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

Optimal. Leaf size=259 \[ \frac {(-2 B+5 i A) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} d}+\frac {(B+i A) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{4 \sqrt {2} a^{5/2} d}-\frac {7 (3 A+i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a^3 d}+\frac {(41 A+15 i B) \cot (c+d x)}{12 a^2 d \sqrt {a+i a \tan (c+d x)}}+\frac {(19 A+9 i B) \cot (c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac {(A+i B) \cot (c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}} \]

[Out]

(5*I*A-2*B)*arctanh((a+I*a*tan(d*x+c))^(1/2)/a^(1/2))/a^(5/2)/d+1/8*(I*A+B)*arctanh(1/2*(a+I*a*tan(d*x+c))^(1/
2)*2^(1/2)/a^(1/2))/a^(5/2)/d*2^(1/2)+1/12*(41*A+15*I*B)*cot(d*x+c)/a^2/d/(a+I*a*tan(d*x+c))^(1/2)-7/4*(3*A+I*
B)*cot(d*x+c)*(a+I*a*tan(d*x+c))^(1/2)/a^3/d+1/5*(A+I*B)*cot(d*x+c)/d/(a+I*a*tan(d*x+c))^(5/2)+1/30*(19*A+9*I*
B)*cot(d*x+c)/a/d/(a+I*a*tan(d*x+c))^(3/2)

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Rubi [A]  time = 1.05, antiderivative size = 259, normalized size of antiderivative = 1.00, 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 = {3596, 3598, 3600, 3480, 206, 3599, 63, 208} \[ \frac {(-2 B+5 i A) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} d}+\frac {(B+i A) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{4 \sqrt {2} a^{5/2} d}-\frac {7 (3 A+i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a^3 d}+\frac {(41 A+15 i B) \cot (c+d x)}{12 a^2 d \sqrt {a+i a \tan (c+d x)}}+\frac {(19 A+9 i B) \cot (c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac {(A+i B) \cot (c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((5*I)*A - 2*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(a^(5/2)*d) + ((I*A + B)*ArcTanh[Sqrt[a + I*a*Ta
n[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(4*Sqrt[2]*a^(5/2)*d) + ((A + I*B)*Cot[c + d*x])/(5*d*(a + I*a*Tan[c + d*x])^(
5/2)) + ((19*A + (9*I)*B)*Cot[c + d*x])/(30*a*d*(a + I*a*Tan[c + d*x])^(3/2)) + ((41*A + (15*I)*B)*Cot[c + d*x
])/(12*a^2*d*Sqrt[a + I*a*Tan[c + d*x]]) - (7*(3*A + I*B)*Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(4*a^3*d)

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 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 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 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 3596

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*A + b*B)*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1))/(2
*f*m*(b*c - a*d)), x] + Dist[1/(2*a*m*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Si
mp[A*(b*c*m - a*d*(2*m + n + 1)) + B*(a*c*m - b*d*(n + 1)) + d*(A*b - a*B)*(m + n + 1)*Tan[e + f*x], x], x], x
] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0] &&  !GtQ[n,
0]

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 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 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]

Rubi steps

\begin {align*} \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{5/2}} \, dx &=\frac {(A+i B) \cot (c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac {\int \frac {\cot ^2(c+d x) \left (a (6 A+i B)-\frac {7}{2} a (i A-B) \tan (c+d x)\right )}{(a+i a \tan (c+d x))^{3/2}} \, dx}{5 a^2}\\ &=\frac {(A+i B) \cot (c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac {(19 A+9 i B) \cot (c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac {\int \frac {\cot ^2(c+d x) \left (\frac {5}{2} a^2 (11 A+3 i B)-\frac {5}{4} a^2 (19 i A-9 B) \tan (c+d x)\right )}{\sqrt {a+i a \tan (c+d x)}} \, dx}{15 a^4}\\ &=\frac {(A+i B) \cot (c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac {(19 A+9 i B) \cot (c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac {(41 A+15 i B) \cot (c+d x)}{12 a^2 d \sqrt {a+i a \tan (c+d x)}}+\frac {\int \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)} \left (\frac {105}{4} a^3 (3 A+i B)-\frac {15}{8} a^3 (41 i A-15 B) \tan (c+d x)\right ) \, dx}{15 a^6}\\ &=\frac {(A+i B) \cot (c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac {(19 A+9 i B) \cot (c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac {(41 A+15 i B) \cot (c+d x)}{12 a^2 d \sqrt {a+i a \tan (c+d x)}}-\frac {7 (3 A+i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a^3 d}+\frac {\int \cot (c+d x) \sqrt {a+i a \tan (c+d x)} \left (-\frac {15}{2} a^4 (5 i A-2 B)-\frac {105}{8} a^4 (3 A+i B) \tan (c+d x)\right ) \, dx}{15 a^7}\\ &=\frac {(A+i B) \cot (c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac {(19 A+9 i B) \cot (c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac {(41 A+15 i B) \cot (c+d x)}{12 a^2 d \sqrt {a+i a \tan (c+d x)}}-\frac {7 (3 A+i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a^3 d}-\frac {(5 i A-2 B) \int \cot (c+d x) (a-i a \tan (c+d x)) \sqrt {a+i a \tan (c+d x)} \, dx}{2 a^4}-\frac {(A-i B) \int \sqrt {a+i a \tan (c+d x)} \, dx}{8 a^3}\\ &=\frac {(A+i B) \cot (c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac {(19 A+9 i B) \cot (c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac {(41 A+15 i B) \cot (c+d x)}{12 a^2 d \sqrt {a+i a \tan (c+d x)}}-\frac {7 (3 A+i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a^3 d}-\frac {(5 i A-2 B) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+i a x}} \, dx,x,\tan (c+d x)\right )}{2 a^2 d}+\frac {(i A+B) \operatorname {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{4 a^2 d}\\ &=\frac {(i A+B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{4 \sqrt {2} a^{5/2} d}+\frac {(A+i B) \cot (c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac {(19 A+9 i B) \cot (c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac {(41 A+15 i B) \cot (c+d x)}{12 a^2 d \sqrt {a+i a \tan (c+d x)}}-\frac {7 (3 A+i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a^3 d}-\frac {(5 A+2 i B) \operatorname {Subst}\left (\int \frac {1}{i-\frac {i x^2}{a}} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{a^3 d}\\ &=\frac {(5 i A-2 B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} d}+\frac {(i A+B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{4 \sqrt {2} a^{5/2} d}+\frac {(A+i B) \cot (c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac {(19 A+9 i B) \cot (c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac {(41 A+15 i B) \cot (c+d x)}{12 a^2 d \sqrt {a+i a \tan (c+d x)}}-\frac {7 (3 A+i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a^3 d}\\ \end {align*}

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Mathematica [A]  time = 8.66, size = 287, normalized size = 1.11 \[ \frac {\sec ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x)) \left (\sqrt {2} e^{2 i (c+d x)} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \left ((B+i A) \sinh ^{-1}\left (e^{i (c+d x)}\right )+4 \sqrt {2} (-2 B+5 i A) \tanh ^{-1}\left (\frac {\sqrt {2} e^{i (c+d x)}}{\sqrt {1+e^{2 i (c+d x)}}}\right )\right )+\frac {14 \sec (c+d x) (2 (9 B-29 i A) \cos (2 (c+d x))-13 i A+3 B)-40 \csc (c+d x) ((20 A+6 i B) \cos (2 (c+d x))-17 A-6 i B)}{15 \sec ^{\frac {3}{2}}(c+d x)}\right )}{8 d (a+i a \tan (c+d x))^{5/2} (A \cos (c+d x)+B \sin (c+d x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sec[c + d*x]^(3/2)*(Sqrt[2]*E^((2*I)*(c + d*x))*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((
2*I)*(c + d*x))]*((I*A + B)*ArcSinh[E^(I*(c + d*x))] + 4*Sqrt[2]*((5*I)*A - 2*B)*ArcTanh[(Sqrt[2]*E^(I*(c + d*
x)))/Sqrt[1 + E^((2*I)*(c + d*x))]]) + (-40*(-17*A - (6*I)*B + (20*A + (6*I)*B)*Cos[2*(c + d*x)])*Csc[c + d*x]
+ 14*((-13*I)*A + 3*B + 2*((-29*I)*A + 9*B)*Cos[2*(c + d*x)])*Sec[c + d*x])/(15*Sec[c + d*x]^(3/2)))*(A + B*T
an[c + d*x]))/(8*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^(5/2))

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fricas [B]  time = 0.79, size = 829, normalized size = 3.20 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(15*sqrt(1/2)*(a^3*d*e^(7*I*d*x + 7*I*c) - a^3*d*e^(5*I*d*x + 5*I*c))*sqrt(-(A^2 - 2*I*A*B - B^2)/(a^5*d
^2))*log((4*sqrt(2)*sqrt(1/2)*(a^3*d*e^(2*I*d*x + 2*I*c) + a^3*d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(-(A^2
- 2*I*A*B - B^2)/(a^5*d^2)) + (4*I*A + 4*B)*a*e^(I*d*x + I*c))*e^(-I*d*x - I*c)/(I*A + B)) - 15*sqrt(1/2)*(a^
3*d*e^(7*I*d*x + 7*I*c) - a^3*d*e^(5*I*d*x + 5*I*c))*sqrt(-(A^2 - 2*I*A*B - B^2)/(a^5*d^2))*log(-(4*sqrt(2)*sq
rt(1/2)*(a^3*d*e^(2*I*d*x + 2*I*c) + a^3*d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(-(A^2 - 2*I*A*B - B^2)/(a^5
*d^2)) - (4*I*A + 4*B)*a*e^(I*d*x + I*c))*e^(-I*d*x - I*c)/(I*A + B)) - 30*(a^3*d*e^(7*I*d*x + 7*I*c) - a^3*d*
e^(5*I*d*x + 5*I*c))*sqrt(-(25*A^2 + 20*I*A*B - 4*B^2)/(a^5*d^2))*log(((-240*I*A + 96*B)*a^2*e^(2*I*d*x + 2*I*
c) + (-80*I*A + 32*B)*a^2 + 32*sqrt(2)*(a^4*d*e^(3*I*d*x + 3*I*c) + a^4*d*e^(I*d*x + I*c))*sqrt(a/(e^(2*I*d*x
+ 2*I*c) + 1))*sqrt(-(25*A^2 + 20*I*A*B - 4*B^2)/(a^5*d^2)))*e^(-2*I*d*x - 2*I*c)/(-5*I*A + 2*B)) + 30*(a^3*d*
e^(7*I*d*x + 7*I*c) - a^3*d*e^(5*I*d*x + 5*I*c))*sqrt(-(25*A^2 + 20*I*A*B - 4*B^2)/(a^5*d^2))*log(((-240*I*A +
96*B)*a^2*e^(2*I*d*x + 2*I*c) + (-80*I*A + 32*B)*a^2 - 32*sqrt(2)*(a^4*d*e^(3*I*d*x + 3*I*c) + a^4*d*e^(I*d*x
+ I*c))*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(-(25*A^2 + 20*I*A*B - 4*B^2)/(a^5*d^2)))*e^(-2*I*d*x - 2*I*c)/
(-5*I*A + 2*B)) + sqrt(2)*((-403*I*A + 123*B)*e^(8*I*d*x + 8*I*c) + (-151*I*A + 21*B)*e^(6*I*d*x + 6*I*c) + (2
80*I*A - 120*B)*e^(4*I*d*x + 4*I*c) + (31*I*A - 21*B)*e^(2*I*d*x + 2*I*c) + 3*I*A - 3*B)*sqrt(a/(e^(2*I*d*x +
2*I*c) + 1)))/(a^3*d*e^(7*I*d*x + 7*I*c) - a^3*d*e^(5*I*d*x + 5*I*c))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \cot \left (d x + c\right )^{2}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 3.65, size = 2858, normalized size = 11.03 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/240/d*(a*(I*sin(d*x+c)+cos(d*x+c))/cos(d*x+c))^(1/2)*(15*A*2^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arc
tan(1/2*(I*cos(d*x+c)-I-sin(d*x+c))/sin(d*x+c)/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))*sin(d*x+c)*cos(d*
x+c)^2+300*B*cos(d*x+c)^2+668*A*cos(d*x+c)^3*sin(d*x+c)-120*B*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/(-
2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))-300*A*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x
+c)))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))*cos(d*x+c)-1260*A*cos(d*x+c)*sin(d*x+c)-204*B*cos(d*x+c)^4-30
0*A*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/s
in(d*x+c))-15*I*B*2^(1/2)*sin(d*x+c)*cos(d*x+c)^2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/2*(I*cos(d*x+c
)-I-sin(d*x+c))/sin(d*x+c)/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))+300*A*sin(d*x+c)*(-2*cos(d*x+c)/(1+co
s(d*x+c)))^(1/2)*arctan(1/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))-120*B*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c))
)^(1/2)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))+300*A*(-2*cos(d*x+c)/(
1+cos(d*x+c)))^(1/2)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))*cos(d*x+c
)^3+120*B*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))*cos(d*x+c)^3+300
*A*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/si
n(d*x+c))*cos(d*x+c)^2+120*B*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2
))*cos(d*x+c)^2-120*B*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))*cos(
d*x+c)-15*B*2^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/2*(I*cos(d*x+c)-I-sin(d*x+c))/sin(d*x+c)/(-2
*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))+192*I*B*sin(d*x+c)*cos(d*x+c)^7+228*I*B*sin(d*x+c)*cos(d*x+c)^3-420
*I*B*sin(d*x+c)*cos(d*x+c)+300*I*A*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/(-2*cos(d*x+c)/(1+cos(d*x+c))
)^(1/2))-120*I*B*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+co
s(d*x+c)-1)/sin(d*x+c))+160*A*cos(d*x+c)^5*sin(d*x+c)+192*I*A*cos(d*x+c)^8+64*I*A*cos(d*x+c)^6+564*I*A*cos(d*x
+c)^4-820*I*A*cos(d*x+c)^2+192*A*sin(d*x+c)*cos(d*x+c)^7-300*I*A*cos(d*x+c)^2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(
1/2)*arctan(1/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))+120*I*B*cos(d*x+c)^2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*
ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))+120*I*B*sin(d*x+c)*(-2*cos(d*x
+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))-15*I*A*2^(1/2)*(-2*cos(d*x+c)/(1+cos(
d*x+c)))^(1/2)*arctan(1/2*(I*cos(d*x+c)-I-sin(d*x+c))/sin(d*x+c)/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))
+300*I*A*cos(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))-120*I*
B*cos(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+cos(d*
x+c)-1)/sin(d*x+c))+300*I*A*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)
))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))-300*I*A*cos(d*x+c)^3*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan
(1/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))+120*I*B*cos(d*x+c)^3*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*ln(-(-(-2*c
os(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))+15*B*2^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c)
))^(1/2)*arctan(1/2*(I*cos(d*x+c)-I-sin(d*x+c))/sin(d*x+c)/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))*cos(d
*x+c)^3+15*B*2^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/2*(I*cos(d*x+c)-I-sin(d*x+c))/sin(d*x+c)/(-
2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))*cos(d*x+c)^2-15*B*2^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arc
tan(1/2*(I*cos(d*x+c)-I-sin(d*x+c))/sin(d*x+c)/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))*cos(d*x+c)-15*A*2
^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/2*(I*cos(d*x+c)-I-sin(d*x+c))/sin(d*x+c)/(-2*cos(d*x+c)/(
1+cos(d*x+c)))^(1/2)*2^(1/2))*sin(d*x+c)+96*B*cos(d*x+c)^6-300*A*cos(d*x+c)^2*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos
(d*x+c)))^(1/2)*arctan(1/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))+120*B*cos(d*x+c)^2*sin(d*x+c)*(-2*cos(d*x+c)/(1
+cos(d*x+c)))^(1/2)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))-192*B*cos(
d*x+c)^8+15*I*B*2^(1/2)*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/2*(I*cos(d*x+c)-I-sin(d*x+c))
/sin(d*x+c)/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))-300*I*A*sin(d*x+c)*cos(d*x+c)^2*(-2*cos(d*x+c)/(1+co
s(d*x+c)))^(1/2)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))-120*I*B*sin(d
*x+c)*cos(d*x+c)^2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))+15*I*A*
2^(1/2)*cos(d*x+c)^3*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/2*(I*cos(d*x+c)-I-sin(d*x+c))/sin(d*x+c)/(-
2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))+15*I*A*2^(1/2)*cos(d*x+c)^2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*a
rctan(1/2*(I*cos(d*x+c)-I-sin(d*x+c))/sin(d*x+c)/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))-15*I*A*2^(1/2)*
cos(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/2*(I*cos(d*x+c)-I-sin(d*x+c))/sin(d*x+c)/(-2*cos(d*x+
c)/(1+cos(d*x+c)))^(1/2)*2^(1/2)))/(cos(d*x+c)^2-1)/a^3

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maxima [A]  time = 0.76, size = 240, normalized size = 0.93 \[ -\frac {i \, a {\left (\frac {4 \, {\left (105 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} {\left (3 \, A + i \, B\right )} - 5 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} {\left (41 \, A + 15 i \, B\right )} a - 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} {\left (19 \, A + 9 i \, B\right )} a^{2} - 12 \, {\left (A + i \, B\right )} a^{3}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a^{3} - {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{4}} + \frac {15 \, \sqrt {2} {\left (A - i \, B\right )} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right )}{a^{\frac {7}{2}}} + \frac {120 \, {\left (5 \, A + 2 i \, B\right )} \log \left (\frac {\sqrt {i \, a \tan \left (d x + c\right ) + a} - \sqrt {a}}{\sqrt {i \, a \tan \left (d x + c\right ) + a} + \sqrt {a}}\right )}{a^{\frac {7}{2}}}\right )}}{240 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/240*I*a*(4*(105*(I*a*tan(d*x + c) + a)^3*(3*A + I*B) - 5*(I*a*tan(d*x + c) + a)^2*(41*A + 15*I*B)*a - 2*(I*
a*tan(d*x + c) + a)*(19*A + 9*I*B)*a^2 - 12*(A + I*B)*a^3)/((I*a*tan(d*x + c) + a)^(7/2)*a^3 - (I*a*tan(d*x +
c) + a)^(5/2)*a^4) + 15*sqrt(2)*(A - I*B)*log(-(sqrt(2)*sqrt(a) - sqrt(I*a*tan(d*x + c) + a))/(sqrt(2)*sqrt(a)
+ sqrt(I*a*tan(d*x + c) + a)))/a^(7/2) + 120*(5*A + 2*I*B)*log((sqrt(I*a*tan(d*x + c) + a) - sqrt(a))/(sqrt(I
*a*tan(d*x + c) + a) + sqrt(a)))/a^(7/2))/d

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mupad [B]  time = 8.25, size = 3002, normalized size = 11.59 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cot(c + d*x)^2*(A + B*tan(c + d*x)))/(a + a*tan(c + d*x)*1i)^(5/2),x)

[Out]

2*atanh((12*a*d^4*(a + a*tan(c + d*x)*1i)^(1/2)*((129*B^2)/(256*a^5*d^2) - (801*A^2)/(256*a^5*d^2) - ((638401*
A^4*a^2)/(16*d^4) + (16129*B^4*a^2)/(16*d^4) - (307555*A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*40767i)/(4*d^4) + (A^
3*B*a^2*256479i)/(4*d^4))^(1/2)/(64*a^6) - (A*B*319i)/(128*a^5*d^2))^(1/2)*((638401*A^4*a^2)/(16*d^4) + (16129
*B^4*a^2)/(16*d^4) - (307555*A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*40767i)/(4*d^4) + (A^3*B*a^2*256479i)/(4*d^4))^
(1/2))/((A^3*d*31161i)/8 + (2159*B^3*d)/8 - (A*B^2*d*15867i)/8 - (38621*A^2*B*d)/8 + (A*d^3*((638401*A^4*a^2)/
(16*d^4) + (16129*B^4*a^2)/(16*d^4) - (307555*A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*40767i)/(4*d^4) + (A^3*B*a^2*2
56479i)/(4*d^4))^(1/2)*41i)/(2*a) - (15*B*d^3*((638401*A^4*a^2)/(16*d^4) + (16129*B^4*a^2)/(16*d^4) - (307555*
A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*40767i)/(4*d^4) + (A^3*B*a^2*256479i)/(4*d^4))^(1/2))/(2*a)) + (799*A^2*a^2*
d^2*(a + a*tan(c + d*x)*1i)^(1/2)*((129*B^2)/(256*a^5*d^2) - (801*A^2)/(256*a^5*d^2) - ((638401*A^4*a^2)/(16*d
^4) + (16129*B^4*a^2)/(16*d^4) - (307555*A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*40767i)/(4*d^4) + (A^3*B*a^2*256479
i)/(4*d^4))^(1/2)/(64*a^6) - (A*B*319i)/(128*a^5*d^2))^(1/2))/((A^3*d*31161i)/8 + (2159*B^3*d)/8 - (A*B^2*d*15
867i)/8 - (38621*A^2*B*d)/8 + (A*d^3*((638401*A^4*a^2)/(16*d^4) + (16129*B^4*a^2)/(16*d^4) - (307555*A^2*B^2*a
^2)/(8*d^4) - (A*B^3*a^2*40767i)/(4*d^4) + (A^3*B*a^2*256479i)/(4*d^4))^(1/2)*41i)/(2*a) - (15*B*d^3*((638401*
A^4*a^2)/(16*d^4) + (16129*B^4*a^2)/(16*d^4) - (307555*A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*40767i)/(4*d^4) + (A^
3*B*a^2*256479i)/(4*d^4))^(1/2))/(2*a)) - (127*B^2*a^2*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*((129*B^2)/(256*a^5*d
^2) - (801*A^2)/(256*a^5*d^2) - ((638401*A^4*a^2)/(16*d^4) + (16129*B^4*a^2)/(16*d^4) - (307555*A^2*B^2*a^2)/(
8*d^4) - (A*B^3*a^2*40767i)/(4*d^4) + (A^3*B*a^2*256479i)/(4*d^4))^(1/2)/(64*a^6) - (A*B*319i)/(128*a^5*d^2))^
(1/2))/((A^3*d*31161i)/8 + (2159*B^3*d)/8 - (A*B^2*d*15867i)/8 - (38621*A^2*B*d)/8 + (A*d^3*((638401*A^4*a^2)/
(16*d^4) + (16129*B^4*a^2)/(16*d^4) - (307555*A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*40767i)/(4*d^4) + (A^3*B*a^2*2
56479i)/(4*d^4))^(1/2)*41i)/(2*a) - (15*B*d^3*((638401*A^4*a^2)/(16*d^4) + (16129*B^4*a^2)/(16*d^4) - (307555*
A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*40767i)/(4*d^4) + (A^3*B*a^2*256479i)/(4*d^4))^(1/2))/(2*a)) + (A*B*a^2*d^2*
(a + a*tan(c + d*x)*1i)^(1/2)*((129*B^2)/(256*a^5*d^2) - (801*A^2)/(256*a^5*d^2) - ((638401*A^4*a^2)/(16*d^4)
+ (16129*B^4*a^2)/(16*d^4) - (307555*A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*40767i)/(4*d^4) + (A^3*B*a^2*256479i)/(
4*d^4))^(1/2)/(64*a^6) - (A*B*319i)/(128*a^5*d^2))^(1/2)*642i)/((A^3*d*31161i)/8 + (2159*B^3*d)/8 - (A*B^2*d*1
5867i)/8 - (38621*A^2*B*d)/8 + (A*d^3*((638401*A^4*a^2)/(16*d^4) + (16129*B^4*a^2)/(16*d^4) - (307555*A^2*B^2*
a^2)/(8*d^4) - (A*B^3*a^2*40767i)/(4*d^4) + (A^3*B*a^2*256479i)/(4*d^4))^(1/2)*41i)/(2*a) - (15*B*d^3*((638401
*A^4*a^2)/(16*d^4) + (16129*B^4*a^2)/(16*d^4) - (307555*A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*40767i)/(4*d^4) + (A
^3*B*a^2*256479i)/(4*d^4))^(1/2))/(2*a)))*(-(4*d^2*((((801*A^2*a)/4 - (129*B^2*a)/4)/d^2 + (A*B*a*319i)/(2*d^2
))^2 + 128*a^6*((((3*A*B^3)/4 + (15*A^3*B)/8)*1i)/(a^4*d^4) - ((25*A^4)/16 + (11*A^2*B^2)/16 + B^4/4)/(a^4*d^4
)))^(1/2) + 801*A^2*a - 129*B^2*a + A*B*a*638i)/(256*a^6*d^2))^(1/2) - (((A*a + B*a*1i)*1i)/(5*d) + ((19*A + B
*9i)*(a + a*tan(c + d*x)*1i)*1i)/(30*d) - ((3*A + B*1i)*(a + a*tan(c + d*x)*1i)^3*7i)/(4*a^2*d) + ((41*A + B*1
5i)*(a + a*tan(c + d*x)*1i)^2*1i)/(12*a*d))/(a*(a + a*tan(c + d*x)*1i)^(5/2) - (a + a*tan(c + d*x)*1i)^(7/2))
- 2*atanh((12*a*d^4*(a + a*tan(c + d*x)*1i)^(1/2)*(((638401*A^4*a^2)/(16*d^4) + (16129*B^4*a^2)/(16*d^4) - (30
7555*A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*40767i)/(4*d^4) + (A^3*B*a^2*256479i)/(4*d^4))^(1/2)/(64*a^6) - (801*A^
2)/(256*a^5*d^2) + (129*B^2)/(256*a^5*d^2) - (A*B*319i)/(128*a^5*d^2))^(1/2)*((638401*A^4*a^2)/(16*d^4) + (161
29*B^4*a^2)/(16*d^4) - (307555*A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*40767i)/(4*d^4) + (A^3*B*a^2*256479i)/(4*d^4)
)^(1/2))/((A^3*d*31161i)/8 + (2159*B^3*d)/8 - (A*B^2*d*15867i)/8 - (38621*A^2*B*d)/8 - (A*d^3*((638401*A^4*a^2
)/(16*d^4) + (16129*B^4*a^2)/(16*d^4) - (307555*A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*40767i)/(4*d^4) + (A^3*B*a^2
*256479i)/(4*d^4))^(1/2)*41i)/(2*a) + (15*B*d^3*((638401*A^4*a^2)/(16*d^4) + (16129*B^4*a^2)/(16*d^4) - (30755
5*A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*40767i)/(4*d^4) + (A^3*B*a^2*256479i)/(4*d^4))^(1/2))/(2*a)) - (799*A^2*a^
2*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*(((638401*A^4*a^2)/(16*d^4) + (16129*B^4*a^2)/(16*d^4) - (307555*A^2*B^2*a
^2)/(8*d^4) - (A*B^3*a^2*40767i)/(4*d^4) + (A^3*B*a^2*256479i)/(4*d^4))^(1/2)/(64*a^6) - (801*A^2)/(256*a^5*d^
2) + (129*B^2)/(256*a^5*d^2) - (A*B*319i)/(128*a^5*d^2))^(1/2))/((A^3*d*31161i)/8 + (2159*B^3*d)/8 - (A*B^2*d*
15867i)/8 - (38621*A^2*B*d)/8 - (A*d^3*((638401*A^4*a^2)/(16*d^4) + (16129*B^4*a^2)/(16*d^4) - (307555*A^2*B^2
*a^2)/(8*d^4) - (A*B^3*a^2*40767i)/(4*d^4) + (A^3*B*a^2*256479i)/(4*d^4))^(1/2)*41i)/(2*a) + (15*B*d^3*((63840
1*A^4*a^2)/(16*d^4) + (16129*B^4*a^2)/(16*d^4) - (307555*A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*40767i)/(4*d^4) + (
A^3*B*a^2*256479i)/(4*d^4))^(1/2))/(2*a)) + (127*B^2*a^2*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*(((638401*A^4*a^2)/
(16*d^4) + (16129*B^4*a^2)/(16*d^4) - (307555*A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*40767i)/(4*d^4) + (A^3*B*a^2*2
56479i)/(4*d^4))^(1/2)/(64*a^6) - (801*A^2)/(256*a^5*d^2) + (129*B^2)/(256*a^5*d^2) - (A*B*319i)/(128*a^5*d^2)
)^(1/2))/((A^3*d*31161i)/8 + (2159*B^3*d)/8 - (A*B^2*d*15867i)/8 - (38621*A^2*B*d)/8 - (A*d^3*((638401*A^4*a^2
)/(16*d^4) + (16129*B^4*a^2)/(16*d^4) - (307555*A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*40767i)/(4*d^4) + (A^3*B*a^2
*256479i)/(4*d^4))^(1/2)*41i)/(2*a) + (15*B*d^3*((638401*A^4*a^2)/(16*d^4) + (16129*B^4*a^2)/(16*d^4) - (30755
5*A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*40767i)/(4*d^4) + (A^3*B*a^2*256479i)/(4*d^4))^(1/2))/(2*a)) - (A*B*a^2*d^
2*(a + a*tan(c + d*x)*1i)^(1/2)*(((638401*A^4*a^2)/(16*d^4) + (16129*B^4*a^2)/(16*d^4) - (307555*A^2*B^2*a^2)/
(8*d^4) - (A*B^3*a^2*40767i)/(4*d^4) + (A^3*B*a^2*256479i)/(4*d^4))^(1/2)/(64*a^6) - (801*A^2)/(256*a^5*d^2) +
(129*B^2)/(256*a^5*d^2) - (A*B*319i)/(128*a^5*d^2))^(1/2)*642i)/((A^3*d*31161i)/8 + (2159*B^3*d)/8 - (A*B^2*d
*15867i)/8 - (38621*A^2*B*d)/8 - (A*d^3*((638401*A^4*a^2)/(16*d^4) + (16129*B^4*a^2)/(16*d^4) - (307555*A^2*B^
2*a^2)/(8*d^4) - (A*B^3*a^2*40767i)/(4*d^4) + (A^3*B*a^2*256479i)/(4*d^4))^(1/2)*41i)/(2*a) + (15*B*d^3*((6384
01*A^4*a^2)/(16*d^4) + (16129*B^4*a^2)/(16*d^4) - (307555*A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*40767i)/(4*d^4) +
(A^3*B*a^2*256479i)/(4*d^4))^(1/2))/(2*a)))*((4*d^2*((((801*A^2*a)/4 - (129*B^2*a)/4)/d^2 + (A*B*a*319i)/(2*d^
2))^2 + 128*a^6*((((3*A*B^3)/4 + (15*A^3*B)/8)*1i)/(a^4*d^4) - ((25*A^4)/16 + (11*A^2*B^2)/16 + B^4/4)/(a^4*d^
4)))^(1/2) - 801*A^2*a + 129*B^2*a - A*B*a*638i)/(256*a^6*d^2))^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (A + B \tan {\left (c + d x \right )}\right ) \cot ^{2}{\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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