Integrand size = 26, antiderivative size = 91 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^2 \, dx=\frac {4 \sqrt [4]{-1} a^2 \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}+\frac {4 a^2 \sqrt {\cot (c+d x)}}{d}-\frac {4 i a^2 \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d} \] Output:
4*(-1)^(1/4)*a^2*arctanh((-1)^(3/4)*cot(d*x+c)^(1/2))/d+4*a^2*cot(d*x+c)^( 1/2)/d-4/3*I*a^2*cot(d*x+c)^(3/2)/d-2/5*a^2*cot(d*x+c)^(5/2)/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.42 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.56 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^2 \, dx=-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) \left (3 \cot (c+d x)+10 i \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},i \tan (c+d x)\right )\right )}{15 d} \] Input:
Integrate[Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^2,x]
Output:
(-2*a^2*Cot[c + d*x]^(3/2)*(3*Cot[c + d*x] + (10*I)*Hypergeometric2F1[-3/2 , 1, -1/2, I*Tan[c + d*x]]))/(15*d)
Time = 0.62 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {3042, 4156, 3042, 4026, 3042, 4011, 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.
| \(\displaystyle \int \cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cot (c+d x)^{7/2} (a+i a \tan (c+d x))^2dx\) |
| \(\Big \downarrow \) 4156 |
| \(\displaystyle \int \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+i a)^2dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (-a \tan \left (c+d x+\frac {\pi }{2}\right )+i a\right )^2dx\) |
| \(\Big \downarrow \) 4026 |
| \(\displaystyle -\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}+\int \cot ^{\frac {3}{2}}(c+d x) \left (2 i a^2 \cot (c+d x)-2 a^2\right )dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}+\int \left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (-2 i \tan \left (c+d x+\frac {\pi }{2}\right ) a^2-2 a^2\right )dx\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \int \sqrt {\cot (c+d x)} \left (-2 \cot (c+d x) a^2-2 i a^2\right )dx-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {4 i a^2 \cot ^{\frac {3}{2}}(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (2 a^2 \tan \left (c+d x+\frac {\pi }{2}\right )-2 i a^2\right )dx-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {4 i a^2 \cot ^{\frac {3}{2}}(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \int \frac {2 a^2-2 i a^2 \cot (c+d x)}{\sqrt {\cot (c+d x)}}dx-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {4 i a^2 \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {4 a^2 \sqrt {\cot (c+d x)}}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {2 i \tan \left (c+d x+\frac {\pi }{2}\right ) a^2+2 a^2}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {4 i a^2 \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {4 a^2 \sqrt {\cot (c+d x)}}{d}\) |
| \(\Big \downarrow \) 4016 |
| \(\displaystyle \frac {8 a^4 \int \frac {1}{-2 i \cot (c+d x) a^2-2 a^2}d\sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {4 i a^2 \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {4 a^2 \sqrt {\cot (c+d x)}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {4 \sqrt [4]{-1} a^2 \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {4 i a^2 \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {4 a^2 \sqrt {\cot (c+d x)}}{d}\) |
Input:
Int[Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^2,x]
Output:
(4*(-1)^(1/4)*a^2*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d + (4*a^2*Sqrt[ Cot[c + d*x]])/d - (((4*I)/3)*a^2*Cot[c + d*x]^(3/2))/d - (2*a^2*Cot[c + d *x]^(5/2))/(5*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. 210 vs. \(2 (74 ) = 148\).
Time = 0.50 (sec) , antiderivative size = 211, normalized size of antiderivative = 2.32
| method | result | size |
| derivativedivides | \(\frac {a^{2} \left (4 \sqrt {\cot \left (d x +c \right )}-\frac {2 \cot \left (d x +c \right )^{\frac {5}{2}}}{5}-\frac {4 i \cot \left (d x +c \right )^{\frac {3}{2}}}{3}-\frac {\sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{2}+\frac {i \sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{2}\right )}{d}\) | \(211\) |
| default | \(\frac {a^{2} \left (4 \sqrt {\cot \left (d x +c \right )}-\frac {2 \cot \left (d x +c \right )^{\frac {5}{2}}}{5}-\frac {4 i \cot \left (d x +c \right )^{\frac {3}{2}}}{3}-\frac {\sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{2}+\frac {i \sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{2}\right )}{d}\) | \(211\) |
Input:
int(cot(d*x+c)^(7/2)*(a+I*a*tan(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/d*a^2*(4*cot(d*x+c)^(1/2)-2/5*cot(d*x+c)^(5/2)-4/3*I*cot(d*x+c)^(3/2)-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)))+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))))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 340 vs. \(2 (73) = 146\).
Time = 0.10 (sec) , antiderivative size = 340, normalized size of antiderivative = 3.74 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^2 \, dx=-\frac {15 \, \sqrt {\frac {16 i \, a^{4}}{d^{2}}} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {{\left (4 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt {\frac {16 i \, a^{4}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{2 \, a^{2}}\right ) - 15 \, \sqrt {\frac {16 i \, a^{4}}{d^{2}}} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {{\left (4 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - \sqrt {\frac {16 i \, a^{4}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{2 \, a^{2}}\right ) - 8 \, {\left (43 \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} - 54 \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + 23 \, a^{2}\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}}{60 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \] Input:
integrate(cot(d*x+c)^(7/2)*(a+I*a*tan(d*x+c))^2,x, algorithm="fricas")
Output:
-1/60*(15*sqrt(16*I*a^4/d^2)*(d*e^(4*I*d*x + 4*I*c) - 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)*(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^2) - 15*sqrt(16*I*a^4/d^2)*(d*e^(4*I*d*x + 4*I*c) - 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)*(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^2) - 8*(43*a^2 *e^(4*I*d*x + 4*I*c) - 54*a^2*e^(2*I*d*x + 2*I*c) + 23*a^2)*sqrt((I*e^(2*I *d*x + 2*I*c) + I)/(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)
Timed out. \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^2 \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**(7/2)*(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. 158 vs. \(2 (73) = 146\).
Time = 0.12 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.74 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^2 \, dx=-\frac {15 \, {\left (-\left (2 i - 2\right ) \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - \left (2 i - 2\right ) \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + \left (i + 1\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \left (i + 1\right ) \, \sqrt {2} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )\right )} a^{2} - \frac {120 \, a^{2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {40 i \, a^{2}}{\tan \left (d x + c\right )^{\frac {3}{2}}} + \frac {12 \, a^{2}}{\tan \left (d x + c\right )^{\frac {5}{2}}}}{30 \, d} \] Input:
integrate(cot(d*x+c)^(7/2)*(a+I*a*tan(d*x+c))^2,x, algorithm="maxima")
Output:
-1/30*(15*(-(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 - 120*a^2/sqrt(tan(d*x + c)) + 40*I*a^2/tan(d*x + c)^(3/2) + 12*a ^2/tan(d*x + c)^(5/2))/d
Time = 0.23 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.64 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^2 \, dx=-\frac {2 \, {\left (\left (15 i - 15\right ) \, \sqrt {2} \arctan \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right ) - \frac {30 \, \tan \left (d x + c\right )^{2} - 10 i \, \tan \left (d x + c\right ) - 3}{\tan \left (d x + c\right )^{\frac {5}{2}}}\right )} a^{2}}{15 \, d} \] Input:
integrate(cot(d*x+c)^(7/2)*(a+I*a*tan(d*x+c))^2,x, algorithm="giac")
Output:
-2/15*((15*I - 15)*sqrt(2)*arctan(-(1/2*I - 1/2)*sqrt(2)*sqrt(tan(d*x + c) )) - (30*tan(d*x + c)^2 - 10*I*tan(d*x + c) - 3)/tan(d*x + c)^(5/2))*a^2/d
Timed out. \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^2 \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{7/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2 \,d x \] Input:
int(cot(c + d*x)^(7/2)*(a + a*tan(c + d*x)*1i)^2,x)
Output:
int(cot(c + d*x)^(7/2)*(a + a*tan(c + d*x)*1i)^2, x)
\[ \int \cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^2 \, dx=\frac {a^{2} \left (-2 \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{2}+10 \sqrt {\cot \left (d x +c \right )}+5 \left (\int \frac {\sqrt {\cot \left (d x +c \right )}}{\cot \left (d x +c \right )}d x \right ) d -5 \left (\int \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{3} \tan \left (d x +c \right )^{2}d x \right ) d +10 \left (\int \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{3} \tan \left (d x +c \right )d x \right ) d i \right )}{5 d} \] Input:
int(cot(d*x+c)^(7/2)*(a+I*a*tan(d*x+c))^2,x)
Output:
(a**2*( - 2*sqrt(cot(c + d*x))*cot(c + d*x)**2 + 10*sqrt(cot(c + d*x)) + 5 *int(sqrt(cot(c + d*x))/cot(c + d*x),x)*d - 5*int(sqrt(cot(c + d*x))*cot(c + d*x)**3*tan(c + d*x)**2,x)*d + 10*int(sqrt(cot(c + d*x))*cot(c + d*x)** 3*tan(c + d*x),x)*d*i))/(5*d)