Integrand size = 26, antiderivative size = 71 \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2 \, dx=-\frac {4 (-1)^{3/4} a^2 \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}-\frac {4 i a^2 \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x)}{3 d} \] Output:
-4*(-1)^(3/4)*a^2*arctanh((-1)^(3/4)*cot(d*x+c)^(1/2))/d-4*I*a^2*cot(d*x+c )^(1/2)/d-2/3*a^2*cot(d*x+c)^(3/2)/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.28 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.69 \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2 \, dx=-\frac {2 a^2 \sqrt {\cot (c+d x)} \left (\cot (c+d x)+6 i \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},i \tan (c+d x)\right )\right )}{3 d} \] Input:
Integrate[Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^2,x]
Output:
(-2*a^2*Sqrt[Cot[c + d*x]]*(Cot[c + d*x] + (6*I)*Hypergeometric2F1[-1/2, 1 , 1/2, I*Tan[c + d*x]]))/(3*d)
Time = 0.49 (sec) , antiderivative size = 71, 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 = {3042, 4156, 3042, 4026, 3042, 4011, 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.
\(\displaystyle \int \cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \cot (c+d x)^{5/2} (a+i a \tan (c+d x))^2dx\) |
\(\Big \downarrow \) 4156 |
\(\displaystyle \int \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (-a \tan \left (c+d x+\frac {\pi }{2}\right )+i a\right )^2dx\) |
\(\Big \downarrow \) 4026 |
\(\displaystyle -\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\int \sqrt {\cot (c+d x)} \left (2 i a^2 \cot (c+d x)-2 a^2\right )dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\int \sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (-2 i \tan \left (c+d x+\frac {\pi }{2}\right ) a^2-2 a^2\right )dx\) |
\(\Big \downarrow \) 4011 |
\(\displaystyle \int \frac {-2 \cot (c+d x) a^2-2 i a^2}{\sqrt {\cot (c+d x)}}dx-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {4 i a^2 \sqrt {\cot (c+d x)}}{d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {2 a^2 \tan \left (c+d x+\frac {\pi }{2}\right )-2 i a^2}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {4 i a^2 \sqrt {\cot (c+d x)}}{d}\) |
\(\Big \downarrow \) 4016 |
\(\displaystyle -\frac {8 a^4 \int \frac {1}{2 i a^2-2 a^2 \cot (c+d x)}d\sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {4 i a^2 \sqrt {\cot (c+d x)}}{d}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {4 (-1)^{3/4} a^2 \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {4 i a^2 \sqrt {\cot (c+d x)}}{d}\) |
Input:
Int[Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^2,x]
Output:
(-4*(-1)^(3/4)*a^2*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d - ((4*I)*a^2* Sqrt[Cot[c + d*x]])/d - (2*a^2*Cot[c + d*x]^(3/2))/(3*d)
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[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*((a + b*Tan[e + f*x])^m/(f*m)), x] + Int [(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x], x] , x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]
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_)])^2, x_Symbol] :> Simp[d^2*((a + b*Tan[e + f*x])^(m + 1)/(b*f*( m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Simp[c^2 - d^2 + 2*c*d*Tan[e + f* x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && !LeQ [m, -1] && !(EqQ[m, 2] && EqQ[a, 0])
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. 200 vs. \(2 (58 ) = 116\).
Time = 0.36 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.83
method | result | size |
derivativedivides | \(\frac {a^{2} \left (-4 i \sqrt {\cot \left (d x +c \right )}-\frac {2 \cot \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {i \sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{2}+\frac {\sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{2}\right )}{d}\) | \(201\) |
default | \(\frac {a^{2} \left (-4 i \sqrt {\cot \left (d x +c \right )}-\frac {2 \cot \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {i \sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{2}+\frac {\sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{2}\right )}{d}\) | \(201\) |
Input:
int(cot(d*x+c)^(5/2)*(a+I*a*tan(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/d*a^2*(-4*I*cot(d*x+c)^(1/2)-2/3*cot(d*x+c)^(3/2)+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*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*cot(d*x+c)^(1/ 2)))+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))))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 297 vs. \(2 (57) = 114\).
Time = 0.09 (sec) , antiderivative size = 297, normalized size of antiderivative = 4.18 \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2 \, dx=-\frac {3 \, \sqrt {-\frac {16 i \, a^{4}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \log \left (\frac {{\left (4 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt {-\frac {16 i \, a^{4}}{d^{2}}} {\left (i \, d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{2 \, a^{2}}\right ) - 3 \, \sqrt {-\frac {16 i \, a^{4}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \log \left (\frac {{\left (4 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt {-\frac {16 i \, a^{4}}{d^{2}}} {\left (-i \, d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{2 \, a^{2}}\right ) + 8 \, {\left (7 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - 5 i \, a^{2}\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}}{12 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )}} \] Input:
integrate(cot(d*x+c)^(5/2)*(a+I*a*tan(d*x+c))^2,x, algorithm="fricas")
Output:
-1/12*(3*sqrt(-16*I*a^4/d^2)*(d*e^(2*I*d*x + 2*I*c) - d)*log(1/2*(4*I*a^2* e^(2*I*d*x + 2*I*c) + sqrt(-16*I*a^4/d^2)*(I*d*e^(2*I*d*x + 2*I*c) - I*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^2) - 3*sqrt(-16*I*a^4/d^2)*(d*e^(2*I*d*x + 2*I*c) - d)*log(1/2*( 4*I*a^2*e^(2*I*d*x + 2*I*c) + sqrt(-16*I*a^4/d^2)*(-I*d*e^(2*I*d*x + 2*I*c ) + I*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^2) + 8*(7*I*a^2*e^(2*I*d*x + 2*I*c) - 5*I*a^2)*sqrt((I* e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1)))/(d*e^(2*I*d*x + 2*I*c ) - d)
Timed out. \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2 \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**(5/2)*(a+I*a*tan(d*x+c))**2,x)
Output:
Timed out
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 145 vs. \(2 (57) = 114\).
Time = 0.23 (sec) , antiderivative size = 145, normalized size of antiderivative = 2.04 \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2 \, dx=\frac {3 \, {\left (\left (2 i + 2\right ) \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + \left (2 i + 2\right ) \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + \left (i - 1\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \left (i - 1\right ) \, \sqrt {2} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )\right )} a^{2} - \frac {24 i \, a^{2}}{\sqrt {\tan \left (d x + c\right )}} - \frac {4 \, a^{2}}{\tan \left (d x + c\right )^{\frac {3}{2}}}}{6 \, d} \] Input:
integrate(cot(d*x+c)^(5/2)*(a+I*a*tan(d*x+c))^2,x, algorithm="maxima")
Output:
1/6*(3*((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^2 - 24*I*a^2/sqrt(tan(d*x + c)) - 4*a^2/tan(d*x + c)^(3/2))/d
Time = 0.20 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.66 \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2 \, dx=-\frac {2 \, {\left (\left (3 i + 3\right ) \, \sqrt {2} \arctan \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right ) + \frac {6 i \, \tan \left (d x + c\right ) + 1}{\tan \left (d x + c\right )^{\frac {3}{2}}}\right )} a^{2}}{3 \, d} \] Input:
integrate(cot(d*x+c)^(5/2)*(a+I*a*tan(d*x+c))^2,x, algorithm="giac")
Output:
-2/3*((3*I + 3)*sqrt(2)*arctan(-(1/2*I - 1/2)*sqrt(2)*sqrt(tan(d*x + c))) + (6*I*tan(d*x + c) + 1)/tan(d*x + c)^(3/2))*a^2/d
Timed out. \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2 \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{5/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2 \,d x \] Input:
int(cot(c + d*x)^(5/2)*(a + a*tan(c + d*x)*1i)^2,x)
Output:
int(cot(c + d*x)^(5/2)*(a + a*tan(c + d*x)*1i)^2, x)
\[ \int \cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2 \, dx=a^{2} \left (-\left (\int \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{2} \tan \left (d x +c \right )^{2}d x \right )+2 \left (\int \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{2} \tan \left (d x +c \right )d x \right ) i +\int \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{2}d x \right ) \] Input:
int(cot(d*x+c)^(5/2)*(a+I*a*tan(d*x+c))^2,x)
Output:
a**2*( - int(sqrt(cot(c + d*x))*cot(c + d*x)**2*tan(c + d*x)**2,x) + 2*int (sqrt(cot(c + d*x))*cot(c + d*x)**2*tan(c + d*x),x)*i + int(sqrt(cot(c + d *x))*cot(c + d*x)**2,x))