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

Optimal. Leaf size=16 \[ -\frac{\csc (x)}{a}-\frac{\log (\sin (x))}{a} \]

[Out]

-(Csc[x]/a) - Log[Sin[x]]/a

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Rubi [A]  time = 0.0415825, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3879, 43} \[ -\frac{\csc (x)}{a}-\frac{\log (\sin (x))}{a} \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]^3/(a + a*Csc[x]),x]

[Out]

-(Csc[x]/a) - Log[Sin[x]]/a

Rule 3879

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]

Rule 43

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])

Rubi steps

\begin{align*} \int \frac{\cot ^3(x)}{a+a \csc (x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a-a x}{x^2} \, dx,x,\sin (x)\right )}{a^2}\\ &=\frac{\operatorname{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]  time = 0.0203576, size = 11, normalized size = 0.69 \[ -\frac{\csc (x)+\log (\sin (x))}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]^3/(a + a*Csc[x]),x]

[Out]

-((Csc[x] + Log[Sin[x]])/a)

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Maple [A]  time = 0.036, size = 16, normalized size = 1. \begin{align*} -{\frac{\csc \left ( x \right ) }{a}}+{\frac{\ln \left ( \csc \left ( x \right ) \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.04973, size = 24, normalized size = 1.5 \begin{align*} -\frac{\log \left (\sin \left (x\right )\right )}{a} - \frac{1}{a \sin \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(sin(x))/a - 1/(a*sin(x))

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Fricas [A]  time = 0.491916, size = 57, normalized size = 3.56 \begin{align*} -\frac{\log \left (\frac{1}{2} \, \sin \left (x\right )\right ) \sin \left (x\right ) + 1}{a \sin \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\cot ^{3}{\left (x \right )}}{\csc{\left (x \right )} + 1}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cot(x)**3/(csc(x) + 1), x)/a

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Giac [A]  time = 1.35568, size = 26, normalized size = 1.62 \begin{align*} -\frac{\log \left ({\left | \sin \left (x\right ) \right |}\right )}{a} - \frac{1}{a \sin \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(abs(sin(x)))/a - 1/(a*sin(x))