3.28 \(\int \frac{\csc ^3(x)}{(a \cos (x)+b \sin (x))^3} \, dx\)

Optimal. Leaf size=117 \[ \frac{\left (a^2+b^2\right )^2}{2 a^3 b^2 (a+b \tan (x))^2}-\frac{\left (a^2-3 b^2\right ) \left (a^2+b^2\right )}{a^4 b^2 (a+b \tan (x))}+\frac{2 \left (a^2+3 b^2\right ) \log (\tan (x))}{a^5}-\frac{2 \left (a^2+3 b^2\right ) \log (a+b \tan (x))}{a^5}+\frac{3 b \cot (x)}{a^4}-\frac{\cot ^2(x)}{2 a^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.136647, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3087, 894} \[ \frac{\left (a^2+b^2\right )^2}{2 a^3 b^2 (a+b \tan (x))^2}-\frac{\left (a^2-3 b^2\right ) \left (a^2+b^2\right )}{a^4 b^2 (a+b \tan (x))}+\frac{2 \left (a^2+3 b^2\right ) \log (\tan (x))}{a^5}-\frac{2 \left (a^2+3 b^2\right ) \log (a+b \tan (x))}{a^5}+\frac{3 b \cot (x)}{a^4}-\frac{\cot ^2(x)}{2 a^3} \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]^3/(a*Cos[x] + b*Sin[x])^3,x]

[Out]

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

Rule 3087

Int[sin[(c_.) + (d_.)*(x_)]^(m_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symb
ol] :> Dist[1/d, Subst[Int[(x^m*(a + b*x)^n)/(1 + x^2)^((m + n + 2)/2), x], x, Tan[c + d*x]], x] /; FreeQ[{a,
b, c, d}, x] && IntegerQ[n] && IntegerQ[(m + n)/2] && NeQ[n, -1] &&  !(GtQ[n, 0] && GtQ[m, 1])

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]))

Rubi steps

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

Mathematica [A]  time = 0.760654, size = 208, normalized size = 1.78 \[ \frac{2 b^2 \left (2 \left (a^2+3 b^2\right ) \log (\sin (x))-2 \left (a^2+3 b^2\right ) \log (a \cos (x)+b \sin (x))-3 \left (a^2+b^2\right )\right )+\cot ^2(x) \left (4 a^2 \left (\left (a^2+3 b^2\right ) \log (\sin (x))-\left (a^2+3 b^2\right ) \log (a \cos (x)+b \sin (x))+3 b^2\right )-a^4 \csc ^2(x)\right )-2 a b \cot (x) \left (-4 \left (a^2+3 b^2\right ) \log (\sin (x))+4 a^2 \log (a \cos (x)+b \sin (x))+a^2 \csc ^2(x)+3 a^2+12 b^2 \log (a \cos (x)+b \sin (x))\right )+6 a^3 b \cot ^3(x)+a^4 \csc ^2(x)}{2 a^5 (a \cot (x)+b)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.143, size = 151, normalized size = 1.3 \begin{align*}{\frac{a}{2\,{b}^{2} \left ( a+b\tan \left ( x \right ) \right ) ^{2}}}+{\frac{1}{a \left ( a+b\tan \left ( x \right ) \right ) ^{2}}}+{\frac{{b}^{2}}{2\,{a}^{3} \left ( a+b\tan \left ( x \right ) \right ) ^{2}}}-{\frac{1}{{b}^{2} \left ( a+b\tan \left ( x \right ) \right ) }}+2\,{\frac{1}{{a}^{2} \left ( a+b\tan \left ( x \right ) \right ) }}+3\,{\frac{{b}^{2}}{{a}^{4} \left ( a+b\tan \left ( x \right ) \right ) }}-2\,{\frac{\ln \left ( a+b\tan \left ( x \right ) \right ) }{{a}^{3}}}-6\,{\frac{\ln \left ( a+b\tan \left ( x \right ) \right ){b}^{2}}{{a}^{5}}}-{\frac{1}{2\,{a}^{3} \left ( \tan \left ( x \right ) \right ) ^{2}}}+2\,{\frac{\ln \left ( \tan \left ( x \right ) \right ) }{{a}^{3}}}+6\,{\frac{\ln \left ( \tan \left ( x \right ) \right ){b}^{2}}{{a}^{5}}}+3\,{\frac{b}{{a}^{4}\tan \left ( x \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.26935, size = 416, normalized size = 3.56 \begin{align*} -\frac{a^{4} - \frac{8 \, a^{3} b \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac{2 \,{\left (a^{4} + 22 \, a^{2} b^{2}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{4 \,{\left (21 \, a^{3} b + 4 \, a b^{3}\right )} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - \frac{{\left (15 \, a^{4} - 144 \, a^{2} b^{2} - 112 \, b^{4}\right )} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} - \frac{4 \,{\left (19 \, a^{3} b + 16 \, a b^{3}\right )} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}}}{8 \,{\left (\frac{a^{7} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{4 \, a^{6} b \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - \frac{4 \, a^{6} b \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac{a^{7} \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} - \frac{2 \,{\left (a^{7} - 2 \, a^{5} b^{2}\right )} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}}\right )}} - \frac{\frac{12 \, b \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}}{8 \, a^{4}} - \frac{2 \,{\left (a^{2} + 3 \, b^{2}\right )} \log \left (-a - \frac{2 \, b \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 )}{a^{5}} + \frac{2 \,{\left (a^{2} + 3 \, b^{2}\right )} \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(a^4 - 8*a^3*b*sin(x)/(cos(x) + 1) - 2*(a^4 + 22*a^2*b^2)*sin(x)^2/(cos(x) + 1)^2 + 4*(21*a^3*b + 4*a*b^3
)*sin(x)^3/(cos(x) + 1)^3 - (15*a^4 - 144*a^2*b^2 - 112*b^4)*sin(x)^4/(cos(x) + 1)^4 - 4*(19*a^3*b + 16*a*b^3)
*sin(x)^5/(cos(x) + 1)^5)/(a^7*sin(x)^2/(cos(x) + 1)^2 + 4*a^6*b*sin(x)^3/(cos(x) + 1)^3 - 4*a^6*b*sin(x)^5/(c
os(x) + 1)^5 + a^7*sin(x)^6/(cos(x) + 1)^6 - 2*(a^7 - 2*a^5*b^2)*sin(x)^4/(cos(x) + 1)^4) - 1/8*(12*b*sin(x)/(
cos(x) + 1) + a*sin(x)^2/(cos(x) + 1)^2)/a^4 - 2*(a^2 + 3*b^2)*log(-a - 2*b*sin(x)/(cos(x) + 1) + a*sin(x)^2/(
cos(x) + 1)^2)/a^5 + 2*(a^2 + 3*b^2)*log(sin(x)/(cos(x) + 1))/a^5

________________________________________________________________________________________

Fricas [B]  time = 0.592269, size = 887, normalized size = 7.58 \begin{align*} -\frac{24 \, a^{2} b^{2} \cos \left (x\right )^{4} - a^{4} + 6 \, a^{2} b^{2} + 2 \,{\left (a^{4} - 15 \, a^{2} b^{2}\right )} \cos \left (x\right )^{2} - 2 \,{\left ({\left (a^{4} + 2 \, a^{2} b^{2} - 3 \, b^{4}\right )} \cos \left (x\right )^{4} - a^{2} b^{2} - 3 \, b^{4} -{\left (a^{4} + a^{2} b^{2} - 6 \, b^{4}\right )} \cos \left (x\right )^{2} + 2 \,{\left ({\left (a^{3} b + 3 \, a b^{3}\right )} \cos \left (x\right )^{3} -{\left (a^{3} b + 3 \, a b^{3}\right )} \cos \left (x\right )\right )} \sin \left (x\right )\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} + b^{2}\right ) + 2 \,{\left ({\left (a^{4} + 2 \, a^{2} b^{2} - 3 \, b^{4}\right )} \cos \left (x\right )^{4} - a^{2} b^{2} - 3 \, b^{4} -{\left (a^{4} + a^{2} b^{2} - 6 \, b^{4}\right )} \cos \left (x\right )^{2} + 2 \,{\left ({\left (a^{3} b + 3 \, a b^{3}\right )} \cos \left (x\right )^{3} -{\left (a^{3} b + 3 \, a b^{3}\right )} \cos \left (x\right )\right )} \sin \left (x\right )\right )} \log \left (-\frac{1}{4} \, \cos \left (x\right )^{2} + \frac{1}{4}\right ) - 4 \,{\left (3 \,{\left (a^{3} b - a b^{3}\right )} \cos \left (x\right )^{3} -{\left (2 \, a^{3} b - 3 \, a b^{3}\right )} \cos \left (x\right )\right )} \sin \left (x\right )}{2 \,{\left (a^{5} b^{2} -{\left (a^{7} - a^{5} b^{2}\right )} \cos \left (x\right )^{4} +{\left (a^{7} - 2 \, a^{5} b^{2}\right )} \cos \left (x\right )^{2} - 2 \,{\left (a^{6} b \cos \left (x\right )^{3} - a^{6} b \cos \left (x\right )\right )} \sin \left (x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(24*a^2*b^2*cos(x)^4 - a^4 + 6*a^2*b^2 + 2*(a^4 - 15*a^2*b^2)*cos(x)^2 - 2*((a^4 + 2*a^2*b^2 - 3*b^4)*cos
(x)^4 - a^2*b^2 - 3*b^4 - (a^4 + a^2*b^2 - 6*b^4)*cos(x)^2 + 2*((a^3*b + 3*a*b^3)*cos(x)^3 - (a^3*b + 3*a*b^3)
*cos(x))*sin(x))*log(2*a*b*cos(x)*sin(x) + (a^2 - b^2)*cos(x)^2 + b^2) + 2*((a^4 + 2*a^2*b^2 - 3*b^4)*cos(x)^4
- a^2*b^2 - 3*b^4 - (a^4 + a^2*b^2 - 6*b^4)*cos(x)^2 + 2*((a^3*b + 3*a*b^3)*cos(x)^3 - (a^3*b + 3*a*b^3)*cos(
x))*sin(x))*log(-1/4*cos(x)^2 + 1/4) - 4*(3*(a^3*b - a*b^3)*cos(x)^3 - (2*a^3*b - 3*a*b^3)*cos(x))*sin(x))/(a^
5*b^2 - (a^7 - a^5*b^2)*cos(x)^4 + (a^7 - 2*a^5*b^2)*cos(x)^2 - 2*(a^6*b*cos(x)^3 - a^6*b*cos(x))*sin(x))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.19542, size = 197, normalized size = 1.68 \begin{align*} \frac{2 \,{\left (a^{2} + 3 \, b^{2}\right )} \log \left ({\left | \tan \left (x\right ) \right |}\right )}{a^{5}} - \frac{2 \,{\left (a^{2} b + 3 \, b^{3}\right )} \log \left ({\left | b \tan \left (x\right ) + a \right |}\right )}{a^{5} b} - \frac{2 \, a^{4} b \tan \left (x\right )^{3} - 4 \, a^{2} b^{3} \tan \left (x\right )^{3} - 12 \, b^{5} \tan \left (x\right )^{3} + a^{5} \tan \left (x\right )^{2} - 6 \, a^{3} b^{2} \tan \left (x\right )^{2} - 18 \, a b^{4} \tan \left (x\right )^{2} - 4 \, a^{2} b^{3} \tan \left (x\right ) + a^{3} b^{2}}{2 \,{\left (b \tan \left (x\right )^{2} + a \tan \left (x\right )\right )}^{2} a^{4} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(a^2 + 3*b^2)*log(abs(tan(x)))/a^5 - 2*(a^2*b + 3*b^3)*log(abs(b*tan(x) + a))/(a^5*b) - 1/2*(2*a^4*b*tan(x)^
3 - 4*a^2*b^3*tan(x)^3 - 12*b^5*tan(x)^3 + a^5*tan(x)^2 - 6*a^3*b^2*tan(x)^2 - 18*a*b^4*tan(x)^2 - 4*a^2*b^3*t
an(x) + a^3*b^2)/((b*tan(x)^2 + a*tan(x))^2*a^4*b^2)