\(\int \frac {\cot ^7(x)}{a+a \csc (x)} \, dx\) [11]

Optimal result

Integrand size = 13, antiderivative size = 58 \[ \int \frac {\cot ^7(x)}{a+a \csc (x)} \, dx=-\frac {\csc (x)}{a}-\frac {\csc ^2(x)}{a}+\frac {2 \csc ^3(x)}{3 a}+\frac {\csc ^4(x)}{4 a}-\frac {\csc ^5(x)}{5 a}-\frac {\log (\sin (x))}{a} \]

[Out]

-csc(x)/a-csc(x)^2/a+2/3*csc(x)^3/a+1/4*csc(x)^4/a-1/5*csc(x)^5/a-ln(sin(x))/a

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3964, 90} \[ \int \frac {\cot ^7(x)}{a+a \csc (x)} \, dx=-\frac {\csc ^5(x)}{5 a}+\frac {\csc ^4(x)}{4 a}+\frac {2 \csc ^3(x)}{3 a}-\frac {\csc ^2(x)}{a}-\frac {\csc (x)}{a}-\frac {\log (\sin (x))}{a} \]

[In]

Int[Cot[x]^7/(a + a*Csc[x]),x]

[Out]

-(Csc[x]/a) - Csc[x]^2/a + (2*Csc[x]^3)/(3*a) + Csc[x]^4/(4*a) - Csc[x]^5/(5*a) - Log[Sin[x]]/a

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 3964

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_.), x_Symbol] :> Dist[1/(a^(m - n
- 1)*b^n*d), Subst[Int[(a - b*x)^((m - 1)/2)*((a + b*x)^((m - 1)/2 + n)/x^(m + n)), x], x, Sin[c + d*x]], x] /
; FreeQ[{a, b, c, d}, x] && IntegerQ[(m - 1)/2] && EqQ[a^2 - b^2, 0] && IntegerQ[n]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a-a x)^3 (a+a x)^2}{x^6} \, dx,x,\sin (x)\right )}{a^6} \\ & = \frac {\text {Subst}\left (\int \left (\frac {a^5}{x^6}-\frac {a^5}{x^5}-\frac {2 a^5}{x^4}+\frac {2 a^5}{x^3}+\frac {a^5}{x^2}-\frac {a^5}{x}\right ) \, dx,x,\sin (x)\right )}{a^6} \\ & = -\frac {\csc (x)}{a}-\frac {\csc ^2(x)}{a}+\frac {2 \csc ^3(x)}{3 a}+\frac {\csc ^4(x)}{4 a}-\frac {\csc ^5(x)}{5 a}-\frac {\log (\sin (x))}{a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.67 \[ \int \frac {\cot ^7(x)}{a+a \csc (x)} \, dx=-\frac {\csc (x)+\csc ^2(x)-\frac {2 \csc ^3(x)}{3}-\frac {\csc ^4(x)}{4}+\frac {\csc ^5(x)}{5}+\log (\sin (x))}{a} \]

[In]

Integrate[Cot[x]^7/(a + a*Csc[x]),x]

[Out]

-((Csc[x] + Csc[x]^2 - (2*Csc[x]^3)/3 - Csc[x]^4/4 + Csc[x]^5/5 + Log[Sin[x]])/a)

Maple [A] (verified)

Time = 2.52 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.71

method	result	size
default	\(\frac {-\frac {1}{\sin \left (x \right )}+\frac {1}{4 \sin \left (x \right )^{4}}-\ln \left (\sin \left (x \right )\right )-\frac {1}{\sin \left (x \right )^{2}}+\frac {2}{3 \sin \left (x \right )^{3}}-\frac {1}{5 \sin \left (x \right )^{5}}}{a}\)	\(41\)
risch	\(\frac {i x}{a}-\frac {2 i \left (30 i {\mathrm e}^{8 i x}+15 \,{\mathrm e}^{9 i x}-60 i {\mathrm e}^{6 i x}-20 \,{\mathrm e}^{7 i x}+60 i {\mathrm e}^{4 i x}+58 \,{\mathrm e}^{5 i x}-30 i {\mathrm e}^{2 i x}-20 \,{\mathrm e}^{3 i x}+15 \,{\mathrm e}^{i x}\right )}{15 a \left ({\mathrm e}^{2 i x}-1\right )^{5}}-\frac {\ln \left ({\mathrm e}^{2 i x}-1\right )}{a}\)	\(105\)

[In]

int(cot(x)^7/(a+a*csc(x)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.21 \[ \int \frac {\cot ^7(x)}{a+a \csc (x)} \, dx=-\frac {60 \, \cos \left (x\right )^{4} + 60 \, {\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \sin \left (x\right )\right ) \sin \left (x\right ) - 80 \, \cos \left (x\right )^{2} - 15 \, {\left (4 \, \cos \left (x\right )^{2} - 3\right )} \sin \left (x\right ) + 32}{60 \, {\left (a \cos \left (x\right )^{4} - 2 \, a \cos \left (x\right )^{2} + a\right )} \sin \left (x\right )} \]

[In]

integrate(cot(x)^7/(a+a*csc(x)),x, algorithm="fricas")

[Out]

-1/60*(60*cos(x)^4 + 60*(cos(x)^4 - 2*cos(x)^2 + 1)*log(1/2*sin(x))*sin(x) - 80*cos(x)^2 - 15*(4*cos(x)^2 - 3)
*sin(x) + 32)/((a*cos(x)^4 - 2*a*cos(x)^2 + a)*sin(x))

Sympy [F]

\[ \int \frac {\cot ^7(x)}{a+a \csc (x)} \, dx=\frac {\int \frac {\cot ^{7}{\left (x \right )}}{\csc {\left (x \right )} + 1}\, dx}{a} \]

[In]

integrate(cot(x)**7/(a+a*csc(x)),x)

[Out]

Integral(cot(x)**7/(csc(x) + 1), x)/a

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.72 \[ \int \frac {\cot ^7(x)}{a+a \csc (x)} \, dx=-\frac {\log \left (\sin \left (x\right )\right )}{a} - \frac {60 \, \sin \left (x\right )^{4} + 60 \, \sin \left (x\right )^{3} - 40 \, \sin \left (x\right )^{2} - 15 \, \sin \left (x\right ) + 12}{60 \, a \sin \left (x\right )^{5}} \]

[In]

integrate(cot(x)^7/(a+a*csc(x)),x, algorithm="maxima")

[Out]

-log(sin(x))/a - 1/60*(60*sin(x)^4 + 60*sin(x)^3 - 40*sin(x)^2 - 15*sin(x) + 12)/(a*sin(x)^5)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.71 \[ \int \frac {\cot ^7(x)}{a+a \csc (x)} \, dx=-\frac {\frac {60 \, \sin \left (x\right )^{4} + 60 \, \sin \left (x\right )^{3} - 40 \, \sin \left (x\right )^{2} - 15 \, \sin \left (x\right ) + 12}{\sin \left (x\right )^{5}} + 60 \, \log \left ({\left | \sin \left (x\right ) \right |}\right )}{60 \, a} \]

[In]

integrate(cot(x)^7/(a+a*csc(x)),x, algorithm="giac")

[Out]

-1/60*((60*sin(x)^4 + 60*sin(x)^3 - 40*sin(x)^2 - 15*sin(x) + 12)/sin(x)^5 + 60*log(abs(sin(x))))/a

Mupad [B] (verification not implemented)

Time = 19.06 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.95 \[ \int \frac {\cot ^7(x)}{a+a \csc (x)} \, dx=-\frac {180\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3-960\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5\,\ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )-50\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2-15\,\mathrm {tan}\left (\frac {x}{2}\right )+300\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+300\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6+180\,{\mathrm {tan}\left (\frac {x}{2}\right )}^7-50\,{\mathrm {tan}\left (\frac {x}{2}\right )}^8-15\,{\mathrm {tan}\left (\frac {x}{2}\right )}^9+6\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{10}+960\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )+6}{960\,a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5} \]

[In]

int(cot(x)^7/(a + a/sin(x)),x)

[Out]

-(180*tan(x/2)^3 - 960*tan(x/2)^5*log(tan(x/2)^2 + 1) - 50*tan(x/2)^2 - 15*tan(x/2) + 300*tan(x/2)^4 + 300*tan
(x/2)^6 + 180*tan(x/2)^7 - 50*tan(x/2)^8 - 15*tan(x/2)^9 + 6*tan(x/2)^10 + 960*tan(x/2)^5*log(tan(x/2)) + 6)/(
960*a*tan(x/2)^5)