\(\int \cot ^3(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx\) [80]

Optimal result

Integrand size = 36, antiderivative size = 171 \[ \int \cot ^3(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\frac {a^{3/2} (11 A-12 i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{4 d}-\frac {2 \sqrt {2} a^{3/2} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {a (5 i A+4 B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 d}-\frac {a A \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d} \]

[Out]

1/4*a^(3/2)*(11*A-12*I*B)*arctanh((a+I*a*tan(d*x+c))^(1/2)/a^(1/2))/d-2*a^(3/2)*(A-I*B)*arctanh(1/2*(a+I*a*tan
(d*x+c))^(1/2)*2^(1/2)/a^(1/2))*2^(1/2)/d-1/4*a*(5*I*A+4*B)*cot(d*x+c)*(a+I*a*tan(d*x+c))^(1/2)/d-1/2*a*A*cot(
d*x+c)^2*(a+I*a*tan(d*x+c))^(1/2)/d

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3674, 3679, 3681, 3561, 212, 3680, 65, 214} \[ \int \cot ^3(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\frac {a^{3/2} (11 A-12 i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{4 d}-\frac {2 \sqrt {2} a^{3/2} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {a (4 B+5 i A) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 d}-\frac {a A \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d} \]

[In]

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

[Out]

(a^(3/2)*(11*A - (12*I)*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(4*d) - (2*Sqrt[2]*a^(3/2)*(A - I*B)*A
rcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d - (a*((5*I)*A + 4*B)*Cot[c + d*x]*Sqrt[a + I*a*Tan[c +
d*x]])/(4*d) - (a*A*Cot[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(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 3674

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] && E
qQ[a^2 + b^2, 0] && GtQ[m, 1] && LtQ[n, -1]

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

Mathematica [A] (verified)

Time = 3.15 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.81 \[ \int \cot ^3(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\frac {a^{3/2} (11 A-12 i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )-8 \sqrt {2} a^{3/2} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )-a \cot (c+d x) (5 i A+4 B+2 A \cot (c+d x)) \sqrt {a+i a \tan (c+d x)}}{4 d} \]

[In]

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

[Out]

(a^(3/2)*(11*A - (12*I)*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]] - 8*Sqrt[2]*a^(3/2)*(A - I*B)*ArcTanh[S
qrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])] - a*Cot[c + d*x]*((5*I)*A + 4*B + 2*A*Cot[c + d*x])*Sqrt[a + I*a*
Tan[c + d*x]])/(4*d)

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.87

method	result	size
derivativedivides	\(\frac {2 a^{3} \left (-\frac {\left (-2 i B +2 A \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{2 a^{\frac {3}{2}}}+\frac {-\frac {\left (-\frac {i B}{2}+\frac {5 A}{8}\right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}+\left (\frac {1}{2} i a B -\frac {3}{8} a A \right ) \sqrt {a +i a \tan \left (d x +c \right )}}{a^{2} \tan \left (d x +c \right )^{2}}+\frac {\left (-12 i B +11 A \right ) \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}}{\sqrt {a}}\right )}{8 \sqrt {a}}}{a}\right )}{d}\)	\(148\)
default	\(\frac {2 a^{3} \left (-\frac {\left (-2 i B +2 A \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{2 a^{\frac {3}{2}}}+\frac {-\frac {\left (-\frac {i B}{2}+\frac {5 A}{8}\right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}+\left (\frac {1}{2} i a B -\frac {3}{8} a A \right ) \sqrt {a +i a \tan \left (d x +c \right )}}{a^{2} \tan \left (d x +c \right )^{2}}+\frac {\left (-12 i B +11 A \right ) \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}}{\sqrt {a}}\right )}{8 \sqrt {a}}}{a}\right )}{d}\)	\(148\)

[In]

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

[Out]

2/d*a^3*(-1/2*(2*A-2*I*B)/a^(3/2)*2^(1/2)*arctanh(1/2*(a+I*a*tan(d*x+c))^(1/2)*2^(1/2)/a^(1/2))+1/a*(-((-1/2*I
*B+5/8*A)*(a+I*a*tan(d*x+c))^(3/2)+(1/2*I*a*B-3/8*a*A)*(a+I*a*tan(d*x+c))^(1/2))/a^2/tan(d*x+c)^2+1/8*(11*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. 762 vs. \(2 (132) = 264\).

Time = 0.27 (sec) , antiderivative size = 762, normalized size of antiderivative = 4.46 \[ \int \cot ^3(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=-\frac {16 \, \sqrt {2} \sqrt {\frac {{\left (A^{2} - 2 i \, A B - B^{2}\right )} a^{3}}{d^{2}}} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {4 \, {\left ({\left (-i \, A - B\right )} a^{2} e^{\left (i \, d x + i \, c\right )} - \sqrt {\frac {{\left (A^{2} - 2 i \, A B - B^{2}\right )} a^{3}}{d^{2}}} {\left (i \, d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{{\left (-i \, A - B\right )} a}\right ) - 16 \, \sqrt {2} \sqrt {\frac {{\left (A^{2} - 2 i \, A B - B^{2}\right )} a^{3}}{d^{2}}} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {4 \, {\left ({\left (-i \, A - B\right )} a^{2} e^{\left (i \, d x + i \, c\right )} - \sqrt {\frac {{\left (A^{2} - 2 i \, A B - B^{2}\right )} a^{3}}{d^{2}}} {\left (-i \, d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{{\left (-i \, A - B\right )} a}\right ) + \sqrt {\frac {{\left (121 \, A^{2} - 264 i \, A B - 144 \, B^{2}\right )} a^{3}}{d^{2}}} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (-\frac {16 \, {\left (3 \, {\left (-11 i \, A - 12 \, B\right )} a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (-11 i \, A - 12 \, B\right )} a^{2} + 2 \, \sqrt {2} \sqrt {\frac {{\left (121 \, A^{2} - 264 i \, A B - 144 \, B^{2}\right )} a^{3}}{d^{2}}} {\left (i \, d e^{\left (3 i \, d x + 3 i \, c\right )} + i \, d e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{11 i \, A + 12 \, B}\right ) - \sqrt {\frac {{\left (121 \, A^{2} - 264 i \, A B - 144 \, B^{2}\right )} a^{3}}{d^{2}}} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (-\frac {16 \, {\left (3 \, {\left (-11 i \, A - 12 \, B\right )} a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (-11 i \, A - 12 \, B\right )} a^{2} + 2 \, \sqrt {2} \sqrt {\frac {{\left (121 \, A^{2} - 264 i \, A B - 144 \, B^{2}\right )} a^{3}}{d^{2}}} {\left (-i \, d e^{\left (3 i \, d x + 3 i \, c\right )} - i \, d e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{11 i \, A + 12 \, B}\right ) - 4 \, \sqrt {2} {\left ({\left (7 \, A - 4 i \, B\right )} a e^{\left (5 i \, d x + 5 i \, c\right )} + 4 \, A a e^{\left (3 i \, d x + 3 i \, c\right )} - {\left (3 \, A - 4 i \, B\right )} a e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{16 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]

[In]

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

[Out]

-1/16*(16*sqrt(2)*sqrt((A^2 - 2*I*A*B - B^2)*a^3/d^2)*(d*e^(4*I*d*x + 4*I*c) - 2*d*e^(2*I*d*x + 2*I*c) + d)*lo
g(4*((-I*A - B)*a^2*e^(I*d*x + I*c) - sqrt((A^2 - 2*I*A*B - B^2)*a^3/d^2)*(I*d*e^(2*I*d*x + 2*I*c) + I*d)*sqrt
(a/(e^(2*I*d*x + 2*I*c) + 1)))*e^(-I*d*x - I*c)/((-I*A - B)*a)) - 16*sqrt(2)*sqrt((A^2 - 2*I*A*B - B^2)*a^3/d^
2)*(d*e^(4*I*d*x + 4*I*c) - 2*d*e^(2*I*d*x + 2*I*c) + d)*log(4*((-I*A - B)*a^2*e^(I*d*x + I*c) - sqrt((A^2 - 2
*I*A*B - B^2)*a^3/d^2)*(-I*d*e^(2*I*d*x + 2*I*c) - I*d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1)))*e^(-I*d*x - I*c)/((
-I*A - B)*a)) + sqrt((121*A^2 - 264*I*A*B - 144*B^2)*a^3/d^2)*(d*e^(4*I*d*x + 4*I*c) - 2*d*e^(2*I*d*x + 2*I*c)
+ d)*log(-16*(3*(-11*I*A - 12*B)*a^2*e^(2*I*d*x + 2*I*c) + (-11*I*A - 12*B)*a^2 + 2*sqrt(2)*sqrt((121*A^2 - 2
64*I*A*B - 144*B^2)*a^3/d^2)*(I*d*e^(3*I*d*x + 3*I*c) + I*d*e^(I*d*x + I*c))*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))
)*e^(-2*I*d*x - 2*I*c)/(11*I*A + 12*B)) - sqrt((121*A^2 - 264*I*A*B - 144*B^2)*a^3/d^2)*(d*e^(4*I*d*x + 4*I*c)
- 2*d*e^(2*I*d*x + 2*I*c) + d)*log(-16*(3*(-11*I*A - 12*B)*a^2*e^(2*I*d*x + 2*I*c) + (-11*I*A - 12*B)*a^2 + 2
*sqrt(2)*sqrt((121*A^2 - 264*I*A*B - 144*B^2)*a^3/d^2)*(-I*d*e^(3*I*d*x + 3*I*c) - I*d*e^(I*d*x + I*c))*sqrt(a
/(e^(2*I*d*x + 2*I*c) + 1)))*e^(-2*I*d*x - 2*I*c)/(11*I*A + 12*B)) - 4*sqrt(2)*((7*A - 4*I*B)*a*e^(5*I*d*x + 5
*I*c) + 4*A*a*e^(3*I*d*x + 3*I*c) - (3*A - 4*I*B)*a*e^(I*d*x + I*c))*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1)))/(d*e^(
4*I*d*x + 4*I*c) - 2*d*e^(2*I*d*x + 2*I*c) + d)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.19 \[ \int \cot ^3(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\frac {{\left (\frac {8 \, \sqrt {2} {\left (A - i \, B\right )} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right )}{\sqrt {a}} - \frac {{\left (11 \, A - 12 i \, B\right )} \log \left (\frac {\sqrt {i \, a \tan \left (d x + c\right ) + a} - \sqrt {a}}{\sqrt {i \, a \tan \left (d x + c\right ) + a} + \sqrt {a}}\right )}{\sqrt {a}} + \frac {2 \, {\left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} {\left (5 \, A - 4 i \, B\right )} - \sqrt {i \, a \tan \left (d x + c\right ) + a} {\left (3 \, A - 4 i \, B\right )} a\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} - 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a + a^{2}}\right )} a^{2}}{8 \, d} \]

[In]

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

[Out]

1/8*(8*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)))/sqrt(a) - (11*A - 12*I*B)*log((sqrt(I*a*tan(d*x + c) + a) - sqrt(a))/(sqrt(I*a*tan(d*x + c) + a
) + sqrt(a)))/sqrt(a) + 2*((I*a*tan(d*x + c) + a)^(3/2)*(5*A - 4*I*B) - sqrt(I*a*tan(d*x + c) + a)*(3*A - 4*I*
B)*a)/((I*a*tan(d*x + c) + a)^2 - 2*(I*a*tan(d*x + c) + a)*a + a^2))*a^2/d

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.40 (sec) , antiderivative size = 3027, normalized size of antiderivative = 17.70 \[ \int \cot ^3(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[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((3*d^4*(a + a*tan(c + d*x)*1i)^(1/2)*((249*A^2*a^3)/(128*d^2) - ((49*A^4*a^18)/(4*d^4) + (64*B^4*a^18)
/d^4 + (40*A^2*B^2*a^18)/d^4 + (A*B^3*a^18*64i)/d^4 + (A^3*B*a^18*28i)/d^4)^(1/2)/(64*a^6) - (17*B^2*a^3)/(8*d
^2) - (A*B*a^3*65i)/(16*d^2))^(1/2)*((49*A^4*a^18)/(4*d^4) + (64*B^4*a^18)/d^4 + (40*A^2*B^2*a^18)/d^4 + (A*B^
3*a^18*64i)/d^4 + (A^3*B*a^18*28i)/d^4)^(1/2))/(2*((133*A^3*a^11*d)/16 - B^3*a^11*d*20i + 29*A*B^2*a^11*d + (A
^2*B*a^11*d*3i)/4 + (3*A*a^2*d^3*((49*A^4*a^18)/(4*d^4) + (64*B^4*a^18)/d^4 + (40*A^2*B^2*a^18)/d^4 + (A*B^3*a
^18*64i)/d^4 + (A^3*B*a^18*28i)/d^4)^(1/2))/8 - (B*a^2*d^3*((49*A^4*a^18)/(4*d^4) + (64*B^4*a^18)/d^4 + (40*A^
2*B^2*a^18)/d^4 + (A*B^3*a^18*64i)/d^4 + (A^3*B*a^18*28i)/d^4)^(1/2)*1i)/2)) + (7*A^2*a^6*d^2*(a + a*tan(c + d
*x)*1i)^(1/2)*((249*A^2*a^3)/(128*d^2) - ((49*A^4*a^18)/(4*d^4) + (64*B^4*a^18)/d^4 + (40*A^2*B^2*a^18)/d^4 +
(A*B^3*a^18*64i)/d^4 + (A^3*B*a^18*28i)/d^4)^(1/2)/(64*a^6) - (17*B^2*a^3)/(8*d^2) - (A*B*a^3*65i)/(16*d^2))^(
1/2))/(4*((133*A^3*a^8*d)/16 - B^3*a^8*d*20i + 29*A*B^2*a^8*d + (A^2*B*a^8*d*3i)/4 + (3*A*d^3*((49*A^4*a^18)/(
4*d^4) + (64*B^4*a^18)/d^4 + (40*A^2*B^2*a^18)/d^4 + (A*B^3*a^18*64i)/d^4 + (A^3*B*a^18*28i)/d^4)^(1/2))/(8*a)
- (B*d^3*((49*A^4*a^18)/(4*d^4) + (64*B^4*a^18)/d^4 + (40*A^2*B^2*a^18)/d^4 + (A*B^3*a^18*64i)/d^4 + (A^3*B*a
^18*28i)/d^4)^(1/2)*1i)/(2*a))) + (4*B^2*a^6*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*((249*A^2*a^3)/(128*d^2) - ((49
*A^4*a^18)/(4*d^4) + (64*B^4*a^18)/d^4 + (40*A^2*B^2*a^18)/d^4 + (A*B^3*a^18*64i)/d^4 + (A^3*B*a^18*28i)/d^4)^
(1/2)/(64*a^6) - (17*B^2*a^3)/(8*d^2) - (A*B*a^3*65i)/(16*d^2))^(1/2))/((133*A^3*a^8*d)/16 - B^3*a^8*d*20i + 2
9*A*B^2*a^8*d + (A^2*B*a^8*d*3i)/4 + (3*A*d^3*((49*A^4*a^18)/(4*d^4) + (64*B^4*a^18)/d^4 + (40*A^2*B^2*a^18)/d
^4 + (A*B^3*a^18*64i)/d^4 + (A^3*B*a^18*28i)/d^4)^(1/2))/(8*a) - (B*d^3*((49*A^4*a^18)/(4*d^4) + (64*B^4*a^18)
/d^4 + (40*A^2*B^2*a^18)/d^4 + (A*B^3*a^18*64i)/d^4 + (A^3*B*a^18*28i)/d^4)^(1/2)*1i)/(2*a)) + (A*B*a^6*d^2*(a
+ a*tan(c + d*x)*1i)^(1/2)*((249*A^2*a^3)/(128*d^2) - ((49*A^4*a^18)/(4*d^4) + (64*B^4*a^18)/d^4 + (40*A^2*B^
2*a^18)/d^4 + (A*B^3*a^18*64i)/d^4 + (A^3*B*a^18*28i)/d^4)^(1/2)/(64*a^6) - (17*B^2*a^3)/(8*d^2) - (A*B*a^3*65
i)/(16*d^2))^(1/2)*2i)/((133*A^3*a^8*d)/16 - B^3*a^8*d*20i + 29*A*B^2*a^8*d + (A^2*B*a^8*d*3i)/4 + (3*A*d^3*((
49*A^4*a^18)/(4*d^4) + (64*B^4*a^18)/d^4 + (40*A^2*B^2*a^18)/d^4 + (A*B^3*a^18*64i)/d^4 + (A^3*B*a^18*28i)/d^4
)^(1/2))/(8*a) - (B*d^3*((49*A^4*a^18)/(4*d^4) + (64*B^4*a^18)/d^4 + (40*A^2*B^2*a^18)/d^4 + (A*B^3*a^18*64i)/
d^4 + (A^3*B*a^18*28i)/d^4)^(1/2)*1i)/(2*a)))*((249*A^2*a^3)/(128*d^2) - ((49*A^4*a^18)/(4*d^4) + (64*B^4*a^18
)/d^4 + (40*A^2*B^2*a^18)/d^4 + (A*B^3*a^18*64i)/d^4 + (A^3*B*a^18*28i)/d^4)^(1/2)/(64*a^6) - (17*B^2*a^3)/(8*
d^2) - (A*B*a^3*65i)/(16*d^2))^(1/2) - (((3*A*a^3 - B*a^3*4i)*(a + a*tan(c + d*x)*1i)^(1/2))/(4*d) - ((5*A*a^2
- B*a^2*4i)*(a + a*tan(c + d*x)*1i)^(3/2))/(4*d))/((a + a*tan(c + d*x)*1i)^2 - 2*a*(a + a*tan(c + d*x)*1i) +
a^2) + 2*atanh((7*A^2*a^6*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*(((49*A^4*a^18)/(4*d^4) + (64*B^4*a^18)/d^4 + (40*
A^2*B^2*a^18)/d^4 + (A*B^3*a^18*64i)/d^4 + (A^3*B*a^18*28i)/d^4)^(1/2)/(64*a^6) + (249*A^2*a^3)/(128*d^2) - (1
7*B^2*a^3)/(8*d^2) - (A*B*a^3*65i)/(16*d^2))^(1/2))/(4*((133*A^3*a^8*d)/16 - B^3*a^8*d*20i + 29*A*B^2*a^8*d +
(A^2*B*a^8*d*3i)/4 - (3*A*d^3*((49*A^4*a^18)/(4*d^4) + (64*B^4*a^18)/d^4 + (40*A^2*B^2*a^18)/d^4 + (A*B^3*a^18
*64i)/d^4 + (A^3*B*a^18*28i)/d^4)^(1/2))/(8*a) + (B*d^3*((49*A^4*a^18)/(4*d^4) + (64*B^4*a^18)/d^4 + (40*A^2*B
^2*a^18)/d^4 + (A*B^3*a^18*64i)/d^4 + (A^3*B*a^18*28i)/d^4)^(1/2)*1i)/(2*a))) - (3*d^4*(a + a*tan(c + d*x)*1i)
^(1/2)*(((49*A^4*a^18)/(4*d^4) + (64*B^4*a^18)/d^4 + (40*A^2*B^2*a^18)/d^4 + (A*B^3*a^18*64i)/d^4 + (A^3*B*a^1
8*28i)/d^4)^(1/2)/(64*a^6) + (249*A^2*a^3)/(128*d^2) - (17*B^2*a^3)/(8*d^2) - (A*B*a^3*65i)/(16*d^2))^(1/2)*((
49*A^4*a^18)/(4*d^4) + (64*B^4*a^18)/d^4 + (40*A^2*B^2*a^18)/d^4 + (A*B^3*a^18*64i)/d^4 + (A^3*B*a^18*28i)/d^4
)^(1/2))/(2*((133*A^3*a^11*d)/16 - B^3*a^11*d*20i + 29*A*B^2*a^11*d + (A^2*B*a^11*d*3i)/4 - (3*A*a^2*d^3*((49*
A^4*a^18)/(4*d^4) + (64*B^4*a^18)/d^4 + (40*A^2*B^2*a^18)/d^4 + (A*B^3*a^18*64i)/d^4 + (A^3*B*a^18*28i)/d^4)^(
1/2))/8 + (B*a^2*d^3*((49*A^4*a^18)/(4*d^4) + (64*B^4*a^18)/d^4 + (40*A^2*B^2*a^18)/d^4 + (A*B^3*a^18*64i)/d^4
+ (A^3*B*a^18*28i)/d^4)^(1/2)*1i)/2)) + (4*B^2*a^6*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*(((49*A^4*a^18)/(4*d^4)
+ (64*B^4*a^18)/d^4 + (40*A^2*B^2*a^18)/d^4 + (A*B^3*a^18*64i)/d^4 + (A^3*B*a^18*28i)/d^4)^(1/2)/(64*a^6) + (2
49*A^2*a^3)/(128*d^2) - (17*B^2*a^3)/(8*d^2) - (A*B*a^3*65i)/(16*d^2))^(1/2))/((133*A^3*a^8*d)/16 - B^3*a^8*d*
20i + 29*A*B^2*a^8*d + (A^2*B*a^8*d*3i)/4 - (3*A*d^3*((49*A^4*a^18)/(4*d^4) + (64*B^4*a^18)/d^4 + (40*A^2*B^2*
a^18)/d^4 + (A*B^3*a^18*64i)/d^4 + (A^3*B*a^18*28i)/d^4)^(1/2))/(8*a) + (B*d^3*((49*A^4*a^18)/(4*d^4) + (64*B^
4*a^18)/d^4 + (40*A^2*B^2*a^18)/d^4 + (A*B^3*a^18*64i)/d^4 + (A^3*B*a^18*28i)/d^4)^(1/2)*1i)/(2*a)) + (A*B*a^6
*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*(((49*A^4*a^18)/(4*d^4) + (64*B^4*a^18)/d^4 + (40*A^2*B^2*a^18)/d^4 + (A*B^
3*a^18*64i)/d^4 + (A^3*B*a^18*28i)/d^4)^(1/2)/(64*a^6) + (249*A^2*a^3)/(128*d^2) - (17*B^2*a^3)/(8*d^2) - (A*B
*a^3*65i)/(16*d^2))^(1/2)*2i)/((133*A^3*a^8*d)/16 - B^3*a^8*d*20i + 29*A*B^2*a^8*d + (A^2*B*a^8*d*3i)/4 - (3*A
*d^3*((49*A^4*a^18)/(4*d^4) + (64*B^4*a^18)/d^4 + (40*A^2*B^2*a^18)/d^4 + (A*B^3*a^18*64i)/d^4 + (A^3*B*a^18*2
8i)/d^4)^(1/2))/(8*a) + (B*d^3*((49*A^4*a^18)/(4*d^4) + (64*B^4*a^18)/d^4 + (40*A^2*B^2*a^18)/d^4 + (A*B^3*a^1
8*64i)/d^4 + (A^3*B*a^18*28i)/d^4)^(1/2)*1i)/(2*a)))*(((49*A^4*a^18)/(4*d^4) + (64*B^4*a^18)/d^4 + (40*A^2*B^2
*a^18)/d^4 + (A*B^3*a^18*64i)/d^4 + (A^3*B*a^18*28i)/d^4)^(1/2)/(64*a^6) + (249*A^2*a^3)/(128*d^2) - (17*B^2*a
^3)/(8*d^2) - (A*B*a^3*65i)/(16*d^2))^(1/2)