\(\int \frac {\csc ^3(x)}{(a+a \sin (x))^2} \, dx\) [19]

Optimal result

Integrand size = 13, antiderivative size = 64 \[ \int \frac {\csc ^3(x)}{(a+a \sin (x))^2} \, dx=-\frac {7 \text {arctanh}(\cos (x))}{2 a^2}+\frac {16 \cot (x)}{3 a^2}-\frac {7 \cot (x) \csc (x)}{2 a^2}+\frac {8 \cot (x) \csc (x)}{3 a^2 (1+\sin (x))}+\frac {\cot (x) \csc (x)}{3 (a+a \sin (x))^2} \]

[Out]

-7/2*arctanh(cos(x))/a^2+16/3*cot(x)/a^2-7/2*cot(x)*csc(x)/a^2+8/3*cot(x)*csc(x)/a^2/(1+sin(x))+1/3*cot(x)*csc
(x)/(a+a*sin(x))^2

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {2845, 3057, 2827, 3853, 3855, 3852, 8} \[ \int \frac {\csc ^3(x)}{(a+a \sin (x))^2} \, dx=-\frac {7 \text {arctanh}(\cos (x))}{2 a^2}+\frac {16 \cot (x)}{3 a^2}-\frac {7 \cot (x) \csc (x)}{2 a^2}+\frac {8 \cot (x) \csc (x)}{3 a^2 (\sin (x)+1)}+\frac {\cot (x) \csc (x)}{3 (a \sin (x)+a)^2} \]

[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 8

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

Rule 2827

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 2845

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 3057

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 3852

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 3853

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 3855

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

Rubi steps \begin{align*} \text {integral}& = \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 \text {Subst}(\int 1 \, dx,x,\cot (x))}{3 a^2} \\ & = -\frac {7 \text {arctanh}(\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] (verified)

Leaf count is larger than twice the leaf count of optimal. \(203\) vs. \(2(64)=128\).

Time = 0.83 (sec) , antiderivative size = 203, normalized size of antiderivative = 3.17 \[ \int \frac {\csc ^3(x)}{(a+a \sin (x))^2} \, dx=\frac {\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \left (-16 \sin \left (\frac {x}{2}\right )-3 \left (1+\cot \left (\frac {x}{2}\right )\right )^3 \sin \left (\frac {x}{2}\right )+8 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )-160 \sin \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^2+24 \cot \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^3-84 \log \left (\cos \left (\frac {x}{2}\right )\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^3+84 \log \left (\sin \left (\frac {x}{2}\right )\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^3-24 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^3 \tan \left (\frac {x}{2}\right )+3 \cos \left (\frac {x}{2}\right ) \left (1+\tan \left (\frac {x}{2}\right )\right )^3\right )}{24 a^2 (1+\sin (x))^2} \]

[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)

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.09

method	result	size
parallelrisch	\(\frac {-151+84 \left (-1+\cos \left (2 x \right )\right ) \ln \left (\csc \left (x \right )-\cot \left (x \right )\right )-32 \left (\sec ^{3}\left (x \right )\right )+32 \tan \left (x \right ) \left (\sec ^{2}\left (x \right )\right )+168 \cos \left (x \right )-112 \sec \left (x \right )+128 \tan \left (x \right )+151 \cos \left (2 x \right )-128 \sin \left (2 x \right )}{24 a^{2} \left (-1+\cos \left (2 x \right )\right )}\)	\(70\)
default	\(\frac {\frac {\left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2}-4 \tan \left (\frac {x}{2}\right )-\frac {1}{2 \tan \left (\frac {x}{2}\right )^{2}}+\frac {4}{\tan \left (\frac {x}{2}\right )}+14 \ln \left (\tan \left (\frac {x}{2}\right )\right )+\frac {16}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {8}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {32}{\tan \left (\frac {x}{2}\right )+1}}{4 a^{2}}\)	\(74\)
risch	\(\frac {63 i {\mathrm e}^{5 i x}+21 \,{\mathrm e}^{6 i x}-126 i {\mathrm e}^{3 i x}-98 \,{\mathrm e}^{4 i x}+75 i {\mathrm e}^{i x}+97 \,{\mathrm e}^{2 i x}-32}{3 \left ({\mathrm e}^{2 i x}-1\right )^{2} \left ({\mathrm e}^{i x}+i\right )^{3} a^{2}}+\frac {7 \ln \left ({\mathrm e}^{i x}-1\right )}{2 a^{2}}-\frac {7 \ln \left ({\mathrm e}^{i x}+1\right )}{2 a^{2}}\)	\(99\)
norman	\(\frac {\frac {14 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{a}-\frac {1}{8 a}+\frac {5 \tan \left (\frac {x}{2}\right )}{8 a}-\frac {5 \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{8 a}+\frac {\tan ^{7}\left (\frac {x}{2}\right )}{8 a}+\frac {95 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{4 a}+\frac {151 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{12 a}}{\tan \left (\frac {x}{2}\right )^{2} a \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {7 \ln \left (\tan \left (\frac {x}{2}\right )\right )}{2 a^{2}}\)	\(100\)

[In]

int(csc(x)^3/(a+a*sin(x))^2,x,method=_RETURNVERBOSE)

[Out]

1/24*(-151+84*(-1+cos(2*x))*ln(csc(x)-cot(x))-32*sec(x)^3+32*tan(x)*sec(x)^2+168*cos(x)-112*sec(x)+128*tan(x)+
151*cos(2*x)-128*sin(2*x))/a^2/(-1+cos(2*x))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 220 vs. \(2 (54) = 108\).

Time = 0.27 (sec) , antiderivative size = 220, normalized size of antiderivative = 3.44 \[ \int \frac {\csc ^3(x)}{(a+a \sin (x))^2} \, dx=-\frac {64 \, \cos \left (x\right )^{4} + 86 \, \cos \left (x\right )^{3} - 54 \, \cos \left (x\right )^{2} + 21 \, {\left (\cos \left (x\right )^{4} - \cos \left (x\right )^{3} - 3 \, \cos \left (x\right )^{2} - {\left (\cos \left (x\right )^{3} + 2 \, \cos \left (x\right )^{2} - \cos \left (x\right ) - 2\right )} \sin \left (x\right ) + \cos \left (x\right ) + 2\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - 21 \, {\left (\cos \left (x\right )^{4} - \cos \left (x\right )^{3} - 3 \, \cos \left (x\right )^{2} - {\left (\cos \left (x\right )^{3} + 2 \, \cos \left (x\right )^{2} - \cos \left (x\right ) - 2\right )} \sin \left (x\right ) + \cos \left (x\right ) + 2\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 2 \, {\left (32 \, \cos \left (x\right )^{3} - 11 \, \cos \left (x\right )^{2} - 38 \, \cos \left (x\right ) + 2\right )} \sin \left (x\right ) - 80 \, \cos \left (x\right ) - 4}{12 \, {\left (a^{2} \cos \left (x\right )^{4} - a^{2} \cos \left (x\right )^{3} - 3 \, a^{2} \cos \left (x\right )^{2} + a^{2} \cos \left (x\right ) + 2 \, a^{2} - {\left (a^{2} \cos \left (x\right )^{3} + 2 \, a^{2} \cos \left (x\right )^{2} - a^{2} \cos \left (x\right ) - 2 \, a^{2}\right )} \sin \left (x\right )\right )}} \]

[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))

Sympy [F]

\[ \int \frac {\csc ^3(x)}{(a+a \sin (x))^2} \, dx=\frac {\int \frac {\csc ^{3}{\left (x \right )}}{\sin ^{2}{\left (x \right )} + 2 \sin {\left (x \right )} + 1}\, dx}{a^{2}} \]

[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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (54) = 108\).

Time = 0.22 (sec) , antiderivative size = 155, normalized size of antiderivative = 2.42 \[ \int \frac {\csc ^3(x)}{(a+a \sin (x))^2} \, dx=\frac {\frac {15 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {239 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {405 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {216 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} - 3}{24 \, {\left (\frac {a^{2} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {3 \, a^{2} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {3 \, a^{2} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {a^{2} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}}\right )}} - \frac {\frac {8 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}}{8 \, a^{2}} + \frac {7 \, \log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{2 \, a^{2}} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.45 \[ \int \frac {\csc ^3(x)}{(a+a \sin (x))^2} \, dx=\frac {7 \, \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{2 \, a^{2}} + \frac {a^{2} \tan \left (\frac {1}{2} \, x\right )^{2} - 8 \, a^{2} \tan \left (\frac {1}{2} \, x\right )}{8 \, a^{4}} - \frac {42 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 8 \, \tan \left (\frac {1}{2} \, x\right ) + 1}{8 \, a^{2} \tan \left (\frac {1}{2} \, x\right )^{2}} + \frac {2 \, {\left (12 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 21 \, \tan \left (\frac {1}{2} \, x\right ) + 11\right )}}{3 \, a^{2} {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{3}} \]

[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)

Mupad [B] (verification not implemented)

Time = 6.37 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.73 \[ \int \frac {\csc ^3(x)}{(a+a \sin (x))^2} \, dx=\frac {36\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+\frac {135\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{2}+\frac {239\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{6}+\frac {5\,\mathrm {tan}\left (\frac {x}{2}\right )}{2}-\frac {1}{2}}{4\,a^2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5+12\,a^2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+12\,a^2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+4\,a^2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}-\frac {\mathrm {tan}\left (\frac {x}{2}\right )}{a^2}+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{8\,a^2}+\frac {7\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{2\,a^2} \]

[In]

int(1/(sin(x)^3*(a + a*sin(x))^2),x)

[Out]

((5*tan(x/2))/2 + (239*tan(x/2)^2)/6 + (135*tan(x/2)^3)/2 + 36*tan(x/2)^4 - 1/2)/(4*a^2*tan(x/2)^2 + 12*a^2*ta
n(x/2)^3 + 12*a^2*tan(x/2)^4 + 4*a^2*tan(x/2)^5) - tan(x/2)/a^2 + tan(x/2)^2/(8*a^2) + (7*log(tan(x/2)))/(2*a^
2)