3.263 \(\int \frac {\sec ^3(c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^2} \, dx\)

Optimal. Leaf size=231 \[ \frac {2 a^3 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{d (a-b)^{5/2} (a+b)^{5/2}}-\frac {a^4 \sin (c+d x)}{b d \left (a^2-b^2\right )^2 (a \cos (c+d x)+b)}-\frac {2 a^3 \left (a^2-3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^2 d (a-b)^{5/2} (a+b)^{5/2}}-\frac {\sin (c+d x)}{2 d (a+b)^2 (1-\cos (c+d x))}-\frac {\sin (c+d x)}{2 d (a-b)^2 (\cos (c+d x)+1)}+\frac {\tanh ^{-1}(\sin (c+d x))}{b^2 d} \]

[Out]

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

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Rubi [A]  time = 0.44, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4397, 2897, 2648, 2664, 12, 2659, 208, 3770} \[ -\frac {a^4 \sin (c+d x)}{b d \left (a^2-b^2\right )^2 (a \cos (c+d x)+b)}-\frac {2 a^3 \left (a^2-3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^2 d (a-b)^{5/2} (a+b)^{5/2}}+\frac {2 a^3 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{d (a-b)^{5/2} (a+b)^{5/2}}-\frac {\sin (c+d x)}{2 d (a+b)^2 (1-\cos (c+d x))}-\frac {\sin (c+d x)}{2 d (a-b)^2 (\cos (c+d x)+1)}+\frac {\tanh ^{-1}(\sin (c+d x))}{b^2 d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^3/(a*Sin[c + d*x] + b*Tan[c + d*x])^2,x]

[Out]

ArcTanh[Sin[c + d*x]]/(b^2*d) + (2*a^3*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a
+ b)^(5/2)*d) - (2*a^3*(a^2 - 3*b^2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^2*(
a + b)^(5/2)*d) - Sin[c + d*x]/(2*(a + b)^2*d*(1 - Cos[c + d*x])) - Sin[c + d*x]/(2*(a - b)^2*d*(1 + Cos[c + d
*x])) - (a^4*Sin[c + d*x])/(b*(a^2 - b^2)^2*d*(b + a*Cos[c + d*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 2648

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

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 3770

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

Rule 4397

Int[u_, x_Symbol] :> Int[TrigSimplify[u], x] /; TrigSimplifyQ[u]

Rubi steps

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

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Mathematica [A]  time = 1.97, size = 196, normalized size = 0.85 \[ -\frac {\frac {2 a^4 \sin (c+d x)}{b (a-b)^2 (a+b)^2 (a \cos (c+d x)+b)}-\frac {4 \left (a^5-4 a^3 b^2\right ) \tanh ^{-1}\left (\frac {(b-a) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^2 \left (a^2-b^2\right )^{5/2}}+\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{(a-b)^2}+\frac {\cot \left (\frac {1}{2} (c+d x)\right )}{(a+b)^2}+\frac {2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{b^2}-\frac {2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{b^2}}{2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^3/(a*Sin[c + d*x] + b*Tan[c + d*x])^2,x]

[Out]

-1/2*((-4*(a^5 - 4*a^3*b^2)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^2*(a^2 - b^2)^(5/2)) + Co
t[(c + d*x)/2]/(a + b)^2 + (2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/b^2 - (2*Log[Cos[(c + d*x)/2] + Sin[(c
+ d*x)/2]])/b^2 + (2*a^4*Sin[c + d*x])/((a - b)^2*b*(a + b)^2*(b + a*Cos[c + d*x])) + Tan[(c + d*x)/2]/(a - b
)^2)/d

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fricas [A]  time = 1.66, size = 864, normalized size = 3.74 \[ \left [-\frac {2 \, a^{6} b - 2 \, b^{7} + {\left (a^{5} b - 4 \, a^{3} b^{3} + {\left (a^{6} - 4 \, a^{4} b^{2}\right )} \cos \left (d x + c\right )\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 ) \sin \left (d x + c\right ) - 2 \, {\left (a^{6} b + a^{4} b^{3} - 2 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2} - {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7} + {\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7} + {\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 2 \, {\left (a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right )}{2 \, {\left ({\left (a^{7} b^{2} - 3 \, a^{5} b^{4} + 3 \, a^{3} b^{6} - a b^{8}\right )} d \cos \left (d x + c\right ) + {\left (a^{6} b^{3} - 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} - b^{9}\right )} d\right )} \sin \left (d x + c\right )}, -\frac {2 \, a^{6} b - 2 \, b^{7} + 2 \, {\left (a^{5} b - 4 \, a^{3} b^{3} + {\left (a^{6} - 4 \, a^{4} b^{2}\right )} \cos \left (d x + c\right )\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 ) \sin \left (d x + c\right ) - 2 \, {\left (a^{6} b + a^{4} b^{3} - 2 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2} - {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7} + {\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7} + {\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 2 \, {\left (a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right )}{2 \, {\left ({\left (a^{7} b^{2} - 3 \, a^{5} b^{4} + 3 \, a^{3} b^{6} - a b^{8}\right )} d \cos \left (d x + c\right ) + {\left (a^{6} b^{3} - 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} - b^{9}\right )} d\right )} \sin \left (d x + c\right )}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3/(a*sin(d*x+c)+b*tan(d*x+c))^2,x, algorithm="fricas")

[Out]

[-1/2*(2*a^6*b - 2*b^7 + (a^5*b - 4*a^3*b^3 + (a^6 - 4*a^4*b^2)*cos(d*x + c))*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))*sin(d*x + c) - 2*(a^6*b + a^4*b^3 - 2*a^2*b^5)*cos(d*x + c)^2 -
(a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7 + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c))*log(sin(d*x + c) +
1)*sin(d*x + c) + (a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7 + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c))*
log(-sin(d*x + c) + 1)*sin(d*x + c) + 2*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*cos(d*x + c))/(((a^7*b^2 - 3*a^5*b^4 + 3
*a^3*b^6 - a*b^8)*d*cos(d*x + c) + (a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^7 - b^9)*d)*sin(d*x + c)), -1/2*(2*a^6*b - 2
*b^7 + 2*(a^5*b - 4*a^3*b^3 + (a^6 - 4*a^4*b^2)*cos(d*x + c))*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)))*sin(d*x + c) - 2*(a^6*b + a^4*b^3 - 2*a^2*b^5)*cos(d*x + c)^2 - (a^
6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7 + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c))*log(sin(d*x + c) + 1)*
sin(d*x + c) + (a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7 + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c))*log(
-sin(d*x + c) + 1)*sin(d*x + c) + 2*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*cos(d*x + c))/(((a^7*b^2 - 3*a^5*b^4 + 3*a^3
*b^6 - a*b^8)*d*cos(d*x + c) + (a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^7 - b^9)*d)*sin(d*x + c))]

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giac [A]  time = 0.68, size = 354, normalized size = 1.53 \[ \frac {\frac {4 \, {\left (a^{5} - 4 \, a^{3} b^{2}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a - 2 \, b\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \sqrt {-a^{2} + b^{2}}} - \frac {\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{2} - 2 \, a b + b^{2}} + \frac {4 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a^{3} b - a^{2} b^{2} - a b^{3} + b^{4}}{{\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}} + \frac {2 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{b^{2}} - \frac {2 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{b^{2}}}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3/(a*sin(d*x+c)+b*tan(d*x+c))^2,x, algorithm="giac")

[Out]

1/2*(4*(a^5 - 4*a^3*b^2)*(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)))/((a^4*b^2 - 2*a^2*b^4 + b^6)*sqrt(-a^2 + b^2)) - tan(1/2*d*x + 1/2*c
)/(a^2 - 2*a*b + b^2) + (4*a^4*tan(1/2*d*x + 1/2*c)^2 - a^3*b*tan(1/2*d*x + 1/2*c)^2 + 3*a^2*b^2*tan(1/2*d*x +
1/2*c)^2 - 3*a*b^3*tan(1/2*d*x + 1/2*c)^2 + b^4*tan(1/2*d*x + 1/2*c)^2 + a^3*b - a^2*b^2 - a*b^3 + b^4)/((a^4
*b - 2*a^2*b^3 + b^5)*(a*tan(1/2*d*x + 1/2*c)^3 - b*tan(1/2*d*x + 1/2*c)^3 - a*tan(1/2*d*x + 1/2*c) - b*tan(1/
2*d*x + 1/2*c))) + 2*log(abs(tan(1/2*d*x + 1/2*c) + 1))/b^2 - 2*log(abs(tan(1/2*d*x + 1/2*c) - 1))/b^2)/d

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maple [A]  time = 0.24, size = 276, normalized size = 1.19 \[ -\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d \left (a^{2}-2 a b +b^{2}\right )}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d \,b^{2}}+\frac {2 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \left (a -b \right )^{2} b \left (a +b \right )^{2} \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )}-\frac {2 a^{5} \arctanh \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{d \left (a -b \right )^{2} b^{2} \left (a +b \right )^{2} \sqrt {\left (a +b \right ) \left (a -b \right )}}+\frac {8 a^{3} \arctanh \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{d \left (a -b \right )^{2} \left (a +b \right )^{2} \sqrt {\left (a +b \right ) \left (a -b \right )}}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d \,b^{2}}-\frac {1}{2 d \left (a +b \right )^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^3/(a*sin(d*x+c)+b*tan(d*x+c))^2,x)

[Out]

-1/2/d/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)-1/d/b^2*ln(tan(1/2*d*x+1/2*c)-1)+2/d*a^4/(a-b)^2/b/(a+b)^2*tan(1/2*d
*x+1/2*c)/(tan(1/2*d*x+1/2*c)^2*a-b*tan(1/2*d*x+1/2*c)^2-a-b)-2/d*a^5/(a-b)^2/b^2/(a+b)^2/((a+b)*(a-b))^(1/2)*
arctanh(tan(1/2*d*x+1/2*c)*(a-b)/((a+b)*(a-b))^(1/2))+8/d*a^3/(a-b)^2/(a+b)^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/d/b^2*ln(tan(1/2*d*x+1/2*c)+1)-1/2/d/(a+b)^2/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(sec(d*x+c)^3/(a*sin(d*x+c)+b*tan(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 = 5.21, size = 6056, normalized size = 26.22 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(c + d*x)^3*(a*sin(c + d*x) + b*tan(c + d*x))^2),x)

[Out]

((a^2 - 2*a*b + b^2)/(a + b) + (tan(c/2 + (d*x)/2)^2*(4*a^4 - a^3*b - 3*a*b^3 + b^4 + 3*a^2*b^2))/(b*(a + b)^2
))/(d*(tan(c/2 + (d*x)/2)^3*(6*a*b^2 - 6*a^2*b + 2*a^3 - 2*b^3) + tan(c/2 + (d*x)/2)*(2*a*b^2 + 2*a^2*b - 2*a^
3 - 2*b^3))) - (atan(-(((tan(c/2 + (d*x)/2)*(32*b^26 - 96*a*b^25 - 224*a^2*b^24 + 928*a^3*b^23 + 480*a^4*b^22
- 4000*a^5*b^21 + 992*a^6*b^20 + 9568*a^7*b^19 - 8128*a^8*b^18 - 12992*a^9*b^17 + 21344*a^10*b^16 + 8224*a^11*
b^15 - 31744*a^12*b^14 + 2176*a^13*b^13 + 29600*a^14*b^12 - 8480*a^15*b^11 - 17632*a^16*b^10 + 7072*a^17*b^9 +
6528*a^18*b^8 - 3008*a^19*b^7 - 1376*a^20*b^6 + 672*a^21*b^5 + 128*a^22*b^4 - 64*a^23*b^3) + (32*b^28 - 32*a*
b^27 - 352*a^2*b^26 + 480*a^3*b^25 + 1504*a^4*b^24 - 2688*a^5*b^23 - 3168*a^6*b^22 + 8064*a^7*b^21 + 2880*a^8*
b^20 - 14784*a^9*b^19 + 1344*a^10*b^18 + 17472*a^11*b^17 - 6720*a^12*b^16 - 13440*a^13*b^15 + 8256*a^14*b^14 +
6528*a^15*b^13 - 5472*a^16*b^12 - 1824*a^17*b^11 + 2080*a^18*b^10 + 224*a^19*b^9 - 416*a^20*b^8 + 32*a^22*b^6
- (tan(c/2 + (d*x)/2)*(128*a^2*b^28 - 64*a*b^29 + 576*a^3*b^27 - 1280*a^4*b^26 - 2240*a^5*b^25 + 5760*a^6*b^2
4 + 4800*a^7*b^23 - 15360*a^8*b^22 - 5760*a^9*b^21 + 26880*a^10*b^20 + 2688*a^11*b^19 - 32256*a^12*b^18 + 2688
*a^13*b^17 + 26880*a^14*b^16 - 5760*a^15*b^15 - 15360*a^16*b^14 + 4800*a^17*b^13 + 5760*a^18*b^12 - 2240*a^19*
b^11 - 1280*a^20*b^10 + 576*a^21*b^9 + 128*a^22*b^8 - 64*a^23*b^7))/b^2)/b^2)*1i)/b^2 + ((tan(c/2 + (d*x)/2)*(
32*b^26 - 96*a*b^25 - 224*a^2*b^24 + 928*a^3*b^23 + 480*a^4*b^22 - 4000*a^5*b^21 + 992*a^6*b^20 + 9568*a^7*b^1
9 - 8128*a^8*b^18 - 12992*a^9*b^17 + 21344*a^10*b^16 + 8224*a^11*b^15 - 31744*a^12*b^14 + 2176*a^13*b^13 + 296
00*a^14*b^12 - 8480*a^15*b^11 - 17632*a^16*b^10 + 7072*a^17*b^9 + 6528*a^18*b^8 - 3008*a^19*b^7 - 1376*a^20*b^
6 + 672*a^21*b^5 + 128*a^22*b^4 - 64*a^23*b^3) - (32*b^28 - 32*a*b^27 - 352*a^2*b^26 + 480*a^3*b^25 + 1504*a^4
*b^24 - 2688*a^5*b^23 - 3168*a^6*b^22 + 8064*a^7*b^21 + 2880*a^8*b^20 - 14784*a^9*b^19 + 1344*a^10*b^18 + 1747
2*a^11*b^17 - 6720*a^12*b^16 - 13440*a^13*b^15 + 8256*a^14*b^14 + 6528*a^15*b^13 - 5472*a^16*b^12 - 1824*a^17*
b^11 + 2080*a^18*b^10 + 224*a^19*b^9 - 416*a^20*b^8 + 32*a^22*b^6 + (tan(c/2 + (d*x)/2)*(128*a^2*b^28 - 64*a*b
^29 + 576*a^3*b^27 - 1280*a^4*b^26 - 2240*a^5*b^25 + 5760*a^6*b^24 + 4800*a^7*b^23 - 15360*a^8*b^22 - 5760*a^9
*b^21 + 26880*a^10*b^20 + 2688*a^11*b^19 - 32256*a^12*b^18 + 2688*a^13*b^17 + 26880*a^14*b^16 - 5760*a^15*b^15
- 15360*a^16*b^14 + 4800*a^17*b^13 + 5760*a^18*b^12 - 2240*a^19*b^11 - 1280*a^20*b^10 + 576*a^21*b^9 + 128*a^
22*b^8 - 64*a^23*b^7))/b^2)/b^2)*1i)/b^2)/(256*a^3*b^21 - 512*a^4*b^20 - 1856*a^5*b^19 + 3200*a^6*b^18 + 6592*
a^7*b^17 - 8704*a^8*b^16 - 15104*a^9*b^15 + 13760*a^10*b^14 + 23872*a^11*b^13 - 14464*a^12*b^12 - 25856*a^13*b
^11 + 11072*a^14*b^10 + 18496*a^15*b^9 - 6400*a^16*b^8 - 8192*a^17*b^7 + 2624*a^18*b^6 + 1984*a^19*b^5 - 640*a
^20*b^4 - 192*a^21*b^3 + 64*a^22*b^2 - (tan(c/2 + (d*x)/2)*(32*b^26 - 96*a*b^25 - 224*a^2*b^24 + 928*a^3*b^23
+ 480*a^4*b^22 - 4000*a^5*b^21 + 992*a^6*b^20 + 9568*a^7*b^19 - 8128*a^8*b^18 - 12992*a^9*b^17 + 21344*a^10*b^
16 + 8224*a^11*b^15 - 31744*a^12*b^14 + 2176*a^13*b^13 + 29600*a^14*b^12 - 8480*a^15*b^11 - 17632*a^16*b^10 +
7072*a^17*b^9 + 6528*a^18*b^8 - 3008*a^19*b^7 - 1376*a^20*b^6 + 672*a^21*b^5 + 128*a^22*b^4 - 64*a^23*b^3) + (
32*b^28 - 32*a*b^27 - 352*a^2*b^26 + 480*a^3*b^25 + 1504*a^4*b^24 - 2688*a^5*b^23 - 3168*a^6*b^22 + 8064*a^7*b
^21 + 2880*a^8*b^20 - 14784*a^9*b^19 + 1344*a^10*b^18 + 17472*a^11*b^17 - 6720*a^12*b^16 - 13440*a^13*b^15 + 8
256*a^14*b^14 + 6528*a^15*b^13 - 5472*a^16*b^12 - 1824*a^17*b^11 + 2080*a^18*b^10 + 224*a^19*b^9 - 416*a^20*b^
8 + 32*a^22*b^6 - (tan(c/2 + (d*x)/2)*(128*a^2*b^28 - 64*a*b^29 + 576*a^3*b^27 - 1280*a^4*b^26 - 2240*a^5*b^25
+ 5760*a^6*b^24 + 4800*a^7*b^23 - 15360*a^8*b^22 - 5760*a^9*b^21 + 26880*a^10*b^20 + 2688*a^11*b^19 - 32256*a
^12*b^18 + 2688*a^13*b^17 + 26880*a^14*b^16 - 5760*a^15*b^15 - 15360*a^16*b^14 + 4800*a^17*b^13 + 5760*a^18*b^
12 - 2240*a^19*b^11 - 1280*a^20*b^10 + 576*a^21*b^9 + 128*a^22*b^8 - 64*a^23*b^7))/b^2)/b^2)/b^2 + (tan(c/2 +
(d*x)/2)*(32*b^26 - 96*a*b^25 - 224*a^2*b^24 + 928*a^3*b^23 + 480*a^4*b^22 - 4000*a^5*b^21 + 992*a^6*b^20 + 95
68*a^7*b^19 - 8128*a^8*b^18 - 12992*a^9*b^17 + 21344*a^10*b^16 + 8224*a^11*b^15 - 31744*a^12*b^14 + 2176*a^13*
b^13 + 29600*a^14*b^12 - 8480*a^15*b^11 - 17632*a^16*b^10 + 7072*a^17*b^9 + 6528*a^18*b^8 - 3008*a^19*b^7 - 13
76*a^20*b^6 + 672*a^21*b^5 + 128*a^22*b^4 - 64*a^23*b^3) - (32*b^28 - 32*a*b^27 - 352*a^2*b^26 + 480*a^3*b^25
+ 1504*a^4*b^24 - 2688*a^5*b^23 - 3168*a^6*b^22 + 8064*a^7*b^21 + 2880*a^8*b^20 - 14784*a^9*b^19 + 1344*a^10*b
^18 + 17472*a^11*b^17 - 6720*a^12*b^16 - 13440*a^13*b^15 + 8256*a^14*b^14 + 6528*a^15*b^13 - 5472*a^16*b^12 -
1824*a^17*b^11 + 2080*a^18*b^10 + 224*a^19*b^9 - 416*a^20*b^8 + 32*a^22*b^6 + (tan(c/2 + (d*x)/2)*(128*a^2*b^2
8 - 64*a*b^29 + 576*a^3*b^27 - 1280*a^4*b^26 - 2240*a^5*b^25 + 5760*a^6*b^24 + 4800*a^7*b^23 - 15360*a^8*b^22
- 5760*a^9*b^21 + 26880*a^10*b^20 + 2688*a^11*b^19 - 32256*a^12*b^18 + 2688*a^13*b^17 + 26880*a^14*b^16 - 5760
*a^15*b^15 - 15360*a^16*b^14 + 4800*a^17*b^13 + 5760*a^18*b^12 - 2240*a^19*b^11 - 1280*a^20*b^10 + 576*a^21*b^
9 + 128*a^22*b^8 - 64*a^23*b^7))/b^2)/b^2)/b^2))*2i)/(b^2*d) - tan(c/2 + (d*x)/2)/(2*d*(a - b)^2) + (a^3*atan(
((a^3*(a - 2*b)*(a + 2*b)*((a + b)^5*(a - b)^5)^(1/2)*(tan(c/2 + (d*x)/2)*(32*b^26 - 96*a*b^25 - 224*a^2*b^24
+ 928*a^3*b^23 + 480*a^4*b^22 - 4000*a^5*b^21 + 992*a^6*b^20 + 9568*a^7*b^19 - 8128*a^8*b^18 - 12992*a^9*b^17
+ 21344*a^10*b^16 + 8224*a^11*b^15 - 31744*a^12*b^14 + 2176*a^13*b^13 + 29600*a^14*b^12 - 8480*a^15*b^11 - 176
32*a^16*b^10 + 7072*a^17*b^9 + 6528*a^18*b^8 - 3008*a^19*b^7 - 1376*a^20*b^6 + 672*a^21*b^5 + 128*a^22*b^4 - 6
4*a^23*b^3) + (a^3*(a - 2*b)*(a + 2*b)*((a + b)^5*(a - b)^5)^(1/2)*(32*b^28 - 32*a*b^27 - 352*a^2*b^26 + 480*a
^3*b^25 + 1504*a^4*b^24 - 2688*a^5*b^23 - 3168*a^6*b^22 + 8064*a^7*b^21 + 2880*a^8*b^20 - 14784*a^9*b^19 + 134
4*a^10*b^18 + 17472*a^11*b^17 - 6720*a^12*b^16 - 13440*a^13*b^15 + 8256*a^14*b^14 + 6528*a^15*b^13 - 5472*a^16
*b^12 - 1824*a^17*b^11 + 2080*a^18*b^10 + 224*a^19*b^9 - 416*a^20*b^8 + 32*a^22*b^6 - (a^3*tan(c/2 + (d*x)/2)*
(a - 2*b)*(a + 2*b)*((a + b)^5*(a - b)^5)^(1/2)*(128*a^2*b^28 - 64*a*b^29 + 576*a^3*b^27 - 1280*a^4*b^26 - 224
0*a^5*b^25 + 5760*a^6*b^24 + 4800*a^7*b^23 - 15360*a^8*b^22 - 5760*a^9*b^21 + 26880*a^10*b^20 + 2688*a^11*b^19
- 32256*a^12*b^18 + 2688*a^13*b^17 + 26880*a^14*b^16 - 5760*a^15*b^15 - 15360*a^16*b^14 + 4800*a^17*b^13 + 57
60*a^18*b^12 - 2240*a^19*b^11 - 1280*a^20*b^10 + 576*a^21*b^9 + 128*a^22*b^8 - 64*a^23*b^7))/(b^12 - 5*a^2*b^1
0 + 10*a^4*b^8 - 10*a^6*b^6 + 5*a^8*b^4 - a^10*b^2)))/(b^12 - 5*a^2*b^10 + 10*a^4*b^8 - 10*a^6*b^6 + 5*a^8*b^4
- a^10*b^2))*1i)/(b^12 - 5*a^2*b^10 + 10*a^4*b^8 - 10*a^6*b^6 + 5*a^8*b^4 - a^10*b^2) + (a^3*(a - 2*b)*(a + 2
*b)*((a + b)^5*(a - b)^5)^(1/2)*(tan(c/2 + (d*x)/2)*(32*b^26 - 96*a*b^25 - 224*a^2*b^24 + 928*a^3*b^23 + 480*a
^4*b^22 - 4000*a^5*b^21 + 992*a^6*b^20 + 9568*a^7*b^19 - 8128*a^8*b^18 - 12992*a^9*b^17 + 21344*a^10*b^16 + 82
24*a^11*b^15 - 31744*a^12*b^14 + 2176*a^13*b^13 + 29600*a^14*b^12 - 8480*a^15*b^11 - 17632*a^16*b^10 + 7072*a^
17*b^9 + 6528*a^18*b^8 - 3008*a^19*b^7 - 1376*a^20*b^6 + 672*a^21*b^5 + 128*a^22*b^4 - 64*a^23*b^3) - (a^3*(a
- 2*b)*(a + 2*b)*((a + b)^5*(a - b)^5)^(1/2)*(32*b^28 - 32*a*b^27 - 352*a^2*b^26 + 480*a^3*b^25 + 1504*a^4*b^2
4 - 2688*a^5*b^23 - 3168*a^6*b^22 + 8064*a^7*b^21 + 2880*a^8*b^20 - 14784*a^9*b^19 + 1344*a^10*b^18 + 17472*a^
11*b^17 - 6720*a^12*b^16 - 13440*a^13*b^15 + 8256*a^14*b^14 + 6528*a^15*b^13 - 5472*a^16*b^12 - 1824*a^17*b^11
+ 2080*a^18*b^10 + 224*a^19*b^9 - 416*a^20*b^8 + 32*a^22*b^6 + (a^3*tan(c/2 + (d*x)/2)*(a - 2*b)*(a + 2*b)*((
a + b)^5*(a - b)^5)^(1/2)*(128*a^2*b^28 - 64*a*b^29 + 576*a^3*b^27 - 1280*a^4*b^26 - 2240*a^5*b^25 + 5760*a^6*
b^24 + 4800*a^7*b^23 - 15360*a^8*b^22 - 5760*a^9*b^21 + 26880*a^10*b^20 + 2688*a^11*b^19 - 32256*a^12*b^18 + 2
688*a^13*b^17 + 26880*a^14*b^16 - 5760*a^15*b^15 - 15360*a^16*b^14 + 4800*a^17*b^13 + 5760*a^18*b^12 - 2240*a^
19*b^11 - 1280*a^20*b^10 + 576*a^21*b^9 + 128*a^22*b^8 - 64*a^23*b^7))/(b^12 - 5*a^2*b^10 + 10*a^4*b^8 - 10*a^
6*b^6 + 5*a^8*b^4 - a^10*b^2)))/(b^12 - 5*a^2*b^10 + 10*a^4*b^8 - 10*a^6*b^6 + 5*a^8*b^4 - a^10*b^2))*1i)/(b^1
2 - 5*a^2*b^10 + 10*a^4*b^8 - 10*a^6*b^6 + 5*a^8*b^4 - a^10*b^2))/(256*a^3*b^21 - 512*a^4*b^20 - 1856*a^5*b^19
+ 3200*a^6*b^18 + 6592*a^7*b^17 - 8704*a^8*b^16 - 15104*a^9*b^15 + 13760*a^10*b^14 + 23872*a^11*b^13 - 14464*
a^12*b^12 - 25856*a^13*b^11 + 11072*a^14*b^10 + 18496*a^15*b^9 - 6400*a^16*b^8 - 8192*a^17*b^7 + 2624*a^18*b^6
+ 1984*a^19*b^5 - 640*a^20*b^4 - 192*a^21*b^3 + 64*a^22*b^2 - (a^3*(a - 2*b)*(a + 2*b)*((a + b)^5*(a - b)^5)^
(1/2)*(tan(c/2 + (d*x)/2)*(32*b^26 - 96*a*b^25 - 224*a^2*b^24 + 928*a^3*b^23 + 480*a^4*b^22 - 4000*a^5*b^21 +
992*a^6*b^20 + 9568*a^7*b^19 - 8128*a^8*b^18 - 12992*a^9*b^17 + 21344*a^10*b^16 + 8224*a^11*b^15 - 31744*a^12*
b^14 + 2176*a^13*b^13 + 29600*a^14*b^12 - 8480*a^15*b^11 - 17632*a^16*b^10 + 7072*a^17*b^9 + 6528*a^18*b^8 - 3
008*a^19*b^7 - 1376*a^20*b^6 + 672*a^21*b^5 + 128*a^22*b^4 - 64*a^23*b^3) + (a^3*(a - 2*b)*(a + 2*b)*((a + b)^
5*(a - b)^5)^(1/2)*(32*b^28 - 32*a*b^27 - 352*a^2*b^26 + 480*a^3*b^25 + 1504*a^4*b^24 - 2688*a^5*b^23 - 3168*a
^6*b^22 + 8064*a^7*b^21 + 2880*a^8*b^20 - 14784*a^9*b^19 + 1344*a^10*b^18 + 17472*a^11*b^17 - 6720*a^12*b^16 -
13440*a^13*b^15 + 8256*a^14*b^14 + 6528*a^15*b^13 - 5472*a^16*b^12 - 1824*a^17*b^11 + 2080*a^18*b^10 + 224*a^
19*b^9 - 416*a^20*b^8 + 32*a^22*b^6 - (a^3*tan(c/2 + (d*x)/2)*(a - 2*b)*(a + 2*b)*((a + b)^5*(a - b)^5)^(1/2)*
(128*a^2*b^28 - 64*a*b^29 + 576*a^3*b^27 - 1280*a^4*b^26 - 2240*a^5*b^25 + 5760*a^6*b^24 + 4800*a^7*b^23 - 153
60*a^8*b^22 - 5760*a^9*b^21 + 26880*a^10*b^20 + 2688*a^11*b^19 - 32256*a^12*b^18 + 2688*a^13*b^17 + 26880*a^14
*b^16 - 5760*a^15*b^15 - 15360*a^16*b^14 + 4800*a^17*b^13 + 5760*a^18*b^12 - 2240*a^19*b^11 - 1280*a^20*b^10 +
576*a^21*b^9 + 128*a^22*b^8 - 64*a^23*b^7))/(b^12 - 5*a^2*b^10 + 10*a^4*b^8 - 10*a^6*b^6 + 5*a^8*b^4 - a^10*b
^2)))/(b^12 - 5*a^2*b^10 + 10*a^4*b^8 - 10*a^6*b^6 + 5*a^8*b^4 - a^10*b^2)))/(b^12 - 5*a^2*b^10 + 10*a^4*b^8 -
10*a^6*b^6 + 5*a^8*b^4 - a^10*b^2) + (a^3*(a - 2*b)*(a + 2*b)*((a + b)^5*(a - b)^5)^(1/2)*(tan(c/2 + (d*x)/2)
*(32*b^26 - 96*a*b^25 - 224*a^2*b^24 + 928*a^3*b^23 + 480*a^4*b^22 - 4000*a^5*b^21 + 992*a^6*b^20 + 9568*a^7*b
^19 - 8128*a^8*b^18 - 12992*a^9*b^17 + 21344*a^10*b^16 + 8224*a^11*b^15 - 31744*a^12*b^14 + 2176*a^13*b^13 + 2
9600*a^14*b^12 - 8480*a^15*b^11 - 17632*a^16*b^10 + 7072*a^17*b^9 + 6528*a^18*b^8 - 3008*a^19*b^7 - 1376*a^20*
b^6 + 672*a^21*b^5 + 128*a^22*b^4 - 64*a^23*b^3) - (a^3*(a - 2*b)*(a + 2*b)*((a + b)^5*(a - b)^5)^(1/2)*(32*b^
28 - 32*a*b^27 - 352*a^2*b^26 + 480*a^3*b^25 + 1504*a^4*b^24 - 2688*a^5*b^23 - 3168*a^6*b^22 + 8064*a^7*b^21 +
2880*a^8*b^20 - 14784*a^9*b^19 + 1344*a^10*b^18 + 17472*a^11*b^17 - 6720*a^12*b^16 - 13440*a^13*b^15 + 8256*a
^14*b^14 + 6528*a^15*b^13 - 5472*a^16*b^12 - 1824*a^17*b^11 + 2080*a^18*b^10 + 224*a^19*b^9 - 416*a^20*b^8 + 3
2*a^22*b^6 + (a^3*tan(c/2 + (d*x)/2)*(a - 2*b)*(a + 2*b)*((a + b)^5*(a - b)^5)^(1/2)*(128*a^2*b^28 - 64*a*b^29
+ 576*a^3*b^27 - 1280*a^4*b^26 - 2240*a^5*b^25 + 5760*a^6*b^24 + 4800*a^7*b^23 - 15360*a^8*b^22 - 5760*a^9*b^
21 + 26880*a^10*b^20 + 2688*a^11*b^19 - 32256*a^12*b^18 + 2688*a^13*b^17 + 26880*a^14*b^16 - 5760*a^15*b^15 -
15360*a^16*b^14 + 4800*a^17*b^13 + 5760*a^18*b^12 - 2240*a^19*b^11 - 1280*a^20*b^10 + 576*a^21*b^9 + 128*a^22*
b^8 - 64*a^23*b^7))/(b^12 - 5*a^2*b^10 + 10*a^4*b^8 - 10*a^6*b^6 + 5*a^8*b^4 - a^10*b^2)))/(b^12 - 5*a^2*b^10
+ 10*a^4*b^8 - 10*a^6*b^6 + 5*a^8*b^4 - a^10*b^2)))/(b^12 - 5*a^2*b^10 + 10*a^4*b^8 - 10*a^6*b^6 + 5*a^8*b^4 -
a^10*b^2)))*(a - 2*b)*(a + 2*b)*((a + b)^5*(a - b)^5)^(1/2)*2i)/(d*(b^12 - 5*a^2*b^10 + 10*a^4*b^8 - 10*a^6*b
^6 + 5*a^8*b^4 - a^10*b^2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{3}{\left (c + d x \right )}}{\left (a \sin {\left (c + d x \right )} + b \tan {\left (c + d x \right )}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**3/(a*sin(d*x+c)+b*tan(d*x+c))**2,x)

[Out]

Integral(sec(c + d*x)**3/(a*sin(c + d*x) + b*tan(c + d*x))**2, x)

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