\(\int \frac {\cot ^3(x)}{a+a \csc (x)} \, dx\) [7]

Optimal result

Integrand size = 13, antiderivative size = 16 \[ \int \frac {\cot ^3(x)}{a+a \csc (x)} \, dx=-\frac {\csc (x)}{a}-\frac {\log (\sin (x))}{a} \]

[Out]

-csc(x)/a-ln(sin(x))/a

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 16, 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, 45} \[ \int \frac {\cot ^3(x)}{a+a \csc (x)} \, dx=-\frac {\csc (x)}{a}-\frac {\log (\sin (x))}{a} \]

[In]

Int[Cot[x]^3/(a + a*Csc[x]),x]

[Out]

-(Csc[x]/a) - Log[Sin[x]]/a

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 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}{x^2} \, dx,x,\sin (x)\right )}{a^2} \\ & = \frac {\text {Subst}\left (\int \left (\frac {a}{x^2}-\frac {a}{x}\right ) \, dx,x,\sin (x)\right )}{a^2} \\ & = -\frac {\csc (x)}{a}-\frac {\log (\sin (x))}{a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.69 \[ \int \frac {\cot ^3(x)}{a+a \csc (x)} \, dx=-\frac {\csc (x)+\log (\sin (x))}{a} \]

[In]

Integrate[Cot[x]^3/(a + a*Csc[x]),x]

[Out]

-((Csc[x] + Log[Sin[x]])/a)

Maple [A] (verified)

Time = 0.61 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81

method	result	size
derivativedivides	\(\frac {-\csc \left (x \right )+\ln \left (\csc \left (x \right )\right )}{a}\)	\(13\)
default	\(\frac {-\csc \left (x \right )+\ln \left (\csc \left (x \right )\right )}{a}\)	\(13\)
risch	\(\frac {i x}{a}-\frac {2 i {\mathrm e}^{i x}}{a \left ({\mathrm e}^{2 i x}-1\right )}-\frac {\ln \left ({\mathrm e}^{2 i x}-1\right )}{a}\)	\(42\)

[In]

int(cot(x)^3/(a+a*csc(x)),x,method=_RETURNVERBOSE)

[Out]

1/a*(-csc(x)+ln(csc(x)))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19 \[ \int \frac {\cot ^3(x)}{a+a \csc (x)} \, dx=-\frac {\log \left (\frac {1}{2} \, \sin \left (x\right )\right ) \sin \left (x\right ) + 1}{a \sin \left (x\right )} \]

[In]

integrate(cot(x)^3/(a+a*csc(x)),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {\cot ^3(x)}{a+a \csc (x)} \, dx=\frac {\int \frac {\cot ^{3}{\left (x \right )}}{\csc {\left (x \right )} + 1}\, dx}{a} \]

[In]

integrate(cot(x)**3/(a+a*csc(x)),x)

[Out]

Integral(cot(x)**3/(csc(x) + 1), x)/a

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\cot ^3(x)}{a+a \csc (x)} \, dx=-\frac {\log \left (\sin \left (x\right )\right )}{a} - \frac {1}{a \sin \left (x\right )} \]

[In]

integrate(cot(x)^3/(a+a*csc(x)),x, algorithm="maxima")

[Out]

-log(sin(x))/a - 1/(a*sin(x))

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {\cot ^3(x)}{a+a \csc (x)} \, dx=-\frac {\frac {1}{\sin \left (x\right )} + \log \left ({\left | \sin \left (x\right ) \right |}\right )}{a} \]

[In]

integrate(cot(x)^3/(a+a*csc(x)),x, algorithm="giac")

[Out]

-(1/sin(x) + log(abs(sin(x))))/a

Mupad [B] (verification not implemented)

Time = 18.85 (sec) , antiderivative size = 36, normalized size of antiderivative = 2.25 \[ \int \frac {\cot ^3(x)}{a+a \csc (x)} \, dx=-\frac {\frac {\mathrm {tan}\left (\frac {x}{2}\right )}{2}-\ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )+\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )+\frac {1}{2\,\mathrm {tan}\left (\frac {x}{2}\right )}}{a} \]

[In]

int(cot(x)^3/(a + a/sin(x)),x)

[Out]

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