\frac {x}{a^2}-\frac {a \tanh ^{-1}(\sin (c+d x))}{b^3 d}-\frac {2 a \left (2 a^2-3 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{b^5 d}+\frac {4 (a-b)^{3/2} (a+b)^{3/2} \left (2 a^2+b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^2 b^5 d}+\frac {\left (a^2-b^2\right )^2 \sin (c+d x)}{a b^4 d (b+a \cos (c+d x))}+\frac {\tan (c+d x)}{b^2 d}+\frac {3 \left (a^2-b^2\right ) \tan (c+d x)}{b^4 d}-\frac {a \sec (c+d x) \tan (c+d x)}{b^3 d}+\frac {\tan ^3(c+d x)}{3 b^2 d}-\frac {\left (a^2-b^2\right )^2 \int \frac {1}{b+a \cos (c+d x)} \, dx}{a^2 b^3}\\ &=-\frac {x}{a^2}-\frac {a \tanh ^{-1}(\sin (c+d x))}{b^3 d}-\frac {2 a \left (2 a^2-3 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{b^5 d}+\frac {4 (a-b)^{3/2} (a+b)^{3/2} \left (2 a^2+b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^2 b^5 d}+\frac {\left (a^2-b^2\right )^2 \sin (c+d x)}{a b^4 d (b+a \cos (c+d x))}+\frac {\tan (c+d x)}{b^2 d}+\frac {3 \left (a^2-b^2\right ) \tan (c+d x)}{b^4 d}-\frac {a \sec (c+d x) \tan (c+d x)}{b^3 d}+\frac {\tan ^3(c+d x)}{3 b^2 d}-\frac {\left (2 \left (a^2-b^2\right )^2\right ) \operatorname {Subst}\left (\int \frac {1}{a+b+(-a+b) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^2 b^3 d}\\ &=-\frac {x}{a^2}-\frac {a \tanh ^{-1}(\sin (c+d x))}{b^3 d}-\frac {2 a \left (2 a^2-3 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{b^5 d}-\frac {2 (a-b)^{3/2} (a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^2 b^3 d}+\frac {4 (a-b)^{3/2} (a+b)^{3/2} \left (2 a^2+b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^2 b^5 d}+\frac {\left (a^2-b^2\right )^2 \sin (c+d x)}{a b^4 d (b+a \cos (c+d x))}+\frac {\tan (c+d x)}{b^2 d}+\frac {3 \left (a^2-b^2\right ) \tan (c+d x)}{b^4 d}-\frac {a \sec (c+d x) \tan (c+d x)}{b^3 d}+\frac {\tan ^3(c+d x)}{3 b^2 d}\\ \end {align*}

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Mathematica [B]  time = 6.25, size = 865, normalized size = 4.32 \[ \frac {(b+a \cos (c+d x))^2 \sin \left (\frac {1}{2} (c+d x)\right ) \sec ^2(c+d x)}{6 b^2 d (a+b \sec (c+d x))^2 \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {(b+a \cos (c+d x))^2 \left (9 a^2 \sin \left (\frac {1}{2} (c+d x)\right )-7 b^2 \sin \left (\frac {1}{2} (c+d x)\right )\right ) \sec ^2(c+d x)}{3 b^4 d (a+b \sec (c+d x))^2 \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {(b+a \cos (c+d x))^2 \left (9 a^2 \sin \left (\frac {1}{2} (c+d x)\right )-7 b^2 \sin \left (\frac {1}{2} (c+d x)\right )\right ) \sec ^2(c+d x)}{3 b^4 d (a+b \sec (c+d x))^2 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {(b+a \cos (c+d x)) \left (\sin (c+d x) a^4-2 b^2 \sin (c+d x) a^2+b^4 \sin (c+d x)\right ) \sec ^2(c+d x)}{a b^4 d (a+b \sec (c+d x))^2}-\frac {(c+d x) (b+a \cos (c+d x))^2 \sec ^2(c+d x)}{a^2 d (a+b \sec (c+d x))^2}-\frac {2 \left (b^2-a^2\right )^2 \left (4 a^2+b^2\right ) \tanh ^{-1}\left (\frac {(b-a) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right ) (b+a \cos (c+d x))^2 \sec ^2(c+d x)}{a^2 b^5 \sqrt {a^2-b^2} d (a+b \sec (c+d x))^2}+\frac {\left (4 a^3-5 a b^2\right ) (b+a \cos (c+d x))^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sec ^2(c+d x)}{b^5 d (a+b \sec (c+d x))^2}+\frac {\left (5 a b^2-4 a^3\right ) (b+a \cos (c+d x))^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sec ^2(c+d x)}{b^5 d (a+b \sec (c+d x))^2}+\frac {(b-6 a) (b+a \cos (c+d x))^2 \sec ^2(c+d x)}{12 b^3 d (a+b \sec (c+d x))^2 \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {(6 a-b) (b+a \cos (c+d x))^2 \sec ^2(c+d x)}{12 b^3 d (a+b \sec (c+d x))^2 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {(b+a \cos (c+d x))^2 \sin \left (\frac {1}{2} (c+d x)\right ) \sec ^2(c+d x)}{6 b^2 d (a+b \sec (c+d x))^2 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^6/(a + b*Sec[c + d*x])^2,x]

[Out]

-(((c + d*x)*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^2)/(a^2*d*(a + b*Sec[c + d*x])^2)) - (2*(-a^2 + b^2)^2*(4*a^2
+ b^2)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^2)/(a^2*b^5*S
qrt[a^2 - b^2]*d*(a + b*Sec[c + d*x])^2) + ((4*a^3 - 5*a*b^2)*(b + a*Cos[c + d*x])^2*Log[Cos[(c + d*x)/2] - Si
n[(c + d*x)/2]]*Sec[c + d*x]^2)/(b^5*d*(a + b*Sec[c + d*x])^2) + ((-4*a^3 + 5*a*b^2)*(b + a*Cos[c + d*x])^2*Lo
g[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sec[c + d*x]^2)/(b^5*d*(a + b*Sec[c + d*x])^2) + ((-6*a + b)*(b + a*Cos
[c + d*x])^2*Sec[c + d*x]^2)/(12*b^3*d*(a + b*Sec[c + d*x])^2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + ((b +
a*Cos[c + d*x])^2*Sec[c + d*x]^2*Sin[(c + d*x)/2])/(6*b^2*d*(a + b*Sec[c + d*x])^2*(Cos[(c + d*x)/2] - Sin[(c
+ d*x)/2])^3) + ((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*Sin[(c + d*x)/2])/(6*b^2*d*(a + b*Sec[c + d*x])^2*(Cos
[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + ((6*a - b)*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^2)/(12*b^3*d*(a + b*Sec[
c + d*x])^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + ((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*(9*a^2*Sin[(c +
d*x)/2] - 7*b^2*Sin[(c + d*x)/2]))/(3*b^4*d*(a + b*Sec[c + d*x])^2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + ((
b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*(9*a^2*Sin[(c + d*x)/2] - 7*b^2*Sin[(c + d*x)/2]))/(3*b^4*d*(a + b*Sec[c
+ d*x])^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + ((b + a*Cos[c + d*x])*Sec[c + d*x]^2*(a^4*Sin[c + d*x] - 2*
a^2*b^2*Sin[c + d*x] + b^4*Sin[c + d*x]))/(a*b^4*d*(a + b*Sec[c + d*x])^2)

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fricas [B]  time = 0.92, size = 843, normalized size = 4.22 \[ \left [-\frac {6 \, a b^{5} d x \cos \left (d x + c\right )^{4} + 6 \, b^{6} d x \cos \left (d x + c\right )^{3} + 3 \, {\left ({\left (4 \, a^{5} - 3 \, a^{3} b^{2} - a b^{4}\right )} \cos \left (d x + c\right )^{4} + {\left (4 \, a^{4} b - 3 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{3}\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + b^{2}}\right ) + 3 \, {\left ({\left (4 \, a^{6} - 5 \, a^{4} b^{2}\right )} \cos \left (d x + c\right )^{4} + {\left (4 \, a^{5} b - 5 \, a^{3} b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left ({\left (4 \, a^{6} - 5 \, a^{4} b^{2}\right )} \cos \left (d x + c\right )^{4} + {\left (4 \, a^{5} b - 5 \, a^{3} b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, a^{3} b^{3} \cos \left (d x + c\right ) - a^{2} b^{4} - {\left (12 \, a^{5} b - 13 \, a^{3} b^{3} + 3 \, a b^{5}\right )} \cos \left (d x + c\right )^{3} - {\left (6 \, a^{4} b^{2} - 7 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{6 \, {\left (a^{3} b^{5} d \cos \left (d x + c\right )^{4} + a^{2} b^{6} d \cos \left (d x + c\right )^{3}\right )}}, -\frac {6 \, a b^{5} d x \cos \left (d x + c\right )^{4} + 6 \, b^{6} d x \cos \left (d x + c\right )^{3} - 6 \, {\left ({\left (4 \, a^{5} - 3 \, a^{3} b^{2} - a b^{4}\right )} \cos \left (d x + c\right )^{4} + {\left (4 \, a^{4} b - 3 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{3}\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )}\right ) + 3 \, {\left ({\left (4 \, a^{6} - 5 \, a^{4} b^{2}\right )} \cos \left (d x + c\right )^{4} + {\left (4 \, a^{5} b - 5 \, a^{3} b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left ({\left (4 \, a^{6} - 5 \, a^{4} b^{2}\right )} \cos \left (d x + c\right )^{4} + {\left (4 \, a^{5} b - 5 \, a^{3} b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, a^{3} b^{3} \cos \left (d x + c\right ) - a^{2} b^{4} - {\left (12 \, a^{5} b - 13 \, a^{3} b^{3} + 3 \, a b^{5}\right )} \cos \left (d x + c\right )^{3} - {\left (6 \, a^{4} b^{2} - 7 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{6 \, {\left (a^{3} b^{5} d \cos \left (d x + c\right )^{4} + a^{2} b^{6} d \cos \left (d x + c\right )^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^6/(a+b*sec(d*x+c))^2,x, algorithm="fricas")

[Out]

[-1/6*(6*a*b^5*d*x*cos(d*x + c)^4 + 6*b^6*d*x*cos(d*x + c)^3 + 3*((4*a^5 - 3*a^3*b^2 - a*b^4)*cos(d*x + c)^4 +
(4*a^4*b - 3*a^2*b^3 - b^5)*cos(d*x + c)^3)*sqrt(a^2 - b^2)*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*cos(d*x +
c)^2 - 2*sqrt(a^2 - b^2)*(b*cos(d*x + c) + a)*sin(d*x + c) + 2*a^2 - b^2)/(a^2*cos(d*x + c)^2 + 2*a*b*cos(d*x
+ c) + b^2)) + 3*((4*a^6 - 5*a^4*b^2)*cos(d*x + c)^4 + (4*a^5*b - 5*a^3*b^3)*cos(d*x + c)^3)*log(sin(d*x + c)
+ 1) - 3*((4*a^6 - 5*a^4*b^2)*cos(d*x + c)^4 + (4*a^5*b - 5*a^3*b^3)*cos(d*x + c)^3)*log(-sin(d*x + c) + 1) +
2*(2*a^3*b^3*cos(d*x + c) - a^2*b^4 - (12*a^5*b - 13*a^3*b^3 + 3*a*b^5)*cos(d*x + c)^3 - (6*a^4*b^2 - 7*a^2*b
^4)*cos(d*x + c)^2)*sin(d*x + c))/(a^3*b^5*d*cos(d*x + c)^4 + a^2*b^6*d*cos(d*x + c)^3), -1/6*(6*a*b^5*d*x*cos
(d*x + c)^4 + 6*b^6*d*x*cos(d*x + c)^3 - 6*((4*a^5 - 3*a^3*b^2 - a*b^4)*cos(d*x + c)^4 + (4*a^4*b - 3*a^2*b^3
- b^5)*cos(d*x + c)^3)*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cos(d*x + c) + a)/((a^2 - b^2)*sin(d*x + c
))) + 3*((4*a^6 - 5*a^4*b^2)*cos(d*x + c)^4 + (4*a^5*b - 5*a^3*b^3)*cos(d*x + c)^3)*log(sin(d*x + c) + 1) - 3*
((4*a^6 - 5*a^4*b^2)*cos(d*x + c)^4 + (4*a^5*b - 5*a^3*b^3)*cos(d*x + c)^3)*log(-sin(d*x + c) + 1) + 2*(2*a^3*
b^3*cos(d*x + c) - a^2*b^4 - (12*a^5*b - 13*a^3*b^3 + 3*a*b^5)*cos(d*x + c)^3 - (6*a^4*b^2 - 7*a^2*b^4)*cos(d*
x + c)^2)*sin(d*x + c))/(a^3*b^5*d*cos(d*x + c)^4 + a^2*b^6*d*cos(d*x + c)^3)]

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giac [B]  time = 4.34, size = 411, normalized size = 2.06 \[ -\frac {\frac {3 \, {\left (d x + c\right )}}{a^{2}} + \frac {3 \, {\left (4 \, a^{3} - 5 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{b^{5}} - \frac {3 \, {\left (4 \, a^{3} - 5 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{b^{5}} + \frac {6 \, {\left (a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a - b\right )} a b^{4}} - \frac {6 \, {\left (4 \, a^{6} - 7 \, a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{\sqrt {-a^{2} + b^{2}} a^{2} b^{5}} + \frac {2 \, {\left (9 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 18 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 16 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3} b^{4}}}{3 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^6/(a+b*sec(d*x+c))^2,x, algorithm="giac")

[Out]

-1/3*(3*(d*x + c)/a^2 + 3*(4*a^3 - 5*a*b^2)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/b^5 - 3*(4*a^3 - 5*a*b^2)*log(a
bs(tan(1/2*d*x + 1/2*c) - 1))/b^5 + 6*(a^4*tan(1/2*d*x + 1/2*c) - 2*a^2*b^2*tan(1/2*d*x + 1/2*c) + b^4*tan(1/2
*d*x + 1/2*c))/((a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 - a - b)*a*b^4) - 6*(4*a^6 - 7*a^4*b^2 +
2*a^2*b^4 + b^6)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(-2*a + 2*b) + arctan(-(a*tan(1/2*d*x + 1/2*c) - b*tan(1
/2*d*x + 1/2*c))/sqrt(-a^2 + b^2)))/(sqrt(-a^2 + b^2)*a^2*b^5) + 2*(9*a^2*tan(1/2*d*x + 1/2*c)^5 + 3*a*b*tan(1
/2*d*x + 1/2*c)^5 - 6*b^2*tan(1/2*d*x + 1/2*c)^5 - 18*a^2*tan(1/2*d*x + 1/2*c)^3 + 16*b^2*tan(1/2*d*x + 1/2*c)
^3 + 9*a^2*tan(1/2*d*x + 1/2*c) - 3*a*b*tan(1/2*d*x + 1/2*c) - 6*b^2*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2
*c)^2 - 1)^3*b^4))/d

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maple [B]  time = 0.51, size = 723, normalized size = 3.62 \[ -\frac {2 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,b^{4} \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b -a -b \right )}+\frac {4 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,b^{2} \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b -a -b \right )}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b -a -b \right )}+\frac {8 a^{4} \arctanh \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{d \,b^{5} \sqrt {\left (a -b \right ) \left (a +b \right )}}-\frac {14 a^{2} \arctanh \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{d \,b^{3} \sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {4 \arctanh \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{d b \sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {2 b \arctanh \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{d \,a^{2} \sqrt {\left (a -b \right ) \left (a +b \right )}}-\frac {1}{3 d \,b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {a}{d \,b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {1}{2 d \,b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {3 a^{2}}{d \,b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {a}{d \,b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {2}{d \,b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {4 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d \,b^{5}}-\frac {5 a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d \,b^{3}}-\frac {1}{3 d \,b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {a}{d \,b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {1}{2 d \,b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {3 a^{2}}{d \,b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {a}{d \,b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {2}{d \,b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {4 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d \,b^{5}}+\frac {5 a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d \,b^{3}}-\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^6/(a+b*sec(d*x+c))^2,x)

[Out]

-2/d/b^4*a^3*tan(1/2*d*x+1/2*c)/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)+4/d/b^2*a*tan(1/2*d*x+1/2*
c)/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)-2/d/a*tan(1/2*d*x+1/2*c)/(a*tan(1/2*d*x+1/2*c)^2-tan(1/
2*d*x+1/2*c)^2*b-a-b)+8/d/b^5*a^4/((a-b)*(a+b))^(1/2)*arctanh(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))-14
/d/b^3*a^2/((a-b)*(a+b))^(1/2)*arctanh(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))+4/d/b/((a-b)*(a+b))^(1/2)
*arctanh(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))+2/d*b/a^2/((a-b)*(a+b))^(1/2)*arctanh(tan(1/2*d*x+1/2*c
)*(a-b)/((a-b)*(a+b))^(1/2))-1/3/d/b^2/(tan(1/2*d*x+1/2*c)-1)^3-1/d/b^3/(tan(1/2*d*x+1/2*c)-1)^2*a-1/2/d/b^2/(
tan(1/2*d*x+1/2*c)-1)^2-3/d/b^4/(tan(1/2*d*x+1/2*c)-1)*a^2-1/d/b^3/(tan(1/2*d*x+1/2*c)-1)*a+2/d/b^2/(tan(1/2*d
*x+1/2*c)-1)+4/d*a^3/b^5*ln(tan(1/2*d*x+1/2*c)-1)-5/d*a/b^3*ln(tan(1/2*d*x+1/2*c)-1)-1/3/d/b^2/(tan(1/2*d*x+1/
2*c)+1)^3+1/d/b^3/(tan(1/2*d*x+1/2*c)+1)^2*a+1/2/d/b^2/(tan(1/2*d*x+1/2*c)+1)^2-3/d/b^4/(tan(1/2*d*x+1/2*c)+1)
*a^2-1/d/b^3/(tan(1/2*d*x+1/2*c)+1)*a+2/d/b^2/(tan(1/2*d*x+1/2*c)+1)-4/d*a^3/b^5*ln(tan(1/2*d*x+1/2*c)+1)+5/d*
a/b^3*ln(tan(1/2*d*x+1/2*c)+1)-2/d/a^2*arctan(tan(1/2*d*x+1/2*c))

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^6/(a+b*sec(d*x+c))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?`
for more details)Is 4*a^2-4*b^2 positive or negative?

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mupad [B]  time = 4.82, size = 9452, normalized size = 47.26 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(c + d*x)^6/(a + b/cos(c + d*x))^2,x)

[Out]

((2*tan(c/2 + (d*x)/2)^5*(10*a*b^3 - 6*a^3*b + 36*a^4 + 9*b^4 - 37*a^2*b^2))/(3*a*b^4) - (2*tan(c/2 + (d*x)/2)
^3*(6*a^3*b - 10*a*b^3 + 36*a^4 + 9*b^4 - 37*a^2*b^2))/(3*a*b^4) + (2*tan(c/2 + (d*x)/2)*(2*a^3*b - 2*a*b^3 +
4*a^4 + b^4 - 5*a^2*b^2))/(a*b^4) + (2*tan(c/2 + (d*x)/2)^7*(a - b)*(3*a*b^2 - 2*a^2*b - 4*a^3 + b^3))/(a*b^4)
)/(d*(a + b + tan(c/2 + (d*x)/2)^8*(a - b) - tan(c/2 + (d*x)/2)^2*(4*a + 2*b) - tan(c/2 + (d*x)/2)^6*(4*a - 2*
b) + 6*a*tan(c/2 + (d*x)/2)^4)) - (2*atan((((((((((8192*(9*a^5*b^21 - 3*a^4*b^22 - 13*a^6*b^20 + 6*a^7*b^19 +
25*a^8*b^18 - 41*a^9*b^17 + 3*a^10*b^16 + 26*a^11*b^15 - 12*a^12*b^14))/(a^3*b^16) - (tan(c/2 + (d*x)/2)*(2*a^
6*b^23 - 6*a^7*b^22 + 8*a^8*b^21 - 8*a^9*b^20 + 6*a^10*b^19 - 2*a^11*b^18)*8192i)/(a^6*b^16))*1i)/a^2 - (8192*
tan(c/2 + (d*x)/2)*(2*a^2*b^23 - 6*a^3*b^22 + 12*a^4*b^21 - 12*a^5*b^20 - 8*a^6*b^19 + 12*a^7*b^18 - 60*a^8*b^
17 + 160*a^9*b^16 - 60*a^10*b^15 - 100*a^11*b^14 + 82*a^12*b^13 - 118*a^13*b^12 + 128*a^14*b^11 + 32*a^15*b^10
- 96*a^16*b^9 + 32*a^17*b^8))/(a^4*b^16))*1i)/a^2 + (8192*(4*a*b^21 - 3*b^22 - 8*a^2*b^20 + 16*a^3*b^19 + 20*
a^4*b^18 - 26*a^5*b^17 + 74*a^6*b^16 - 280*a^7*b^15 + 192*a^8*b^14 + 332*a^9*b^13 - 1088*a^10*b^12 + 1040*a^11
*b^11 + 1129*a^12*b^10 - 2366*a^13*b^9 + 20*a^14*b^8 + 1696*a^15*b^7 - 528*a^16*b^6 - 416*a^17*b^5 + 192*a^18*
b^4))/(a^3*b^16))*1i)/a^2 - (8192*tan(c/2 + (d*x)/2)*(a*b^20 - 256*a^20*b + 256*a^21 - b^21 - 4*a^2*b^19 + 4*a
^3*b^18 - 40*a^4*b^17 + 140*a^5*b^16 - 250*a^6*b^15 + 90*a^7*b^14 + 588*a^8*b^13 - 624*a^9*b^12 + 132*a^10*b^1
1 + 28*a^11*b^10 - 2361*a^12*b^9 + 2297*a^13*b^8 + 4320*a^14*b^7 - 4320*a^15*b^6 - 3680*a^16*b^5 + 3680*a^17*b
^4 + 1536*a^18*b^3 - 1536*a^19*b^2))/(a^4*b^16))/a^2 - ((((((((8192*(9*a^5*b^21 - 3*a^4*b^22 - 13*a^6*b^20 + 6
*a^7*b^19 + 25*a^8*b^18 - 41*a^9*b^17 + 3*a^10*b^16 + 26*a^11*b^15 - 12*a^12*b^14))/(a^3*b^16) + (tan(c/2 + (d
*x)/2)*(2*a^6*b^23 - 6*a^7*b^22 + 8*a^8*b^21 - 8*a^9*b^20 + 6*a^10*b^19 - 2*a^11*b^18)*8192i)/(a^6*b^16))*1i)/
a^2 + (8192*tan(c/2 + (d*x)/2)*(2*a^2*b^23 - 6*a^3*b^22 + 12*a^4*b^21 - 12*a^5*b^20 - 8*a^6*b^19 + 12*a^7*b^18
- 60*a^8*b^17 + 160*a^9*b^16 - 60*a^10*b^15 - 100*a^11*b^14 + 82*a^12*b^13 - 118*a^13*b^12 + 128*a^14*b^11 +
32*a^15*b^10 - 96*a^16*b^9 + 32*a^17*b^8))/(a^4*b^16))*1i)/a^2 + (8192*(4*a*b^21 - 3*b^22 - 8*a^2*b^20 + 16*a^
3*b^19 + 20*a^4*b^18 - 26*a^5*b^17 + 74*a^6*b^16 - 280*a^7*b^15 + 192*a^8*b^14 + 332*a^9*b^13 - 1088*a^10*b^12
+ 1040*a^11*b^11 + 1129*a^12*b^10 - 2366*a^13*b^9 + 20*a^14*b^8 + 1696*a^15*b^7 - 528*a^16*b^6 - 416*a^17*b^5
+ 192*a^18*b^4))/(a^3*b^16))*1i)/a^2 + (8192*tan(c/2 + (d*x)/2)*(a*b^20 - 256*a^20*b + 256*a^21 - b^21 - 4*a^
2*b^19 + 4*a^3*b^18 - 40*a^4*b^17 + 140*a^5*b^16 - 250*a^6*b^15 + 90*a^7*b^14 + 588*a^8*b^13 - 624*a^9*b^12 +
132*a^10*b^11 + 28*a^11*b^10 - 2361*a^12*b^9 + 2297*a^13*b^8 + 4320*a^14*b^7 - 4320*a^15*b^6 - 3680*a^16*b^5 +
3680*a^17*b^4 + 1536*a^18*b^3 - 1536*a^19*b^2))/(a^4*b^16))/a^2)/((((((((((8192*(9*a^5*b^21 - 3*a^4*b^22 - 13
*a^6*b^20 + 6*a^7*b^19 + 25*a^8*b^18 - 41*a^9*b^17 + 3*a^10*b^16 + 26*a^11*b^15 - 12*a^12*b^14))/(a^3*b^16) -
(tan(c/2 + (d*x)/2)*(2*a^6*b^23 - 6*a^7*b^22 + 8*a^8*b^21 - 8*a^9*b^20 + 6*a^10*b^19 - 2*a^11*b^18)*8192i)/(a^
6*b^16))*1i)/a^2 - (8192*tan(c/2 + (d*x)/2)*(2*a^2*b^23 - 6*a^3*b^22 + 12*a^4*b^21 - 12*a^5*b^20 - 8*a^6*b^19
+ 12*a^7*b^18 - 60*a^8*b^17 + 160*a^9*b^16 - 60*a^10*b^15 - 100*a^11*b^14 + 82*a^12*b^13 - 118*a^13*b^12 + 128
*a^14*b^11 + 32*a^15*b^10 - 96*a^16*b^9 + 32*a^17*b^8))/(a^4*b^16))*1i)/a^2 + (8192*(4*a*b^21 - 3*b^22 - 8*a^2
*b^20 + 16*a^3*b^19 + 20*a^4*b^18 - 26*a^5*b^17 + 74*a^6*b^16 - 280*a^7*b^15 + 192*a^8*b^14 + 332*a^9*b^13 - 1
088*a^10*b^12 + 1040*a^11*b^11 + 1129*a^12*b^10 - 2366*a^13*b^9 + 20*a^14*b^8 + 1696*a^15*b^7 - 528*a^16*b^6 -
416*a^17*b^5 + 192*a^18*b^4))/(a^3*b^16))*1i)/a^2 - (8192*tan(c/2 + (d*x)/2)*(a*b^20 - 256*a^20*b + 256*a^21
- b^21 - 4*a^2*b^19 + 4*a^3*b^18 - 40*a^4*b^17 + 140*a^5*b^16 - 250*a^6*b^15 + 90*a^7*b^14 + 588*a^8*b^13 - 62
4*a^9*b^12 + 132*a^10*b^11 + 28*a^11*b^10 - 2361*a^12*b^9 + 2297*a^13*b^8 + 4320*a^14*b^7 - 4320*a^15*b^6 - 36
80*a^16*b^5 + 3680*a^17*b^4 + 1536*a^18*b^3 - 1536*a^19*b^2))/(a^4*b^16))*1i)/a^2 + (((((((((8192*(9*a^5*b^21
- 3*a^4*b^22 - 13*a^6*b^20 + 6*a^7*b^19 + 25*a^8*b^18 - 41*a^9*b^17 + 3*a^10*b^16 + 26*a^11*b^15 - 12*a^12*b^1
4))/(a^3*b^16) + (tan(c/2 + (d*x)/2)*(2*a^6*b^23 - 6*a^7*b^22 + 8*a^8*b^21 - 8*a^9*b^20 + 6*a^10*b^19 - 2*a^11
*b^18)*8192i)/(a^6*b^16))*1i)/a^2 + (8192*tan(c/2 + (d*x)/2)*(2*a^2*b^23 - 6*a^3*b^22 + 12*a^4*b^21 - 12*a^5*b
^20 - 8*a^6*b^19 + 12*a^7*b^18 - 60*a^8*b^17 + 160*a^9*b^16 - 60*a^10*b^15 - 100*a^11*b^14 + 82*a^12*b^13 - 11
8*a^13*b^12 + 128*a^14*b^11 + 32*a^15*b^10 - 96*a^16*b^9 + 32*a^17*b^8))/(a^4*b^16))*1i)/a^2 + (8192*(4*a*b^21
- 3*b^22 - 8*a^2*b^20 + 16*a^3*b^19 + 20*a^4*b^18 - 26*a^5*b^17 + 74*a^6*b^16 - 280*a^7*b^15 + 192*a^8*b^14 +
332*a^9*b^13 - 1088*a^10*b^12 + 1040*a^11*b^11 + 1129*a^12*b^10 - 2366*a^13*b^9 + 20*a^14*b^8 + 1696*a^15*b^7
- 528*a^16*b^6 - 416*a^17*b^5 + 192*a^18*b^4))/(a^3*b^16))*1i)/a^2 + (8192*tan(c/2 + (d*x)/2)*(a*b^20 - 256*a
^20*b + 256*a^21 - b^21 - 4*a^2*b^19 + 4*a^3*b^18 - 40*a^4*b^17 + 140*a^5*b^16 - 250*a^6*b^15 + 90*a^7*b^14 +
588*a^8*b^13 - 624*a^9*b^12 + 132*a^10*b^11 + 28*a^11*b^10 - 2361*a^12*b^9 + 2297*a^13*b^8 + 4320*a^14*b^7 - 4
320*a^15*b^6 - 3680*a^16*b^5 + 3680*a^17*b^4 + 1536*a^18*b^3 - 1536*a^19*b^2))/(a^4*b^16))*1i)/a^2 - (16384*(5
*a*b^17 - 256*a^17*b + 256*a^18 - 5*b^18 - 91*a^2*b^16 + 116*a^3*b^15 - 14*a^4*b^14 + 174*a^5*b^13 + 582*a^6*b
^12 - 1036*a^7*b^11 - 1133*a^8*b^10 + 101*a^9*b^9 + 2245*a^10*b^8 + 2624*a^11*b^7 - 3792*a^12*b^6 - 3264*a^13*
b^5 + 3488*a^14*b^4 + 1536*a^15*b^3 - 1536*a^16*b^2))/(a^3*b^16))))/(a^2*d) - (a*atan(((a*((8192*tan(c/2 + (d*
x)/2)*(a*b^20 - 256*a^20*b + 256*a^21 - b^21 - 4*a^2*b^19 + 4*a^3*b^18 - 40*a^4*b^17 + 140*a^5*b^16 - 250*a^6*
b^15 + 90*a^7*b^14 + 588*a^8*b^13 - 624*a^9*b^12 + 132*a^10*b^11 + 28*a^11*b^10 - 2361*a^12*b^9 + 2297*a^13*b^
8 + 4320*a^14*b^7 - 4320*a^15*b^6 - 3680*a^16*b^5 + 3680*a^17*b^4 + 1536*a^18*b^3 - 1536*a^19*b^2))/(a^4*b^16)
+ (a*((8192*(4*a*b^21 - 3*b^22 - 8*a^2*b^20 + 16*a^3*b^19 + 20*a^4*b^18 - 26*a^5*b^17 + 74*a^6*b^16 - 280*a^7
*b^15 + 192*a^8*b^14 + 332*a^9*b^13 - 1088*a^10*b^12 + 1040*a^11*b^11 + 1129*a^12*b^10 - 2366*a^13*b^9 + 20*a^
14*b^8 + 1696*a^15*b^7 - 528*a^16*b^6 - 416*a^17*b^5 + 192*a^18*b^4))/(a^3*b^16) + (a*((8192*tan(c/2 + (d*x)/2
)*(2*a^2*b^23 - 6*a^3*b^22 + 12*a^4*b^21 - 12*a^5*b^20 - 8*a^6*b^19 + 12*a^7*b^18 - 60*a^8*b^17 + 160*a^9*b^16
- 60*a^10*b^15 - 100*a^11*b^14 + 82*a^12*b^13 - 118*a^13*b^12 + 128*a^14*b^11 + 32*a^15*b^10 - 96*a^16*b^9 +
32*a^17*b^8))/(a^4*b^16) + (a*((8192*(9*a^5*b^21 - 3*a^4*b^22 - 13*a^6*b^20 + 6*a^7*b^19 + 25*a^8*b^18 - 41*a^
9*b^17 + 3*a^10*b^16 + 26*a^11*b^15 - 12*a^12*b^14))/(a^3*b^16) + (8192*tan(c/2 + (d*x)/2)*(4*a^2 - 5*b^2)*(2*
a^6*b^23 - 6*a^7*b^22 + 8*a^8*b^21 - 8*a^9*b^20 + 6*a^10*b^19 - 2*a^11*b^18))/(a^3*b^21))*(4*a^2 - 5*b^2))/b^5
)*(4*a^2 - 5*b^2))/b^5)*(4*a^2 - 5*b^2))/b^5)*(4*a^2 - 5*b^2)*1i)/b^5 + (a*((8192*tan(c/2 + (d*x)/2)*(a*b^20 -
256*a^20*b + 256*a^21 - b^21 - 4*a^2*b^19 + 4*a^3*b^18 - 40*a^4*b^17 + 140*a^5*b^16 - 250*a^6*b^15 + 90*a^7*b
^14 + 588*a^8*b^13 - 624*a^9*b^12 + 132*a^10*b^11 + 28*a^11*b^10 - 2361*a^12*b^9 + 2297*a^13*b^8 + 4320*a^14*b
^7 - 4320*a^15*b^6 - 3680*a^16*b^5 + 3680*a^17*b^4 + 1536*a^18*b^3 - 1536*a^19*b^2))/(a^4*b^16) - (a*((8192*(4
*a*b^21 - 3*b^22 - 8*a^2*b^20 + 16*a^3*b^19 + 20*a^4*b^18 - 26*a^5*b^17 + 74*a^6*b^16 - 280*a^7*b^15 + 192*a^8
*b^14 + 332*a^9*b^13 - 1088*a^10*b^12 + 1040*a^11*b^11 + 1129*a^12*b^10 - 2366*a^13*b^9 + 20*a^14*b^8 + 1696*a
^15*b^7 - 528*a^16*b^6 - 416*a^17*b^5 + 192*a^18*b^4))/(a^3*b^16) - (a*((8192*tan(c/2 + (d*x)/2)*(2*a^2*b^23 -
6*a^3*b^22 + 12*a^4*b^21 - 12*a^5*b^20 - 8*a^6*b^19 + 12*a^7*b^18 - 60*a^8*b^17 + 160*a^9*b^16 - 60*a^10*b^15
- 100*a^11*b^14 + 82*a^12*b^13 - 118*a^13*b^12 + 128*a^14*b^11 + 32*a^15*b^10 - 96*a^16*b^9 + 32*a^17*b^8))/(
a^4*b^16) - (a*((8192*(9*a^5*b^21 - 3*a^4*b^22 - 13*a^6*b^20 + 6*a^7*b^19 + 25*a^8*b^18 - 41*a^9*b^17 + 3*a^10
*b^16 + 26*a^11*b^15 - 12*a^12*b^14))/(a^3*b^16) - (8192*tan(c/2 + (d*x)/2)*(4*a^2 - 5*b^2)*(2*a^6*b^23 - 6*a^
7*b^22 + 8*a^8*b^21 - 8*a^9*b^20 + 6*a^10*b^19 - 2*a^11*b^18))/(a^3*b^21))*(4*a^2 - 5*b^2))/b^5)*(4*a^2 - 5*b^
2))/b^5)*(4*a^2 - 5*b^2))/b^5)*(4*a^2 - 5*b^2)*1i)/b^5)/((16384*(5*a*b^17 - 256*a^17*b + 256*a^18 - 5*b^18 - 9
1*a^2*b^16 + 116*a^3*b^15 - 14*a^4*b^14 + 174*a^5*b^13 + 582*a^6*b^12 - 1036*a^7*b^11 - 1133*a^8*b^10 + 101*a^
9*b^9 + 2245*a^10*b^8 + 2624*a^11*b^7 - 3792*a^12*b^6 - 3264*a^13*b^5 + 3488*a^14*b^4 + 1536*a^15*b^3 - 1536*a
^16*b^2))/(a^3*b^16) - (a*((8192*tan(c/2 + (d*x)/2)*(a*b^20 - 256*a^20*b + 256*a^21 - b^21 - 4*a^2*b^19 + 4*a^
3*b^18 - 40*a^4*b^17 + 140*a^5*b^16 - 250*a^6*b^15 + 90*a^7*b^14 + 588*a^8*b^13 - 624*a^9*b^12 + 132*a^10*b^11
+ 28*a^11*b^10 - 2361*a^12*b^9 + 2297*a^13*b^8 + 4320*a^14*b^7 - 4320*a^15*b^6 - 3680*a^16*b^5 + 3680*a^17*b^
4 + 1536*a^18*b^3 - 1536*a^19*b^2))/(a^4*b^16) + (a*((8192*(4*a*b^21 - 3*b^22 - 8*a^2*b^20 + 16*a^3*b^19 + 20*
a^4*b^18 - 26*a^5*b^17 + 74*a^6*b^16 - 280*a^7*b^15 + 192*a^8*b^14 + 332*a^9*b^13 - 1088*a^10*b^12 + 1040*a^11
*b^11 + 1129*a^12*b^10 - 2366*a^13*b^9 + 20*a^14*b^8 + 1696*a^15*b^7 - 528*a^16*b^6 - 416*a^17*b^5 + 192*a^18*
b^4))/(a^3*b^16) + (a*((8192*tan(c/2 + (d*x)/2)*(2*a^2*b^23 - 6*a^3*b^22 + 12*a^4*b^21 - 12*a^5*b^20 - 8*a^6*b
^19 + 12*a^7*b^18 - 60*a^8*b^17 + 160*a^9*b^16 - 60*a^10*b^15 - 100*a^11*b^14 + 82*a^12*b^13 - 118*a^13*b^12 +
128*a^14*b^11 + 32*a^15*b^10 - 96*a^16*b^9 + 32*a^17*b^8))/(a^4*b^16) + (a*((8192*(9*a^5*b^21 - 3*a^4*b^22 -
13*a^6*b^20 + 6*a^7*b^19 + 25*a^8*b^18 - 41*a^9*b^17 + 3*a^10*b^16 + 26*a^11*b^15 - 12*a^12*b^14))/(a^3*b^16)
+ (8192*tan(c/2 + (d*x)/2)*(4*a^2 - 5*b^2)*(2*a^6*b^23 - 6*a^7*b^22 + 8*a^8*b^21 - 8*a^9*b^20 + 6*a^10*b^19 -
2*a^11*b^18))/(a^3*b^21))*(4*a^2 - 5*b^2))/b^5)*(4*a^2 - 5*b^2))/b^5)*(4*a^2 - 5*b^2))/b^5)*(4*a^2 - 5*b^2))/b
^5 + (a*((8192*tan(c/2 + (d*x)/2)*(a*b^20 - 256*a^20*b + 256*a^21 - b^21 - 4*a^2*b^19 + 4*a^3*b^18 - 40*a^4*b^
17 + 140*a^5*b^16 - 250*a^6*b^15 + 90*a^7*b^14 + 588*a^8*b^13 - 624*a^9*b^12 + 132*a^10*b^11 + 28*a^11*b^10 -
2361*a^12*b^9 + 2297*a^13*b^8 + 4320*a^14*b^7 - 4320*a^15*b^6 - 3680*a^16*b^5 + 3680*a^17*b^4 + 1536*a^18*b^3
- 1536*a^19*b^2))/(a^4*b^16) - (a*((8192*(4*a*b^21 - 3*b^22 - 8*a^2*b^20 + 16*a^3*b^19 + 20*a^4*b^18 - 26*a^5*
b^17 + 74*a^6*b^16 - 280*a^7*b^15 + 192*a^8*b^14 + 332*a^9*b^13 - 1088*a^10*b^12 + 1040*a^11*b^11 + 1129*a^12*
b^10 - 2366*a^13*b^9 + 20*a^14*b^8 + 1696*a^15*b^7 - 528*a^16*b^6 - 416*a^17*b^5 + 192*a^18*b^4))/(a^3*b^16) -
(a*((8192*tan(c/2 + (d*x)/2)*(2*a^2*b^23 - 6*a^3*b^22 + 12*a^4*b^21 - 12*a^5*b^20 - 8*a^6*b^19 + 12*a^7*b^18
- 60*a^8*b^17 + 160*a^9*b^16 - 60*a^10*b^15 - 100*a^11*b^14 + 82*a^12*b^13 - 118*a^13*b^12 + 128*a^14*b^11 + 3
2*a^15*b^10 - 96*a^16*b^9 + 32*a^17*b^8))/(a^4*b^16) - (a*((8192*(9*a^5*b^21 - 3*a^4*b^22 - 13*a^6*b^20 + 6*a^
7*b^19 + 25*a^8*b^18 - 41*a^9*b^17 + 3*a^10*b^16 + 26*a^11*b^15 - 12*a^12*b^14))/(a^3*b^16) - (8192*tan(c/2 +
(d*x)/2)*(4*a^2 - 5*b^2)*(2*a^6*b^23 - 6*a^7*b^22 + 8*a^8*b^21 - 8*a^9*b^20 + 6*a^10*b^19 - 2*a^11*b^18))/(a^3
*b^21))*(4*a^2 - 5*b^2))/b^5)*(4*a^2 - 5*b^2))/b^5)*(4*a^2 - 5*b^2))/b^5)*(4*a^2 - 5*b^2))/b^5))*(4*a^2 - 5*b^
2)*2i)/(b^5*d) - (atan((((4*a^2 + b^2)*((8192*tan(c/2 + (d*x)/2)*(a*b^20 - 256*a^20*b + 256*a^21 - b^21 - 4*a^
2*b^19 + 4*a^3*b^18 - 40*a^4*b^17 + 140*a^5*b^16 - 250*a^6*b^15 + 90*a^7*b^14 + 588*a^8*b^13 - 624*a^9*b^12 +
132*a^10*b^11 + 28*a^11*b^10 - 2361*a^12*b^9 + 2297*a^13*b^8 + 4320*a^14*b^7 - 4320*a^15*b^6 - 3680*a^16*b^5 +
3680*a^17*b^4 + 1536*a^18*b^3 - 1536*a^19*b^2))/(a^4*b^16) + ((4*a^2 + b^2)*((8192*(4*a*b^21 - 3*b^22 - 8*a^2
*b^20 + 16*a^3*b^19 + 20*a^4*b^18 - 26*a^5*b^17 + 74*a^6*b^16 - 280*a^7*b^15 + 192*a^8*b^14 + 332*a^9*b^13 - 1
088*a^10*b^12 + 1040*a^11*b^11 + 1129*a^12*b^10 - 2366*a^13*b^9 + 20*a^14*b^8 + 1696*a^15*b^7 - 528*a^16*b^6 -
416*a^17*b^5 + 192*a^18*b^4))/(a^3*b^16) + (((8192*tan(c/2 + (d*x)/2)*(2*a^2*b^23 - 6*a^3*b^22 + 12*a^4*b^21
- 12*a^5*b^20 - 8*a^6*b^19 + 12*a^7*b^18 - 60*a^8*b^17 + 160*a^9*b^16 - 60*a^10*b^15 - 100*a^11*b^14 + 82*a^12
*b^13 - 118*a^13*b^12 + 128*a^14*b^11 + 32*a^15*b^10 - 96*a^16*b^9 + 32*a^17*b^8))/(a^4*b^16) + ((4*a^2 + b^2)
*((8192*(9*a^5*b^21 - 3*a^4*b^22 - 13*a^6*b^20 + 6*a^7*b^19 + 25*a^8*b^18 - 41*a^9*b^17 + 3*a^10*b^16 + 26*a^1
1*b^15 - 12*a^12*b^14))/(a^3*b^16) + (8192*tan(c/2 + (d*x)/2)*(4*a^2 + b^2)*((a + b)^3*(a - b)^3)^(1/2)*(2*a^6
*b^23 - 6*a^7*b^22 + 8*a^8*b^21 - 8*a^9*b^20 + 6*a^10*b^19 - 2*a^11*b^18))/(a^6*b^21))*((a + b)^3*(a - b)^3)^(
1/2))/(a^2*b^5))*(4*a^2 + b^2)*((a + b)^3*(a - b)^3)^(1/2))/(a^2*b^5))*((a + b)^3*(a - b)^3)^(1/2))/(a^2*b^5))
*((a + b)^3*(a - b)^3)^(1/2)*1i)/(a^2*b^5) + ((4*a^2 + b^2)*((8192*tan(c/2 + (d*x)/2)*(a*b^20 - 256*a^20*b + 2
56*a^21 - b^21 - 4*a^2*b^19 + 4*a^3*b^18 - 40*a^4*b^17 + 140*a^5*b^16 - 250*a^6*b^15 + 90*a^7*b^14 + 588*a^8*b
^13 - 624*a^9*b^12 + 132*a^10*b^11 + 28*a^11*b^10 - 2361*a^12*b^9 + 2297*a^13*b^8 + 4320*a^14*b^7 - 4320*a^15*
b^6 - 3680*a^16*b^5 + 3680*a^17*b^4 + 1536*a^18*b^3 - 1536*a^19*b^2))/(a^4*b^16) - ((4*a^2 + b^2)*((8192*(4*a*
b^21 - 3*b^22 - 8*a^2*b^20 + 16*a^3*b^19 + 20*a^4*b^18 - 26*a^5*b^17 + 74*a^6*b^16 - 280*a^7*b^15 + 192*a^8*b^
14 + 332*a^9*b^13 - 1088*a^10*b^12 + 1040*a^11*b^11 + 1129*a^12*b^10 - 2366*a^13*b^9 + 20*a^14*b^8 + 1696*a^15
*b^7 - 528*a^16*b^6 - 416*a^17*b^5 + 192*a^18*b^4))/(a^3*b^16) - (((8192*tan(c/2 + (d*x)/2)*(2*a^2*b^23 - 6*a^
3*b^22 + 12*a^4*b^21 - 12*a^5*b^20 - 8*a^6*b^19 + 12*a^7*b^18 - 60*a^8*b^17 + 160*a^9*b^16 - 60*a^10*b^15 - 10
0*a^11*b^14 + 82*a^12*b^13 - 118*a^13*b^12 + 128*a^14*b^11 + 32*a^15*b^10 - 96*a^16*b^9 + 32*a^17*b^8))/(a^4*b
^16) - ((4*a^2 + b^2)*((8192*(9*a^5*b^21 - 3*a^4*b^22 - 13*a^6*b^20 + 6*a^7*b^19 + 25*a^8*b^18 - 41*a^9*b^17 +
3*a^10*b^16 + 26*a^11*b^15 - 12*a^12*b^14))/(a^3*b^16) - (8192*tan(c/2 + (d*x)/2)*(4*a^2 + b^2)*((a + b)^3*(a
- b)^3)^(1/2)*(2*a^6*b^23 - 6*a^7*b^22 + 8*a^8*b^21 - 8*a^9*b^20 + 6*a^10*b^19 - 2*a^11*b^18))/(a^6*b^21))*((
a + b)^3*(a - b)^3)^(1/2))/(a^2*b^5))*(4*a^2 + b^2)*((a + b)^3*(a - b)^3)^(1/2))/(a^2*b^5))*((a + b)^3*(a - b)
^3)^(1/2))/(a^2*b^5))*((a + b)^3*(a - b)^3)^(1/2)*1i)/(a^2*b^5))/((16384*(5*a*b^17 - 256*a^17*b + 256*a^18 - 5
*b^18 - 91*a^2*b^16 + 116*a^3*b^15 - 14*a^4*b^14 + 174*a^5*b^13 + 582*a^6*b^12 - 1036*a^7*b^11 - 1133*a^8*b^10
+ 101*a^9*b^9 + 2245*a^10*b^8 + 2624*a^11*b^7 - 3792*a^12*b^6 - 3264*a^13*b^5 + 3488*a^14*b^4 + 1536*a^15*b^3
- 1536*a^16*b^2))/(a^3*b^16) - ((4*a^2 + b^2)*((8192*tan(c/2 + (d*x)/2)*(a*b^20 - 256*a^20*b + 256*a^21 - b^2
1 - 4*a^2*b^19 + 4*a^3*b^18 - 40*a^4*b^17 + 140*a^5*b^16 - 250*a^6*b^15 + 90*a^7*b^14 + 588*a^8*b^13 - 624*a^9
*b^12 + 132*a^10*b^11 + 28*a^11*b^10 - 2361*a^12*b^9 + 2297*a^13*b^8 + 4320*a^14*b^7 - 4320*a^15*b^6 - 3680*a^
16*b^5 + 3680*a^17*b^4 + 1536*a^18*b^3 - 1536*a^19*b^2))/(a^4*b^16) + ((4*a^2 + b^2)*((8192*(4*a*b^21 - 3*b^22
- 8*a^2*b^20 + 16*a^3*b^19 + 20*a^4*b^18 - 26*a^5*b^17 + 74*a^6*b^16 - 280*a^7*b^15 + 192*a^8*b^14 + 332*a^9*
b^13 - 1088*a^10*b^12 + 1040*a^11*b^11 + 1129*a^12*b^10 - 2366*a^13*b^9 + 20*a^14*b^8 + 1696*a^15*b^7 - 528*a^
16*b^6 - 416*a^17*b^5 + 192*a^18*b^4))/(a^3*b^16) + (((8192*tan(c/2 + (d*x)/2)*(2*a^2*b^23 - 6*a^3*b^22 + 12*a
^4*b^21 - 12*a^5*b^20 - 8*a^6*b^19 + 12*a^7*b^18 - 60*a^8*b^17 + 160*a^9*b^16 - 60*a^10*b^15 - 100*a^11*b^14 +
82*a^12*b^13 - 118*a^13*b^12 + 128*a^14*b^11 + 32*a^15*b^10 - 96*a^16*b^9 + 32*a^17*b^8))/(a^4*b^16) + ((4*a^
2 + b^2)*((8192*(9*a^5*b^21 - 3*a^4*b^22 - 13*a^6*b^20 + 6*a^7*b^19 + 25*a^8*b^18 - 41*a^9*b^17 + 3*a^10*b^16
+ 26*a^11*b^15 - 12*a^12*b^14))/(a^3*b^16) + (8192*tan(c/2 + (d*x)/2)*(4*a^2 + b^2)*((a + b)^3*(a - b)^3)^(1/2
)*(2*a^6*b^23 - 6*a^7*b^22 + 8*a^8*b^21 - 8*a^9*b^20 + 6*a^10*b^19 - 2*a^11*b^18))/(a^6*b^21))*((a + b)^3*(a -
b)^3)^(1/2))/(a^2*b^5))*(4*a^2 + b^2)*((a + b)^3*(a - b)^3)^(1/2))/(a^2*b^5))*((a + b)^3*(a - b)^3)^(1/2))/(a
^2*b^5))*((a + b)^3*(a - b)^3)^(1/2))/(a^2*b^5) + ((4*a^2 + b^2)*((8192*tan(c/2 + (d*x)/2)*(a*b^20 - 256*a^20*
b + 256*a^21 - b^21 - 4*a^2*b^19 + 4*a^3*b^18 - 40*a^4*b^17 + 140*a^5*b^16 - 250*a^6*b^15 + 90*a^7*b^14 + 588*
a^8*b^13 - 624*a^9*b^12 + 132*a^10*b^11 + 28*a^11*b^10 - 2361*a^12*b^9 + 2297*a^13*b^8 + 4320*a^14*b^7 - 4320*
a^15*b^6 - 3680*a^16*b^5 + 3680*a^17*b^4 + 1536*a^18*b^3 - 1536*a^19*b^2))/(a^4*b^16) - ((4*a^2 + b^2)*((8192*
(4*a*b^21 - 3*b^22 - 8*a^2*b^20 + 16*a^3*b^19 + 20*a^4*b^18 - 26*a^5*b^17 + 74*a^6*b^16 - 280*a^7*b^15 + 192*a
^8*b^14 + 332*a^9*b^13 - 1088*a^10*b^12 + 1040*a^11*b^11 + 1129*a^12*b^10 - 2366*a^13*b^9 + 20*a^14*b^8 + 1696
*a^15*b^7 - 528*a^16*b^6 - 416*a^17*b^5 + 192*a^18*b^4))/(a^3*b^16) - (((8192*tan(c/2 + (d*x)/2)*(2*a^2*b^23 -
6*a^3*b^22 + 12*a^4*b^21 - 12*a^5*b^20 - 8*a^6*b^19 + 12*a^7*b^18 - 60*a^8*b^17 + 160*a^9*b^16 - 60*a^10*b^15
- 100*a^11*b^14 + 82*a^12*b^13 - 118*a^13*b^12 + 128*a^14*b^11 + 32*a^15*b^10 - 96*a^16*b^9 + 32*a^17*b^8))/(
a^4*b^16) - ((4*a^2 + b^2)*((8192*(9*a^5*b^21 - 3*a^4*b^22 - 13*a^6*b^20 + 6*a^7*b^19 + 25*a^8*b^18 - 41*a^9*b
^17 + 3*a^10*b^16 + 26*a^11*b^15 - 12*a^12*b^14))/(a^3*b^16) - (8192*tan(c/2 + (d*x)/2)*(4*a^2 + b^2)*((a + b)
^3*(a - b)^3)^(1/2)*(2*a^6*b^23 - 6*a^7*b^22 + 8*a^8*b^21 - 8*a^9*b^20 + 6*a^10*b^19 - 2*a^11*b^18))/(a^6*b^21
))*((a + b)^3*(a - b)^3)^(1/2))/(a^2*b^5))*(4*a^2 + b^2)*((a + b)^3*(a - b)^3)^(1/2))/(a^2*b^5))*((a + b)^3*(a
- b)^3)^(1/2))/(a^2*b^5))*((a + b)^3*(a - b)^3)^(1/2))/(a^2*b^5)))*(4*a^2 + b^2)*((a + b)^3*(a - b)^3)^(1/2)*
2i)/(a^2*b^5*d)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^{6}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)**6/(a+b*sec(d*x+c))**2,x)

[Out]

Integral(tan(c + d*x)**6/(a + b*sec(c + d*x))**2, x)

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