\(\int \frac {\cot ^5(x)}{a+a \csc (x)} \, dx\) [9]

Optimal result

Integrand size = 13, antiderivative size = 36 \[ \int \frac {\cot ^5(x)}{a+a \csc (x)} \, dx=\frac {\csc (x)}{a}+\frac {\csc ^2(x)}{2 a}-\frac {\csc ^3(x)}{3 a}+\frac {\log (\sin (x))}{a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 36, 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, 76} \[ \int \frac {\cot ^5(x)}{a+a \csc (x)} \, dx=-\frac {\csc ^3(x)}{3 a}+\frac {\csc ^2(x)}{2 a}+\frac {\csc (x)}{a}+\frac {\log (\sin (x))}{a} \]

[In]

Int[Cot[x]^5/(a + a*Csc[x]),x]

[Out]

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

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.72 \[ \int \frac {\cot ^5(x)}{a+a \csc (x)} \, dx=\frac {\csc (x)+\frac {\csc ^2(x)}{2}-\frac {\csc ^3(x)}{3}+\log (\sin (x))}{a} \]

[In]

Integrate[Cot[x]^5/(a + a*Csc[x]),x]

[Out]

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

Maple [A] (verified)

Time = 1.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.69

method	result	size
default	\(\frac {\frac {1}{\sin \left (x \right )}+\frac {1}{2 \sin \left (x \right )^{2}}+\ln \left (\sin \left (x \right )\right )-\frac {1}{3 \sin \left (x \right )^{3}}}{a}\)	\(25\)
risch	\(-\frac {i x}{a}+\frac {2 i \left (3 i {\mathrm e}^{4 i x}+3 \,{\mathrm e}^{5 i x}-3 i {\mathrm e}^{2 i x}-2 \,{\mathrm e}^{3 i x}+3 \,{\mathrm e}^{i x}\right )}{3 a \left ({\mathrm e}^{2 i x}-1\right )^{3}}+\frac {\ln \left ({\mathrm e}^{2 i x}-1\right )}{a}\)	\(74\)

[In]

int(cot(x)^5/(a+a*csc(x)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.25 \[ \int \frac {\cot ^5(x)}{a+a \csc (x)} \, dx=\frac {6 \, {\left (\cos \left (x\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \sin \left (x\right )\right ) \sin \left (x\right ) + 6 \, \cos \left (x\right )^{2} - 3 \, \sin \left (x\right ) - 4}{6 \, {\left (a \cos \left (x\right )^{2} - a\right )} \sin \left (x\right )} \]

[In]

integrate(cot(x)^5/(a+a*csc(x)),x, algorithm="fricas")

[Out]

1/6*(6*(cos(x)^2 - 1)*log(1/2*sin(x))*sin(x) + 6*cos(x)^2 - 3*sin(x) - 4)/((a*cos(x)^2 - a)*sin(x))

Sympy [F]

\[ \int \frac {\cot ^5(x)}{a+a \csc (x)} \, dx=\frac {\int \frac {\cot ^{5}{\left (x \right )}}{\csc {\left (x \right )} + 1}\, dx}{a} \]

[In]

integrate(cot(x)**5/(a+a*csc(x)),x)

[Out]

Integral(cot(x)**5/(csc(x) + 1), x)/a

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.81 \[ \int \frac {\cot ^5(x)}{a+a \csc (x)} \, dx=\frac {\log \left (\sin \left (x\right )\right )}{a} + \frac {6 \, \sin \left (x\right )^{2} + 3 \, \sin \left (x\right ) - 2}{6 \, a \sin \left (x\right )^{3}} \]

[In]

integrate(cot(x)^5/(a+a*csc(x)),x, algorithm="maxima")

[Out]

log(sin(x))/a + 1/6*(6*sin(x)^2 + 3*sin(x) - 2)/(a*sin(x)^3)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.81 \[ \int \frac {\cot ^5(x)}{a+a \csc (x)} \, dx=\frac {\frac {6 \, \sin \left (x\right )^{2} + 3 \, \sin \left (x\right ) - 2}{\sin \left (x\right )^{3}} + 6 \, \log \left ({\left | \sin \left (x\right ) \right |}\right )}{6 \, a} \]

[In]

integrate(cot(x)^5/(a+a*csc(x)),x, algorithm="giac")

[Out]

1/6*((6*sin(x)^2 + 3*sin(x) - 2)/sin(x)^3 + 6*log(abs(sin(x))))/a

Mupad [B] (verification not implemented)

Time = 18.69 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.22 \[ \int \frac {\cot ^5(x)}{a+a \csc (x)} \, dx=\frac {3\,\mathrm {tan}\left (\frac {x}{2}\right )}{8\,a}-\frac {\ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )}{a}+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{8\,a}-\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{24\,a}+\frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{a}+\frac {3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+\mathrm {tan}\left (\frac {x}{2}\right )-\frac {1}{3}}{8\,a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3} \]

[In]

int(cot(x)^5/(a + a/sin(x)),x)

[Out]

(3*tan(x/2))/(8*a) - log(tan(x/2)^2 + 1)/a + tan(x/2)^2/(8*a) - tan(x/2)^3/(24*a) + log(tan(x/2))/a + (tan(x/2
) + 3*tan(x/2)^2 - 1/3)/(8*a*tan(x/2)^3)