3.6.37 \(\int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx\) [537]

Optimal. Leaf size=282 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{3/2} d}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{3/2} d}-\frac {2 a^2 \tan ^3(c+d x)}{b \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {2 \left (16 a^4+6 a^2 b^2-5 b^4\right ) \sqrt {a+b \tan (c+d x)}}{5 b^4 \left (a^2+b^2\right ) d}-\frac {2 a \left (8 a^2+3 b^2\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{5 b^3 \left (a^2+b^2\right ) d}+\frac {2 \left (6 a^2+b^2\right ) \tan ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{5 b^2 \left (a^2+b^2\right ) d} \]

[Out]

-arctanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))/(a-I*b)^(3/2)/d-arctanh((a+b*tan(d*x+c))^(1/2)/(a+I*b)^(1/2))/(
a+I*b)^(3/2)/d+2/5*(16*a^4+6*a^2*b^2-5*b^4)*(a+b*tan(d*x+c))^(1/2)/b^4/(a^2+b^2)/d-2/5*a*(8*a^2+3*b^2)*(a+b*ta
n(d*x+c))^(1/2)*tan(d*x+c)/b^3/(a^2+b^2)/d+2/5*(6*a^2+b^2)*(a+b*tan(d*x+c))^(1/2)*tan(d*x+c)^2/b^2/(a^2+b^2)/d
-2*a^2*tan(d*x+c)^3/b/(a^2+b^2)/d/(a+b*tan(d*x+c))^(1/2)

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Rubi [A]
time = 0.49, antiderivative size = 282, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3646, 3728, 3711, 3620, 3618, 65, 214} \begin {gather*} -\frac {2 a^2 \tan ^3(c+d x)}{b d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {2 \left (6 a^2+b^2\right ) \tan ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{5 b^2 d \left (a^2+b^2\right )}-\frac {2 a \left (8 a^2+3 b^2\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{5 b^3 d \left (a^2+b^2\right )}+\frac {2 \left (16 a^4+6 a^2 b^2-5 b^4\right ) \sqrt {a+b \tan (c+d x)}}{5 b^4 d \left (a^2+b^2\right )}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d (a-i b)^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d (a+i b)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^5/(a + b*Tan[c + d*x])^(3/2),x]

[Out]

-(ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]]/((a - I*b)^(3/2)*d)) - ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt
[a + I*b]]/((a + I*b)^(3/2)*d) - (2*a^2*Tan[c + d*x]^3)/(b*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]]) + (2*(16*a^
4 + 6*a^2*b^2 - 5*b^4)*Sqrt[a + b*Tan[c + d*x]])/(5*b^4*(a^2 + b^2)*d) - (2*a*(8*a^2 + 3*b^2)*Tan[c + d*x]*Sqr
t[a + b*Tan[c + d*x]])/(5*b^3*(a^2 + b^2)*d) + (2*(6*a^2 + b^2)*Tan[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/(5*b^
2*(a^2 + b^2)*d)

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

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

Rule 3618

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c*(
d/f), Subst[Int[(a + (b/d)*x)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] &&
NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]

Rule 3620

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c
+ I*d)/2, Int[(a + b*Tan[e + f*x])^m*(1 - I*Tan[e + f*x]), x], x] + Dist[(c - I*d)/2, Int[(a + b*Tan[e + f*x]
)^m*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0]
&& NeQ[c^2 + d^2, 0] &&  !IntegerQ[m]

Rule 3646

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - D
ist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(
m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3
*a*b^2*d)*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*(n + 1)))*Tan[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && Gt
Q[m, 2] && LtQ[n, -1] && IntegerQ[2*m]

Rule 3711

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])
^m*Simp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0]
&&  !LeQ[m, -1]

Rule 3728

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*
tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*(a + b*Tan[e + f*x])^m*((c + d
*Tan[e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Tan[e + f*x])^(m - 1)*(c +
d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C*(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f
*x] - (C*m*(b*c - a*d) - b*B*d*(m + n + 1))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}
, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !Intege
rQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rubi steps

\begin {align*} \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx &=-\frac {2 a^2 \tan ^3(c+d x)}{b \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {2 \int \frac {\tan ^2(c+d x) \left (3 a^2-\frac {1}{2} a b \tan (c+d x)+\frac {1}{2} \left (6 a^2+b^2\right ) \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{b \left (a^2+b^2\right )}\\ &=-\frac {2 a^2 \tan ^3(c+d x)}{b \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {2 \left (6 a^2+b^2\right ) \tan ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{5 b^2 \left (a^2+b^2\right ) d}+\frac {4 \int \frac {\tan (c+d x) \left (-a \left (6 a^2+b^2\right )-\frac {5}{4} b^3 \tan (c+d x)-\frac {3}{4} a \left (8 a^2+3 b^2\right ) \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{5 b^2 \left (a^2+b^2\right )}\\ &=-\frac {2 a^2 \tan ^3(c+d x)}{b \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}-\frac {2 a \left (8 a^2+3 b^2\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{5 b^3 \left (a^2+b^2\right ) d}+\frac {2 \left (6 a^2+b^2\right ) \tan ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{5 b^2 \left (a^2+b^2\right ) d}+\frac {8 \int \frac {\frac {3}{4} a^2 \left (8 a^2+3 b^2\right )+\frac {15}{8} a b^3 \tan (c+d x)+\frac {3}{8} \left (16 a^4+6 a^2 b^2-5 b^4\right ) \tan ^2(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{15 b^3 \left (a^2+b^2\right )}\\ &=-\frac {2 a^2 \tan ^3(c+d x)}{b \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {2 \left (16 a^4+6 a^2 b^2-5 b^4\right ) \sqrt {a+b \tan (c+d x)}}{5 b^4 \left (a^2+b^2\right ) d}-\frac {2 a \left (8 a^2+3 b^2\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{5 b^3 \left (a^2+b^2\right ) d}+\frac {2 \left (6 a^2+b^2\right ) \tan ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{5 b^2 \left (a^2+b^2\right ) d}+\frac {8 \int \frac {\frac {15 b^4}{8}+\frac {15}{8} a b^3 \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{15 b^3 \left (a^2+b^2\right )}\\ &=-\frac {2 a^2 \tan ^3(c+d x)}{b \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {2 \left (16 a^4+6 a^2 b^2-5 b^4\right ) \sqrt {a+b \tan (c+d x)}}{5 b^4 \left (a^2+b^2\right ) d}-\frac {2 a \left (8 a^2+3 b^2\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{5 b^3 \left (a^2+b^2\right ) d}+\frac {2 \left (6 a^2+b^2\right ) \tan ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{5 b^2 \left (a^2+b^2\right ) d}-\frac {\int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 (i a-b)}+\frac {\int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 (i a+b)}\\ &=-\frac {2 a^2 \tan ^3(c+d x)}{b \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {2 \left (16 a^4+6 a^2 b^2-5 b^4\right ) \sqrt {a+b \tan (c+d x)}}{5 b^4 \left (a^2+b^2\right ) d}-\frac {2 a \left (8 a^2+3 b^2\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{5 b^3 \left (a^2+b^2\right ) d}+\frac {2 \left (6 a^2+b^2\right ) \tan ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{5 b^2 \left (a^2+b^2\right ) d}+\frac {\text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a-i b x}} \, dx,x,i \tan (c+d x)\right )}{2 (a-i b) d}+\frac {\text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a+i b x}} \, dx,x,-i \tan (c+d x)\right )}{2 (a+i b) d}\\ &=-\frac {2 a^2 \tan ^3(c+d x)}{b \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {2 \left (16 a^4+6 a^2 b^2-5 b^4\right ) \sqrt {a+b \tan (c+d x)}}{5 b^4 \left (a^2+b^2\right ) d}-\frac {2 a \left (8 a^2+3 b^2\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{5 b^3 \left (a^2+b^2\right ) d}+\frac {2 \left (6 a^2+b^2\right ) \tan ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{5 b^2 \left (a^2+b^2\right ) d}+\frac {\text {Subst}\left (\int \frac {1}{-1+\frac {i a}{b}-\frac {i x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{(i a-b) b d}-\frac {\text {Subst}\left (\int \frac {1}{-1-\frac {i a}{b}+\frac {i x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{b (i a+b) d}\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{3/2} d}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{3/2} d}-\frac {2 a^2 \tan ^3(c+d x)}{b \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {2 \left (16 a^4+6 a^2 b^2-5 b^4\right ) \sqrt {a+b \tan (c+d x)}}{5 b^4 \left (a^2+b^2\right ) d}-\frac {2 a \left (8 a^2+3 b^2\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{5 b^3 \left (a^2+b^2\right ) d}+\frac {2 \left (6 a^2+b^2\right ) \tan ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{5 b^2 \left (a^2+b^2\right ) d}\\ \end {align*}

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Mathematica [A]
time = 6.00, size = 213, normalized size = 0.76 \begin {gather*} \frac {-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{3/2}}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{3/2}}+\frac {2 \left (16 a^5+5 a^3 b^2-6 a b^4+2 b \left (4 a^4+a^2 b^2-3 b^4\right ) \tan (c+d x)-3 a b^2 \left (a^2+b^2\right ) \tan ^2(c+d x)+b^2 \left (a^2+b^2\right ) \sec ^2(c+d x) (a+b \tan (c+d x))\right )}{b^4 \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{5 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^5/(a + b*Tan[c + d*x])^(3/2),x]

[Out]

((-5*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/(a - I*b)^(3/2) - (5*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sq
rt[a + I*b]])/(a + I*b)^(3/2) + (2*(16*a^5 + 5*a^3*b^2 - 6*a*b^4 + 2*b*(4*a^4 + a^2*b^2 - 3*b^4)*Tan[c + d*x]
- 3*a*b^2*(a^2 + b^2)*Tan[c + d*x]^2 + b^2*(a^2 + b^2)*Sec[c + d*x]^2*(a + b*Tan[c + d*x])))/(b^4*(a^2 + b^2)*
Sqrt[a + b*Tan[c + d*x]]))/(5*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(877\) vs. \(2(252)=504\).
time = 0.17, size = 878, normalized size = 3.11

method	result	size
derivativedivides	\(\frac {\frac {2 \left (a +b \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5}-2 a \left (a +b \tan \left (d x +c \right )\right )^{\frac {3}{2}}+6 a^{2} \sqrt {a +b \tan \left (d x +c \right )}-2 b^{2} \sqrt {a +b \tan \left (d x +c \right )}-\frac {2 b^{4} \left (\frac {\frac {\left (-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{2}+2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}+2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{2}\right ) \ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right )}{2}+\frac {2 \left (2 a^{4}-2 b^{4}-\frac {\left (-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{2}+2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}+2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{2}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{2}\right ) \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{4 \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}+\frac {-\frac {\left (-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{2}+2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}+2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{2}\right ) \ln \left (-b \tan \left (d x +c \right )-a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-\sqrt {a^{2}+b^{2}}\right )}{2}+\frac {2 \left (-2 a^{4}+2 b^{4}+\frac {\left (-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{2}+2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}+2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{2}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{2}\right ) \arctan \left (\frac {-2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{4 \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{a^{2}+b^{2}}+\frac {2 a^{5}}{\left (a^{2}+b^{2}\right ) \sqrt {a +b \tan \left (d x +c \right )}}}{d \,b^{4}}\)	\(878\)
default	\(\frac {\frac {2 \left (a +b \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5}-2 a \left (a +b \tan \left (d x +c \right )\right )^{\frac {3}{2}}+6 a^{2} \sqrt {a +b \tan \left (d x +c \right )}-2 b^{2} \sqrt {a +b \tan \left (d x +c \right )}-\frac {2 b^{4} \left (\frac {\frac {\left (-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{2}+2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}+2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{2}\right ) \ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right )}{2}+\frac {2 \left (2 a^{4}-2 b^{4}-\frac {\left (-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{2}+2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}+2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{2}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{2}\right ) \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{4 \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}+\frac {-\frac {\left (-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{2}+2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}+2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{2}\right ) \ln \left (-b \tan \left (d x +c \right )-a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-\sqrt {a^{2}+b^{2}}\right )}{2}+\frac {2 \left (-2 a^{4}+2 b^{4}+\frac {\left (-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{2}+2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}+2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{2}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{2}\right ) \arctan \left (\frac {-2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{4 \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{a^{2}+b^{2}}+\frac {2 a^{5}}{\left (a^{2}+b^{2}\right ) \sqrt {a +b \tan \left (d x +c \right )}}}{d \,b^{4}}\)	\(878\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^5/(a+b*tan(d*x+c))^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^5/(a+b*tan(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

Timed out

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 5852 vs. \(2 (248) = 496\).
time = 1.16, size = 5852, normalized size = 20.75 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^5/(a+b*tan(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

-1/20*(20*sqrt(2)*((a^10*b^4 + 3*a^8*b^6 + 2*a^6*b^8 - 2*a^4*b^10 - 3*a^2*b^12 - b^14)*d^5*cos(d*x + c)^4 + 2*
(a^9*b^5 + 4*a^7*b^7 + 6*a^5*b^9 + 4*a^3*b^11 + a*b^13)*d^5*cos(d*x + c)^3*sin(d*x + c) + (a^8*b^6 + 4*a^6*b^8
+ 6*a^4*b^10 + 4*a^2*b^12 + b^14)*d^5*cos(d*x + c)^2)*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 - (a^9 - 6*a^5*
b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*a^4*b^2 - 6*a^2*b^4 + b^6
))*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10
+ b^12)*d^4))*(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(3/4)*arctan(-((3*a^12 + 14*a^10*b^2 + 25*a^8*b^4
+ 20*a^6*b^6 + 5*a^4*b^8 - 2*a^2*b^10 - b^12)*d^4*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*
a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4))*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))
+ (3*a^9 + 8*a^7*b^2 + 6*a^5*b^4 - a*b^8)*d^2*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8
*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + sqrt(2)*((a^14 + 5*a^12*b^2 + 9*a^10*b^4 + 5*a^8*b
^6 - 5*a^6*b^8 - 9*a^4*b^10 - 5*a^2*b^12 - b^14)*d^7*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 +
15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4))*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^
4)) + (a^11 + 5*a^9*b^2 + 10*a^7*b^4 + 10*a^5*b^6 + 5*a^3*b^8 + a*b^10)*d^5*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)
/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))*sqrt((a^6 + 3*a^4*b^2
+ 3*a^2*b^4 + b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^
4)))/(9*a^4*b^2 - 6*a^2*b^4 + b^6))*sqrt(((9*a^8 + 12*a^6*b^2 - 2*a^4*b^4 - 4*a^2*b^6 + b^8)*d^2*sqrt(1/((a^6
+ 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*cos(d*x + c) + sqrt(2)*((9*a^9 + 12*a^7*b^2 - 2*a^5*b^4 - 4*a^3*b^6 + a*b
^8)*d^3*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*cos(d*x + c) + (9*a^6 - 15*a^4*b^2 + 7*a^2*b^4 - b^6
)*d*cos(d*x + c))*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(1
/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*a^4*b^2 - 6*a^2*b^4 + b^6))*sqrt((a*cos(d*x + c) + b*sin(d*x +
c))/cos(d*x + c))*(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(1/4) + (9*a^5 - 6*a^3*b^2 + a*b^4)*cos(d*x +
c) + (9*a^4*b - 6*a^2*b^3 + b^5)*sin(d*x + c))/cos(d*x + c))*(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(3
/4) + sqrt(2)*((3*a^16 + 14*a^14*b^2 + 22*a^12*b^4 + 6*a^10*b^6 - 20*a^8*b^8 - 22*a^6*b^10 - 6*a^4*b^12 + 2*a^
2*b^14 + b^16)*d^7*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b
^8 + 6*a^2*b^10 + b^12)*d^4))*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)) + (3*a^13 + 14*a^11*b^2 + 25*a
^9*b^4 + 20*a^7*b^6 + 5*a^5*b^8 - 2*a^3*b^10 - a*b^12)*d^5*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*
b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6
- (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*a^4*b^2 -
6*a^2*b^4 + b^6))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6
)*d^4))^(3/4))/(9*a^4*b^2 - 6*a^2*b^4 + b^6)) + 20*sqrt(2)*((a^10*b^4 + 3*a^8*b^6 + 2*a^6*b^8 - 2*a^4*b^10 - 3
*a^2*b^12 - b^14)*d^5*cos(d*x + c)^4 + 2*(a^9*b^5 + 4*a^7*b^7 + 6*a^5*b^9 + 4*a^3*b^11 + a*b^13)*d^5*cos(d*x +
c)^3*sin(d*x + c) + (a^8*b^6 + 4*a^6*b^8 + 6*a^4*b^10 + 4*a^2*b^12 + b^14)*d^5*cos(d*x + c)^2)*sqrt((a^6 + 3*
a^4*b^2 + 3*a^2*b^4 + b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 +
b^6)*d^4)))/(9*a^4*b^2 - 6*a^2*b^4 + b^6))*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^
4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4))*(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(3/4)*arc
tan(((3*a^12 + 14*a^10*b^2 + 25*a^8*b^4 + 20*a^6*b^6 + 5*a^4*b^8 - 2*a^2*b^10 - b^12)*d^4*sqrt((9*a^4*b^2 - 6*
a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4))*sqrt(1/((
a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)) + (3*a^9 + 8*a^7*b^2 + 6*a^5*b^4 - a*b^8)*d^2*sqrt((9*a^4*b^2 - 6*a^2
*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) - sqrt(2)*((
a^14 + 5*a^12*b^2 + 9*a^10*b^4 + 5*a^8*b^6 - 5*a^6*b^8 - 9*a^4*b^10 - 5*a^2*b^12 - b^14)*d^7*sqrt((9*a^4*b^2 -
6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4))*sqrt(1
/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)) + (a^11 + 5*a^9*b^2 + 10*a^7*b^4 + 10*a^5*b^6 + 5*a^3*b^8 + a*b^10
)*d^5*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^
10 + b^12)*d^4)))*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(1
/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*a^4*b^2 - 6*a^2*b^4 + b^6))*sqrt(((9*a^8 + 12*a^6*b^2 - 2*a^4*
b^4 - 4*a^2*b^6 + b^8)*d^2*sqrt(1/((a^6 + 3*a^4...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tan ^{5}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)**5/(a+b*tan(d*x+c))**(3/2),x)

[Out]

Integral(tan(c + d*x)**5/(a + b*tan(c + d*x))**(3/2), x)

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^5/(a+b*tan(d*x+c))^(3/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [B]
time = 14.85, size = 2930, normalized size = 10.39 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(c + d*x)^5/(a + b*tan(c + d*x))^(3/2),x)

[Out]

((8*a^2)/(b^4*d) - (2*(a^2 + b^2))/(b^4*d))*(a + b*tan(c + d*x))^(1/2) - atan(((1/(a^3*d^2 + b^3*d^2*1i - 3*a*
b^2*d^2 - a^2*b*d^2*3i))^(1/2)*(((1/(a^3*d^2 + b^3*d^2*1i - 3*a*b^2*d^2 - a^2*b*d^2*3i))^(1/2)*(32*a^6*b^6*d^4
- 48*a^2*b^10*d^4 - 32*a^4*b^8*d^4 - 16*b^12*d^4 + 48*a^8*b^4*d^4 + 16*a^10*b^2*d^4 + ((1/(a^3*d^2 + b^3*d^2*
1i - 3*a*b^2*d^2 - a^2*b*d^2*3i))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^12*d^5 + 320*a^3*b^10*d^5 + 640*a^5
*b^8*d^5 + 640*a^7*b^6*d^5 + 320*a^9*b^4*d^5 + 64*a^11*b^2*d^5))/4))/2 + ((a + b*tan(c + d*x))^(1/2)*(16*b^10*
d^3 + 32*a^2*b^8*d^3 - 32*a^6*b^4*d^3 - 16*a^8*b^2*d^3))/2)*1i + (1/(a^3*d^2 + b^3*d^2*1i - 3*a*b^2*d^2 - a^2*
b*d^2*3i))^(1/2)*(((1/(a^3*d^2 + b^3*d^2*1i - 3*a*b^2*d^2 - a^2*b*d^2*3i))^(1/2)*(16*b^12*d^4 + 48*a^2*b^10*d^
4 + 32*a^4*b^8*d^4 - 32*a^6*b^6*d^4 - 48*a^8*b^4*d^4 - 16*a^10*b^2*d^4 + ((1/(a^3*d^2 + b^3*d^2*1i - 3*a*b^2*d
^2 - a^2*b*d^2*3i))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^12*d^5 + 320*a^3*b^10*d^5 + 640*a^5*b^8*d^5 + 640
*a^7*b^6*d^5 + 320*a^9*b^4*d^5 + 64*a^11*b^2*d^5))/4))/2 + ((a + b*tan(c + d*x))^(1/2)*(16*b^10*d^3 + 32*a^2*b
^8*d^3 - 32*a^6*b^4*d^3 - 16*a^8*b^2*d^3))/2)*1i)/((1/(a^3*d^2 + b^3*d^2*1i - 3*a*b^2*d^2 - a^2*b*d^2*3i))^(1/
2)*(((1/(a^3*d^2 + b^3*d^2*1i - 3*a*b^2*d^2 - a^2*b*d^2*3i))^(1/2)*(16*b^12*d^4 + 48*a^2*b^10*d^4 + 32*a^4*b^8
*d^4 - 32*a^6*b^6*d^4 - 48*a^8*b^4*d^4 - 16*a^10*b^2*d^4 + ((1/(a^3*d^2 + b^3*d^2*1i - 3*a*b^2*d^2 - a^2*b*d^2
*3i))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^12*d^5 + 320*a^3*b^10*d^5 + 640*a^5*b^8*d^5 + 640*a^7*b^6*d^5 +
320*a^9*b^4*d^5 + 64*a^11*b^2*d^5))/4))/2 + ((a + b*tan(c + d*x))^(1/2)*(16*b^10*d^3 + 32*a^2*b^8*d^3 - 32*a^
6*b^4*d^3 - 16*a^8*b^2*d^3))/2) - (1/(a^3*d^2 + b^3*d^2*1i - 3*a*b^2*d^2 - a^2*b*d^2*3i))^(1/2)*(((1/(a^3*d^2
+ b^3*d^2*1i - 3*a*b^2*d^2 - a^2*b*d^2*3i))^(1/2)*(32*a^6*b^6*d^4 - 48*a^2*b^10*d^4 - 32*a^4*b^8*d^4 - 16*b^12
*d^4 + 48*a^8*b^4*d^4 + 16*a^10*b^2*d^4 + ((1/(a^3*d^2 + b^3*d^2*1i - 3*a*b^2*d^2 - a^2*b*d^2*3i))^(1/2)*(a +
b*tan(c + d*x))^(1/2)*(64*a*b^12*d^5 + 320*a^3*b^10*d^5 + 640*a^5*b^8*d^5 + 640*a^7*b^6*d^5 + 320*a^9*b^4*d^5
+ 64*a^11*b^2*d^5))/4))/2 + ((a + b*tan(c + d*x))^(1/2)*(16*b^10*d^3 + 32*a^2*b^8*d^3 - 32*a^6*b^4*d^3 - 16*a^
8*b^2*d^3))/2) + 16*a*b^8*d^2 + 48*a^3*b^6*d^2 + 48*a^5*b^4*d^2 + 16*a^7*b^2*d^2))*(1/(a^3*d^2 + b^3*d^2*1i -
3*a*b^2*d^2 - a^2*b*d^2*3i))^(1/2)*1i - atan((((1i/(4*(a^3*d^2*1i + b^3*d^2 - a*b^2*d^2*3i - 3*a^2*b*d^2)))^(1
/2)*((1i/(4*(a^3*d^2*1i + b^3*d^2 - a*b^2*d^2*3i - 3*a^2*b*d^2)))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^12*
d^5 + 320*a^3*b^10*d^5 + 640*a^5*b^8*d^5 + 640*a^7*b^6*d^5 + 320*a^9*b^4*d^5 + 64*a^11*b^2*d^5) - 32*b^12*d^4
- 96*a^2*b^10*d^4 - 64*a^4*b^8*d^4 + 64*a^6*b^6*d^4 + 96*a^8*b^4*d^4 + 32*a^10*b^2*d^4) + (a + b*tan(c + d*x))
^(1/2)*(16*b^10*d^3 + 32*a^2*b^8*d^3 - 32*a^6*b^4*d^3 - 16*a^8*b^2*d^3))*(1i/(4*(a^3*d^2*1i + b^3*d^2 - a*b^2*
d^2*3i - 3*a^2*b*d^2)))^(1/2)*1i + ((1i/(4*(a^3*d^2*1i + b^3*d^2 - a*b^2*d^2*3i - 3*a^2*b*d^2)))^(1/2)*(32*b^1
2*d^4 + (1i/(4*(a^3*d^2*1i + b^3*d^2 - a*b^2*d^2*3i - 3*a^2*b*d^2)))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^
12*d^5 + 320*a^3*b^10*d^5 + 640*a^5*b^8*d^5 + 640*a^7*b^6*d^5 + 320*a^9*b^4*d^5 + 64*a^11*b^2*d^5) + 96*a^2*b^
10*d^4 + 64*a^4*b^8*d^4 - 64*a^6*b^6*d^4 - 96*a^8*b^4*d^4 - 32*a^10*b^2*d^4) + (a + b*tan(c + d*x))^(1/2)*(16*
b^10*d^3 + 32*a^2*b^8*d^3 - 32*a^6*b^4*d^3 - 16*a^8*b^2*d^3))*(1i/(4*(a^3*d^2*1i + b^3*d^2 - a*b^2*d^2*3i - 3*
a^2*b*d^2)))^(1/2)*1i)/(((1i/(4*(a^3*d^2*1i + b^3*d^2 - a*b^2*d^2*3i - 3*a^2*b*d^2)))^(1/2)*(32*b^12*d^4 + (1i
/(4*(a^3*d^2*1i + b^3*d^2 - a*b^2*d^2*3i - 3*a^2*b*d^2)))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^12*d^5 + 32
0*a^3*b^10*d^5 + 640*a^5*b^8*d^5 + 640*a^7*b^6*d^5 + 320*a^9*b^4*d^5 + 64*a^11*b^2*d^5) + 96*a^2*b^10*d^4 + 64
*a^4*b^8*d^4 - 64*a^6*b^6*d^4 - 96*a^8*b^4*d^4 - 32*a^10*b^2*d^4) + (a + b*tan(c + d*x))^(1/2)*(16*b^10*d^3 +
32*a^2*b^8*d^3 - 32*a^6*b^4*d^3 - 16*a^8*b^2*d^3))*(1i/(4*(a^3*d^2*1i + b^3*d^2 - a*b^2*d^2*3i - 3*a^2*b*d^2))
)^(1/2) - ((1i/(4*(a^3*d^2*1i + b^3*d^2 - a*b^2*d^2*3i - 3*a^2*b*d^2)))^(1/2)*((1i/(4*(a^3*d^2*1i + b^3*d^2 -
a*b^2*d^2*3i - 3*a^2*b*d^2)))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^12*d^5 + 320*a^3*b^10*d^5 + 640*a^5*b^8
*d^5 + 640*a^7*b^6*d^5 + 320*a^9*b^4*d^5 + 64*a^11*b^2*d^5) - 32*b^12*d^4 - 96*a^2*b^10*d^4 - 64*a^4*b^8*d^4 +
64*a^6*b^6*d^4 + 96*a^8*b^4*d^4 + 32*a^10*b^2*d^4) + (a + b*tan(c + d*x))^(1/2)*(16*b^10*d^3 + 32*a^2*b^8*d^3
- 32*a^6*b^4*d^3 - 16*a^8*b^2*d^3))*(1i/(4*(a^3*d^2*1i + b^3*d^2 - a*b^2*d^2*3i - 3*a^2*b*d^2)))^(1/2) + 16*a
*b^8*d^2 + 48*a^3*b^6*d^2 + 48*a^5*b^4*d^2 + 16*a^7*b^2*d^2))*(1i/(4*(a^3*d^2*1i + b^3*d^2 - a*b^2*d^2*3i - 3*
a^2*b*d^2)))^(1/2)*2i + (2*(a + b*tan(c + d*x))^(5/2))/(5*b^4*d) - (2*a*(a + b*tan(c + d*x))^(3/2))/(b^4*d) +
(2*a^5)/(b^4*d*(a^2 + b^2)*(a + b*tan(c + d*x))^(1/2))

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