Integrand size = 24, antiderivative size = 74 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^2 \, dx=2 a^2 x+\frac {2 a^2 \cot (c+d x)}{d}-\frac {i a^2 \cot ^2(c+d x)}{d}-\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {2 i a^2 \log (\sin (c+d x))}{d} \] Output:
2*a^2*x+2*a^2*cot(d*x+c)/d-I*a^2*cot(d*x+c)^2/d-1/3*a^2*cot(d*x+c)^3/d-2*I *a^2*ln(sin(d*x+c))/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.04 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.39 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^2 \, dx=-\frac {i a^2 \csc ^2(c+d x)}{d}-\frac {a^2 \cot ^3(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-\tan ^2(c+d x)\right )}{3 d}+\frac {a^2 \cot (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-\tan ^2(c+d x)\right )}{d}-\frac {2 i a^2 \log (\sin (c+d x))}{d} \] Input:
Integrate[Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])^2,x]
Output:
((-I)*a^2*Csc[c + d*x]^2)/d - (a^2*Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2])/(3*d) + (a^2*Cot[c + d*x]*Hypergeometric2F1[-1/ 2, 1, 1/2, -Tan[c + d*x]^2])/d - ((2*I)*a^2*Log[Sin[c + d*x]])/d
Time = 0.59 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.07, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {3042, 4025, 27, 3042, 4012, 25, 3042, 4012, 3042, 4014, 3042, 25, 3956}
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.
\(\displaystyle \int \cot ^4(c+d x) (a+i a \tan (c+d x))^2 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a+i a \tan (c+d x))^2}{\tan (c+d x)^4}dx\) |
\(\Big \downarrow \) 4025 |
\(\displaystyle -\frac {a^2 \cot ^3(c+d x)}{3 d}+\int 2 \cot ^3(c+d x) \left (i a^2-a^2 \tan (c+d x)\right )dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {a^2 \cot ^3(c+d x)}{3 d}+2 \int \cot ^3(c+d x) \left (i a^2-a^2 \tan (c+d x)\right )dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a^2 \cot ^3(c+d x)}{3 d}+2 \int \frac {i a^2-a^2 \tan (c+d x)}{\tan (c+d x)^3}dx\) |
\(\Big \downarrow \) 4012 |
\(\displaystyle -\frac {a^2 \cot ^3(c+d x)}{3 d}+2 \left (\int -\cot ^2(c+d x) \left (i \tan (c+d x) a^2+a^2\right )dx-\frac {i a^2 \cot ^2(c+d x)}{2 d}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {a^2 \cot ^3(c+d x)}{3 d}+2 \left (-\int \cot ^2(c+d x) \left (i \tan (c+d x) a^2+a^2\right )dx-\frac {i a^2 \cot ^2(c+d x)}{2 d}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a^2 \cot ^3(c+d x)}{3 d}+2 \left (-\int \frac {i \tan (c+d x) a^2+a^2}{\tan (c+d x)^2}dx-\frac {i a^2 \cot ^2(c+d x)}{2 d}\right )\) |
\(\Big \downarrow \) 4012 |
\(\displaystyle -\frac {a^2 \cot ^3(c+d x)}{3 d}+2 \left (-\int \cot (c+d x) \left (i a^2-a^2 \tan (c+d x)\right )dx-\frac {i a^2 \cot ^2(c+d x)}{2 d}+\frac {a^2 \cot (c+d x)}{d}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a^2 \cot ^3(c+d x)}{3 d}+2 \left (-\int \frac {i a^2-a^2 \tan (c+d x)}{\tan (c+d x)}dx-\frac {i a^2 \cot ^2(c+d x)}{2 d}+\frac {a^2 \cot (c+d x)}{d}\right )\) |
\(\Big \downarrow \) 4014 |
\(\displaystyle -\frac {a^2 \cot ^3(c+d x)}{3 d}+2 \left (-i a^2 \int \cot (c+d x)dx-\frac {i a^2 \cot ^2(c+d x)}{2 d}+\frac {a^2 \cot (c+d x)}{d}+a^2 x\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a^2 \cot ^3(c+d x)}{3 d}+2 \left (-i a^2 \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx-\frac {i a^2 \cot ^2(c+d x)}{2 d}+\frac {a^2 \cot (c+d x)}{d}+a^2 x\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {a^2 \cot ^3(c+d x)}{3 d}+2 \left (i a^2 \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx-\frac {i a^2 \cot ^2(c+d x)}{2 d}+\frac {a^2 \cot (c+d x)}{d}+a^2 x\right )\) |
\(\Big \downarrow \) 3956 |
\(\displaystyle -\frac {a^2 \cot ^3(c+d x)}{3 d}+2 \left (-\frac {i a^2 \cot ^2(c+d x)}{2 d}+\frac {a^2 \cot (c+d x)}{d}-\frac {i a^2 \log (-\sin (c+d x))}{d}+a^2 x\right )\) |
Input:
Int[Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])^2,x]
Output:
-1/3*(a^2*Cot[c + d*x]^3)/d + 2*(a^2*x + (a^2*Cot[c + d*x])/d - ((I/2)*a^2 *Cot[c + d*x]^2)/d - (I*a^2*Log[-Sin[c + d*x]])/d)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/ (f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x] )^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a , b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1 ]
Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[(a*c + b*d)*(x/(a^2 + b^2)), x] + Simp[(b*c - a *d)/(a^2 + b^2) Int[(b - a*Tan[e + f*x])/(a + b*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] && N eQ[a*c + b*d, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(b*c - a*d)^2*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[a*c^2 + 2*b*c*d - a*d^2 - (b*c^2 - 2*a*c*d - b*d^2)*Ta n[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0]
Time = 1.32 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.86
method | result | size |
parallelrisch | \(\frac {a^{2} \left (-\frac {\cot \left (d x +c \right )^{3}}{3}-2 i \ln \left (\tan \left (d x +c \right )\right )+i \ln \left (\sec \left (d x +c \right )^{2}\right )+2 d x -i \cot \left (d x +c \right )^{2}+2 \cot \left (d x +c \right )\right )}{d}\) | \(64\) |
derivativedivides | \(\frac {-a^{2} \left (-\cot \left (d x +c \right )-d x -c \right )+2 i a^{2} \left (-\frac {\cot \left (d x +c \right )^{2}}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+a^{2} \left (-\frac {\cot \left (d x +c \right )^{3}}{3}+\cot \left (d x +c \right )+d x +c \right )}{d}\) | \(78\) |
default | \(\frac {-a^{2} \left (-\cot \left (d x +c \right )-d x -c \right )+2 i a^{2} \left (-\frac {\cot \left (d x +c \right )^{2}}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+a^{2} \left (-\frac {\cot \left (d x +c \right )^{3}}{3}+\cot \left (d x +c \right )+d x +c \right )}{d}\) | \(78\) |
risch | \(-\frac {4 a^{2} c}{d}+\frac {2 i a^{2} \left (15 \,{\mathrm e}^{4 i \left (d x +c \right )}-18 \,{\mathrm e}^{2 i \left (d x +c \right )}+7\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}-\frac {2 i a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(78\) |
norman | \(\frac {-\frac {a^{2}}{3 d}+2 a^{2} x \tan \left (d x +c \right )^{3}+\frac {2 a^{2} \tan \left (d x +c \right )^{2}}{d}-\frac {i a^{2} \tan \left (d x +c \right )}{d}}{\tan \left (d x +c \right )^{3}}+\frac {i a^{2} \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{d}-\frac {2 i a^{2} \ln \left (\tan \left (d x +c \right )\right )}{d}\) | \(101\) |
Input:
int(cot(d*x+c)^4*(a+I*a*tan(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
a^2*(-1/3*cot(d*x+c)^3-2*I*ln(tan(d*x+c))+I*ln(sec(d*x+c)^2)+2*d*x-I*cot(d *x+c)^2+2*cot(d*x+c))/d
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (68) = 136\).
Time = 0.08 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.88 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^2 \, dx=-\frac {2 \, {\left (-15 i \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} + 18 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - 7 i \, a^{2} + 3 \, {\left (i \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} - 3 i \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} + 3 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{2}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )\right )}}{3 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} - 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )}} \] Input:
integrate(cot(d*x+c)^4*(a+I*a*tan(d*x+c))^2,x, algorithm="fricas")
Output:
-2/3*(-15*I*a^2*e^(4*I*d*x + 4*I*c) + 18*I*a^2*e^(2*I*d*x + 2*I*c) - 7*I*a ^2 + 3*(I*a^2*e^(6*I*d*x + 6*I*c) - 3*I*a^2*e^(4*I*d*x + 4*I*c) + 3*I*a^2* e^(2*I*d*x + 2*I*c) - I*a^2)*log(e^(2*I*d*x + 2*I*c) - 1))/(d*e^(6*I*d*x + 6*I*c) - 3*d*e^(4*I*d*x + 4*I*c) + 3*d*e^(2*I*d*x + 2*I*c) - d)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (66) = 132\).
Time = 0.21 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.84 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^2 \, dx=- \frac {2 i a^{2} \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac {30 i a^{2} e^{4 i c} e^{4 i d x} - 36 i a^{2} e^{2 i c} e^{2 i d x} + 14 i a^{2}}{3 d e^{6 i c} e^{6 i d x} - 9 d e^{4 i c} e^{4 i d x} + 9 d e^{2 i c} e^{2 i d x} - 3 d} \] Input:
integrate(cot(d*x+c)**4*(a+I*a*tan(d*x+c))**2,x)
Output:
-2*I*a**2*log(exp(2*I*d*x) - exp(-2*I*c))/d + (30*I*a**2*exp(4*I*c)*exp(4* I*d*x) - 36*I*a**2*exp(2*I*c)*exp(2*I*d*x) + 14*I*a**2)/(3*d*exp(6*I*c)*ex p(6*I*d*x) - 9*d*exp(4*I*c)*exp(4*I*d*x) + 9*d*exp(2*I*c)*exp(2*I*d*x) - 3 *d)
Time = 0.12 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.12 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^2 \, dx=\frac {6 \, {\left (d x + c\right )} a^{2} + 3 i \, a^{2} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 6 i \, a^{2} \log \left (\tan \left (d x + c\right )\right ) + \frac {6 \, a^{2} \tan \left (d x + c\right )^{2} - 3 i \, a^{2} \tan \left (d x + c\right ) - a^{2}}{\tan \left (d x + c\right )^{3}}}{3 \, d} \] Input:
integrate(cot(d*x+c)^4*(a+I*a*tan(d*x+c))^2,x, algorithm="maxima")
Output:
1/3*(6*(d*x + c)*a^2 + 3*I*a^2*log(tan(d*x + c)^2 + 1) - 6*I*a^2*log(tan(d *x + c)) + (6*a^2*tan(d*x + c)^2 - 3*I*a^2*tan(d*x + c) - a^2)/tan(d*x + c )^3)/d
Time = 0.19 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.04 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^2 \, dx=\frac {2 i \, a^{2} \log \left (\tan \left (d x + c\right ) + i\right )}{d} - \frac {2 i \, a^{2} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{d} + \frac {6 \, a^{2} \tan \left (d x + c\right )^{2} - 3 i \, a^{2} \tan \left (d x + c\right ) - a^{2}}{3 \, d \tan \left (d x + c\right )^{3}} \] Input:
integrate(cot(d*x+c)^4*(a+I*a*tan(d*x+c))^2,x, algorithm="giac")
Output:
2*I*a^2*log(tan(d*x + c) + I)/d - 2*I*a^2*log(abs(tan(d*x + c)))/d + 1/3*( 6*a^2*tan(d*x + c)^2 - 3*I*a^2*tan(d*x + c) - a^2)/(d*tan(d*x + c)^3)
Time = 0.91 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.92 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^2 \, dx=\frac {2\,a^2\,\mathrm {cot}\left (c+d\,x\right )}{d}+\frac {4\,a^2\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{d}-\frac {a^2\,{\mathrm {cot}\left (c+d\,x\right )}^3}{3\,d}-\frac {a^2\,{\mathrm {cot}\left (c+d\,x\right )}^2\,1{}\mathrm {i}}{d} \] Input:
int(cot(c + d*x)^4*(a + a*tan(c + d*x)*1i)^2,x)
Output:
(2*a^2*cot(c + d*x))/d + (4*a^2*atan(2*tan(c + d*x) + 1i))/d - (a^2*cot(c + d*x)^2*1i)/d - (a^2*cot(c + d*x)^3)/(3*d)
Time = 0.18 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.61 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^2 \, dx=\frac {a^{2} \left (14 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-2 \cos \left (d x +c \right )+12 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )^{3} i -12 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{3} i +12 \sin \left (d x +c \right )^{3} d x +3 \sin \left (d x +c \right )^{3} i -6 \sin \left (d x +c \right ) i \right )}{6 \sin \left (d x +c \right )^{3} d} \] Input:
int(cot(d*x+c)^4*(a+I*a*tan(d*x+c))^2,x)
Output:
(a**2*(14*cos(c + d*x)*sin(c + d*x)**2 - 2*cos(c + d*x) + 12*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**3*i - 12*log(tan((c + d*x)/2))*sin(c + d*x)* *3*i + 12*sin(c + d*x)**3*d*x + 3*sin(c + d*x)**3*i - 6*sin(c + d*x)*i))/( 6*sin(c + d*x)**3*d)