\(\int \frac {\cot ^3(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx\) [125]

Optimal result

Integrand size = 26, antiderivative size = 220 \[ \int \frac {\cot ^3(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\frac {23 \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{3/2} d}-\frac {\text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {\cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {17 \cot ^2(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {21 i \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a^2 d}-\frac {11 \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 a^2 d} \]

[Out]

23/4*arctanh((a+I*a*tan(d*x+c))^(1/2)/a^(1/2))/a^(3/2)/d-1/4*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)+17/6*cot(d*x+c)^2/a/d/(a+I*a*tan(d*x+c))^(1/2)+21/4*I*cot(d*x+c)*(a+I*a*tan(d*x+c))^(1/
2)/a^2/d-11/3*cot(d*x+c)^2*(a+I*a*tan(d*x+c))^(1/2)/a^2/d+1/3*cot(d*x+c)^2/d/(a+I*a*tan(d*x+c))^(3/2)

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {3640, 3677, 3679, 3681, 3561, 212, 3680, 65, 214} \[ \int \frac {\cot ^3(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\frac {23 \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{3/2} d}-\frac {\text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {11 \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 a^2 d}+\frac {21 i \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a^2 d}+\frac {17 \cot ^2(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {\cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}} \]

[In]

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

[Out]

(23*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(4*a^(3/2)*d) - ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*S
qrt[a])]/(2*Sqrt[2]*a^(3/2)*d) + Cot[c + d*x]^2/(3*d*(a + I*a*Tan[c + d*x])^(3/2)) + (17*Cot[c + d*x]^2)/(6*a*
d*Sqrt[a + I*a*Tan[c + d*x]]) + (((21*I)/4)*Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(a^2*d) - (11*Cot[c + d*x
]^2*Sqrt[a + I*a*Tan[c + d*x]])/(3*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 3640

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

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 {\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 (5 a-\frac {7}{2} i a \tan (c+d x)\right )}{\sqrt {a+i a \tan (c+d x)}} \, dx}{3 a^2} \\ & = \frac {\cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {17 \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 (22 a^2-\frac {85}{4} i a^2 \tan (c+d x)\right ) \, dx}{3 a^4} \\ & = \frac {\cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {17 \cot ^2(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {11 \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 a^2 d}+\frac {\int \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)} \left (-\frac {63 i a^3}{2}-33 a^3 \tan (c+d x)\right ) \, dx}{6 a^5} \\ & = \frac {\cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {17 \cot ^2(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {21 i \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a^2 d}-\frac {11 \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 a^2 d}+\frac {\int \cot (c+d x) \sqrt {a+i a \tan (c+d x)} \left (-\frac {69 a^4}{4}+\frac {63}{4} i a^4 \tan (c+d x)\right ) \, dx}{6 a^6} \\ & = \frac {\cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {17 \cot ^2(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {21 i \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a^2 d}-\frac {11 \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 a^2 d}-\frac {23 \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 \int \sqrt {a+i a \tan (c+d x)} \, dx}{4 a^2} \\ & = \frac {\cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {17 \cot ^2(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {21 i \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a^2 d}-\frac {11 \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 a^2 d}-\frac {\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 \text {Subst}\left (\int \frac {1}{x \sqrt {a+i a x}} \, dx,x,\tan (c+d x)\right )}{8 a d} \\ & = -\frac {\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 {\cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {17 \cot ^2(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {21 i \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a^2 d}-\frac {11 \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 a^2 d}+\frac {(23 i) \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 \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{3/2} d}-\frac {\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 {\cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {17 \cot ^2(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {21 i \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a^2 d}-\frac {11 \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 a^2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.26 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.66 \[ \int \frac {\cot ^3(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\frac {69 \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )-3 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )-\frac {\sqrt {a} \left (63 i+82 \cot (c+d x)-9 i \cot ^2(c+d x)+6 \cot ^3(c+d x)\right )}{(i+\cot (c+d x)) \sqrt {a+i a \tan (c+d x)}}}{12 a^{3/2} d} \]

[In]

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

[Out]

(69*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]] - 3*Sqrt[2]*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a
])] - (Sqrt[a]*(63*I + 82*Cot[c + d*x] - (9*I)*Cot[c + d*x]^2 + 6*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 = 1.13 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.70

method	result	size
derivativedivides	\(\frac {2 a^{4} \left (\frac {-\frac {-\frac {7 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{8}+\frac {9 a \sqrt {a +i a \tan \left (d x +c \right )}}{8}}{a^{2} \tan \left (d x +c \right )^{2}}+\frac {23 \,\operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}}{\sqrt {a}}\right )}{8 \sqrt {a}}}{a^{5}}-\frac {7}{4 a^{5} \sqrt {a +i a \tan \left (d x +c \right )}}-\frac {1}{6 a^{4} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{8 a^{\frac {11}{2}}}\right )}{d}\)	\(155\)
default	\(\frac {2 a^{4} \left (\frac {-\frac {-\frac {7 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{8}+\frac {9 a \sqrt {a +i a \tan \left (d x +c \right )}}{8}}{a^{2} \tan \left (d x +c \right )^{2}}+\frac {23 \,\operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}}{\sqrt {a}}\right )}{8 \sqrt {a}}}{a^{5}}-\frac {7}{4 a^{5} \sqrt {a +i a \tan \left (d x +c \right )}}-\frac {1}{6 a^{4} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{8 a^{\frac {11}{2}}}\right )}{d}\)	\(155\)

[In]

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

[Out]

2/d*a^4*(1/a^5*(-(-7/8*(a+I*a*tan(d*x+c))^(3/2)+9/8*a*(a+I*a*tan(d*x+c))^(1/2))/a^2/tan(d*x+c)^2+23/8/a^(1/2)*
arctanh((a+I*a*tan(d*x+c))^(1/2)/a^(1/2)))-7/4/a^5/(a+I*a*tan(d*x+c))^(1/2)-1/6/a^4/(a+I*a*tan(d*x+c))^(3/2)-1
/8/a^(11/2)*2^(1/2)*arctanh(1/2*(a+I*a*tan(d*x+c))^(1/2)*2^(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. 678 vs. \(2 (171) = 342\).

Time = 0.25 (sec) , antiderivative size = 678, normalized size of antiderivative = 3.08 \[ \int \frac {\cot ^3(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=-\frac {12 \, \sqrt {\frac {1}{2}} {\left (a^{2} d e^{\left (7 i \, d x + 7 i \, c\right )} - 2 \, 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 {1}{a^{3} d^{2}}} \log \left (4 \, {\left (\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 {1}{a^{3} d^{2}}} + a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) - 12 \, \sqrt {\frac {1}{2}} {\left (a^{2} d e^{\left (7 i \, d x + 7 i \, c\right )} - 2 \, 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 {1}{a^{3} d^{2}}} \log \left (-4 \, {\left (\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 {1}{a^{3} d^{2}}} - a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) - 69 \, {\left (a^{2} d e^{\left (7 i \, d x + 7 i \, c\right )} - 2 \, 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 {1}{a^{3} d^{2}}} \log \left (16 \, {\left (3 \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + 2 \, \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 {1}{a^{3} d^{2}}} + a^{2}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}\right ) + 69 \, {\left (a^{2} d e^{\left (7 i \, d x + 7 i \, c\right )} - 2 \, 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 {1}{a^{3} d^{2}}} \log \left (16 \, {\left (3 \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - 2 \, \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 {1}{a^{3} d^{2}}} + a^{2}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}\right ) + 4 \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (37 \, e^{\left (8 i \, d x + 8 i \, c\right )} - 33 \, e^{\left (6 i \, d x + 6 i \, c\right )} - 50 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 21 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}}{48 \, {\left (a^{2} d e^{\left (7 i \, d x + 7 i \, c\right )} - 2 \, 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 )}} \]

[In]

integrate(cot(d*x+c)^3/(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
(1/(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))*s
qrt(1/(a^3*d^2)) + a*e^(I*d*x + I*c))*e^(-I*d*x - I*c)) - 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(1/(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(1/(a^3*d^2)) - a*e^(I*d*x + I*c))*e^(-I*d*x - I*c)) -
69*(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(1/(a^3*d^2))*log
(16*(3*a^2*e^(2*I*d*x + 2*I*c) + 2*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(1/(a^3*d^2)) + a^2)*e^(-2*I*d*x - 2*I*c)) + 69*(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(1/(a^3*d^2))*log(16*(3*a^2*e^(2*I*d*x + 2*I*c) - 2*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(1/(a^3*d^2)) + a^2
)*e^(-2*I*d*x - 2*I*c)) + 4*sqrt(2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*(37*e^(8*I*d*x + 8*I*c) - 33*e^(6*I*d*x
+ 6*I*c) - 50*e^(4*I*d*x + 4*I*c) + 21*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]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.00 \[ \int \frac {\cot ^3(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=-\frac {a^{2} {\left (\frac {2 \, {\left (63 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} - 107 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} a + 34 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a^{2} + 4 \, a^{3}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a^{3} - 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{4} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{5}} - \frac {3 \, \sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right )}{a^{\frac {7}{2}}} + \frac {69 \, \log \left (\frac {\sqrt {i \, a \tan \left (d x + c\right ) + a} - \sqrt {a}}{\sqrt {i \, a \tan \left (d x + c\right ) + a} + \sqrt {a}}\right )}{a^{\frac {7}{2}}}\right )}}{24 \, d} \]

[In]

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

[Out]

-1/24*a^2*(2*(63*(I*a*tan(d*x + c) + a)^3 - 107*(I*a*tan(d*x + c) + a)^2*a + 34*(I*a*tan(d*x + c) + a)*a^2 + 4
*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)*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) + 69*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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 4.35 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.85 \[ \int \frac {\cot ^3(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=-\frac {\frac {a^2}{3}+\frac {21\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^3}{4\,a}+\frac {17\,a\,\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}{6}-\frac {107\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2}{12}}{d\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{7/2}-2\,a\,d\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2}+a^2\,d\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}}+\frac {23\,\mathrm {atanh}\left (\frac {a\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{\sqrt {a^3}}\right )}{4\,d\,\sqrt {a^3}}-\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,a\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{2\,\sqrt {a^3}}\right )}{4\,d\,\sqrt {a^3}} \]

[In]

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

[Out]

(23*atanh((a*(a + a*tan(c + d*x)*1i)^(1/2))/(a^3)^(1/2)))/(4*d*(a^3)^(1/2)) - ((21*(a + a*tan(c + d*x)*1i)^3)/
(4*a) - (107*(a + a*tan(c + d*x)*1i)^2)/12 + (17*a*(a + a*tan(c + d*x)*1i))/6 + a^2/3)/(d*(a + a*tan(c + d*x)*
1i)^(7/2) - 2*a*d*(a + a*tan(c + d*x)*1i)^(5/2) + a^2*d*(a + a*tan(c + d*x)*1i)^(3/2)) - (2^(1/2)*atanh((2^(1/
2)*a*(a + a*tan(c + d*x)*1i)^(1/2))/(2*(a^3)^(1/2))))/(4*d*(a^3)^(1/2))