\(\int \cot ^6(c+d x) (a+i a \tan (c+d x)) \, dx\) [12]

Optimal result

Integrand size = 22, antiderivative size = 100 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x)) \, dx=-a x-\frac {a \cot (c+d x)}{d}+\frac {i a \cot ^2(c+d x)}{2 d}+\frac {a \cot ^3(c+d x)}{3 d}-\frac {i a \cot ^4(c+d x)}{4 d}-\frac {a \cot ^5(c+d x)}{5 d}+\frac {i a \log (\sin (c+d x))}{d} \]

[Out]

-a*x-a*cot(d*x+c)/d+1/2*I*a*cot(d*x+c)^2/d+1/3*a*cot(d*x+c)^3/d-1/4*I*a*cot(d*x+c)^4/d-1/5*a*cot(d*x+c)^5/d+I*
a*ln(sin(d*x+c))/d

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {3610, 3612, 3556} \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x)) \, dx=-\frac {a \cot ^5(c+d x)}{5 d}-\frac {i a \cot ^4(c+d x)}{4 d}+\frac {a \cot ^3(c+d x)}{3 d}+\frac {i a \cot ^2(c+d x)}{2 d}-\frac {a \cot (c+d x)}{d}+\frac {i a \log (\sin (c+d x))}{d}-a x \]

[In]

Int[Cot[c + d*x]^6*(a + I*a*Tan[c + d*x]),x]

[Out]

-(a*x) - (a*Cot[c + d*x])/d + ((I/2)*a*Cot[c + d*x]^2)/d + (a*Cot[c + d*x]^3)/(3*d) - ((I/4)*a*Cot[c + d*x]^4)
/d - (a*Cot[c + d*x]^5)/(5*d) + (I*a*Log[Sin[c + d*x]])/d

Rule 3556

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

Rule 3610

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] + Dist[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 3612

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] + Dist[(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] && NeQ[a*c + b*d, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {a \cot ^5(c+d x)}{5 d}+\int \cot ^5(c+d x) (i a-a \tan (c+d x)) \, dx \\ & = -\frac {i a \cot ^4(c+d x)}{4 d}-\frac {a \cot ^5(c+d x)}{5 d}+\int \cot ^4(c+d x) (-a-i a \tan (c+d x)) \, dx \\ & = \frac {a \cot ^3(c+d x)}{3 d}-\frac {i a \cot ^4(c+d x)}{4 d}-\frac {a \cot ^5(c+d x)}{5 d}+\int \cot ^3(c+d x) (-i a+a \tan (c+d x)) \, dx \\ & = \frac {i a \cot ^2(c+d x)}{2 d}+\frac {a \cot ^3(c+d x)}{3 d}-\frac {i a \cot ^4(c+d x)}{4 d}-\frac {a \cot ^5(c+d x)}{5 d}+\int \cot ^2(c+d x) (a+i a \tan (c+d x)) \, dx \\ & = -\frac {a \cot (c+d x)}{d}+\frac {i a \cot ^2(c+d x)}{2 d}+\frac {a \cot ^3(c+d x)}{3 d}-\frac {i a \cot ^4(c+d x)}{4 d}-\frac {a \cot ^5(c+d x)}{5 d}+\int \cot (c+d x) (i a-a \tan (c+d x)) \, dx \\ & = -a x-\frac {a \cot (c+d x)}{d}+\frac {i a \cot ^2(c+d x)}{2 d}+\frac {a \cot ^3(c+d x)}{3 d}-\frac {i a \cot ^4(c+d x)}{4 d}-\frac {a \cot ^5(c+d x)}{5 d}+(i a) \int \cot (c+d x) \, dx \\ & = -a x-\frac {a \cot (c+d x)}{d}+\frac {i a \cot ^2(c+d x)}{2 d}+\frac {a \cot ^3(c+d x)}{3 d}-\frac {i a \cot ^4(c+d x)}{4 d}-\frac {a \cot ^5(c+d x)}{5 d}+\frac {i a \log (\sin (c+d x))}{d} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 0.04 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.01 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x)) \, dx=\frac {i a \cot ^2(c+d x)}{2 d}-\frac {i a \cot ^4(c+d x)}{4 d}-\frac {a \cot ^5(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},-\tan ^2(c+d x)\right )}{5 d}+\frac {i a \log (\cos (c+d x))}{d}+\frac {i a \log (\tan (c+d x))}{d} \]

[In]

Integrate[Cot[c + d*x]^6*(a + I*a*Tan[c + d*x]),x]

[Out]

((I/2)*a*Cot[c + d*x]^2)/d - ((I/4)*a*Cot[c + d*x]^4)/d - (a*Cot[c + d*x]^5*Hypergeometric2F1[-5/2, 1, -3/2, -
Tan[c + d*x]^2])/(5*d) + (I*a*Log[Cos[c + d*x]])/d + (I*a*Log[Tan[c + d*x]])/d

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.84

method	result	size
parallelrisch	\(-\frac {\left (i \left (\cot ^{4}\left (d x +c \right )\right )+\frac {4 \left (\cot ^{5}\left (d x +c \right )\right )}{5}-2 i \left (\cot ^{2}\left (d x +c \right )\right )-\frac {4 \left (\cot ^{3}\left (d x +c \right )\right )}{3}-4 i \ln \left (\tan \left (d x +c \right )\right )+2 i \ln \left (\sec ^{2}\left (d x +c \right )\right )+4 d x +4 \cot \left (d x +c \right )\right ) a}{4 d}\)	\(84\)
derivativedivides	\(\frac {a \left (-\frac {1}{5 \tan \left (d x +c \right )^{5}}-\frac {1}{\tan \left (d x +c \right )}+\frac {i}{2 \tan \left (d x +c \right )^{2}}-\frac {i}{4 \tan \left (d x +c \right )^{4}}+i \ln \left (\tan \left (d x +c \right )\right )+\frac {1}{3 \tan \left (d x +c \right )^{3}}-\frac {i \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\)	\(92\)
default	\(\frac {a \left (-\frac {1}{5 \tan \left (d x +c \right )^{5}}-\frac {1}{\tan \left (d x +c \right )}+\frac {i}{2 \tan \left (d x +c \right )^{2}}-\frac {i}{4 \tan \left (d x +c \right )^{4}}+i \ln \left (\tan \left (d x +c \right )\right )+\frac {1}{3 \tan \left (d x +c \right )^{3}}-\frac {i \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\)	\(92\)
risch	\(\frac {2 a c}{d}-\frac {2 i a \left (75 \,{\mathrm e}^{8 i \left (d x +c \right )}-150 \,{\mathrm e}^{6 i \left (d x +c \right )}+200 \,{\mathrm e}^{4 i \left (d x +c \right )}-100 \,{\mathrm e}^{2 i \left (d x +c \right )}+23\right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}+\frac {i a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\)	\(94\)
norman	\(\frac {-\frac {a}{5 d}-a x \left (\tan ^{5}\left (d x +c \right )\right )+\frac {a \left (\tan ^{2}\left (d x +c \right )\right )}{3 d}-\frac {a \left (\tan ^{4}\left (d x +c \right )\right )}{d}-\frac {i a \tan \left (d x +c \right )}{4 d}+\frac {i a \left (\tan ^{3}\left (d x +c \right )\right )}{2 d}}{\tan \left (d x +c \right )^{5}}+\frac {i a \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {i a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\)	\(118\)

[In]

int(cot(d*x+c)^6*(a+I*a*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-1/4*(I*cot(d*x+c)^4+4/5*cot(d*x+c)^5-2*I*cot(d*x+c)^2-4/3*cot(d*x+c)^3-4*I*ln(tan(d*x+c))+2*I*ln(sec(d*x+c)^2
)+4*d*x+4*cot(d*x+c))*a/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. 197 vs. \(2 (86) = 172\).

Time = 0.23 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.97 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x)) \, dx=\frac {-150 i \, a e^{\left (8 i \, d x + 8 i \, c\right )} + 300 i \, a e^{\left (6 i \, d x + 6 i \, c\right )} - 400 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 200 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} - 15 \, {\left (-i \, a e^{\left (10 i \, d x + 10 i \, c\right )} + 5 i \, a e^{\left (8 i \, d x + 8 i \, c\right )} - 10 i \, a e^{\left (6 i \, d x + 6 i \, c\right )} + 10 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} - 5 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right ) - 46 i \, a}{15 \, {\left (d e^{\left (10 i \, d x + 10 i \, c\right )} - 5 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, d e^{\left (6 i \, d x + 6 i \, c\right )} - 10 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )}} \]

[In]

integrate(cot(d*x+c)^6*(a+I*a*tan(d*x+c)),x, algorithm="fricas")

[Out]

1/15*(-150*I*a*e^(8*I*d*x + 8*I*c) + 300*I*a*e^(6*I*d*x + 6*I*c) - 400*I*a*e^(4*I*d*x + 4*I*c) + 200*I*a*e^(2*
I*d*x + 2*I*c) - 15*(-I*a*e^(10*I*d*x + 10*I*c) + 5*I*a*e^(8*I*d*x + 8*I*c) - 10*I*a*e^(6*I*d*x + 6*I*c) + 10*
I*a*e^(4*I*d*x + 4*I*c) - 5*I*a*e^(2*I*d*x + 2*I*c) + I*a)*log(e^(2*I*d*x + 2*I*c) - 1) - 46*I*a)/(d*e^(10*I*d
*x + 10*I*c) - 5*d*e^(8*I*d*x + 8*I*c) + 10*d*e^(6*I*d*x + 6*I*c) - 10*d*e^(4*I*d*x + 4*I*c) + 5*d*e^(2*I*d*x
+ 2*I*c) - d)

Sympy [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (83) = 166\).

Time = 0.29 (sec) , antiderivative size = 206, normalized size of antiderivative = 2.06 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x)) \, dx=\frac {i a \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac {- 150 i a e^{8 i c} e^{8 i d x} + 300 i a e^{6 i c} e^{6 i d x} - 400 i a e^{4 i c} e^{4 i d x} + 200 i a e^{2 i c} e^{2 i d x} - 46 i a}{15 d e^{10 i c} e^{10 i d x} - 75 d e^{8 i c} e^{8 i d x} + 150 d e^{6 i c} e^{6 i d x} - 150 d e^{4 i c} e^{4 i d x} + 75 d e^{2 i c} e^{2 i d x} - 15 d} \]

[In]

integrate(cot(d*x+c)**6*(a+I*a*tan(d*x+c)),x)

[Out]

I*a*log(exp(2*I*d*x) - exp(-2*I*c))/d + (-150*I*a*exp(8*I*c)*exp(8*I*d*x) + 300*I*a*exp(6*I*c)*exp(6*I*d*x) -
400*I*a*exp(4*I*c)*exp(4*I*d*x) + 200*I*a*exp(2*I*c)*exp(2*I*d*x) - 46*I*a)/(15*d*exp(10*I*c)*exp(10*I*d*x) -
75*d*exp(8*I*c)*exp(8*I*d*x) + 150*d*exp(6*I*c)*exp(6*I*d*x) - 150*d*exp(4*I*c)*exp(4*I*d*x) + 75*d*exp(2*I*c)
*exp(2*I*d*x) - 15*d)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.93 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x)) \, dx=-\frac {60 \, {\left (d x + c\right )} a + 30 i \, a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 60 i \, a \log \left (\tan \left (d x + c\right )\right ) + \frac {60 \, a \tan \left (d x + c\right )^{4} - 30 i \, a \tan \left (d x + c\right )^{3} - 20 \, a \tan \left (d x + c\right )^{2} + 15 i \, a \tan \left (d x + c\right ) + 12 \, a}{\tan \left (d x + c\right )^{5}}}{60 \, d} \]

[In]

integrate(cot(d*x+c)^6*(a+I*a*tan(d*x+c)),x, algorithm="maxima")

[Out]

-1/60*(60*(d*x + c)*a + 30*I*a*log(tan(d*x + c)^2 + 1) - 60*I*a*log(tan(d*x + c)) + (60*a*tan(d*x + c)^4 - 30*
I*a*tan(d*x + c)^3 - 20*a*tan(d*x + c)^2 + 15*I*a*tan(d*x + c) + 12*a)/tan(d*x + c)^5)/d

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (86) = 172\).

Time = 0.59 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.86 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x)) \, dx=\frac {6 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 i \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 70 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 180 i \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1920 i \, a \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right ) + 960 i \, a \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 660 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {-2192 i \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 660 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 180 i \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 70 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 i \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{960 \, d} \]

[In]

integrate(cot(d*x+c)^6*(a+I*a*tan(d*x+c)),x, algorithm="giac")

[Out]

1/960*(6*a*tan(1/2*d*x + 1/2*c)^5 - 15*I*a*tan(1/2*d*x + 1/2*c)^4 - 70*a*tan(1/2*d*x + 1/2*c)^3 + 180*I*a*tan(
1/2*d*x + 1/2*c)^2 - 1920*I*a*log(tan(1/2*d*x + 1/2*c) + I) + 960*I*a*log(tan(1/2*d*x + 1/2*c)) + 660*a*tan(1/
2*d*x + 1/2*c) + (-2192*I*a*tan(1/2*d*x + 1/2*c)^5 - 660*a*tan(1/2*d*x + 1/2*c)^4 + 180*I*a*tan(1/2*d*x + 1/2*
c)^3 + 70*a*tan(1/2*d*x + 1/2*c)^2 - 15*I*a*tan(1/2*d*x + 1/2*c) - 6*a)/tan(1/2*d*x + 1/2*c)^5)/d

Mupad [B] (verification not implemented)

Time = 4.76 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.79 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x)) \, dx=-\frac {2\,a\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{d}-\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^4-\frac {1{}\mathrm {i}\,a\,{\mathrm {tan}\left (c+d\,x\right )}^3}{2}-\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^2}{3}+\frac {1{}\mathrm {i}\,a\,\mathrm {tan}\left (c+d\,x\right )}{4}+\frac {a}{5}}{d\,{\mathrm {tan}\left (c+d\,x\right )}^5} \]

[In]

int(cot(c + d*x)^6*(a + a*tan(c + d*x)*1i),x)

[Out]

- (2*a*atan(2*tan(c + d*x) + 1i))/d - (a/5 + (a*tan(c + d*x)*1i)/4 - (a*tan(c + d*x)^2)/3 - (a*tan(c + d*x)^3*
1i)/2 + a*tan(c + d*x)^4)/(d*tan(c + d*x)^5)