3.89 \(\int \cot ^5(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx\)

Optimal. Leaf size=261 \[ -\frac{3 a^{5/2} (121 A-120 i B) \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right )}{64 d}+\frac{4 \sqrt{2} a^{5/2} (A-i B) \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{d}-\frac{a^2 (8 B+11 i A) \cot ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{24 d}+\frac{a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{96 d}+\frac{a^2 (152 B+149 i A) \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{64 d}-\frac{a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d} \]

[Out]

(-3*a^(5/2)*(121*A - (120*I)*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(64*d) + (4*Sqrt[2]*a^(5/2)*(A -
I*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d + (a^2*((149*I)*A + 152*B)*Cot[c + d*x]*Sqrt[a +
I*a*Tan[c + d*x]])/(64*d) + (a^2*(107*A - (104*I)*B)*Cot[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(96*d) - (a^2
*((11*I)*A + 8*B)*Cot[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]])/(24*d) - (a*A*Cot[c + d*x]^4*(a + I*a*Tan[c + d*x
])^(3/2))/(4*d)

________________________________________________________________________________________

Rubi [A]  time = 1.0046, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {3593, 3598, 3600, 3480, 206, 3599, 63, 208} \[ -\frac{3 a^{5/2} (121 A-120 i B) \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right )}{64 d}+\frac{4 \sqrt{2} a^{5/2} (A-i B) \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{d}-\frac{a^2 (8 B+11 i A) \cot ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{24 d}+\frac{a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{96 d}+\frac{a^2 (152 B+149 i A) \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{64 d}-\frac{a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^5*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]

[Out]

(-3*a^(5/2)*(121*A - (120*I)*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(64*d) + (4*Sqrt[2]*a^(5/2)*(A -
I*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d + (a^2*((149*I)*A + 152*B)*Cot[c + d*x]*Sqrt[a +
I*a*Tan[c + d*x]])/(64*d) + (a^2*(107*A - (104*I)*B)*Cot[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(96*d) - (a^2
*((11*I)*A + 8*B)*Cot[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]])/(24*d) - (a*A*Cot[c + d*x]^4*(a + I*a*Tan[c + d*x
])^(3/2))/(4*d)

Rule 3593

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

Rule 3598

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

Rule 3600

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

Rule 3480

Int[Sqrt[(a_) + (b_.)*tan[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[(-2*b)/d, Subst[Int[1/(2*a - x^2), x], x, Sq
rt[a + b*Tan[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 3599

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

Rule 63

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 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]

Rubi steps

\begin{align*} \int \cot ^5(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx &=-\frac{a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}+\frac{1}{4} \int \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2} \left (\frac{1}{2} a (11 i A+8 B)-\frac{1}{2} a (5 A-8 i B) \tan (c+d x)\right ) \, dx\\ &=-\frac{a^2 (11 i A+8 B) \cot ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{24 d}-\frac{a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}+\frac{1}{12} \int \cot ^3(c+d x) \sqrt{a+i a \tan (c+d x)} \left (-\frac{1}{4} a^2 (107 A-104 i B)-\frac{1}{4} a^2 (85 i A+88 B) \tan (c+d x)\right ) \, dx\\ &=\frac{a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{96 d}-\frac{a^2 (11 i A+8 B) \cot ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{24 d}-\frac{a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}+\frac{\int \cot ^2(c+d x) \sqrt{a+i a \tan (c+d x)} \left (-\frac{3}{8} a^3 (149 i A+152 B)+\frac{3}{8} a^3 (107 A-104 i B) \tan (c+d x)\right ) \, dx}{24 a}\\ &=\frac{a^2 (149 i A+152 B) \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{64 d}+\frac{a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{96 d}-\frac{a^2 (11 i A+8 B) \cot ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{24 d}-\frac{a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}+\frac{\int \cot (c+d x) \sqrt{a+i a \tan (c+d x)} \left (\frac{9}{16} a^4 (121 A-120 i B)+\frac{3}{16} a^4 (149 i A+152 B) \tan (c+d x)\right ) \, dx}{24 a^2}\\ &=\frac{a^2 (149 i A+152 B) \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{64 d}+\frac{a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{96 d}-\frac{a^2 (11 i A+8 B) \cot ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{24 d}-\frac{a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}+\frac{1}{128} (3 a (121 A-120 i B)) \int \cot (c+d x) (a-i a \tan (c+d x)) \sqrt{a+i a \tan (c+d x)} \, dx+\left (4 a^2 (i A+B)\right ) \int \sqrt{a+i a \tan (c+d x)} \, dx\\ &=\frac{a^2 (149 i A+152 B) \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{64 d}+\frac{a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{96 d}-\frac{a^2 (11 i A+8 B) \cot ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{24 d}-\frac{a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}+\frac{\left (8 a^3 (A-i B)\right ) \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\sqrt{a+i a \tan (c+d x)}\right )}{d}+\frac{\left (3 a^3 (121 A-120 i B)\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+i a x}} \, dx,x,\tan (c+d x)\right )}{128 d}\\ &=\frac{4 \sqrt{2} a^{5/2} (A-i B) \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{d}+\frac{a^2 (149 i A+152 B) \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{64 d}+\frac{a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{96 d}-\frac{a^2 (11 i A+8 B) \cot ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{24 d}-\frac{a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}-\frac{\left (3 a^2 (121 i A+120 B)\right ) \operatorname{Subst}\left (\int \frac{1}{i-\frac{i x^2}{a}} \, dx,x,\sqrt{a+i a \tan (c+d x)}\right )}{64 d}\\ &=-\frac{3 a^{5/2} (121 A-120 i B) \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right )}{64 d}+\frac{4 \sqrt{2} a^{5/2} (A-i B) \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{d}+\frac{a^2 (149 i A+152 B) \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{64 d}+\frac{a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{96 d}-\frac{a^2 (11 i A+8 B) \cot ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{24 d}-\frac{a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}\\ \end{align*}

Mathematica [B]  time = 8.98414, size = 698, normalized size = 2.67 \[ \frac{\cos ^3(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \left (\csc (c) \left (\frac{1}{24} \cos (2 c)-\frac{1}{24} i \sin (2 c)\right ) \csc ^3(c+d x) (8 B \sin (d x)+17 i A \sin (d x))+\csc (c) \left (\frac{1}{192} \cos (3 c)-\frac{1}{192} i \sin (3 c)\right ) \csc ^2(c+d x) (223 A \sin (2 c)-223 i A \cos (2 c)+87 i A-136 i B \sin (2 c)-136 B \cos (2 c)+72 B)+\csc (c) \left (\frac{1}{192} \cos (2 c)-\frac{1}{192} i \sin (2 c)\right ) \csc (c+d x) (-520 B \sin (d x)-583 i A \sin (d x))+\csc (c) \left (\frac{1}{192} \cos (2 c)-\frac{1}{192} i \sin (2 c)\right ) (-262 A \sin (c)+583 i A \cos (c)+208 i B \sin (c)+520 B \cos (c))+\left (-\frac{1}{4} A \cos (2 c)+\frac{1}{4} i A \sin (2 c)\right ) \csc ^4(c+d x)\right )}{d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}+\frac{e^{-2 i c} \sqrt{e^{i d x}} (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \left (3 \sqrt{2} (121 A-120 i B) \left (\log \left (\left (-1+e^{i (c+d x)}\right )^2\right )-\log \left (\left (1+e^{i (c+d x)}\right )^2\right )+\log \left (-2 e^{i (c+d x)} \left (1+\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}\right )+3 e^{2 i (c+d x)}+2 \sqrt{2} \sqrt{1+e^{2 i (c+d x)}}+3\right )-\log \left (2 e^{i (c+d x)} \left (1+\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}\right )+3 e^{2 i (c+d x)}+2 \sqrt{2} \sqrt{1+e^{2 i (c+d x)}}+3\right )\right )+2048 (A-i B) \sinh ^{-1}\left (e^{i (c+d x)}\right )\right )}{256 \sqrt{2} d \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \sec ^{\frac{7}{2}}(c+d x) (\cos (d x)+i \sin (d x))^{5/2} (A \cos (c+d x)+B \sin (c+d x))} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cot[c + d*x]^5*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]

[Out]

(Sqrt[E^(I*d*x)]*(2048*(A - I*B)*ArcSinh[E^(I*(c + d*x))] + 3*Sqrt[2]*(121*A - (120*I)*B)*(Log[(-1 + E^(I*(c +
d*x)))^2] - Log[(1 + E^(I*(c + d*x)))^2] + Log[3 + 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d
*x))] - 2*E^(I*(c + d*x))*(1 + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] - Log[3 + 3*E^((2*I)*(c + d*x)) + 2*Sqr
t[2]*Sqrt[1 + E^((2*I)*(c + d*x))] + 2*E^(I*(c + d*x))*(1 + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])]))*(a + I*a
*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/(256*Sqrt[2]*d*E^((2*I)*c)*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c +
d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Sec[c + d*x]^(7/2)*(Cos[d*x] + I*Sin[d*x])^(5/2)*(A*Cos[c + d*x] + B*Sin
[c + d*x])) + (Cos[c + d*x]^3*(Csc[c]*((583*I)*A*Cos[c] + 520*B*Cos[c] - 262*A*Sin[c] + (208*I)*B*Sin[c])*(Cos
[2*c]/192 - (I/192)*Sin[2*c]) + Csc[c + d*x]^4*(-(A*Cos[2*c])/4 + (I/4)*A*Sin[2*c]) + Csc[c]*Csc[c + d*x]^2*((
87*I)*A + 72*B - (223*I)*A*Cos[2*c] - 136*B*Cos[2*c] + 223*A*Sin[2*c] - (136*I)*B*Sin[2*c])*(Cos[3*c]/192 - (I
/192)*Sin[3*c]) + Csc[c]*Csc[c + d*x]*(Cos[2*c]/192 - (I/192)*Sin[2*c])*((-583*I)*A*Sin[d*x] - 520*B*Sin[d*x])
+ Csc[c]*Csc[c + d*x]^3*(Cos[2*c]/24 - (I/24)*Sin[2*c])*((17*I)*A*Sin[d*x] + 8*B*Sin[d*x]))*(a + I*a*Tan[c +
d*x])^(5/2)*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x]))

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Maple [B]  time = 0.429, size = 3444, normalized size = 13.2 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^5*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x)

[Out]

1/384/d*a^2*(a*(I*sin(d*x+c)+cos(d*x+c))/cos(d*x+c))^(1/2)*(1536*I*B*cos(d*x+c)^5*(-2*cos(d*x+c)/(cos(d*x+c)+1
))^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*2^(1/2)+1536*I*A*cos(
d*x+c)^4*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*2^(1/2)
+1536*I*B*cos(d*x+c)^4*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))
^(1/2)*sin(d*x+c)/cos(d*x+c))*2^(1/2)-3072*I*A*cos(d*x+c)^3*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/2*2^
(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*2^(1/2)-3072*I*B*cos(d*x+c)^3*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)
*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*2^(1/2)-3072*I*A*cos(d*x+c)^2
*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*2^(1/2)-3072*I*
B*cos(d*x+c)^2*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*s
in(d*x+c)/cos(d*x+c))*2^(1/2)+1536*I*A*cos(d*x+c)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/2*2^(1/2)*(-2*
cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*2^(1/2)+1536*I*B*cos(d*x+c)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(1/2
*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*2^(1/2)-1536*A*2^(1/2)*cos(d*x+c)*(-2*cos
(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))+
1536*B*2^(1/2)*cos(d*x+c)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1
))^(1/2))+3072*A*2^(1/2)*cos(d*x+c)^3*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/
(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))-3072*B*2^(1/2)*cos(d*x+c)^3*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*
arctan(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+3072*A*2^(1/2)*cos(d*x+c)^2*(-2*cos(d*x+c)/(cos(d*x+c
)+1))^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))-3072*B*2^(1/2)*cos
(d*x+c)^2*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))-1690*A
*cos(d*x+c)^4*sin(d*x+c)+1690*I*A*cos(d*x+c)^5+524*I*A*cos(d*x+c)^4-2488*I*A*cos(d*x+c)^3-428*I*A*cos(d*x+c)^2
+894*I*A*cos(d*x+c)+1536*I*A*cos(d*x+c)^5*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/2*2^(1/2)*(-2*cos(d*x+
c)/(cos(d*x+c)+1))^(1/2))*2^(1/2)+894*A*cos(d*x+c)*sin(d*x+c)-1166*A*cos(d*x+c)^3*sin(d*x+c)+1322*A*cos(d*x+c)
^2*sin(d*x+c)-1089*A*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*ln(-(-(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c
)+cos(d*x+c)-1)/sin(d*x+c))+1080*B*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/(-2*cos(d*x+c)/(cos(d*x+c)+1)
)^(1/2))+1456*cos(d*x+c)^5*B+1080*B*cos(d*x+c)^4*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/(-2*cos(d*x+c)/
(cos(d*x+c)+1))^(1/2))+1456*I*B*cos(d*x+c)^4*sin(d*x+c)+1040*I*B*cos(d*x+c)^3*sin(d*x+c)-1328*I*B*sin(d*x+c)*c
os(d*x+c)^2+1089*I*A*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))-912*I
*B*cos(d*x+c)*sin(d*x+c)+1080*I*B*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*ln(-(-(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1
/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))-1089*A*cos(d*x+c)^5*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*ln(-(-(-2*co
s(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))+1080*B*cos(d*x+c)^5*(-2*cos(d*x+c)/(cos(d*
x+c)+1))^(1/2)*arctan(1/(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))-1089*A*cos(d*x+c)^4*(-2*cos(d*x+c)/(cos(d*x+c)+1
))^(1/2)*ln(-(-(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))+2178*A*cos(d*x+c)^3*(
-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*ln(-(-(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*
x+c))-2160*B*cos(d*x+c)^3*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+
2178*A*cos(d*x+c)^2*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*ln(-(-(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)
+cos(d*x+c)-1)/sin(d*x+c))-2160*B*cos(d*x+c)^2*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/(-2*cos(d*x+c)/(c
os(d*x+c)+1))^(1/2))-1536*A*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(c
os(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))-1089*A*cos(d*x+c)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*ln(-(-(-2*co
s(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))+1536*B*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+
1))^(1/2)*arctan(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+1080*B*cos(d*x+c)*(-2*cos(d*x+c)/(cos(d*x+c
)+1))^(1/2)*arctan(1/(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+912*B*cos(d*x+c)-2368*cos(d*x+c)^3*B-1536*A*cos(d*x
+c)^5*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)
/cos(d*x+c))*2^(1/2)+1536*B*cos(d*x+c)^5*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/2*2^(1/2)*(-2*cos(d*x+c
)/(cos(d*x+c)+1))^(1/2))*2^(1/2)-1536*A*cos(d*x+c)^4*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(1/2*2^(1/2)*
(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*2^(1/2)+1536*B*cos(d*x+c)^4*(-2*cos(d*x+c)/(cos(d*
x+c)+1))^(1/2)*arctan(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*2^(1/2)+1089*I*A*cos(d*x+c)^5*(-2*cos(
d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+1080*I*B*cos(d*x+c)^5*(-2*cos(d*x+
c)/(cos(d*x+c)+1))^(1/2)*ln(-(-(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))+1089*
I*A*cos(d*x+c)^4*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+1080*I*B*
cos(d*x+c)^4*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*ln(-(-(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)+cos(d*
x+c)-1)/sin(d*x+c))-2178*I*A*cos(d*x+c)^3*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/(-2*cos(d*x+c)/(cos(d*
x+c)+1))^(1/2))-2160*I*B*cos(d*x+c)^3*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*ln(-(-(-2*cos(d*x+c)/(cos(d*x+c)+1)
)^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))-2178*I*A*cos(d*x+c)^2*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan
(1/(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))-2160*I*B*cos(d*x+c)^2*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*ln(-(-(-2*
cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))+1536*I*A*(-2*cos(d*x+c)/(cos(d*x+c)+1))^
(1/2)*arctan(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*2^(1/2)+1536*I*B*(-2*cos(d*x+c)/(cos(d*x+c)+1))
^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*2^(1/2)+1089*I*A*cos(d*
x+c)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+1080*I*B*cos(d*x+c)*(
-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*ln(-(-(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*
x+c))+416*cos(d*x+c)^4*B-416*B*cos(d*x+c)^2)/(cos(d*x+c)-1)/(I*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c)/(cos(d*x+c)
+1)^2

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^5*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.93416, size = 2722, normalized size = 10.43 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^5*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorithm="fricas")

[Out]

-1/384*(2*sqrt(2)*(13*(65*A - 56*I*B)*a^2*e^(8*I*d*x + 8*I*c) - 2*(215*A - 392*I*B)*a^2*e^(6*I*d*x + 6*I*c) -
4*(35*A - 104*I*B)*a^2*e^(4*I*d*x + 4*I*c) + 2*(407*A - 392*I*B)*a^2*e^(2*I*d*x + 2*I*c) - 3*(107*A - 104*I*B)
*a^2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*e^(I*d*x + I*c) + 3*sqrt((131769*A^2 - 261360*I*A*B - 129600*B^2)*a^5/
d^2)*(d*e^(8*I*d*x + 8*I*c) - 4*d*e^(6*I*d*x + 6*I*c) + 6*d*e^(4*I*d*x + 4*I*c) - 4*d*e^(2*I*d*x + 2*I*c) + d)
*log((sqrt(2)*((363*I*A + 360*B)*a^2*e^(2*I*d*x + 2*I*c) + (363*I*A + 360*B)*a^2)*sqrt(a/(e^(2*I*d*x + 2*I*c)
+ 1))*e^(I*d*x + I*c) + 2*I*sqrt((131769*A^2 - 261360*I*A*B - 129600*B^2)*a^5/d^2)*d*e^(2*I*d*x + 2*I*c))*e^(-
2*I*d*x - 2*I*c)/((363*I*A + 360*B)*a^2)) - 3*sqrt((131769*A^2 - 261360*I*A*B - 129600*B^2)*a^5/d^2)*(d*e^(8*I
*d*x + 8*I*c) - 4*d*e^(6*I*d*x + 6*I*c) + 6*d*e^(4*I*d*x + 4*I*c) - 4*d*e^(2*I*d*x + 2*I*c) + d)*log((sqrt(2)*
((363*I*A + 360*B)*a^2*e^(2*I*d*x + 2*I*c) + (363*I*A + 360*B)*a^2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*e^(I*d*x
+ I*c) - 2*I*sqrt((131769*A^2 - 261360*I*A*B - 129600*B^2)*a^5/d^2)*d*e^(2*I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I*
c)/((363*I*A + 360*B)*a^2)) - 192*sqrt((32*A^2 - 64*I*A*B - 32*B^2)*a^5/d^2)*(d*e^(8*I*d*x + 8*I*c) - 4*d*e^(6
*I*d*x + 6*I*c) + 6*d*e^(4*I*d*x + 4*I*c) - 4*d*e^(2*I*d*x + 2*I*c) + d)*log((sqrt(2)*((4*I*A + 4*B)*a^2*e^(2*
I*d*x + 2*I*c) + (4*I*A + 4*B)*a^2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*e^(I*d*x + I*c) + I*sqrt((32*A^2 - 64*I*
A*B - 32*B^2)*a^5/d^2)*d*e^(2*I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c)/((4*I*A + 4*B)*a^2)) + 192*sqrt((32*A^2 - 6
4*I*A*B - 32*B^2)*a^5/d^2)*(d*e^(8*I*d*x + 8*I*c) - 4*d*e^(6*I*d*x + 6*I*c) + 6*d*e^(4*I*d*x + 4*I*c) - 4*d*e^
(2*I*d*x + 2*I*c) + d)*log((sqrt(2)*((4*I*A + 4*B)*a^2*e^(2*I*d*x + 2*I*c) + (4*I*A + 4*B)*a^2)*sqrt(a/(e^(2*I
*d*x + 2*I*c) + 1))*e^(I*d*x + I*c) - I*sqrt((32*A^2 - 64*I*A*B - 32*B^2)*a^5/d^2)*d*e^(2*I*d*x + 2*I*c))*e^(-
2*I*d*x - 2*I*c)/((4*I*A + 4*B)*a^2)))/(d*e^(8*I*d*x + 8*I*c) - 4*d*e^(6*I*d*x + 6*I*c) + 6*d*e^(4*I*d*x + 4*I
*c) - 4*d*e^(2*I*d*x + 2*I*c) + d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**5*(a+I*a*tan(d*x+c))**(5/2)*(A+B*tan(d*x+c)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \tan \left (d x + c\right ) + A\right )}{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \cot \left (d x + c\right )^{5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^5*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorithm="giac")

[Out]

integrate((B*tan(d*x + c) + A)*(I*a*tan(d*x + c) + a)^(5/2)*cot(d*x + c)^5, x)