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

Optimal. Leaf size=27 \[ -\frac{\cot (x)}{a}+\frac{\tanh ^{-1}(\cos (x))}{a}-\frac{\cot (x)}{a \csc (x)+a} \]

[Out]

ArcTanh[Cos[x]]/a - Cot[x]/a - Cot[x]/(a + a*Csc[x])

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Rubi [A]  time = 0.0867268, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {3790, 3789, 3770, 3794} \[ -\frac{\cot (x)}{a}+\frac{\tanh ^{-1}(\cos (x))}{a}-\frac{\cot (x)}{a \csc (x)+a} \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]^3/(a + a*Csc[x]),x]

[Out]

ArcTanh[Cos[x]]/a - Cot[x]/a - Cot[x]/(a + a*Csc[x])

Rule 3790

Int[csc[(e_.) + (f_.)*(x_)]^3/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> -Simp[Cot[e + f*x]/(b*f), x
] - Dist[a/b, Int[Csc[e + f*x]^2/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x]

Rule 3789

Int[csc[(e_.) + (f_.)*(x_)]^2/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[Csc[e + f*x],
x], x] - Dist[a/b, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x]

Rule 3770

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

Rule 3794

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> -Simp[Cot[e + f*x]/(f*(b + a*
Csc[e + f*x])), x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

\begin{align*} \int \frac{\csc ^3(x)}{a+a \csc (x)} \, dx &=-\frac{\cot (x)}{a}-\int \frac{\csc ^2(x)}{a+a \csc (x)} \, dx\\ &=-\frac{\cot (x)}{a}-\frac{\int \csc (x) \, dx}{a}+\int \frac{\csc (x)}{a+a \csc (x)} \, dx\\ &=\frac{\tanh ^{-1}(\cos (x))}{a}-\frac{\cot (x)}{a}-\frac{\cot (x)}{a+a \csc (x)}\\ \end{align*}

Mathematica [B]  time = 0.164203, size = 63, normalized size = 2.33 \[ \frac{\tan \left (\frac{x}{2}\right )-\cot \left (\frac{x}{2}\right )-2 \log \left (\sin \left (\frac{x}{2}\right )\right )+2 \log \left (\cos \left (\frac{x}{2}\right )\right )+\frac{4 \sin \left (\frac{x}{2}\right )}{\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )}}{2 a} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]^3/(a + a*Csc[x]),x]

[Out]

(-Cot[x/2] + 2*Log[Cos[x/2]] - 2*Log[Sin[x/2]] + (4*Sin[x/2])/(Cos[x/2] + Sin[x/2]) + Tan[x/2])/(2*a)

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Maple [A]  time = 0.028, size = 45, normalized size = 1.7 \begin{align*}{\frac{1}{2\,a}\tan \left ({\frac{x}{2}} \right ) }-2\,{\frac{1}{a \left ( \tan \left ( x/2 \right ) +1 \right ) }}-{\frac{1}{2\,a} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-{\frac{1}{a}\ln \left ( \tan \left ({\frac{x}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/a*tan(1/2*x)-2/a/(tan(1/2*x)+1)-1/2/a/tan(1/2*x)-1/a*ln(tan(1/2*x))

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Maxima [B]  time = 1.0311, size = 92, normalized size = 3.41 \begin{align*} -\frac{\frac{5 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + 1}{2 \,{\left (\frac{a \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right )}} - \frac{\log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a} + \frac{\sin \left (x\right )}{2 \, a{\left (\cos \left (x\right ) + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(5*sin(x)/(cos(x) + 1) + 1)/(a*sin(x)/(cos(x) + 1) + a*sin(x)^2/(cos(x) + 1)^2) - log(sin(x)/(cos(x) + 1)
)/a + 1/2*sin(x)/(a*(cos(x) + 1))

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Fricas [B]  time = 0.486788, size = 301, normalized size = 11.15 \begin{align*} \frac{4 \, \cos \left (x\right )^{2} +{\left (\cos \left (x\right )^{2} -{\left (\cos \left (x\right ) + 1\right )} \sin \left (x\right ) - 1\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) -{\left (\cos \left (x\right )^{2} -{\left (\cos \left (x\right ) + 1\right )} \sin \left (x\right ) - 1\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + 2 \,{\left (2 \, \cos \left (x\right ) + 1\right )} \sin \left (x\right ) + 2 \, \cos \left (x\right ) - 2}{2 \,{\left (a \cos \left (x\right )^{2} -{\left (a \cos \left (x\right ) + a\right )} \sin \left (x\right ) - a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\csc ^{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(csc(x)**3/(a+a*csc(x)),x)

[Out]

Integral(csc(x)**3/(csc(x) + 1), x)/a

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Giac [A]  time = 1.34789, size = 72, normalized size = 2.67 \begin{align*} -\frac{\log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) \right |}\right )}{a} + \frac{\tan \left (\frac{1}{2} \, x\right )}{2 \, a} + \frac{\tan \left (\frac{1}{2} \, x\right )^{2} - 4 \, \tan \left (\frac{1}{2} \, x\right ) - 1}{2 \,{\left (\tan \left (\frac{1}{2} \, x\right )^{2} + \tan \left (\frac{1}{2} \, x\right )\right )} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(abs(tan(1/2*x)))/a + 1/2*tan(1/2*x)/a + 1/2*(tan(1/2*x)^2 - 4*tan(1/2*x) - 1)/((tan(1/2*x)^2 + tan(1/2*x)
)*a)