Integrand size = 26, antiderivative size = 64 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^3 \, dx=-\frac {8 \sqrt [4]{-1} a^3 \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}-\frac {2 \left (i a^3+a^3 \cot (c+d x)\right )}{d \sqrt {\cot (c+d x)}} \] Output:
-8*(-1)^(1/4)*a^3*arctanh((-1)^(3/4)*cot(d*x+c)^(1/2))/d-2*(I*a^3+a^3*cot( d*x+c))/d/cot(d*x+c)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.39 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.88 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^3 \, dx=-\frac {2 a^3 \left (i-3 \cot (c+d x)+4 \cot (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},i \tan (c+d x)\right )\right )}{d \sqrt {\cot (c+d x)}} \] Input:
Integrate[Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^3,x]
Output:
(-2*a^3*(I - 3*Cot[c + d*x] + 4*Cot[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/ 2, I*Tan[c + d*x]]))/(d*Sqrt[Cot[c + d*x]])
Time = 0.43 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3042, 4156, 3042, 4036, 27, 3042, 4016, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cot (c+d x)^{3/2} (a+i a \tan (c+d x))^3dx\) |
| \(\Big \downarrow \) 4156 |
| \(\displaystyle \int \frac {(a \cot (c+d x)+i a)^3}{\cot ^{\frac {3}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (-a \tan \left (c+d x+\frac {\pi }{2}\right )+i a\right )^3}{\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4036 |
| \(\displaystyle -2 \int -\frac {2 i \left (\cot (c+d x) a^3+i a^3\right )}{\sqrt {\cot (c+d x)}}dx-\frac {2 \left (a^3 \cot (c+d x)+i a^3\right )}{d \sqrt {\cot (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 i \int \frac {\cot (c+d x) a^3+i a^3}{\sqrt {\cot (c+d x)}}dx-\frac {2 \left (a^3 \cot (c+d x)+i a^3\right )}{d \sqrt {\cot (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 4 i \int \frac {i a^3-a^3 \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 \left (a^3 \cot (c+d x)+i a^3\right )}{d \sqrt {\cot (c+d x)}}\) |
| \(\Big \downarrow \) 4016 |
| \(\displaystyle -\frac {8 i a^6 \int \frac {1}{a^3 \cot (c+d x)-i a^3}d\sqrt {\cot (c+d x)}}{d}-\frac {2 \left (a^3 \cot (c+d x)+i a^3\right )}{d \sqrt {\cot (c+d x)}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {8 \sqrt [4]{-1} a^3 \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}-\frac {2 \left (a^3 \cot (c+d x)+i a^3\right )}{d \sqrt {\cot (c+d x)}}\) |
Input:
Int[Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^3,x]
Output:
(-8*(-1)^(1/4)*a^3*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d - (2*(I*a^3 + a^3*Cot[c + d*x]))/(d*Sqrt[Cot[c + d*x]])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_ )]], x_Symbol] :> Simp[2*(c^2/f) Subst[Int[1/(b*c - d*x^2), x], x, Sqrt[b *Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && EqQ[c^2 + d^2, 0]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-a^2)*(b*c - a*d)*(a + b*Tan[e + f*x] )^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(b*c + a*d)*(n + 1))), x] + Si mp[a/(d*(b*c + a*d)*(n + 1)) Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[ e + f*x])^(n + 1)*Simp[b*(b*c*(m - 2) - a*d*(m - 2*n - 4)) + (a*b*c*(m - 2) + b^2*d*(n + 1) - a^2*d*(m + n - 1))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 1] && LtQ[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
Int[(cot[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x _)]^(n_.))^(p_.), x_Symbol] :> Simp[d^(n*p) Int[(d*Cot[e + f*x])^(m - n*p )*(b + a*Cot[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x] && !IntegerQ[m] && IntegersQ[n, p]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (55 ) = 110\).
Time = 0.21 (sec) , antiderivative size = 200, normalized size of antiderivative = 3.12
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (-2 \sqrt {\cot \left (d x +c \right )}+\sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )-i \sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )-\frac {2 i}{\sqrt {\cot \left (d x +c \right )}}\right )}{d}\) | \(200\) |
| default | \(\frac {a^{3} \left (-2 \sqrt {\cot \left (d x +c \right )}+\sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )-i \sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )-\frac {2 i}{\sqrt {\cot \left (d x +c \right )}}\right )}{d}\) | \(200\) |
Input:
int(cot(d*x+c)^(3/2)*(a+I*a*tan(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d*a^3*(-2*cot(d*x+c)^(1/2)+2^(1/2)*(ln((cot(d*x+c)+2^(1/2)*cot(d*x+c)^(1 /2)+1)/(cot(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2)+1))+2*arctan(1+2^(1/2)*cot(d*x +c)^(1/2))+2*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2)))-I*2^(1/2)*(ln((cot(d*x+c )-2^(1/2)*cot(d*x+c)^(1/2)+1)/(cot(d*x+c)+2^(1/2)*cot(d*x+c)^(1/2)+1))+2*a rctan(1+2^(1/2)*cot(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2)))-2 *I/cot(d*x+c)^(1/2))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 281 vs. \(2 (54) = 108\).
Time = 0.11 (sec) , antiderivative size = 281, normalized size of antiderivative = 4.39 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^3 \, dx=-\frac {16 \, a^{3} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} e^{\left (2 i \, d x + 2 i \, c\right )} - \sqrt {\frac {64 i \, a^{6}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {{\left (8 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt {\frac {64 i \, a^{6}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{4 \, a^{3}}\right ) + \sqrt {\frac {64 i \, a^{6}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {{\left (8 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - \sqrt {\frac {64 i \, a^{6}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{4 \, a^{3}}\right )}{4 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \] Input:
integrate(cot(d*x+c)^(3/2)*(a+I*a*tan(d*x+c))^3,x, algorithm="fricas")
Output:
-1/4*(16*a^3*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*e ^(2*I*d*x + 2*I*c) - sqrt(64*I*a^6/d^2)*(d*e^(2*I*d*x + 2*I*c) + d)*log(1/ 4*(8*I*a^3*e^(2*I*d*x + 2*I*c) + sqrt(64*I*a^6/d^2)*(d*e^(2*I*d*x + 2*I*c) - d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1)))*e^(-2*I *d*x - 2*I*c)/a^3) + sqrt(64*I*a^6/d^2)*(d*e^(2*I*d*x + 2*I*c) + d)*log(1/ 4*(8*I*a^3*e^(2*I*d*x + 2*I*c) - sqrt(64*I*a^6/d^2)*(d*e^(2*I*d*x + 2*I*c) - d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1)))*e^(-2*I *d*x - 2*I*c)/a^3))/(d*e^(2*I*d*x + 2*I*c) + d)
\[ \int \cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^3 \, dx=- i a^{3} \left (\int i \cot ^{\frac {3}{2}}{\left (c + d x \right )}\, dx + \int \left (- 3 \tan {\left (c + d x \right )} \cot ^{\frac {3}{2}}{\left (c + d x \right )}\right )\, dx + \int \tan ^{3}{\left (c + d x \right )} \cot ^{\frac {3}{2}}{\left (c + d x \right )}\, dx + \int \left (- 3 i \tan ^{2}{\left (c + d x \right )} \cot ^{\frac {3}{2}}{\left (c + d x \right )}\right )\, dx\right ) \] Input:
integrate(cot(d*x+c)**(3/2)*(a+I*a*tan(d*x+c))**3,x)
Output:
-I*a**3*(Integral(I*cot(c + d*x)**(3/2), x) + Integral(-3*tan(c + d*x)*cot (c + d*x)**(3/2), x) + Integral(tan(c + d*x)**3*cot(c + d*x)**(3/2), x) + Integral(-3*I*tan(c + d*x)**2*cot(c + d*x)**(3/2), x))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 144 vs. \(2 (54) = 108\).
Time = 0.25 (sec) , antiderivative size = 144, normalized size of antiderivative = 2.25 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^3 \, dx=-\frac {{\left (\left (2 i - 2\right ) \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + \left (2 i - 2\right ) \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - \left (i + 1\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \left (i + 1\right ) \, \sqrt {2} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )\right )} a^{3} + 2 i \, a^{3} \sqrt {\tan \left (d x + c\right )} + \frac {2 \, a^{3}}{\sqrt {\tan \left (d x + c\right )}}}{d} \] Input:
integrate(cot(d*x+c)^(3/2)*(a+I*a*tan(d*x+c))^3,x, algorithm="maxima")
Output:
-(((2*I - 2)*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2/sqrt(tan(d*x + c)))) + (2*I - 2)*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c)))) - (I + 1)*sqrt(2)*log(sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) + ( I + 1)*sqrt(2)*log(-sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1))*a^3 + 2*I*a^3*sqrt(tan(d*x + c)) + 2*a^3/sqrt(tan(d*x + c)))/d
Time = 0.23 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.72 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^3 \, dx=-\frac {2 \, {\left (-\left (2 i - 2\right ) \, \sqrt {2} \arctan \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right ) + i \, \sqrt {\tan \left (d x + c\right )} + \frac {1}{\sqrt {\tan \left (d x + c\right )}}\right )} a^{3}}{d} \] Input:
integrate(cot(d*x+c)^(3/2)*(a+I*a*tan(d*x+c))^3,x, algorithm="giac")
Output:
-2*(-(2*I - 2)*sqrt(2)*arctan(-(1/2*I - 1/2)*sqrt(2)*sqrt(tan(d*x + c))) + I*sqrt(tan(d*x + c)) + 1/sqrt(tan(d*x + c)))*a^3/d
Timed out. \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^3 \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{3/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^3 \,d x \] Input:
int(cot(c + d*x)^(3/2)*(a + a*tan(c + d*x)*1i)^3,x)
Output:
int(cot(c + d*x)^(3/2)*(a + a*tan(c + d*x)*1i)^3, x)
\[ \int \cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^3 \, dx=\frac {a^{3} \left (-2 \sqrt {\cot \left (d x +c \right )}-\left (\int \frac {\sqrt {\cot \left (d x +c \right )}}{\cot \left (d x +c \right )}d x \right ) d -\left (\int \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right ) \tan \left (d x +c \right )^{3}d x \right ) d i -3 \left (\int \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right ) \tan \left (d x +c \right )^{2}d x \right ) d +3 \left (\int \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right ) \tan \left (d x +c \right )d x \right ) d i \right )}{d} \] Input:
int(cot(d*x+c)^(3/2)*(a+I*a*tan(d*x+c))^3,x)
Output:
(a**3*( - 2*sqrt(cot(c + d*x)) - int(sqrt(cot(c + d*x))/cot(c + d*x),x)*d - int(sqrt(cot(c + d*x))*cot(c + d*x)*tan(c + d*x)**3,x)*d*i - 3*int(sqrt( cot(c + d*x))*cot(c + d*x)*tan(c + d*x)**2,x)*d + 3*int(sqrt(cot(c + d*x)) *cot(c + d*x)*tan(c + d*x),x)*d*i))/d