3.19 \(\int \frac{\csc ^3(x)}{(a+a \sin (x))^2} \, dx\)

Optimal. Leaf size=64 \[ \frac{16 \cot (x)}{3 a^2}-\frac{7 \tanh ^{-1}(\cos (x))}{2 a^2}-\frac{7 \cot (x) \csc (x)}{2 a^2}+\frac{8 \cot (x) \csc (x)}{3 a^2 (\sin (x)+1)}+\frac{\cot (x) \csc (x)}{3 (a \sin (x)+a)^2} \]

[Out]

(-7*ArcTanh[Cos[x]])/(2*a^2) + (16*Cot[x])/(3*a^2) - (7*Cot[x]*Csc[x])/(2*a^2) + (8*Cot[x]*Csc[x])/(3*a^2*(1 +
Sin[x])) + (Cot[x]*Csc[x])/(3*(a + a*Sin[x])^2)

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Rubi [A]  time = 0.145165, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{16 \cot (x)}{3 a^2}-\frac{7 \tanh ^{-1}(\cos (x))}{2 a^2}-\frac{7 \cot (x) \csc (x)}{2 a^2}+\frac{8 \cot (x) \csc (x)}{3 a^2 (\sin (x)+1)}+\frac{\cot (x) \csc (x)}{3 (a \sin (x)+a)^2} \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]^3/(a + a*Sin[x])^2,x]

[Out]

(-7*ArcTanh[Cos[x]])/(2*a^2) + (16*Cot[x])/(3*a^2) - (7*Cot[x]*Csc[x])/(2*a^2) + (8*Cot[x]*Csc[x])/(3*a^2*(1 +
Sin[x])) + (Cot[x]*Csc[x])/(3*(a + a*Sin[x])^2)

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))^2} \, dx &=\frac{\cot (x) \csc (x)}{3 (a+a \sin (x))^2}+\frac{\int \frac{\csc ^3(x) (5 a-3 a \sin (x))}{a+a \sin (x)} \, dx}{3 a^2}\\ &=\frac{8 \cot (x) \csc (x)}{3 a^2 (1+\sin (x))}+\frac{\cot (x) \csc (x)}{3 (a+a \sin (x))^2}+\frac{\int \csc ^3(x) \left (21 a^2-16 a^2 \sin (x)\right ) \, dx}{3 a^4}\\ &=\frac{8 \cot (x) \csc (x)}{3 a^2 (1+\sin (x))}+\frac{\cot (x) \csc (x)}{3 (a+a \sin (x))^2}-\frac{16 \int \csc ^2(x) \, dx}{3 a^2}+\frac{7 \int \csc ^3(x) \, dx}{a^2}\\ &=-\frac{7 \cot (x) \csc (x)}{2 a^2}+\frac{8 \cot (x) \csc (x)}{3 a^2 (1+\sin (x))}+\frac{\cot (x) \csc (x)}{3 (a+a \sin (x))^2}+\frac{7 \int \csc (x) \, dx}{2 a^2}+\frac{16 \operatorname{Subst}(\int 1 \, dx,x,\cot (x))}{3 a^2}\\ &=-\frac{7 \tanh ^{-1}(\cos (x))}{2 a^2}+\frac{16 \cot (x)}{3 a^2}-\frac{7 \cot (x) \csc (x)}{2 a^2}+\frac{8 \cot (x) \csc (x)}{3 a^2 (1+\sin (x))}+\frac{\cot (x) \csc (x)}{3 (a+a \sin (x))^2}\\ \end{align*}

Mathematica [B]  time = 0.597961, size = 203, normalized size = 3.17 \[ \frac{\left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right ) \left (-16 \sin \left (\frac{x}{2}\right )-160 \sin \left (\frac{x}{2}\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^2+8 \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )+3 \cos \left (\frac{x}{2}\right ) \left (\tan \left (\frac{x}{2}\right )+1\right )^3-3 \sin \left (\frac{x}{2}\right ) \left (\cot \left (\frac{x}{2}\right )+1\right )^3-84 \log \left (\cos \left (\frac{x}{2}\right )\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^3+84 \log \left (\sin \left (\frac{x}{2}\right )\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^3-24 \tan \left (\frac{x}{2}\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^3+24 \cot \left (\frac{x}{2}\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^3\right )}{24 a^2 (\sin (x)+1)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]^3/(a + a*Sin[x])^2,x]

[Out]

((Cos[x/2] + Sin[x/2])*(-16*Sin[x/2] - 3*(1 + Cot[x/2])^3*Sin[x/2] + 8*(Cos[x/2] + Sin[x/2]) - 160*Sin[x/2]*(C
os[x/2] + Sin[x/2])^2 + 24*Cot[x/2]*(Cos[x/2] + Sin[x/2])^3 - 84*Log[Cos[x/2]]*(Cos[x/2] + Sin[x/2])^3 + 84*Lo
g[Sin[x/2]]*(Cos[x/2] + Sin[x/2])^3 - 24*(Cos[x/2] + Sin[x/2])^3*Tan[x/2] + 3*Cos[x/2]*(1 + Tan[x/2])^3))/(24*
a^2*(1 + Sin[x])^2)

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Maple [A]  time = 0.061, size = 92, normalized size = 1.4 \begin{align*}{\frac{1}{8\,{a}^{2}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}}-{\frac{1}{{a}^{2}}\tan \left ({\frac{x}{2}} \right ) }+{\frac{4}{3\,{a}^{2}} \left ( \tan \left ({\frac{x}{2}} \right ) +1 \right ) ^{-3}}-2\,{\frac{1}{{a}^{2} \left ( \tan \left ( x/2 \right ) +1 \right ) ^{2}}}+8\,{\frac{1}{{a}^{2} \left ( \tan \left ( x/2 \right ) +1 \right ) }}-{\frac{1}{8\,{a}^{2}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-2}}+{\frac{1}{{a}^{2}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-1}}+{\frac{7}{2\,{a}^{2}}\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))^2,x)

[Out]

1/8/a^2*tan(1/2*x)^2-1/a^2*tan(1/2*x)+4/3/a^2/(tan(1/2*x)+1)^3-2/a^2/(tan(1/2*x)+1)^2+8/a^2/(tan(1/2*x)+1)-1/8
/a^2/tan(1/2*x)^2+1/a^2/tan(1/2*x)+7/2/a^2*ln(tan(1/2*x))

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Maxima [B]  time = 1.84253, size = 209, normalized size = 3.27 \begin{align*} \frac{\frac{15 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{239 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{405 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{216 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} - 3}{24 \,{\left (\frac{a^{2} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{3 \, a^{2} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{3 \, a^{2} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac{a^{2} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}}\right )}} - \frac{\frac{8 \, \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^{2}} + \frac{7 \, \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{2 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^3/(a+a*sin(x))^2,x, algorithm="maxima")

[Out]

1/24*(15*sin(x)/(cos(x) + 1) + 239*sin(x)^2/(cos(x) + 1)^2 + 405*sin(x)^3/(cos(x) + 1)^3 + 216*sin(x)^4/(cos(x
) + 1)^4 - 3)/(a^2*sin(x)^2/(cos(x) + 1)^2 + 3*a^2*sin(x)^3/(cos(x) + 1)^3 + 3*a^2*sin(x)^4/(cos(x) + 1)^4 + a
^2*sin(x)^5/(cos(x) + 1)^5) - 1/8*(8*sin(x)/(cos(x) + 1) - sin(x)^2/(cos(x) + 1)^2)/a^2 + 7/2*log(sin(x)/(cos(
x) + 1))/a^2

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Fricas [B]  time = 1.55372, size = 666, normalized size = 10.41 \begin{align*} -\frac{64 \, \cos \left (x\right )^{4} + 86 \, \cos \left (x\right )^{3} - 54 \, \cos \left (x\right )^{2} + 21 \,{\left (\cos \left (x\right )^{4} - \cos \left (x\right )^{3} - 3 \, \cos \left (x\right )^{2} -{\left (\cos \left (x\right )^{3} + 2 \, \cos \left (x\right )^{2} - \cos \left (x\right ) - 2\right )} \sin \left (x\right ) + \cos \left (x\right ) + 2\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) - 21 \,{\left (\cos \left (x\right )^{4} - \cos \left (x\right )^{3} - 3 \, \cos \left (x\right )^{2} -{\left (\cos \left (x\right )^{3} + 2 \, \cos \left (x\right )^{2} - \cos \left (x\right ) - 2\right )} \sin \left (x\right ) + \cos \left (x\right ) + 2\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + 2 \,{\left (32 \, \cos \left (x\right )^{3} - 11 \, \cos \left (x\right )^{2} - 38 \, \cos \left (x\right ) + 2\right )} \sin \left (x\right ) - 80 \, \cos \left (x\right ) - 4}{12 \,{\left (a^{2} \cos \left (x\right )^{4} - a^{2} \cos \left (x\right )^{3} - 3 \, a^{2} \cos \left (x\right )^{2} + a^{2} \cos \left (x\right ) + 2 \, a^{2} -{\left (a^{2} \cos \left (x\right )^{3} + 2 \, a^{2} \cos \left (x\right )^{2} - a^{2} \cos \left (x\right ) - 2 \, a^{2}\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))^2,x, algorithm="fricas")

[Out]

-1/12*(64*cos(x)^4 + 86*cos(x)^3 - 54*cos(x)^2 + 21*(cos(x)^4 - cos(x)^3 - 3*cos(x)^2 - (cos(x)^3 + 2*cos(x)^2
- cos(x) - 2)*sin(x) + cos(x) + 2)*log(1/2*cos(x) + 1/2) - 21*(cos(x)^4 - cos(x)^3 - 3*cos(x)^2 - (cos(x)^3 +
2*cos(x)^2 - cos(x) - 2)*sin(x) + cos(x) + 2)*log(-1/2*cos(x) + 1/2) + 2*(32*cos(x)^3 - 11*cos(x)^2 - 38*cos(
x) + 2)*sin(x) - 80*cos(x) - 4)/(a^2*cos(x)^4 - a^2*cos(x)^3 - 3*a^2*cos(x)^2 + a^2*cos(x) + 2*a^2 - (a^2*cos(
x)^3 + 2*a^2*cos(x)^2 - a^2*cos(x) - 2*a^2)*sin(x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 2.1323, size = 126, normalized size = 1.97 \begin{align*} \frac{7 \, \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) \right |}\right )}{2 \, a^{2}} + \frac{a^{2} \tan \left (\frac{1}{2} \, x\right )^{2} - 8 \, a^{2} \tan \left (\frac{1}{2} \, x\right )}{8 \, a^{4}} - \frac{42 \, \tan \left (\frac{1}{2} \, x\right )^{2} - 8 \, \tan \left (\frac{1}{2} \, x\right ) + 1}{8 \, a^{2} \tan \left (\frac{1}{2} \, x\right )^{2}} + \frac{2 \,{\left (12 \, \tan \left (\frac{1}{2} \, x\right )^{2} + 21 \, \tan \left (\frac{1}{2} \, x\right ) + 11\right )}}{3 \, a^{2}{\left (\tan \left (\frac{1}{2} \, x\right ) + 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^3/(a+a*sin(x))^2,x, algorithm="giac")

[Out]

7/2*log(abs(tan(1/2*x)))/a^2 + 1/8*(a^2*tan(1/2*x)^2 - 8*a^2*tan(1/2*x))/a^4 - 1/8*(42*tan(1/2*x)^2 - 8*tan(1/
2*x) + 1)/(a^2*tan(1/2*x)^2) + 2/3*(12*tan(1/2*x)^2 + 21*tan(1/2*x) + 11)/(a^2*(tan(1/2*x) + 1)^3)