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

Optimal. Leaf size=58 \[ \frac{22 \cos (x)}{15 \left (a^3 \sin (x)+a^3\right )}-\frac{\tanh ^{-1}(\cos (x))}{a^3}+\frac{7 \cos (x)}{15 a (a \sin (x)+a)^2}+\frac{\cos (x)}{5 (a \sin (x)+a)^3} \]

[Out]

-(ArcTanh[Cos[x]]/a^3) + Cos[x]/(5*(a + a*Sin[x])^3) + (7*Cos[x])/(15*a*(a + a*Sin[x])^2) + (22*Cos[x])/(15*(a
^3 + a^3*Sin[x]))

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Rubi [A]  time = 0.160529, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {2766, 2978, 12, 3770} \[ \frac{22 \cos (x)}{15 \left (a^3 \sin (x)+a^3\right )}-\frac{\tanh ^{-1}(\cos (x))}{a^3}+\frac{7 \cos (x)}{15 a (a \sin (x)+a)^2}+\frac{\cos (x)}{5 (a \sin (x)+a)^3} \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]/(a + a*Sin[x])^3,x]

[Out]

-(ArcTanh[Cos[x]]/a^3) + Cos[x]/(5*(a + a*Sin[x])^3) + (7*Cos[x])/(15*a*(a + a*Sin[x])^2) + (22*Cos[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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3770

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

Rubi steps

\begin{align*} \int \frac{\csc (x)}{(a+a \sin (x))^3} \, dx &=\frac{\cos (x)}{5 (a+a \sin (x))^3}+\frac{\int \frac{\csc (x) (5 a-2 a \sin (x))}{(a+a \sin (x))^2} \, dx}{5 a^2}\\ &=\frac{\cos (x)}{5 (a+a \sin (x))^3}+\frac{7 \cos (x)}{15 a (a+a \sin (x))^2}+\frac{\int \frac{\csc (x) \left (15 a^2-7 a^2 \sin (x)\right )}{a+a \sin (x)} \, dx}{15 a^4}\\ &=\frac{\cos (x)}{5 (a+a \sin (x))^3}+\frac{7 \cos (x)}{15 a (a+a \sin (x))^2}+\frac{22 \cos (x)}{15 \left (a^3+a^3 \sin (x)\right )}+\frac{\int 15 a^3 \csc (x) \, dx}{15 a^6}\\ &=\frac{\cos (x)}{5 (a+a \sin (x))^3}+\frac{7 \cos (x)}{15 a (a+a \sin (x))^2}+\frac{22 \cos (x)}{15 \left (a^3+a^3 \sin (x)\right )}+\frac{\int \csc (x) \, dx}{a^3}\\ &=-\frac{\tanh ^{-1}(\cos (x))}{a^3}+\frac{\cos (x)}{5 (a+a \sin (x))^3}+\frac{7 \cos (x)}{15 a (a+a \sin (x))^2}+\frac{22 \cos (x)}{15 \left (a^3+a^3 \sin (x)\right )}\\ \end{align*}

Mathematica [B]  time = 0.0689821, size = 160, normalized size = 2.76 \[ \frac{\left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right ) \left (-6 \sin \left (\frac{x}{2}\right )-44 \sin \left (\frac{x}{2}\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^4+7 \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^3-14 \sin \left (\frac{x}{2}\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^2+3 \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )-15 \log \left (\cos \left (\frac{x}{2}\right )\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^5+15 \log \left (\sin \left (\frac{x}{2}\right )\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^5\right )}{15 (a \sin (x)+a)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]/(a + a*Sin[x])^3,x]

[Out]

((Cos[x/2] + Sin[x/2])*(-6*Sin[x/2] + 3*(Cos[x/2] + Sin[x/2]) - 14*Sin[x/2]*(Cos[x/2] + Sin[x/2])^2 + 7*(Cos[x
/2] + Sin[x/2])^3 - 44*Sin[x/2]*(Cos[x/2] + Sin[x/2])^4 - 15*Log[Cos[x/2]]*(Cos[x/2] + Sin[x/2])^5 + 15*Log[Si
n[x/2]]*(Cos[x/2] + Sin[x/2])^5))/(15*(a + a*Sin[x])^3)

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Maple [A]  time = 0.055, size = 76, normalized size = 1.3 \begin{align*}{\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{20}{3\,{a}^{3}} \left ( \tan \left ({\frac{x}{2}} \right ) +1 \right ) ^{-3}}-6\,{\frac{1}{{a}^{3} \left ( \tan \left ( x/2 \right ) +1 \right ) ^{2}}}+6\,{\frac{1}{{a}^{3} \left ( \tan \left ( x/2 \right ) +1 \right ) }}+{\frac{1}{{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)/(a+a*sin(x))^3,x)

[Out]

8/5/a^3/(tan(1/2*x)+1)^5-4/a^3/(tan(1/2*x)+1)^4+20/3/a^3/(tan(1/2*x)+1)^3-6/a^3/(tan(1/2*x)+1)^2+6/a^3/(tan(1/
2*x)+1)+1/a^3*ln(tan(1/2*x))

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Maxima [B]  time = 2.05255, size = 193, normalized size = 3.33 \begin{align*} \frac{2 \,{\left (\frac{115 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{185 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{135 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{45 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + 32\right )}}{15 \,{\left (a^{3} + \frac{5 \, a^{3} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{10 \, a^{3} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{10 \, a^{3} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{5 \, a^{3} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac{a^{3} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}}\right )}} + \frac{\log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(115*sin(x)/(cos(x) + 1) + 185*sin(x)^2/(cos(x) + 1)^2 + 135*sin(x)^3/(cos(x) + 1)^3 + 45*sin(x)^4/(cos(x
) + 1)^4 + 32)/(a^3 + 5*a^3*sin(x)/(cos(x) + 1) + 10*a^3*sin(x)^2/(cos(x) + 1)^2 + 10*a^3*sin(x)^3/(cos(x) + 1
)^3 + 5*a^3*sin(x)^4/(cos(x) + 1)^4 + a^3*sin(x)^5/(cos(x) + 1)^5) + log(sin(x)/(cos(x) + 1))/a^3

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Fricas [B]  time = 1.38515, size = 536, normalized size = 9.24 \begin{align*} \frac{44 \, \cos \left (x\right )^{3} - 58 \, \cos \left (x\right )^{2} - 15 \,{\left (\cos \left (x\right )^{3} + 3 \, \cos \left (x\right )^{2} +{\left (\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 ) + 15 \,{\left (\cos \left (x\right )^{3} + 3 \, \cos \left (x\right )^{2} +{\left (\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 (22 \, \cos \left (x\right )^{2} + 51 \, \cos \left (x\right ) - 3\right )} \sin \left (x\right ) - 108 \, \cos \left (x\right ) - 6}{30 \,{\left (a^{3} \cos \left (x\right )^{3} + 3 \, a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3} +{\left (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)/(a+a*sin(x))^3,x, algorithm="fricas")

[Out]

1/30*(44*cos(x)^3 - 58*cos(x)^2 - 15*(cos(x)^3 + 3*cos(x)^2 + (cos(x)^2 - 2*cos(x) - 4)*sin(x) - 2*cos(x) - 4)
*log(1/2*cos(x) + 1/2) + 15*(cos(x)^3 + 3*cos(x)^2 + (cos(x)^2 - 2*cos(x) - 4)*sin(x) - 2*cos(x) - 4)*log(-1/2
*cos(x) + 1/2) - 2*(22*cos(x)^2 + 51*cos(x) - 3)*sin(x) - 108*cos(x) - 6)/(a^3*cos(x)^3 + 3*a^3*cos(x)^2 - 2*a
^3*cos(x) - 4*a^3 + (a^3*cos(x)^2 - 2*a^3*cos(x) - 4*a^3)*sin(x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\csc{\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)/(a+a*sin(x))**3,x)

[Out]

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

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Giac [A]  time = 1.28153, size = 76, normalized size = 1.31 \begin{align*} \frac{\log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) \right |}\right )}{a^{3}} + \frac{2 \,{\left (45 \, \tan \left (\frac{1}{2} \, x\right )^{4} + 135 \, \tan \left (\frac{1}{2} \, x\right )^{3} + 185 \, \tan \left (\frac{1}{2} \, x\right )^{2} + 115 \, \tan \left (\frac{1}{2} \, x\right ) + 32\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)/(a+a*sin(x))^3,x, algorithm="giac")

[Out]

log(abs(tan(1/2*x)))/a^3 + 2/15*(45*tan(1/2*x)^4 + 135*tan(1/2*x)^3 + 185*tan(1/2*x)^2 + 115*tan(1/2*x) + 32)/
(a^3*(tan(1/2*x) + 1)^5)