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

Optimal. Leaf size=38 \[ -\frac {\left (1-\frac {a^2}{b^2}\right ) \log (a+b \csc (x))}{a}-\frac {\log (\sin (x))}{a}-\frac {\csc (x)}{b} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 38, normalized size of antiderivative = 1.00, 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 = {3885, 894} \[ -\frac {\left (1-\frac {a^2}{b^2}\right ) \log (a+b \csc (x))}{a}-\frac {\log (\sin (x))}{a}-\frac {\csc (x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]^3/(a + b*Csc[x]),x]

[Out]

-(Csc[x]/b) - ((1 - a^2/b^2)*Log[a + b*Csc[x]])/a - Log[Sin[x]]/a

Rule 894

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && IntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 3885

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> -Dist[(-1)^((m - 1
)/2)/(d*b^(m - 1)), Subst[Int[((b^2 - x^2)^((m - 1)/2)*(a + x)^n)/x, x], x, b*Csc[c + d*x]], x] /; FreeQ[{a, b
, c, d, n}, x] && IntegerQ[(m - 1)/2] && NeQ[a^2 - b^2, 0]

Rubi steps

\begin {align*} \int \frac {\cot ^3(x)}{a+b \csc (x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {b^2-x^2}{x (a+x)} \, dx,x,b \csc (x)\right )}{b^2}\\ &=\frac {\operatorname {Subst}\left (\int \left (-1+\frac {b^2}{a x}+\frac {a^2-b^2}{a (a+x)}\right ) \, dx,x,b \csc (x)\right )}{b^2}\\ &=-\frac {\csc (x)}{b}-\frac {\left (1-\frac {a^2}{b^2}\right ) \log (a+b \csc (x))}{a}-\frac {\log (\sin (x))}{a}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 39, normalized size = 1.03 \[ \frac {\left (a^2-b^2\right ) \log (a \sin (x)+b)+a^2 (-\log (\sin (x)))-a b \csc (x)}{a b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]^3/(a + b*Csc[x]),x]

[Out]

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

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fricas [A]  time = 0.48, size = 47, normalized size = 1.24 \[ -\frac {a^{2} \log \left (-\frac {1}{2} \, \sin \relax (x)\right ) \sin \relax (x) - {\left (a^{2} - b^{2}\right )} \log \left (a \sin \relax (x) + b\right ) \sin \relax (x) + a b}{a b^{2} \sin \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.65, size = 44, normalized size = 1.16 \[ -\frac {a \log \left ({\left | \sin \relax (x) \right |}\right )}{b^{2}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left ({\left | a \sin \relax (x) + b \right |}\right )}{a b^{2}} - \frac {1}{b \sin \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.38, size = 44, normalized size = 1.16 \[ \frac {a \ln \left (b +a \sin \relax (x )\right )}{b^{2}}-\frac {\ln \left (b +a \sin \relax (x )\right )}{a}-\frac {1}{b \sin \relax (x )}-\frac {a \ln \left (\sin \relax (x )\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.31, size = 42, normalized size = 1.11 \[ -\frac {a \log \left (\sin \relax (x)\right )}{b^{2}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left (a \sin \relax (x) + b\right )}{a b^{2}} - \frac {1}{b \sin \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.46, size = 75, normalized size = 1.97 \[ \ln \left (b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+2\,a\,\mathrm {tan}\left (\frac {x}{2}\right )+b\right )\,\left (\frac {a}{b^2}-\frac {1}{a}\right )-\frac {\mathrm {tan}\left (\frac {x}{2}\right )}{2\,b}+\frac {\ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )}{a}-\frac {1}{2\,b\,\mathrm {tan}\left (\frac {x}{2}\right )}-\frac {a\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{b^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)^3/(a + b/sin(x)),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cot(x)**3/(a + b*csc(x)), x)

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