3.102 \(\int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx\)

Optimal. Leaf size=217 \[ \frac {(-2 B+3 i A) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {(B+i A) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {(7 A+3 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{2 a^2 d}+\frac {(13 A+7 i B) \cot (c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {(A+i B) \cot (c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}} \]

[Out]

(3*I*A-2*B)*arctanh((a+I*a*tan(d*x+c))^(1/2)/a^(1/2))/a^(3/2)/d+1/4*(I*A+B)*arctanh(1/2*(a+I*a*tan(d*x+c))^(1/
2)*2^(1/2)/a^(1/2))/a^(3/2)/d*2^(1/2)+1/6*(13*A+7*I*B)*cot(d*x+c)/a/d/(a+I*a*tan(d*x+c))^(1/2)-1/2*(7*A+3*I*B)
*cot(d*x+c)*(a+I*a*tan(d*x+c))^(1/2)/a^2/d+1/3*(A+I*B)*cot(d*x+c)/d/(a+I*a*tan(d*x+c))^(3/2)

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Rubi [A]  time = 0.80, antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ \frac {(-2 B+3 i A) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {(B+i A) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {(7 A+3 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{2 a^2 d}+\frac {(13 A+7 i B) \cot (c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {(A+i B) \cot (c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(3/2),x]

[Out]

(((3*I)*A - 2*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) + ((I*A + B)*ArcTanh[Sqrt[a + I*a*Ta
n[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(2*Sqrt[2]*a^(3/2)*d) + ((A + I*B)*Cot[c + d*x])/(3*d*(a + I*a*Tan[c + d*x])^(
3/2)) + ((13*A + (7*I)*B)*Cot[c + d*x])/(6*a*d*Sqrt[a + I*a*Tan[c + d*x]]) - ((7*A + (3*I)*B)*Cot[c + d*x]*Sqr
t[a + I*a*Tan[c + d*x]])/(2*a^2*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))^{3/2}} \, dx &=\frac {(A+i B) \cot (c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {\int \frac {\cot ^2(c+d x) \left (a (4 A+i B)-\frac {5}{2} a (i A-B) \tan (c+d x)\right )}{\sqrt {a+i a \tan (c+d x)}} \, dx}{3 a^2}\\ &=\frac {(A+i B) \cot (c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(13 A+7 i B) \cot (c+d x)}{6 a 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 {3}{2} a^2 (7 A+3 i B)-\frac {3}{4} a^2 (13 i A-7 B) \tan (c+d x)\right ) \, dx}{3 a^4}\\ &=\frac {(A+i B) \cot (c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(13 A+7 i B) \cot (c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {(7 A+3 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{2 a^2 d}+\frac {\int \cot (c+d x) \sqrt {a+i a \tan (c+d x)} \left (-\frac {3}{2} a^3 (3 i A-2 B)-\frac {3}{4} a^3 (7 A+3 i B) \tan (c+d x)\right ) \, dx}{3 a^5}\\ &=\frac {(A+i B) \cot (c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(13 A+7 i B) \cot (c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {(7 A+3 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{2 a^2 d}-\frac {(3 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^3}-\frac {(A-i B) \int \sqrt {a+i a \tan (c+d x)} \, dx}{4 a^2}\\ &=\frac {(A+i B) \cot (c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(13 A+7 i B) \cot (c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {(7 A+3 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{2 a^2 d}-\frac {(3 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 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 )}{2 a d}\\ &=\frac {(i A+B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {(A+i B) \cot (c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(13 A+7 i B) \cot (c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {(7 A+3 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{2 a^2 d}-\frac {(3 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^2 d}\\ &=\frac {(3 i A-2 B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {(i A+B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {(A+i B) \cot (c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(13 A+7 i B) \cot (c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {(7 A+3 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{2 a^2 d}\\ \end {align*}

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Mathematica [A]  time = 5.39, size = 259, normalized size = 1.19 \[ \frac {\sqrt {\sec (c+d x)} (A+B \tan (c+d x)) \left (\sqrt {2} \left (\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{3/2} \left (1+e^{2 i (c+d x)}\right )^{3/2} \left ((B+i A) \sinh ^{-1}\left (e^{i (c+d x)}\right )+2 \sqrt {2} (-2 B+3 i A) \tanh ^{-1}\left (\frac {\sqrt {2} e^{i (c+d x)}}{\sqrt {1+e^{2 i (c+d x)}}}\right )\right )-\frac {\csc (c+d x) ((-11 B+29 i A) \sin (2 (c+d x))+9 (3 A+i B) \cos (2 (c+d x))-3 (5 A+3 i B))}{3 \sqrt {\sec (c+d x)}}\right )}{4 d (a+i a \tan (c+d x))^{3/2} (A \cos (c+d x)+B \sin (c+d x))} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(3/2),x]

[Out]

(Sqrt[Sec[c + d*x]]*(Sqrt[2]*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(3/2)*(1 + E^((2*I)*(c + d*x)))^(3/2)
*((I*A + B)*ArcSinh[E^(I*(c + d*x))] + 2*Sqrt[2]*((3*I)*A - 2*B)*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[1 + E^
((2*I)*(c + d*x))]]) - (Csc[c + d*x]*(-3*(5*A + (3*I)*B) + 9*(3*A + I*B)*Cos[2*(c + d*x)] + ((29*I)*A - 11*B)*
Sin[2*(c + d*x)]))/(3*Sqrt[Sec[c + d*x]]))*(A + B*Tan[c + d*x]))/(4*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I
*a*Tan[c + d*x])^(3/2))

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fricas [B]  time = 0.99, size = 810, normalized size = 3.73 \[ \frac {3 \, \sqrt {\frac {1}{2}} {\left (a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} - a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )}\right )} \sqrt {-\frac {A^{2} - 2 i \, A B - B^{2}}{a^{3} d^{2}}} \log \left (\frac {{\left (4 \, \sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {-\frac {A^{2} - 2 i \, A B - B^{2}}{a^{3} d^{2}}} + {\left (4 i \, A + 4 \, B\right )} a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{i \, A + B}\right ) - 3 \, \sqrt {\frac {1}{2}} {\left (a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} - a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )}\right )} \sqrt {-\frac {A^{2} - 2 i \, A B - B^{2}}{a^{3} d^{2}}} \log \left (-\frac {{\left (4 \, \sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {-\frac {A^{2} - 2 i \, A B - B^{2}}{a^{3} d^{2}}} - {\left (4 i \, A + 4 \, B\right )} a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{i \, A + B}\right ) - 3 \, {\left (a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} - a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )}\right )} \sqrt {-\frac {9 \, A^{2} + 12 i \, A B - 4 \, B^{2}}{a^{3} d^{2}}} \log \left (\frac {{\left ({\left (-144 i \, A + 96 \, B\right )} a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (-48 i \, A + 32 \, B\right )} a^{2} + 32 \, \sqrt {2} {\left (a^{3} d e^{\left (3 i \, d x + 3 i \, c\right )} + a^{3} d e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {-\frac {9 \, A^{2} + 12 i \, A B - 4 \, B^{2}}{a^{3} d^{2}}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{-3 i \, A + 2 \, B}\right ) + 3 \, {\left (a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} - a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )}\right )} \sqrt {-\frac {9 \, A^{2} + 12 i \, A B - 4 \, B^{2}}{a^{3} d^{2}}} \log \left (\frac {{\left ({\left (-144 i \, A + 96 \, B\right )} a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (-48 i \, A + 32 \, B\right )} a^{2} - 32 \, \sqrt {2} {\left (a^{3} d e^{\left (3 i \, d x + 3 i \, c\right )} + a^{3} d e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {-\frac {9 \, A^{2} + 12 i \, A B - 4 \, B^{2}}{a^{3} d^{2}}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{-3 i \, A + 2 \, B}\right ) + \sqrt {2} {\left ({\left (-28 i \, A + 10 \, B\right )} e^{\left (6 i \, d x + 6 i \, c\right )} + {\left (-13 i \, A + B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + {\left (16 i \, A - 10 \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, A - B\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{12 \, {\left (a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} - a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )}\right )}} \]

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))^(3/2),x, algorithm="fricas")

[Out]

1/12*(3*sqrt(1/2)*(a^2*d*e^(5*I*d*x + 5*I*c) - a^2*d*e^(3*I*d*x + 3*I*c))*sqrt(-(A^2 - 2*I*A*B - B^2)/(a^3*d^2
))*log((4*sqrt(2)*sqrt(1/2)*(a^2*d*e^(2*I*d*x + 2*I*c) + a^2*d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(-(A^2 -
2*I*A*B - B^2)/(a^3*d^2)) + (4*I*A + 4*B)*a*e^(I*d*x + I*c))*e^(-I*d*x - I*c)/(I*A + B)) - 3*sqrt(1/2)*(a^2*d
*e^(5*I*d*x + 5*I*c) - a^2*d*e^(3*I*d*x + 3*I*c))*sqrt(-(A^2 - 2*I*A*B - B^2)/(a^3*d^2))*log(-(4*sqrt(2)*sqrt(
1/2)*(a^2*d*e^(2*I*d*x + 2*I*c) + a^2*d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(-(A^2 - 2*I*A*B - B^2)/(a^3*d^
2)) - (4*I*A + 4*B)*a*e^(I*d*x + I*c))*e^(-I*d*x - I*c)/(I*A + B)) - 3*(a^2*d*e^(5*I*d*x + 5*I*c) - a^2*d*e^(3
*I*d*x + 3*I*c))*sqrt(-(9*A^2 + 12*I*A*B - 4*B^2)/(a^3*d^2))*log(((-144*I*A + 96*B)*a^2*e^(2*I*d*x + 2*I*c) +
(-48*I*A + 32*B)*a^2 + 32*sqrt(2)*(a^3*d*e^(3*I*d*x + 3*I*c) + a^3*d*e^(I*d*x + I*c))*sqrt(a/(e^(2*I*d*x + 2*I
*c) + 1))*sqrt(-(9*A^2 + 12*I*A*B - 4*B^2)/(a^3*d^2)))*e^(-2*I*d*x - 2*I*c)/(-3*I*A + 2*B)) + 3*(a^2*d*e^(5*I*
d*x + 5*I*c) - a^2*d*e^(3*I*d*x + 3*I*c))*sqrt(-(9*A^2 + 12*I*A*B - 4*B^2)/(a^3*d^2))*log(((-144*I*A + 96*B)*a
^2*e^(2*I*d*x + 2*I*c) + (-48*I*A + 32*B)*a^2 - 32*sqrt(2)*(a^3*d*e^(3*I*d*x + 3*I*c) + a^3*d*e^(I*d*x + I*c))
*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(-(9*A^2 + 12*I*A*B - 4*B^2)/(a^3*d^2)))*e^(-2*I*d*x - 2*I*c)/(-3*I*A +
2*B)) + sqrt(2)*((-28*I*A + 10*B)*e^(6*I*d*x + 6*I*c) + (-13*I*A + B)*e^(4*I*d*x + 4*I*c) + (16*I*A - 10*B)*e
^(2*I*d*x + 2*I*c) + I*A - B)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1)))/(a^2*d*e^(5*I*d*x + 5*I*c) - a^2*d*e^(3*I*d*x
+ 3*I*c))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \cot \left (d x + c\right )^{2}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]

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))^(3/2),x, algorithm="giac")

[Out]

integrate((B*tan(d*x + c) + A)*cot(d*x + c)^2/(I*a*tan(d*x + c) + a)^(3/2), x)

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maple [B]  time = 3.55, size = 2818, normalized size = 12.99 \[ \text {output too large to display} \]

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))^(3/2),x)

[Out]

-1/24/d*(a*(I*sin(d*x+c)+cos(d*x+c))/cos(d*x+c))^(1/2)*(3*A*2^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arcta
n(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+28*B*cos(d*x+c)^2+44*A*cos(d*x+c)^3*sin(d*x+c)-12*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))-18*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)-84*A*cos(d*x+c)*sin(d*x+c)-12*B*cos(d*x+c)^4-18*A*(-2*co
s(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))
+18*A*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))-12*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))+3*I*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))*sin(d*x+c)+3*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)
)-3*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))-18*I*A*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))*sin(d*x+c)+18*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+12*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+18*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)^2+12*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))*c
os(d*x+c)^2-12*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*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))-36*I*B*cos(d*x+c)*sin(d*x+c)+16*I*B*cos(d*x+c)^5*sin(d*x+c)+20*I*B*cos(d*
x+c)^3*sin(d*x+c)+18*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))-1
2*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)+cos(d*x+c)-1)
/sin(d*x+c))+18*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))-12*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))-18*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))+12*I*B*cos(d*x+c)^3*(-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))-3*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))+18*I*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))*sin(d*x+
c)+12*I*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))*sin(d*x+c)-18*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))+12*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))+16*A*cos(d*x+c)^5*sin(d*x+c)+16*I*A*cos(d*x+c)^6+36*I*A*cos(d*x+c)^4-52*I*A*cos(d*x+c)^2-3*I*B
*2^(1/2)*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))*sin(d*x+c)+3*I*A*2^(1/2)*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))-12*I
*B*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))*sin(d*x+c)
+3*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+3*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)^2-3*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*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-s
in(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)-16*B*cos(d*x+c)^6-18*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))+12*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)))/(cos(d*x+c)^2-1)/a^2

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maxima [A]  time = 1.68, size = 215, normalized size = 0.99 \[ -\frac {i \, a {\left (\frac {4 \, {\left (3 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} {\left (7 \, A + 3 i \, B\right )} - {\left (i \, a \tan \left (d x + c\right ) + a\right )} {\left (13 \, A + 7 i \, B\right )} a - 2 \, {\left (A + i \, B\right )} a^{2}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{2} - {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{3}} + \frac {3 \, \sqrt {2} {\left (A - i \, B\right )} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right )}{a^{\frac {5}{2}}} + \frac {12 \, {\left (3 \, A + 2 i \, B\right )} \log \left (\frac {\sqrt {i \, a \tan \left (d x + c\right ) + a} - \sqrt {a}}{\sqrt {i \, a \tan \left (d x + c\right ) + a} + \sqrt {a}}\right )}{a^{\frac {5}{2}}}\right )}}{24 \, d} \]

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))^(3/2),x, algorithm="maxima")

[Out]

-1/24*I*a*(4*(3*(I*a*tan(d*x + c) + a)^2*(7*A + 3*I*B) - (I*a*tan(d*x + c) + a)*(13*A + 7*I*B)*a - 2*(A + I*B)
*a^2)/((I*a*tan(d*x + c) + a)^(5/2)*a^2 - (I*a*tan(d*x + c) + a)^(3/2)*a^3) + 3*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^(5/2) + 12*(3*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^(5/2))/d

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mupad [B]  time = 8.23, size = 3051, normalized size = 14.06 \[ \text {result too large to display} \]

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)^(3/2),x)

[Out]

2*atanh((3*d^4*(a + a*tan(c + d*x)*1i)^(1/2)*((33*B^2)/(64*a^3*d^2) - (73*A^2)/(64*a^3*d^2) - ((5041*A^4*a^6)/
d^4 + (961*B^4*a^6)/d^4 - (14006*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*6076i)/d^4 + (A^3*B*a^6*13916i)/d^4)^(1/2)/(64*
a^6) - (A*B*47i)/(32*a^3*d^2))^(1/2)*((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/d^4 - (14006*A^2*B^2*a^6)/d^4 - (A*B^
3*a^6*6076i)/d^4 + (A^3*B*a^6*13916i)/d^4)^(1/2))/((A^3*a^2*d*781i)/4 + (279*B^3*a^2*d)/4 - (A*B^2*a^2*d*1223i
)/4 - (1717*A^2*B*a^2*d)/4 + (A*d^3*((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/d^4 - (14006*A^2*B^2*a^6)/d^4 - (A*B^3
*a^6*6076i)/d^4 + (A^3*B*a^6*13916i)/d^4)^(1/2)*13i)/(4*a) - (7*B*d^3*((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/d^4
- (14006*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*6076i)/d^4 + (A^3*B*a^6*13916i)/d^4)^(1/2))/(4*a)) + (71*A^2*d^2*(a + a
*tan(c + d*x)*1i)^(1/2)*((33*B^2)/(64*a^3*d^2) - (73*A^2)/(64*a^3*d^2) - ((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/d
^4 - (14006*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*6076i)/d^4 + (A^3*B*a^6*13916i)/d^4)^(1/2)/(64*a^6) - (A*B*47i)/(32*
a^3*d^2))^(1/2))/((A^3*d*781i)/(4*a) + (279*B^3*d)/(4*a) - (A*B^2*d*1223i)/(4*a) - (1717*A^2*B*d)/(4*a) + (A*d
^3*((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/d^4 - (14006*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*6076i)/d^4 + (A^3*B*a^6*1391
6i)/d^4)^(1/2)*13i)/(4*a^4) - (7*B*d^3*((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/d^4 - (14006*A^2*B^2*a^6)/d^4 - (A*
B^3*a^6*6076i)/d^4 + (A^3*B*a^6*13916i)/d^4)^(1/2))/(4*a^4)) - (31*B^2*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*((33*
B^2)/(64*a^3*d^2) - (73*A^2)/(64*a^3*d^2) - ((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/d^4 - (14006*A^2*B^2*a^6)/d^4
- (A*B^3*a^6*6076i)/d^4 + (A^3*B*a^6*13916i)/d^4)^(1/2)/(64*a^6) - (A*B*47i)/(32*a^3*d^2))^(1/2))/((A^3*d*781i
)/(4*a) + (279*B^3*d)/(4*a) - (A*B^2*d*1223i)/(4*a) - (1717*A^2*B*d)/(4*a) + (A*d^3*((5041*A^4*a^6)/d^4 + (961
*B^4*a^6)/d^4 - (14006*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*6076i)/d^4 + (A^3*B*a^6*13916i)/d^4)^(1/2)*13i)/(4*a^4) -
(7*B*d^3*((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/d^4 - (14006*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*6076i)/d^4 + (A^3*B*a
^6*13916i)/d^4)^(1/2))/(4*a^4)) + (A*B*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*((33*B^2)/(64*a^3*d^2) - (73*A^2)/(64
*a^3*d^2) - ((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/d^4 - (14006*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*6076i)/d^4 + (A^3*B
*a^6*13916i)/d^4)^(1/2)/(64*a^6) - (A*B*47i)/(32*a^3*d^2))^(1/2)*98i)/((A^3*d*781i)/(4*a) + (279*B^3*d)/(4*a)
- (A*B^2*d*1223i)/(4*a) - (1717*A^2*B*d)/(4*a) + (A*d^3*((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/d^4 - (14006*A^2*B
^2*a^6)/d^4 - (A*B^3*a^6*6076i)/d^4 + (A^3*B*a^6*13916i)/d^4)^(1/2)*13i)/(4*a^4) - (7*B*d^3*((5041*A^4*a^6)/d^
4 + (961*B^4*a^6)/d^4 - (14006*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*6076i)/d^4 + (A^3*B*a^6*13916i)/d^4)^(1/2))/(4*a^
4)))*(-(d^2*(((73*A^2*a^3 - 33*B^2*a^3)/d^2 + (A*B*a^3*94i)/d^2)^2 + 128*a^6*(((A*B^3 + (3*A^3*B)/2)*1i)/d^4 -
((9*A^4)/4 + (11*A^2*B^2)/4 + B^4)/d^4))^(1/2) + 73*A^2*a^3 - 33*B^2*a^3 + A*B*a^3*94i)/(64*a^6*d^2))^(1/2) -
(((A*a + B*a*1i)*1i)/(3*d) + ((13*A + B*7i)*(a + a*tan(c + d*x)*1i)*1i)/(6*d) - ((7*A + B*3i)*(a + a*tan(c +
d*x)*1i)^2*1i)/(2*a*d))/(a*(a + a*tan(c + d*x)*1i)^(3/2) - (a + a*tan(c + d*x)*1i)^(5/2)) - 2*atanh((3*d^4*(a
+ a*tan(c + d*x)*1i)^(1/2)*(((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/d^4 - (14006*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*607
6i)/d^4 + (A^3*B*a^6*13916i)/d^4)^(1/2)/(64*a^6) - (73*A^2)/(64*a^3*d^2) + (33*B^2)/(64*a^3*d^2) - (A*B*47i)/(
32*a^3*d^2))^(1/2)*((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/d^4 - (14006*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*6076i)/d^4 +
(A^3*B*a^6*13916i)/d^4)^(1/2))/((A^3*a^2*d*781i)/4 + (279*B^3*a^2*d)/4 - (A*B^2*a^2*d*1223i)/4 - (1717*A^2*B*
a^2*d)/4 - (A*d^3*((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/d^4 - (14006*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*6076i)/d^4 +
(A^3*B*a^6*13916i)/d^4)^(1/2)*13i)/(4*a) + (7*B*d^3*((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/d^4 - (14006*A^2*B^2*a
^6)/d^4 - (A*B^3*a^6*6076i)/d^4 + (A^3*B*a^6*13916i)/d^4)^(1/2))/(4*a)) - (71*A^2*d^2*(a + a*tan(c + d*x)*1i)^
(1/2)*(((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/d^4 - (14006*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*6076i)/d^4 + (A^3*B*a^6*
13916i)/d^4)^(1/2)/(64*a^6) - (73*A^2)/(64*a^3*d^2) + (33*B^2)/(64*a^3*d^2) - (A*B*47i)/(32*a^3*d^2))^(1/2))/(
(A^3*d*781i)/(4*a) + (279*B^3*d)/(4*a) - (A*B^2*d*1223i)/(4*a) - (1717*A^2*B*d)/(4*a) - (A*d^3*((5041*A^4*a^6)
/d^4 + (961*B^4*a^6)/d^4 - (14006*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*6076i)/d^4 + (A^3*B*a^6*13916i)/d^4)^(1/2)*13i
)/(4*a^4) + (7*B*d^3*((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/d^4 - (14006*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*6076i)/d^4
+ (A^3*B*a^6*13916i)/d^4)^(1/2))/(4*a^4)) + (31*B^2*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*(((5041*A^4*a^6)/d^4 +
(961*B^4*a^6)/d^4 - (14006*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*6076i)/d^4 + (A^3*B*a^6*13916i)/d^4)^(1/2)/(64*a^6) -
(73*A^2)/(64*a^3*d^2) + (33*B^2)/(64*a^3*d^2) - (A*B*47i)/(32*a^3*d^2))^(1/2))/((A^3*d*781i)/(4*a) + (279*B^3
*d)/(4*a) - (A*B^2*d*1223i)/(4*a) - (1717*A^2*B*d)/(4*a) - (A*d^3*((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/d^4 - (1
4006*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*6076i)/d^4 + (A^3*B*a^6*13916i)/d^4)^(1/2)*13i)/(4*a^4) + (7*B*d^3*((5041*A
^4*a^6)/d^4 + (961*B^4*a^6)/d^4 - (14006*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*6076i)/d^4 + (A^3*B*a^6*13916i)/d^4)^(1
/2))/(4*a^4)) - (A*B*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*(((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/d^4 - (14006*A^2*B
^2*a^6)/d^4 - (A*B^3*a^6*6076i)/d^4 + (A^3*B*a^6*13916i)/d^4)^(1/2)/(64*a^6) - (73*A^2)/(64*a^3*d^2) + (33*B^2
)/(64*a^3*d^2) - (A*B*47i)/(32*a^3*d^2))^(1/2)*98i)/((A^3*d*781i)/(4*a) + (279*B^3*d)/(4*a) - (A*B^2*d*1223i)/
(4*a) - (1717*A^2*B*d)/(4*a) - (A*d^3*((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/d^4 - (14006*A^2*B^2*a^6)/d^4 - (A*B
^3*a^6*6076i)/d^4 + (A^3*B*a^6*13916i)/d^4)^(1/2)*13i)/(4*a^4) + (7*B*d^3*((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/
d^4 - (14006*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*6076i)/d^4 + (A^3*B*a^6*13916i)/d^4)^(1/2))/(4*a^4)))*((d^2*(((73*A
^2*a^3 - 33*B^2*a^3)/d^2 + (A*B*a^3*94i)/d^2)^2 + 128*a^6*(((A*B^3 + (3*A^3*B)/2)*1i)/d^4 - ((9*A^4)/4 + (11*A
^2*B^2)/4 + B^4)/d^4))^(1/2) - 73*A^2*a^3 + 33*B^2*a^3 - A*B*a^3*94i)/(64*a^6*d^2))^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (A + B \tan {\left (c + d x \right )}\right ) \cot ^{2}{\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}}}\, dx \]

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))**(3/2),x)

[Out]

Integral((A + B*tan(c + d*x))*cot(c + d*x)**2/(I*a*(tan(c + d*x) - I))**(3/2), x)

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