\(\int \cot ^5(c+d x) (a+i a \tan (c+d x))^2 \, dx\) [22]

Optimal result

Integrand size = 24, antiderivative size = 93 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x))^2 \, dx=2 i a^2 x+\frac {2 i a^2 \cot (c+d x)}{d}+\frac {a^2 \cot ^2(c+d x)}{d}-\frac {2 i a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot ^4(c+d x)}{4 d}+\frac {2 a^2 \log (\sin (c+d x))}{d} \] Output:

2*I*a^2*x+2*I*a^2*cot(d*x+c)/d+a^2*cot(d*x+c)^2/d-2/3*I*a^2*cot(d*x+c)^3/d 
-1/4*a^2*cot(d*x+c)^4/d+2*a^2*ln(sin(d*x+c))/d
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.03 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.97 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x))^2 \, dx=\frac {3 a^2 \csc ^2(c+d x)}{2 d}-\frac {a^2 \csc ^4(c+d x)}{4 d}-\frac {2 i a^2 \cot ^3(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-\tan ^2(c+d x)\right )}{3 d}+\frac {2 a^2 \log (\sin (c+d x))}{d} \] Input:

Integrate[Cot[c + d*x]^5*(a + I*a*Tan[c + d*x])^2,x]
 

Output:

(3*a^2*Csc[c + d*x]^2)/(2*d) - (a^2*Csc[c + d*x]^4)/(4*d) - (((2*I)/3)*a^2 
*Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2])/d + (2* 
a^2*Log[Sin[c + d*x]])/d
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.08, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 4025, 27, 3042, 4012, 25, 3042, 4012, 3042, 4012, 25, 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.

\(\displaystyle \int \cot ^5(c+d x) (a+i a \tan (c+d x))^2 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a+i a \tan (c+d x))^2}{\tan (c+d x)^5}dx\)

\(\Big \downarrow \) 4025

\(\displaystyle -\frac {a^2 \cot ^4(c+d x)}{4 d}+\int 2 \cot ^4(c+d x) \left (i a^2-a^2 \tan (c+d x)\right )dx\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {a^2 \cot ^4(c+d x)}{4 d}+2 \int \cot ^4(c+d x) \left (i a^2-a^2 \tan (c+d x)\right )dx\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a^2 \cot ^4(c+d x)}{4 d}+2 \int \frac {i a^2-a^2 \tan (c+d x)}{\tan (c+d x)^4}dx\)

\(\Big \downarrow \) 4012

\(\displaystyle -\frac {a^2 \cot ^4(c+d x)}{4 d}+2 \left (\int -\cot ^3(c+d x) \left (i \tan (c+d x) a^2+a^2\right )dx-\frac {i a^2 \cot ^3(c+d x)}{3 d}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {a^2 \cot ^4(c+d x)}{4 d}+2 \left (-\int \cot ^3(c+d x) \left (i \tan (c+d x) a^2+a^2\right )dx-\frac {i a^2 \cot ^3(c+d x)}{3 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a^2 \cot ^4(c+d x)}{4 d}+2 \left (-\int \frac {i \tan (c+d x) a^2+a^2}{\tan (c+d x)^3}dx-\frac {i a^2 \cot ^3(c+d x)}{3 d}\right )\)

\(\Big \downarrow \) 4012

\(\displaystyle -\frac {a^2 \cot ^4(c+d x)}{4 d}+2 \left (-\int \cot ^2(c+d x) \left (i a^2-a^2 \tan (c+d x)\right )dx-\frac {i a^2 \cot ^3(c+d x)}{3 d}+\frac {a^2 \cot ^2(c+d x)}{2 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a^2 \cot ^4(c+d x)}{4 d}+2 \left (-\int \frac {i a^2-a^2 \tan (c+d x)}{\tan (c+d x)^2}dx-\frac {i a^2 \cot ^3(c+d x)}{3 d}+\frac {a^2 \cot ^2(c+d x)}{2 d}\right )\)

\(\Big \downarrow \) 4012

\(\displaystyle -\frac {a^2 \cot ^4(c+d x)}{4 d}+2 \left (-\int -\cot (c+d x) \left (i \tan (c+d x) a^2+a^2\right )dx-\frac {i a^2 \cot ^3(c+d x)}{3 d}+\frac {a^2 \cot ^2(c+d x)}{2 d}+\frac {i a^2 \cot (c+d x)}{d}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {a^2 \cot ^4(c+d x)}{4 d}+2 \left (\int \cot (c+d x) \left (i \tan (c+d x) a^2+a^2\right )dx-\frac {i a^2 \cot ^3(c+d x)}{3 d}+\frac {a^2 \cot ^2(c+d x)}{2 d}+\frac {i a^2 \cot (c+d x)}{d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a^2 \cot ^4(c+d x)}{4 d}+2 \left (\int \frac {i \tan (c+d x) a^2+a^2}{\tan (c+d x)}dx-\frac {i a^2 \cot ^3(c+d x)}{3 d}+\frac {a^2 \cot ^2(c+d x)}{2 d}+\frac {i a^2 \cot (c+d x)}{d}\right )\)

\(\Big \downarrow \) 4014

\(\displaystyle -\frac {a^2 \cot ^4(c+d x)}{4 d}+2 \left (a^2 \int \cot (c+d x)dx-\frac {i a^2 \cot ^3(c+d x)}{3 d}+\frac {a^2 \cot ^2(c+d x)}{2 d}+\frac {i a^2 \cot (c+d x)}{d}+i a^2 x\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a^2 \cot ^4(c+d x)}{4 d}+2 \left (a^2 \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx-\frac {i a^2 \cot ^3(c+d x)}{3 d}+\frac {a^2 \cot ^2(c+d x)}{2 d}+\frac {i a^2 \cot (c+d x)}{d}+i a^2 x\right )\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {a^2 \cot ^4(c+d x)}{4 d}+2 \left (-a^2 \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx-\frac {i a^2 \cot ^3(c+d x)}{3 d}+\frac {a^2 \cot ^2(c+d x)}{2 d}+\frac {i a^2 \cot (c+d x)}{d}+i a^2 x\right )\)

\(\Big \downarrow \) 3956

\(\displaystyle -\frac {a^2 \cot ^4(c+d x)}{4 d}+2 \left (-\frac {i a^2 \cot ^3(c+d x)}{3 d}+\frac {a^2 \cot ^2(c+d x)}{2 d}+\frac {i a^2 \cot (c+d x)}{d}+\frac {a^2 \log (-\sin (c+d x))}{d}+i a^2 x\right )\)

Input:

Int[Cot[c + d*x]^5*(a + I*a*Tan[c + d*x])^2,x]
 

Output:

-1/4*(a^2*Cot[c + d*x]^4)/d + 2*(I*a^2*x + (I*a^2*Cot[c + d*x])/d + (a^2*C 
ot[c + d*x]^2)/(2*d) - ((I/3)*a^2*Cot[c + d*x]^3)/d + (a^2*Log[-Sin[c + d* 
x]])/d)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3956

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d 
*x], x]]/d, x] /; FreeQ[{c, d}, x]
 

rule 4012

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 
]
 

rule 4014

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]
 

rule 4025

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]
 

Maple [A] (verified)

Time = 1.42 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.81

method	result	size
parallelrisch	\(-\frac {a^{2} \left (3 \cot \left (d x +c \right )^{4}+8 i \cot \left (d x +c \right )^{3}-24 i d x -12 \cot \left (d x +c \right )^{2}-24 i \cot \left (d x +c \right )-24 \ln \left (\tan \left (d x +c \right )\right )+12 \ln \left (\sec \left (d x +c \right )^{2}\right )\right )}{12 d}\)	\(75\)
risch	\(-\frac {4 i a^{2} c}{d}-\frac {2 a^{2} \left (21 \,{\mathrm e}^{6 i \left (d x +c \right )}-36 \,{\mathrm e}^{4 i \left (d x +c \right )}+29 \,{\mathrm e}^{2 i \left (d x +c \right )}-8\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {2 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\)	\(88\)
derivativedivides	\(\frac {-a^{2} \left (-\frac {\cot \left (d x +c \right )^{2}}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+2 i a^{2} \left (-\frac {\cot \left (d x +c \right )^{3}}{3}+\cot \left (d x +c \right )+d x +c \right )+a^{2} \left (-\frac {\cot \left (d x +c \right )^{4}}{4}+\frac {\cot \left (d x +c \right )^{2}}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\)	\(90\)
default	\(\frac {-a^{2} \left (-\frac {\cot \left (d x +c \right )^{2}}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+2 i a^{2} \left (-\frac {\cot \left (d x +c \right )^{3}}{3}+\cot \left (d x +c \right )+d x +c \right )+a^{2} \left (-\frac {\cot \left (d x +c \right )^{4}}{4}+\frac {\cot \left (d x +c \right )^{2}}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\)	\(90\)
norman	\(\frac {\frac {a^{2} \tan \left (d x +c \right )^{2}}{d}-\frac {a^{2}}{4 d}-\frac {2 i a^{2} \tan \left (d x +c \right )}{3 d}+\frac {2 i a^{2} \tan \left (d x +c \right )^{3}}{d}+2 i a^{2} x \tan \left (d x +c \right )^{4}}{\tan \left (d x +c \right )^{4}}+\frac {2 a^{2} \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {a^{2} \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{d}\)	\(116\)

Input:

int(cot(d*x+c)^5*(a+I*a*tan(d*x+c))^2,x,method=_RETURNVERBOSE)
 

Output:

-1/12*a^2*(3*cot(d*x+c)^4+8*I*cot(d*x+c)^3-24*I*d*x-12*cot(d*x+c)^2-24*I*c 
ot(d*x+c)-24*ln(tan(d*x+c))+12*ln(sec(d*x+c)^2))/d
 

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. 174 vs. \(2 (83) = 166\).

Time = 0.08 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.87 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x))^2 \, dx=-\frac {2 \, {\left (21 \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} - 36 \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} + 29 \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - 8 \, a^{2} - 3 \, {\left (a^{2} e^{\left (8 i \, d x + 8 i \, c\right )} - 4 \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} - 4 \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )\right )}}{3 \, {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} - 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} - 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \] Input:

integrate(cot(d*x+c)^5*(a+I*a*tan(d*x+c))^2,x, algorithm="fricas")
 

Output:

-2/3*(21*a^2*e^(6*I*d*x + 6*I*c) - 36*a^2*e^(4*I*d*x + 4*I*c) + 29*a^2*e^( 
2*I*d*x + 2*I*c) - 8*a^2 - 3*(a^2*e^(8*I*d*x + 8*I*c) - 4*a^2*e^(6*I*d*x + 
 6*I*c) + 6*a^2*e^(4*I*d*x + 4*I*c) - 4*a^2*e^(2*I*d*x + 2*I*c) + a^2)*log 
(e^(2*I*d*x + 2*I*c) - 1))/(d*e^(8*I*d*x + 8*I*c) - 4*d*e^(6*I*d*x + 6*I*c 
) + 6*d*e^(4*I*d*x + 4*I*c) - 4*d*e^(2*I*d*x + 2*I*c) + d)
 

Sympy [A] (verification not implemented)

Time = 0.86 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.81 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x))^2 \, dx=\frac {2 a^{2} \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac {- 42 a^{2} e^{6 i c} e^{6 i d x} + 72 a^{2} e^{4 i c} e^{4 i d x} - 58 a^{2} e^{2 i c} e^{2 i d x} + 16 a^{2}}{3 d e^{8 i c} e^{8 i d x} - 12 d e^{6 i c} e^{6 i d x} + 18 d e^{4 i c} e^{4 i d x} - 12 d e^{2 i c} e^{2 i d x} + 3 d} \] Input:

integrate(cot(d*x+c)**5*(a+I*a*tan(d*x+c))**2,x)
 

Output:

2*a**2*log(exp(2*I*d*x) - exp(-2*I*c))/d + (-42*a**2*exp(6*I*c)*exp(6*I*d* 
x) + 72*a**2*exp(4*I*c)*exp(4*I*d*x) - 58*a**2*exp(2*I*c)*exp(2*I*d*x) + 1 
6*a**2)/(3*d*exp(8*I*c)*exp(8*I*d*x) - 12*d*exp(6*I*c)*exp(6*I*d*x) + 18*d 
*exp(4*I*c)*exp(4*I*d*x) - 12*d*exp(2*I*c)*exp(2*I*d*x) + 3*d)
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.04 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x))^2 \, dx=-\frac {-24 i \, {\left (d x + c\right )} a^{2} + 12 \, a^{2} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 24 \, a^{2} \log \left (\tan \left (d x + c\right )\right ) - \frac {24 i \, a^{2} \tan \left (d x + c\right )^{3} + 12 \, a^{2} \tan \left (d x + c\right )^{2} - 8 i \, a^{2} \tan \left (d x + c\right ) - 3 \, a^{2}}{\tan \left (d x + c\right )^{4}}}{12 \, d} \] Input:

integrate(cot(d*x+c)^5*(a+I*a*tan(d*x+c))^2,x, algorithm="maxima")
 

Output:

-1/12*(-24*I*(d*x + c)*a^2 + 12*a^2*log(tan(d*x + c)^2 + 1) - 24*a^2*log(t 
an(d*x + c)) - (24*I*a^2*tan(d*x + c)^3 + 12*a^2*tan(d*x + c)^2 - 8*I*a^2* 
tan(d*x + c) - 3*a^2)/tan(d*x + c)^4)/d
 

Giac [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.97 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x))^2 \, dx=-\frac {2 \, a^{2} \log \left (\tan \left (d x + c\right ) + i\right )}{d} + \frac {2 \, a^{2} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{d} - \frac {-24 i \, a^{2} \tan \left (d x + c\right )^{3} - 12 \, a^{2} \tan \left (d x + c\right )^{2} + 8 i \, a^{2} \tan \left (d x + c\right ) + 3 \, a^{2}}{12 \, d \tan \left (d x + c\right )^{4}} \] Input:

integrate(cot(d*x+c)^5*(a+I*a*tan(d*x+c))^2,x, algorithm="giac")
 

Output:

-2*a^2*log(tan(d*x + c) + I)/d + 2*a^2*log(abs(tan(d*x + c)))/d - 1/12*(-2 
4*I*a^2*tan(d*x + c)^3 - 12*a^2*tan(d*x + c)^2 + 8*I*a^2*tan(d*x + c) + 3* 
a^2)/(d*tan(d*x + c)^4)
 

Mupad [B] (verification not implemented)

Time = 1.08 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.86 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x))^2 \, dx=\frac {a^2\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,4{}\mathrm {i}}{d}-\frac {-a^2\,{\mathrm {tan}\left (c+d\,x\right )}^3\,2{}\mathrm {i}-a^2\,{\mathrm {tan}\left (c+d\,x\right )}^2+\frac {a^2\,\mathrm {tan}\left (c+d\,x\right )\,2{}\mathrm {i}}{3}+\frac {a^2}{4}}{d\,{\mathrm {tan}\left (c+d\,x\right )}^4} \] Input:

int(cot(c + d*x)^5*(a + a*tan(c + d*x)*1i)^2,x)
 

Output:

(a^2*atan(2*tan(c + d*x) + 1i)*4i)/d - ((a^2*tan(c + d*x)*2i)/3 + a^2/4 - 
a^2*tan(c + d*x)^2 - a^2*tan(c + d*x)^3*2i)/(d*tan(c + d*x)^4)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.37 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x))^2 \, dx=\frac {a^{2} \left (256 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} i -64 \cos \left (d x +c \right ) \sin \left (d x +c \right ) i -192 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )^{4}+192 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{4}+192 \sin \left (d x +c \right )^{4} d i x -63 \sin \left (d x +c \right )^{4}+144 \sin \left (d x +c \right )^{2}-24\right )}{96 \sin \left (d x +c \right )^{4} d} \] Input:

int(cot(d*x+c)^5*(a+I*a*tan(d*x+c))^2,x)
 

Output:

(a**2*(256*cos(c + d*x)*sin(c + d*x)**3*i - 64*cos(c + d*x)*sin(c + d*x)*i 
 - 192*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**4 + 192*log(tan((c + d*x 
)/2))*sin(c + d*x)**4 + 192*sin(c + d*x)**4*d*i*x - 63*sin(c + d*x)**4 + 1 
44*sin(c + d*x)**2 - 24))/(96*sin(c + d*x)**4*d)