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 = 0.09, antiderivative size = 27, normalized size of antiderivative = 1.00, 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 3770

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

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Mathematica [B]  time = 0.16, 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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fricas [B]  time = 0.52, size = 91, normalized size = 3.37 \[ \frac {4 \, \cos \relax (x)^{2} + {\left (\cos \relax (x)^{2} - {\left (\cos \relax (x) + 1\right )} \sin \relax (x) - 1\right )} \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) - {\left (\cos \relax (x)^{2} - {\left (\cos \relax (x) + 1\right )} \sin \relax (x) - 1\right )} \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + 2 \, {\left (2 \, \cos \relax (x) + 1\right )} \sin \relax (x) + 2 \, \cos \relax (x) - 2}{2 \, {\left (a \cos \relax (x)^{2} - {\left (a \cos \relax (x) + a\right )} \sin \relax (x) - a\right )}} \]

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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giac [A]  time = 0.56, size = 53, normalized size = 1.96 \[ -\frac {\log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{a} + \frac {\tan \left (\frac {1}{2} \, x\right )}{2 \, a} + \frac {\tan \left (\frac {1}{2} \, x\right )^{2} - 4 \, \tan \left (\frac {1}{2} \, x\right ) - 1}{2 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} + \tan \left (\frac {1}{2} \, x\right )\right )} a} \]

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)

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maple [A]  time = 0.23, size = 45, normalized size = 1.67 \[ \frac {\tan \left (\frac {x}{2}\right )}{2 a}-\frac {1}{2 a \tan \left (\frac {x}{2}\right )}-\frac {\ln \left (\tan \left (\frac {x}{2}\right )\right )}{a}-\frac {2}{a \left (\tan \left (\frac {x}{2}\right )+1\right )} \]

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

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maxima [B]  time = 0.33, size = 68, normalized size = 2.52 \[ -\frac {\frac {5 \, \sin \relax (x)}{\cos \relax (x) + 1} + 1}{2 \, {\left (\frac {a \sin \relax (x)}{\cos \relax (x) + 1} + \frac {a \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}}\right )}} - \frac {\log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right )}{a} + \frac {\sin \relax (x)}{2 \, a {\left (\cos \relax (x) + 1\right )}} \]

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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mupad [B]  time = 0.25, size = 49, normalized size = 1.81 \[ \frac {\mathrm {tan}\left (\frac {x}{2}\right )}{2\,a}-\frac {5\,\mathrm {tan}\left (\frac {x}{2}\right )+1}{2\,a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+2\,a\,\mathrm {tan}\left (\frac {x}{2}\right )}-\frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(x)^3*(a + a/sin(x))),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\csc ^{3}{\relax (x )}}{\csc {\relax (x )} + 1}\, dx}{a} \]

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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