\(\int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx\) [103]

Optimal result

Integrand size = 36, antiderivative size = 268 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx=\frac {(23 A+12 i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{3/2} d}-\frac {(A-i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {(A+i B) \cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(17 A+11 i B) \cot ^2(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {7 (3 i A-2 B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a^2 d}-\frac {(22 A+13 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{6 a^2 d} \]

[Out]

1/4*(23*A+12*I*B)*arctanh((a+I*a*tan(d*x+c))^(1/2)/a^(1/2))/a^(3/2)/d-1/4*(A-I*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*(17*A+11*I*B)*cot(d*x+c)^2/a/d/(a+I*a*tan(d*x+c))^(1/2)+7/4*(3
*I*A-2*B)*cot(d*x+c)*(a+I*a*tan(d*x+c))^(1/2)/a^2/d-1/6*(22*A+13*I*B)*cot(d*x+c)^2*(a+I*a*tan(d*x+c))^(1/2)/a^
2/d+1/3*(A+I*B)*cot(d*x+c)^2/d/(a+I*a*tan(d*x+c))^(3/2)

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3677, 3679, 3681, 3561, 212, 3680, 65, 214} \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx=\frac {(23 A+12 i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{3/2} d}-\frac {(A-i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {(22 A+13 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{6 a^2 d}+\frac {7 (-2 B+3 i A) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a^2 d}+\frac {(17 A+11 i B) \cot ^2(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {(A+i B) \cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}} \]

[In]

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

[Out]

((23*A + (12*I)*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(4*a^(3/2)*d) - ((A - I*B)*ArcTanh[Sqrt[a + I*
a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(2*Sqrt[2]*a^(3/2)*d) + ((A + I*B)*Cot[c + d*x]^2)/(3*d*(a + I*a*Tan[c + d
*x])^(3/2)) + ((17*A + (11*I)*B)*Cot[c + d*x]^2)/(6*a*d*Sqrt[a + I*a*Tan[c + d*x]]) + (7*((3*I)*A - 2*B)*Cot[c
+ d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(4*a^2*d) - ((22*A + (13*I)*B)*Cot[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/
(6*a^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 212

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

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 3561

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 3677

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 3679

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 3680

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 3681

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*} \text {integral}& = \frac {(A+i B) \cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {\int \frac {\cot ^3(c+d x) \left (a (5 A+2 i B)-\frac {7}{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 ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(17 A+11 i B) \cot ^2(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {\int \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)} \left (a^2 (22 A+13 i B)-\frac {5}{4} a^2 (17 i A-11 B) \tan (c+d x)\right ) \, dx}{3 a^4} \\ & = \frac {(A+i B) \cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(17 A+11 i B) \cot ^2(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {(22 A+13 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{6 a^2 d}+\frac {\int \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)} \left (-\frac {21}{2} a^3 (3 i A-2 B)-\frac {3}{2} a^3 (22 A+13 i B) \tan (c+d x)\right ) \, dx}{6 a^5} \\ & = \frac {(A+i B) \cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(17 A+11 i B) \cot ^2(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {7 (3 i A-2 B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a^2 d}-\frac {(22 A+13 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{6 a^2 d}+\frac {\int \cot (c+d x) \sqrt {a+i a \tan (c+d x)} \left (-\frac {3}{4} a^4 (23 A+12 i B)+\frac {21}{4} a^4 (3 i A-2 B) \tan (c+d x)\right ) \, dx}{6 a^6} \\ & = \frac {(A+i B) \cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(17 A+11 i B) \cot ^2(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {7 (3 i A-2 B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a^2 d}-\frac {(22 A+13 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{6 a^2 d}-\frac {(23 A+12 i B) \int \cot (c+d x) (a-i a \tan (c+d x)) \sqrt {a+i a \tan (c+d x)} \, dx}{8 a^3}-\frac {(i A+B) \int \sqrt {a+i a \tan (c+d x)} \, dx}{4 a^2} \\ & = \frac {(A+i B) \cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(17 A+11 i B) \cot ^2(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {7 (3 i A-2 B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a^2 d}-\frac {(22 A+13 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{6 a^2 d}-\frac {(A-i B) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{2 a d}-\frac {(23 A+12 i B) \text {Subst}\left (\int \frac {1}{x \sqrt {a+i a x}} \, dx,x,\tan (c+d x)\right )}{8 a d} \\ & = -\frac {(A-i B) \text {arctanh}\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 ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(17 A+11 i B) \cot ^2(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {7 (3 i A-2 B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a^2 d}-\frac {(22 A+13 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{6 a^2 d}+\frac {(23 i A-12 B) \text {Subst}\left (\int \frac {1}{i-\frac {i x^2}{a}} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{4 a^2 d} \\ & = \frac {(23 A+12 i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{3/2} d}-\frac {(A-i B) \text {arctanh}\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 ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(17 A+11 i B) \cot ^2(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {7 (3 i A-2 B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a^2 d}-\frac {(22 A+13 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{6 a^2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.45 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.68 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx=\frac {3 (23 A+12 i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )-3 \sqrt {2} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )-\frac {\sqrt {a} \left (63 i A-42 B+(82 A+58 i B) \cot (c+d x)+3 (-3 i A+4 B) \cot ^2(c+d x)+6 A \cot ^3(c+d x)\right )}{(i+\cot (c+d x)) \sqrt {a+i a \tan (c+d x)}}}{12 a^{3/2} d} \]

[In]

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

[Out]

(3*(23*A + (12*I)*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]] - 3*Sqrt[2]*(A - I*B)*ArcTanh[Sqrt[a + I*a*Ta
n[c + d*x]]/(Sqrt[2]*Sqrt[a])] - (Sqrt[a]*((63*I)*A - 42*B + (82*A + (58*I)*B)*Cot[c + d*x] + 3*((-3*I)*A + 4*
B)*Cot[c + d*x]^2 + 6*A*Cot[c + d*x]^3))/((I + Cot[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]]))/(12*a^(3/2)*d)

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.74

method	result	size
derivativedivides	\(\frac {2 a^{3} \left (-\frac {5 i B +7 A}{4 a^{4} \sqrt {a +i a \tan \left (d x +c \right )}}-\frac {i B +A}{6 a^{3} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {\left (-i B +A \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{8 a^{\frac {9}{2}}}+\frac {-\frac {\left (-\frac {i B}{2}-\frac {7 A}{8}\right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}+\left (\frac {1}{2} i a B +\frac {9}{8} a A \right ) \sqrt {a +i a \tan \left (d x +c \right )}}{a^{2} \tan \left (d x +c \right )^{2}}+\frac {\left (12 i B +23 A \right ) \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}}{\sqrt {a}}\right )}{8 \sqrt {a}}}{a^{4}}\right )}{d}\)	\(198\)
default	\(\frac {2 a^{3} \left (-\frac {5 i B +7 A}{4 a^{4} \sqrt {a +i a \tan \left (d x +c \right )}}-\frac {i B +A}{6 a^{3} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {\left (-i B +A \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{8 a^{\frac {9}{2}}}+\frac {-\frac {\left (-\frac {i B}{2}-\frac {7 A}{8}\right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}+\left (\frac {1}{2} i a B +\frac {9}{8} a A \right ) \sqrt {a +i a \tan \left (d x +c \right )}}{a^{2} \tan \left (d x +c \right )^{2}}+\frac {\left (12 i B +23 A \right ) \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}}{\sqrt {a}}\right )}{8 \sqrt {a}}}{a^{4}}\right )}{d}\)	\(198\)

[In]

int(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^(3/2),x,method=_RETURNVERBOSE)

[Out]

2/d*a^3*(-1/4/a^4*(7*A+5*I*B)/(a+I*a*tan(d*x+c))^(1/2)-1/6/a^3*(A+I*B)/(a+I*a*tan(d*x+c))^(3/2)-1/8/a^(9/2)*(A
-I*B)*2^(1/2)*arctanh(1/2*(a+I*a*tan(d*x+c))^(1/2)*2^(1/2)/a^(1/2))+1/a^4*(-((-1/2*I*B-7/8*A)*(a+I*a*tan(d*x+c
))^(3/2)+(1/2*I*a*B+9/8*a*A)*(a+I*a*tan(d*x+c))^(1/2))/a^2/tan(d*x+c)^2+1/8*(23*A+12*I*B)/a^(1/2)*arctanh((a+I
*a*tan(d*x+c))^(1/2)/a^(1/2))))

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 903 vs. \(2 (209) = 418\).

Time = 0.28 (sec) , antiderivative size = 903, normalized size of antiderivative = 3.37 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/48*(12*sqrt(1/2)*(a^2*d*e^(7*I*d*x + 7*I*c) - 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)*(I*a^2*d*e^(2*I*d*x + 2*I*c) + I*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)) + (-I*A - B)*a*e^(I*d*x + I*c))*e^(-I*d*x - I*c)/
(I*A + B)) - 12*sqrt(1/2)*(a^2*d*e^(7*I*d*x + 7*I*c) - 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)*(-I*a^2*d*e^(2*I*d*x + 2*I*c) - I*a^2*d)*sqr
t(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((A^2 - 2*I*A*B - B^2)/(a^3*d^2)) + (-I*A - B)*a*e^(I*d*x + I*c))*e^(-I*d*x
- I*c)/(I*A + B)) + 3*(a^2*d*e^(7*I*d*x + 7*I*c) - 2*a^2*d*e^(5*I*d*x + 5*I*c) + a^2*d*e^(3*I*d*x + 3*I*c))*s
qrt((529*A^2 + 552*I*A*B - 144*B^2)/(a^3*d^2))*log(-16*(3*(23*I*A - 12*B)*a^2*e^(2*I*d*x + 2*I*c) + (23*I*A -
12*B)*a^2 + 2*sqrt(2)*(I*a^3*d*e^(3*I*d*x + 3*I*c) + I*a^3*d*e^(I*d*x + I*c))*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1)
)*sqrt((529*A^2 + 552*I*A*B - 144*B^2)/(a^3*d^2)))*e^(-2*I*d*x - 2*I*c)/(-23*I*A + 12*B)) - 3*(a^2*d*e^(7*I*d*
x + 7*I*c) - 2*a^2*d*e^(5*I*d*x + 5*I*c) + a^2*d*e^(3*I*d*x + 3*I*c))*sqrt((529*A^2 + 552*I*A*B - 144*B^2)/(a^
3*d^2))*log(-16*(3*(23*I*A - 12*B)*a^2*e^(2*I*d*x + 2*I*c) + (23*I*A - 12*B)*a^2 + 2*sqrt(2)*(-I*a^3*d*e^(3*I*
d*x + 3*I*c) - I*a^3*d*e^(I*d*x + I*c))*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((529*A^2 + 552*I*A*B - 144*B^2)
/(a^3*d^2)))*e^(-2*I*d*x - 2*I*c)/(-23*I*A + 12*B)) - 4*sqrt(2)*((37*A + 28*I*B)*e^(8*I*d*x + 8*I*c) - 3*(11*A
+ 5*I*B)*e^(6*I*d*x + 6*I*c) - (50*A + 29*I*B)*e^(4*I*d*x + 4*I*c) + 3*(7*A + 5*I*B)*e^(2*I*d*x + 2*I*c) + A
+ I*B)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1)))/(a^2*d*e^(7*I*d*x + 7*I*c) - 2*a^2*d*e^(5*I*d*x + 5*I*c) + a^2*d*e^(
3*I*d*x + 3*I*c))

Sympy [F]

\[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx=\int \frac {\left (A + B \tan {\left (c + d x \right )}\right ) \cot ^{3}{\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}}}\, dx \]

[In]

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

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.97 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx=-\frac {a^{2} {\left (\frac {2 \, {\left (21 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} {\left (3 \, A + 2 i \, B\right )} - {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} {\left (107 \, A + 68 i \, B\right )} a + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} {\left (17 \, A + 11 i \, B\right )} a^{2} + 4 \, {\left (A + i \, B\right )} a^{3}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a^{3} - 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{4} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{5}} - \frac {3 \, \sqrt {2} {\left (A - i \, B\right )} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right )}{a^{\frac {7}{2}}} + \frac {3 \, {\left (23 \, A + 12 i \, B\right )} \log \left (\frac {\sqrt {i \, a \tan \left (d x + c\right ) + a} - \sqrt {a}}{\sqrt {i \, a \tan \left (d x + c\right ) + a} + \sqrt {a}}\right )}{a^{\frac {7}{2}}}\right )}}{24 \, d} \]

[In]

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

[Out]

-1/24*a^2*(2*(21*(I*a*tan(d*x + c) + a)^3*(3*A + 2*I*B) - (I*a*tan(d*x + c) + a)^2*(107*A + 68*I*B)*a + 2*(I*a
*tan(d*x + c) + a)*(17*A + 11*I*B)*a^2 + 4*(A + I*B)*a^3)/((I*a*tan(d*x + c) + a)^(7/2)*a^3 - 2*(I*a*tan(d*x +
c) + a)^(5/2)*a^4 + (I*a*tan(d*x + c) + a)^(3/2)*a^5) - 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^(7/2) + 3*(23*A + 12*I*B)*log((sqrt(I*a*t
an(d*x + c) + a) - sqrt(a))/(sqrt(I*a*tan(d*x + c) + a) + sqrt(a)))/a^(7/2))/d

Giac [F]

\[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx=\int { \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \cot \left (d x + c\right )^{3}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

Mupad [B] (verification not implemented)

Time = 9.93 (sec) , antiderivative size = 3106, normalized size of antiderivative = 11.59 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx=\text {Too large to display} \]

[In]

int((cot(c + d*x)^3*(A + B*tan(c + d*x)))/(a + a*tan(c + d*x)*1i)^(3/2),x)

[Out]

2*atanh((48*d^4*(a + a*tan(c + d*x)*1i)^(1/2)*((531*A^2)/(128*a^3*d^2) - ((277729*A^4*a^6)/(4*d^4) + (5041*B^4
*a^6)/d^4 - (114701*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2)/(64*a^6) - (73*
B^2)/(64*a^3*d^2) + (A*B*137i)/(32*a^3*d^2))^(1/2)*((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (114701*A^
2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2))/(B^3*a^2*d*3124i - 25296*A^3*a^2*d +
19048*A*B^2*a^2*d - A^2*B*a^2*d*38282i + (88*A*d^3*((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (114701*A
^2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2))/a + (B*d^3*((277729*A^4*a^6)/(4*d^4
) + (5041*B^4*a^6)/d^4 - (114701*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2)*52
i)/a) - (4216*A^2*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*((531*A^2)/(128*a^3*d^2) - ((277729*A^4*a^6)/(4*d^4) + (50
41*B^4*a^6)/d^4 - (114701*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2)/(64*a^6)
- (73*B^2)/(64*a^3*d^2) + (A*B*137i)/(32*a^3*d^2))^(1/2))/((B^3*d*3124i)/a - (25296*A^3*d)/a + (19048*A*B^2*d)
/a - (A^2*B*d*38282i)/a + (88*A*d^3*((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (114701*A^2*B^2*a^6)/d^4
- (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2))/a^4 + (B*d^3*((277729*A^4*a^6)/(4*d^4) + (5041*B^4*
a^6)/d^4 - (114701*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2)*52i)/a^4) + (113
6*B^2*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*((531*A^2)/(128*a^3*d^2) - ((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/
d^4 - (114701*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2)/(64*a^6) - (73*B^2)/(
64*a^3*d^2) + (A*B*137i)/(32*a^3*d^2))^(1/2))/((B^3*d*3124i)/a - (25296*A^3*d)/a + (19048*A*B^2*d)/a - (A^2*B*
d*38282i)/a + (88*A*d^3*((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (114701*A^2*B^2*a^6)/d^4 - (A*B^3*a^6
*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2))/a^4 + (B*d^3*((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (
114701*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2)*52i)/a^4) - (A*B*d^2*(a + a*
tan(c + d*x)*1i)^(1/2)*((531*A^2)/(128*a^3*d^2) - ((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (114701*A^2
*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2)/(64*a^6) - (73*B^2)/(64*a^3*d^2) + (A*
B*137i)/(32*a^3*d^2))^(1/2)*4448i)/((B^3*d*3124i)/a - (25296*A^3*d)/a + (19048*A*B^2*d)/a - (A^2*B*d*38282i)/a
+ (88*A*d^3*((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (114701*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^
4 + (A^3*B*a^6*146506i)/d^4)^(1/2))/a^4 + (B*d^3*((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (114701*A^2*
B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2)*52i)/a^4))*(-(2*d^2*((((531*A^2*a^3)/2
- 73*B^2*a^3)/d^2 + (A*B*a^3*274i)/d^2)^2 + 128*a^6*((((33*A*B^3)/8 + (253*A^3*B)/32)*1i)/d^4 - ((529*A^4)/64
+ (431*A^2*B^2)/64 + (9*B^4)/4)/d^4))^(1/2) - 531*A^2*a^3 + 146*B^2*a^3 - A*B*a^3*548i)/(128*a^6*d^2))^(1/2) +
2*atanh((48*d^4*(a + a*tan(c + d*x)*1i)^(1/2)*(((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (114701*A^2*B
^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2)/(64*a^6) + (531*A^2)/(128*a^3*d^2) - (73
*B^2)/(64*a^3*d^2) + (A*B*137i)/(32*a^3*d^2))^(1/2)*((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (114701*A
^2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2))/(25296*A^3*a^2*d - B^3*a^2*d*3124i
- 19048*A*B^2*a^2*d + A^2*B*a^2*d*38282i + (88*A*d^3*((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (114701*
A^2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2))/a + (B*d^3*((277729*A^4*a^6)/(4*d^
4) + (5041*B^4*a^6)/d^4 - (114701*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2)*5
2i)/a) + (4216*A^2*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*(((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (114701
*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2)/(64*a^6) + (531*A^2)/(128*a^3*d^2)
- (73*B^2)/(64*a^3*d^2) + (A*B*137i)/(32*a^3*d^2))^(1/2))/((25296*A^3*d)/a - (B^3*d*3124i)/a - (19048*A*B^2*d
)/a + (A^2*B*d*38282i)/a + (88*A*d^3*((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (114701*A^2*B^2*a^6)/d^4
- (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2))/a^4 + (B*d^3*((277729*A^4*a^6)/(4*d^4) + (5041*B^4
*a^6)/d^4 - (114701*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2)*52i)/a^4) - (11
36*B^2*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*(((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (114701*A^2*B^2*a^6
)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2)/(64*a^6) + (531*A^2)/(128*a^3*d^2) - (73*B^2)/
(64*a^3*d^2) + (A*B*137i)/(32*a^3*d^2))^(1/2))/((25296*A^3*d)/a - (B^3*d*3124i)/a - (19048*A*B^2*d)/a + (A^2*B
*d*38282i)/a + (88*A*d^3*((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (114701*A^2*B^2*a^6)/d^4 - (A*B^3*a^
6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2))/a^4 + (B*d^3*((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 -
(114701*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2)*52i)/a^4) + (A*B*d^2*(a + a
*tan(c + d*x)*1i)^(1/2)*(((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (114701*A^2*B^2*a^6)/d^4 - (A*B^3*a^
6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2)/(64*a^6) + (531*A^2)/(128*a^3*d^2) - (73*B^2)/(64*a^3*d^2) + (A
*B*137i)/(32*a^3*d^2))^(1/2)*4448i)/((25296*A^3*d)/a - (B^3*d*3124i)/a - (19048*A*B^2*d)/a + (A^2*B*d*38282i)/
a + (88*A*d^3*((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (114701*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d
^4 + (A^3*B*a^6*146506i)/d^4)^(1/2))/a^4 + (B*d^3*((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (114701*A^2
*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2)*52i)/a^4))*((2*d^2*((((531*A^2*a^3)/2
- 73*B^2*a^3)/d^2 + (A*B*a^3*274i)/d^2)^2 + 128*a^6*((((33*A*B^3)/8 + (253*A^3*B)/32)*1i)/d^4 - ((529*A^4)/64
+ (431*A^2*B^2)/64 + (9*B^4)/4)/d^4))^(1/2) + 531*A^2*a^3 - 146*B^2*a^3 + A*B*a^3*548i)/(128*a^6*d^2))^(1/2) -
((A*a^2 + B*a^2*1i)/(3*d) - ((107*A + B*68i)*(a + a*tan(c + d*x)*1i)^2)/(12*d) + ((17*A*a + B*a*11i)*(a + a*t
an(c + d*x)*1i))/(6*d) + (7*(3*A + B*2i)*(a + a*tan(c + d*x)*1i)^3)/(4*a*d))/((a + a*tan(c + d*x)*1i)^(7/2) -
2*a*(a + a*tan(c + d*x)*1i)^(5/2) + a^2*(a + a*tan(c + d*x)*1i)^(3/2))