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

Optimal. Leaf size=153 \[ \frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (x)+b^{2/3} \sin ^2(x)\right )}{6 a^{5/3}}-\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (x)\right )}{3 a^{5/3}}+\frac{b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sin (x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{5/3}}+\frac{\log \left (a+b \sin ^3(x)\right )}{3 a}-\frac{\csc ^2(x)}{2 a}-\frac{\log (\sin (x))}{a} \]

[Out]

(b^(2/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[x])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(5/3)) - Csc[x]^2/(2*a) - Log[Sin[
x]]/a - (b^(2/3)*Log[a^(1/3) + b^(1/3)*Sin[x]])/(3*a^(5/3)) + (b^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[x] +
b^(2/3)*Sin[x]^2])/(6*a^(5/3)) + Log[a + b*Sin[x]^3]/(3*a)

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Rubi [A]  time = 0.190871, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {3230, 1834, 1871, 200, 31, 634, 617, 204, 628, 260} \[ \frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (x)+b^{2/3} \sin ^2(x)\right )}{6 a^{5/3}}-\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (x)\right )}{3 a^{5/3}}+\frac{b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sin (x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{5/3}}+\frac{\log \left (a+b \sin ^3(x)\right )}{3 a}-\frac{\csc ^2(x)}{2 a}-\frac{\log (\sin (x))}{a} \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]^3/(a + b*Sin[x]^3),x]

[Out]

(b^(2/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[x])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(5/3)) - Csc[x]^2/(2*a) - Log[Sin[
x]]/a - (b^(2/3)*Log[a^(1/3) + b^(1/3)*Sin[x]])/(3*a^(5/3)) + (b^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[x] +
b^(2/3)*Sin[x]^2])/(6*a^(5/3)) + Log[a + b*Sin[x]^3]/(3*a)

Rule 3230

Int[((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With
[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[(x^m*(a + b*(c*ff*x)^n)^p)/(1 - ff^2*x^2)^(
(m + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && ILtQ[(m - 1)/2, 0]

Rule 1834

Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[((c*x)^m*Pq)/(a + b*
x^n), x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IntegerQ[n] &&  !IGtQ[m, 0]

Rule 1871

Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x,
2]}, Int[(A + B*x)/(a + b*x^3), x] + Dist[C, Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] ||  !Ration
alQ[a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int \frac{\cot ^3(x)}{a+b \sin ^3(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1-x^2}{x^3 \left (a+b x^3\right )} \, dx,x,\sin (x)\right )\\ &=\operatorname{Subst}\left (\int \left (\frac{1}{a x^3}-\frac{1}{a x}+\frac{b \left (-1+x^2\right )}{a \left (a+b x^3\right )}\right ) \, dx,x,\sin (x)\right )\\ &=-\frac{\csc ^2(x)}{2 a}-\frac{\log (\sin (x))}{a}+\frac{b \operatorname{Subst}\left (\int \frac{-1+x^2}{a+b x^3} \, dx,x,\sin (x)\right )}{a}\\ &=-\frac{\csc ^2(x)}{2 a}-\frac{\log (\sin (x))}{a}-\frac{b \operatorname{Subst}\left (\int \frac{1}{a+b x^3} \, dx,x,\sin (x)\right )}{a}+\frac{b \operatorname{Subst}\left (\int \frac{x^2}{a+b x^3} \, dx,x,\sin (x)\right )}{a}\\ &=-\frac{\csc ^2(x)}{2 a}-\frac{\log (\sin (x))}{a}+\frac{\log \left (a+b \sin ^3(x)\right )}{3 a}-\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,\sin (x)\right )}{3 a^{5/3}}-\frac{b \operatorname{Subst}\left (\int \frac{2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sin (x)\right )}{3 a^{5/3}}\\ &=-\frac{\csc ^2(x)}{2 a}-\frac{\log (\sin (x))}{a}-\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (x)\right )}{3 a^{5/3}}+\frac{\log \left (a+b \sin ^3(x)\right )}{3 a}+\frac{b^{2/3} \operatorname{Subst}\left (\int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sin (x)\right )}{6 a^{5/3}}-\frac{b \operatorname{Subst}\left (\int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sin (x)\right )}{2 a^{4/3}}\\ &=-\frac{\csc ^2(x)}{2 a}-\frac{\log (\sin (x))}{a}-\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (x)\right )}{3 a^{5/3}}+\frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (x)+b^{2/3} \sin ^2(x)\right )}{6 a^{5/3}}+\frac{\log \left (a+b \sin ^3(x)\right )}{3 a}-\frac{b^{2/3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} \sin (x)}{\sqrt [3]{a}}\right )}{a^{5/3}}\\ &=\frac{b^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sin (x)}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt{3} a^{5/3}}-\frac{\csc ^2(x)}{2 a}-\frac{\log (\sin (x))}{a}-\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (x)\right )}{3 a^{5/3}}+\frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (x)+b^{2/3} \sin ^2(x)\right )}{6 a^{5/3}}+\frac{\log \left (a+b \sin ^3(x)\right )}{3 a}\\ \end{align*}

Mathematica [A]  time = 0.297401, size = 143, normalized size = 0.93 \[ \frac{2 \left (a^{2/3}-(-1)^{2/3} b^{2/3}\right ) \log \left (-(-1)^{2/3} \sqrt [3]{a}-\sqrt [3]{b} \sin (x)\right )+2 \left (a^{2/3}-b^{2/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (x)\right )+2 \left (a^{2/3}+\sqrt [3]{-1} b^{2/3}\right ) \log \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} \sin (x)\right )-3 a^{2/3} \csc ^2(x)-6 a^{2/3} \log (\sin (x))}{6 a^{5/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]^3/(a + b*Sin[x]^3),x]

[Out]

(-3*a^(2/3)*Csc[x]^2 - 6*a^(2/3)*Log[Sin[x]] + 2*(a^(2/3) - (-1)^(2/3)*b^(2/3))*Log[-((-1)^(2/3)*a^(1/3)) - b^
(1/3)*Sin[x]] + 2*(a^(2/3) - b^(2/3))*Log[a^(1/3) + b^(1/3)*Sin[x]] + 2*(a^(2/3) + (-1)^(1/3)*b^(2/3))*Log[a^(
1/3) + (-1)^(2/3)*b^(1/3)*Sin[x]])/(6*a^(5/3))

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Maple [A]  time = 0.078, size = 126, normalized size = 0.8 \begin{align*} -{\frac{1}{3\,a}\ln \left ( \sin \left ( x \right ) +\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{1}{6\,a}\ln \left ( \left ( \sin \left ( x \right ) \right ) ^{2}-\sqrt [3]{{\frac{a}{b}}}\sin \left ( x \right ) + \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{\sqrt{3}}{3\,a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\sin \left ( x \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\ln \left ( a+b \left ( \sin \left ( x \right ) \right ) ^{3} \right ) }{3\,a}}-{\frac{1}{2\,a \left ( \sin \left ( x \right ) \right ) ^{2}}}-{\frac{\ln \left ( \sin \left ( x \right ) \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/a/(a/b)^(2/3)*ln(sin(x)+(a/b)^(1/3))+1/6/a/(a/b)^(2/3)*ln(sin(x)^2-(a/b)^(1/3)*sin(x)+(a/b)^(2/3))-1/3/a/
(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*sin(x)-1))+1/3*ln(a+b*sin(x)^3)/a-1/2/a/sin(x)^2-ln(sin(
x))/a

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cot(x)**3/(a + b*sin(x)**3), x)

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Giac [A]  time = 1.11203, size = 194, normalized size = 1.27 \begin{align*} \frac{b \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | -\left (-\frac{a}{b}\right )^{\frac{1}{3}} + \sin \left (x\right ) \right |}\right )}{3 \, a^{2}} + \frac{\log \left ({\left | b \sin \left (x\right )^{3} + a \right |}\right )}{3 \, a} - \frac{\log \left ({\left | \sin \left (x\right ) \right |}\right )}{a} - \frac{\sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (\left (-\frac{a}{b}\right )^{\frac{1}{3}} + 2 \, \sin \left (x\right )\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a^{2}} - \frac{\left (-a b^{2}\right )^{\frac{1}{3}} \log \left (\sin \left (x\right )^{2} + \left (-\frac{a}{b}\right )^{\frac{1}{3}} \sin \left (x\right ) + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a^{2}} - \frac{1}{2 \, a \sin \left (x\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b*(-a/b)^(1/3)*log(abs(-(-a/b)^(1/3) + sin(x)))/a^2 + 1/3*log(abs(b*sin(x)^3 + a))/a - log(abs(sin(x)))/a
- 1/3*sqrt(3)*(-a*b^2)^(1/3)*arctan(1/3*sqrt(3)*((-a/b)^(1/3) + 2*sin(x))/(-a/b)^(1/3))/a^2 - 1/6*(-a*b^2)^(1/
3)*log(sin(x)^2 + (-a/b)^(1/3)*sin(x) + (-a/b)^(2/3))/a^2 - 1/2/(a*sin(x)^2)