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

Optimal. Leaf size=86 \[ \frac{152 \cot (x)}{15 a^3}-\frac{13 \tanh ^{-1}(\cos (x))}{2 a^3}-\frac{13 \cot (x) \csc (x)}{2 a^3}+\frac{76 \cot (x) \csc (x)}{15 \left (a^3 \sin (x)+a^3\right )}+\frac{11 \cot (x) \csc (x)}{15 a (a \sin (x)+a)^2}+\frac{\cot (x) \csc (x)}{5 (a \sin (x)+a)^3} \]

[Out]

(-13*ArcTanh[Cos[x]])/(2*a^3) + (152*Cot[x])/(15*a^3) - (13*Cot[x]*Csc[x])/(2*a^3) + (Cot[x]*Csc[x])/(5*(a + a
*Sin[x])^3) + (11*Cot[x]*Csc[x])/(15*a*(a + a*Sin[x])^2) + (76*Cot[x]*Csc[x])/(15*(a^3 + a^3*Sin[x]))

________________________________________________________________________________________

Rubi [A]  time = 0.23502, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {2766, 2978, 2748, 3768, 3770, 3767, 8} \[ \frac{152 \cot (x)}{15 a^3}-\frac{13 \tanh ^{-1}(\cos (x))}{2 a^3}-\frac{13 \cot (x) \csc (x)}{2 a^3}+\frac{76 \cot (x) \csc (x)}{15 \left (a^3 \sin (x)+a^3\right )}+\frac{11 \cot (x) \csc (x)}{15 a (a \sin (x)+a)^2}+\frac{\cot (x) \csc (x)}{5 (a \sin (x)+a)^3} \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]^3/(a + a*Sin[x])^3,x]

[Out]

(-13*ArcTanh[Cos[x]])/(2*a^3) + (152*Cot[x])/(15*a^3) - (13*Cot[x]*Csc[x])/(2*a^3) + (Cot[x]*Csc[x])/(5*(a + a
*Sin[x])^3) + (11*Cot[x]*Csc[x])/(15*a*(a + a*Sin[x])^2) + (76*Cot[x]*Csc[x])/(15*(a^3 + a^3*Sin[x]))

Rule 2766

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(a*f*(2*m + 1)*(b*c - a*d)), x] + Dis
t[1/(a*(2*m + 1)*(b*c - a*d)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[b*c*(m + 1) - a*d*
(2*m + n + 2) + b*d*(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d,
0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] &&  !GtQ[n, 0] && (IntegersQ[2*m, 2*n] || (IntegerQ
[m] && EqQ[c, 0]))

Rule 2978

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*
x])^(n + 1))/(a*f*(2*m + 1)*(b*c - a*d)), x] + Dist[1/(a*(2*m + 1)*(b*c - a*d)), Int[(a + b*Sin[e + f*x])^(m +
1)*(c + d*Sin[e + f*x])^n*Simp[B*(a*c*m + b*d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*
(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2
- b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] &&  !GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c,
0])

Rule 2748

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

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

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{\csc ^3(x)}{(a+a \sin (x))^3} \, dx &=\frac{\cot (x) \csc (x)}{5 (a+a \sin (x))^3}+\frac{\int \frac{\csc ^3(x) (7 a-4 a \sin (x))}{(a+a \sin (x))^2} \, dx}{5 a^2}\\ &=\frac{\cot (x) \csc (x)}{5 (a+a \sin (x))^3}+\frac{11 \cot (x) \csc (x)}{15 a (a+a \sin (x))^2}+\frac{\int \frac{\csc ^3(x) \left (43 a^2-33 a^2 \sin (x)\right )}{a+a \sin (x)} \, dx}{15 a^4}\\ &=\frac{\cot (x) \csc (x)}{5 (a+a \sin (x))^3}+\frac{11 \cot (x) \csc (x)}{15 a (a+a \sin (x))^2}+\frac{76 \cot (x) \csc (x)}{15 \left (a^3+a^3 \sin (x)\right )}+\frac{\int \csc ^3(x) \left (195 a^3-152 a^3 \sin (x)\right ) \, dx}{15 a^6}\\ &=\frac{\cot (x) \csc (x)}{5 (a+a \sin (x))^3}+\frac{11 \cot (x) \csc (x)}{15 a (a+a \sin (x))^2}+\frac{76 \cot (x) \csc (x)}{15 \left (a^3+a^3 \sin (x)\right )}-\frac{152 \int \csc ^2(x) \, dx}{15 a^3}+\frac{13 \int \csc ^3(x) \, dx}{a^3}\\ &=-\frac{13 \cot (x) \csc (x)}{2 a^3}+\frac{\cot (x) \csc (x)}{5 (a+a \sin (x))^3}+\frac{11 \cot (x) \csc (x)}{15 a (a+a \sin (x))^2}+\frac{76 \cot (x) \csc (x)}{15 \left (a^3+a^3 \sin (x)\right )}+\frac{13 \int \csc (x) \, dx}{2 a^3}+\frac{152 \operatorname{Subst}(\int 1 \, dx,x,\cot (x))}{15 a^3}\\ &=-\frac{13 \tanh ^{-1}(\cos (x))}{2 a^3}+\frac{152 \cot (x)}{15 a^3}-\frac{13 \cot (x) \csc (x)}{2 a^3}+\frac{\cot (x) \csc (x)}{5 (a+a \sin (x))^3}+\frac{11 \cot (x) \csc (x)}{15 a (a+a \sin (x))^2}+\frac{76 \cot (x) \csc (x)}{15 \left (a^3+a^3 \sin (x)\right )}\\ \end{align*}

Mathematica [B]  time = 0.406562, size = 247, normalized size = 2.87 \[ \frac{\left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right ) \left (-48 \sin \left (\frac{x}{2}\right )+15 \cos ^3\left (\frac{x}{2}\right ) \left (\tan \left (\frac{x}{2}\right )+1\right )^5-1712 \sin \left (\frac{x}{2}\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^4+136 \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^3-272 \sin \left (\frac{x}{2}\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^2+24 \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )-15 \sin ^3\left (\frac{x}{2}\right ) \left (\cot \left (\frac{x}{2}\right )+1\right )^5-780 \log \left (\cos \left (\frac{x}{2}\right )\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^5+780 \log \left (\sin \left (\frac{x}{2}\right )\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^5-180 \tan \left (\frac{x}{2}\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^5+180 \cot \left (\frac{x}{2}\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^5\right )}{120 a^3 (\sin (x)+1)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((Cos[x/2] + Sin[x/2])*(-48*Sin[x/2] - 15*(1 + Cot[x/2])^5*Sin[x/2]^3 + 24*(Cos[x/2] + Sin[x/2]) - 272*Sin[x/2
]*(Cos[x/2] + Sin[x/2])^2 + 136*(Cos[x/2] + Sin[x/2])^3 - 1712*Sin[x/2]*(Cos[x/2] + Sin[x/2])^4 + 180*Cot[x/2]
*(Cos[x/2] + Sin[x/2])^5 - 780*Log[Cos[x/2]]*(Cos[x/2] + Sin[x/2])^5 + 780*Log[Sin[x/2]]*(Cos[x/2] + Sin[x/2])
^5 - 180*(Cos[x/2] + Sin[x/2])^5*Tan[x/2] + 15*Cos[x/2]^3*(1 + Tan[x/2])^5))/(120*a^3*(1 + Sin[x])^3)

________________________________________________________________________________________

Maple [A]  time = 0.068, size = 119, normalized size = 1.4 \begin{align*}{\frac{1}{8\,{a}^{3}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}}-{\frac{3}{2\,{a}^{3}}\tan \left ({\frac{x}{2}} \right ) }+{\frac{8}{5\,{a}^{3}} \left ( \tan \left ({\frac{x}{2}} \right ) +1 \right ) ^{-5}}-4\,{\frac{1}{{a}^{3} \left ( \tan \left ( x/2 \right ) +1 \right ) ^{4}}}+{\frac{28}{3\,{a}^{3}} \left ( \tan \left ({\frac{x}{2}} \right ) +1 \right ) ^{-3}}-10\,{\frac{1}{{a}^{3} \left ( \tan \left ( x/2 \right ) +1 \right ) ^{2}}}+20\,{\frac{1}{{a}^{3} \left ( \tan \left ( x/2 \right ) +1 \right ) }}-{\frac{1}{8\,{a}^{3}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-2}}+{\frac{3}{2\,{a}^{3}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-1}}+{\frac{13}{2\,{a}^{3}}\ln \left ( \tan \left ({\frac{x}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8/a^3*tan(1/2*x)^2-3/2/a^3*tan(1/2*x)+8/5/a^3/(tan(1/2*x)+1)^5-4/a^3/(tan(1/2*x)+1)^4+28/3/a^3/(tan(1/2*x)+1
)^3-10/a^3/(tan(1/2*x)+1)^2+20/a^3/(tan(1/2*x)+1)-1/8/a^3/tan(1/2*x)^2+3/2/a^3/tan(1/2*x)+13/2/a^3*ln(tan(1/2*
x))

________________________________________________________________________________________

Maxima [B]  time = 1.83054, size = 282, normalized size = 3.28 \begin{align*} \frac{\frac{105 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{2782 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{9410 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{13645 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac{9285 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac{2580 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} - 15}{120 \,{\left (\frac{a^{3} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{5 \, a^{3} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{10 \, a^{3} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac{10 \, a^{3} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac{5 \, a^{3} \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac{a^{3} \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}}\right )}} - \frac{\frac{12 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}}{8 \, a^{3}} + \frac{13 \, \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{2 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(105*sin(x)/(cos(x) + 1) + 2782*sin(x)^2/(cos(x) + 1)^2 + 9410*sin(x)^3/(cos(x) + 1)^3 + 13645*sin(x)^4/
(cos(x) + 1)^4 + 9285*sin(x)^5/(cos(x) + 1)^5 + 2580*sin(x)^6/(cos(x) + 1)^6 - 15)/(a^3*sin(x)^2/(cos(x) + 1)^
2 + 5*a^3*sin(x)^3/(cos(x) + 1)^3 + 10*a^3*sin(x)^4/(cos(x) + 1)^4 + 10*a^3*sin(x)^5/(cos(x) + 1)^5 + 5*a^3*si
n(x)^6/(cos(x) + 1)^6 + a^3*sin(x)^7/(cos(x) + 1)^7) - 1/8*(12*sin(x)/(cos(x) + 1) - sin(x)^2/(cos(x) + 1)^2)/
a^3 + 13/2*log(sin(x)/(cos(x) + 1))/a^3

________________________________________________________________________________________

Fricas [B]  time = 1.55687, size = 861, normalized size = 10.01 \begin{align*} \frac{608 \, \cos \left (x\right )^{5} - 826 \, \cos \left (x\right )^{4} - 2174 \, \cos \left (x\right )^{3} + 784 \, \cos \left (x\right )^{2} - 195 \,{\left (\cos \left (x\right )^{5} + 3 \, \cos \left (x\right )^{4} - 3 \, \cos \left (x\right )^{3} - 7 \, \cos \left (x\right )^{2} +{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{3} - 5 \, \cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 4\right )} \sin \left (x\right ) + 2 \, \cos \left (x\right ) + 4\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + 195 \,{\left (\cos \left (x\right )^{5} + 3 \, \cos \left (x\right )^{4} - 3 \, \cos \left (x\right )^{3} - 7 \, \cos \left (x\right )^{2} +{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{3} - 5 \, \cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 4\right )} \sin \left (x\right ) + 2 \, \cos \left (x\right ) + 4\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) - 2 \,{\left (304 \, \cos \left (x\right )^{4} + 717 \, \cos \left (x\right )^{3} - 370 \, \cos \left (x\right )^{2} - 762 \, \cos \left (x\right ) + 6\right )} \sin \left (x\right ) + 1536 \, \cos \left (x\right ) + 12}{60 \,{\left (a^{3} \cos \left (x\right )^{5} + 3 \, a^{3} \cos \left (x\right )^{4} - 3 \, a^{3} \cos \left (x\right )^{3} - 7 \, a^{3} \cos \left (x\right )^{2} + 2 \, a^{3} \cos \left (x\right ) + 4 \, a^{3} +{\left (a^{3} \cos \left (x\right )^{4} - 2 \, a^{3} \cos \left (x\right )^{3} - 5 \, a^{3} \cos \left (x\right )^{2} + 2 \, a^{3} \cos \left (x\right ) + 4 \, a^{3}\right )} \sin \left (x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(608*cos(x)^5 - 826*cos(x)^4 - 2174*cos(x)^3 + 784*cos(x)^2 - 195*(cos(x)^5 + 3*cos(x)^4 - 3*cos(x)^3 - 7
*cos(x)^2 + (cos(x)^4 - 2*cos(x)^3 - 5*cos(x)^2 + 2*cos(x) + 4)*sin(x) + 2*cos(x) + 4)*log(1/2*cos(x) + 1/2) +
195*(cos(x)^5 + 3*cos(x)^4 - 3*cos(x)^3 - 7*cos(x)^2 + (cos(x)^4 - 2*cos(x)^3 - 5*cos(x)^2 + 2*cos(x) + 4)*si
n(x) + 2*cos(x) + 4)*log(-1/2*cos(x) + 1/2) - 2*(304*cos(x)^4 + 717*cos(x)^3 - 370*cos(x)^2 - 762*cos(x) + 6)*
sin(x) + 1536*cos(x) + 12)/(a^3*cos(x)^5 + 3*a^3*cos(x)^4 - 3*a^3*cos(x)^3 - 7*a^3*cos(x)^2 + 2*a^3*cos(x) + 4
*a^3 + (a^3*cos(x)^4 - 2*a^3*cos(x)^3 - 5*a^3*cos(x)^2 + 2*a^3*cos(x) + 4*a^3)*sin(x))

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\csc ^{3}{\left (x \right )}}{\sin ^{3}{\left (x \right )} + 3 \sin ^{2}{\left (x \right )} + 3 \sin{\left (x \right )} + 1}\, dx}{a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(csc(x)**3/(sin(x)**3 + 3*sin(x)**2 + 3*sin(x) + 1), x)/a**3

________________________________________________________________________________________

Giac [A]  time = 1.46168, size = 147, normalized size = 1.71 \begin{align*} \frac{13 \, \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) \right |}\right )}{2 \, a^{3}} - \frac{78 \, \tan \left (\frac{1}{2} \, x\right )^{2} - 12 \, \tan \left (\frac{1}{2} \, x\right ) + 1}{8 \, a^{3} \tan \left (\frac{1}{2} \, x\right )^{2}} + \frac{a^{3} \tan \left (\frac{1}{2} \, x\right )^{2} - 12 \, a^{3} \tan \left (\frac{1}{2} \, x\right )}{8 \, a^{6}} + \frac{2 \,{\left (150 \, \tan \left (\frac{1}{2} \, x\right )^{4} + 525 \, \tan \left (\frac{1}{2} \, x\right )^{3} + 745 \, \tan \left (\frac{1}{2} \, x\right )^{2} + 485 \, \tan \left (\frac{1}{2} \, x\right ) + 127\right )}}{15 \, a^{3}{\left (\tan \left (\frac{1}{2} \, x\right ) + 1\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

13/2*log(abs(tan(1/2*x)))/a^3 - 1/8*(78*tan(1/2*x)^2 - 12*tan(1/2*x) + 1)/(a^3*tan(1/2*x)^2) + 1/8*(a^3*tan(1/
2*x)^2 - 12*a^3*tan(1/2*x))/a^6 + 2/15*(150*tan(1/2*x)^4 + 525*tan(1/2*x)^3 + 745*tan(1/2*x)^2 + 485*tan(1/2*x
) + 127)/(a^3*(tan(1/2*x) + 1)^5)