3.12 \(\int \frac{\csc ^6(x)}{a+b \cot (x)} \, dx\)

Optimal. Leaf size=81 \[ -\frac{\left (a^2+2 b^2\right ) \cot ^2(x)}{2 b^3}+\frac{a \left (a^2+2 b^2\right ) \cot (x)}{b^4}-\frac{\left (a^2+b^2\right )^2 \log (a+b \cot (x))}{b^5}+\frac{a \cot ^3(x)}{3 b^2}-\frac{\cot ^4(x)}{4 b} \]

[Out]

(a*(a^2 + 2*b^2)*Cot[x])/b^4 - ((a^2 + 2*b^2)*Cot[x]^2)/(2*b^3) + (a*Cot[x]^3)/(3*b^2) - Cot[x]^4/(4*b) - ((a^
2 + b^2)^2*Log[a + b*Cot[x]])/b^5

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Rubi [A]  time = 0.0996165, antiderivative size = 81, 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 = {3506, 697} \[ -\frac{\left (a^2+2 b^2\right ) \cot ^2(x)}{2 b^3}+\frac{a \left (a^2+2 b^2\right ) \cot (x)}{b^4}-\frac{\left (a^2+b^2\right )^2 \log (a+b \cot (x))}{b^5}+\frac{a \cot ^3(x)}{3 b^2}-\frac{\cot ^4(x)}{4 b} \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]^6/(a + b*Cot[x]),x]

[Out]

(a*(a^2 + 2*b^2)*Cot[x])/b^4 - ((a^2 + 2*b^2)*Cot[x]^2)/(2*b^3) + (a*Cot[x]^3)/(3*b^2) - Cot[x]^4/(4*b) - ((a^
2 + b^2)^2*Log[a + b*Cot[x]])/b^5

Rule 3506

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[1/(b*f), Subst
[Int[(a + x)^n*(1 + x^2/b^2)^(m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && NeQ[a^2 + b
^2, 0] && IntegerQ[m/2]

Rule 697

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 0.374664, size = 85, normalized size = 1.05 \[ \frac{-6 b^2 \left (a^2+b^2\right ) \csc ^2(x)+4 a b \cot (x) \left (3 a^2+b^2 \csc ^2(x)+5 b^2\right )+12 \left (a^2+b^2\right )^2 (\log (\sin (x))-\log (a \sin (x)+b \cos (x)))-3 b^4 \csc ^4(x)}{12 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]^6/(a + b*Cot[x]),x]

[Out]

(-6*b^2*(a^2 + b^2)*Csc[x]^2 - 3*b^4*Csc[x]^4 + 4*a*b*Cot[x]*(3*a^2 + 5*b^2 + b^2*Csc[x]^2) + 12*(a^2 + b^2)^2
*(Log[Sin[x]] - Log[b*Cos[x] + a*Sin[x]]))/(12*b^5)

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Maple [A]  time = 0.056, size = 133, normalized size = 1.6 \begin{align*} -{\frac{1}{4\,b \left ( \tan \left ( x \right ) \right ) ^{4}}}-{\frac{{a}^{2}}{2\,{b}^{3} \left ( \tan \left ( x \right ) \right ) ^{2}}}-{\frac{1}{b \left ( \tan \left ( x \right ) \right ) ^{2}}}+{\frac{\ln \left ( \tan \left ( x \right ) \right ){a}^{4}}{{b}^{5}}}+2\,{\frac{\ln \left ( \tan \left ( x \right ) \right ){a}^{2}}{{b}^{3}}}+{\frac{\ln \left ( \tan \left ( x \right ) \right ) }{b}}+{\frac{a}{3\,{b}^{2} \left ( \tan \left ( x \right ) \right ) ^{3}}}+{\frac{{a}^{3}}{{b}^{4}\tan \left ( x \right ) }}+2\,{\frac{a}{{b}^{2}\tan \left ( x \right ) }}-{\frac{\ln \left ( a\tan \left ( x \right ) +b \right ){a}^{4}}{{b}^{5}}}-2\,{\frac{\ln \left ( a\tan \left ( x \right ) +b \right ){a}^{2}}{{b}^{3}}}-{\frac{\ln \left ( a\tan \left ( x \right ) +b \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(x)^6/(a+b*cot(x)),x)

[Out]

-1/4/b/tan(x)^4-1/2/b^3/tan(x)^2*a^2-1/b/tan(x)^2+1/b^5*ln(tan(x))*a^4+2/b^3*ln(tan(x))*a^2+1/b*ln(tan(x))+1/3
*a/b^2/tan(x)^3+1/b^4*a^3/tan(x)+2/b^2*a/tan(x)-1/b^5*ln(a*tan(x)+b)*a^4-2/b^3*ln(a*tan(x)+b)*a^2-1/b*ln(a*tan
(x)+b)

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Maxima [A]  time = 1.21936, size = 143, normalized size = 1.77 \begin{align*} -\frac{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (a \tan \left (x\right ) + b\right )}{b^{5}} + \frac{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (x\right )\right )}{b^{5}} + \frac{4 \, a b^{2} \tan \left (x\right ) + 12 \,{\left (a^{3} + 2 \, a b^{2}\right )} \tan \left (x\right )^{3} - 3 \, b^{3} - 6 \,{\left (a^{2} b + 2 \, b^{3}\right )} \tan \left (x\right )^{2}}{12 \, b^{4} \tan \left (x\right )^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^6/(a+b*cot(x)),x, algorithm="maxima")

[Out]

-(a^4 + 2*a^2*b^2 + b^4)*log(a*tan(x) + b)/b^5 + (a^4 + 2*a^2*b^2 + b^4)*log(tan(x))/b^5 + 1/12*(4*a*b^2*tan(x
) + 12*(a^3 + 2*a*b^2)*tan(x)^3 - 3*b^3 - 6*(a^2*b + 2*b^3)*tan(x)^2)/(b^4*tan(x)^4)

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Fricas [B]  time = 1.88844, size = 599, normalized size = 7.4 \begin{align*} -\frac{6 \, a^{2} b^{2} + 9 \, b^{4} - 6 \,{\left (a^{2} b^{2} + b^{4}\right )} \cos \left (x\right )^{2} + 6 \,{\left ({\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (x\right )^{4} + a^{4} + 2 \, a^{2} b^{2} + b^{4} - 2 \,{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (x\right )^{2}\right )} \log \left (2 \, a b \cos \left (x\right ) \sin \left (x\right ) -{\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + a^{2}\right ) - 6 \,{\left ({\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (x\right )^{4} + a^{4} + 2 \, a^{2} b^{2} + b^{4} - 2 \,{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (x\right )^{2}\right )} \log \left (-\frac{1}{4} \, \cos \left (x\right )^{2} + \frac{1}{4}\right ) + 4 \,{\left ({\left (3 \, a^{3} b + 5 \, a b^{3}\right )} \cos \left (x\right )^{3} - 3 \,{\left (a^{3} b + 2 \, a b^{3}\right )} \cos \left (x\right )\right )} \sin \left (x\right )}{12 \,{\left (b^{5} \cos \left (x\right )^{4} - 2 \, b^{5} \cos \left (x\right )^{2} + b^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^6/(a+b*cot(x)),x, algorithm="fricas")

[Out]

-1/12*(6*a^2*b^2 + 9*b^4 - 6*(a^2*b^2 + b^4)*cos(x)^2 + 6*((a^4 + 2*a^2*b^2 + b^4)*cos(x)^4 + a^4 + 2*a^2*b^2
+ b^4 - 2*(a^4 + 2*a^2*b^2 + b^4)*cos(x)^2)*log(2*a*b*cos(x)*sin(x) - (a^2 - b^2)*cos(x)^2 + a^2) - 6*((a^4 +
2*a^2*b^2 + b^4)*cos(x)^4 + a^4 + 2*a^2*b^2 + b^4 - 2*(a^4 + 2*a^2*b^2 + b^4)*cos(x)^2)*log(-1/4*cos(x)^2 + 1/
4) + 4*((3*a^3*b + 5*a*b^3)*cos(x)^3 - 3*(a^3*b + 2*a*b^3)*cos(x))*sin(x))/(b^5*cos(x)^4 - 2*b^5*cos(x)^2 + b^
5)

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Sympy [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(csc(x)**6/(a+b*cot(x)),x)

[Out]

Timed out

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Giac [B]  time = 1.34744, size = 204, normalized size = 2.52 \begin{align*} \frac{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \log \left ({\left | \tan \left (x\right ) \right |}\right )}{b^{5}} - \frac{{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \log \left ({\left | a \tan \left (x\right ) + b \right |}\right )}{a b^{5}} - \frac{25 \, a^{4} \tan \left (x\right )^{4} + 50 \, a^{2} b^{2} \tan \left (x\right )^{4} + 25 \, b^{4} \tan \left (x\right )^{4} - 12 \, a^{3} b \tan \left (x\right )^{3} - 24 \, a b^{3} \tan \left (x\right )^{3} + 6 \, a^{2} b^{2} \tan \left (x\right )^{2} + 12 \, b^{4} \tan \left (x\right )^{2} - 4 \, a b^{3} \tan \left (x\right ) + 3 \, b^{4}}{12 \, b^{5} \tan \left (x\right )^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^6/(a+b*cot(x)),x, algorithm="giac")

[Out]

(a^4 + 2*a^2*b^2 + b^4)*log(abs(tan(x)))/b^5 - (a^5 + 2*a^3*b^2 + a*b^4)*log(abs(a*tan(x) + b))/(a*b^5) - 1/12
*(25*a^4*tan(x)^4 + 50*a^2*b^2*tan(x)^4 + 25*b^4*tan(x)^4 - 12*a^3*b*tan(x)^3 - 24*a*b^3*tan(x)^3 + 6*a^2*b^2*
tan(x)^2 + 12*b^4*tan(x)^2 - 4*a*b^3*tan(x) + 3*b^4)/(b^5*tan(x)^4)