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

Optimal. Leaf size=84 \[ -\frac{2 b^3 \tan ^{-1}\left (\frac{a \tan \left (\frac{x}{2}\right )+b}{\sqrt{a^2-b^2}}\right )}{a^3 \sqrt{a^2-b^2}}-\frac{\left (a^2+2 b^2\right ) \tanh ^{-1}(\cos (x))}{2 a^3}+\frac{b \cot (x)}{a^2}-\frac{\cot (x) \csc (x)}{2 a} \]

[Out]

(-2*b^3*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^3*Sqrt[a^2 - b^2]) - ((a^2 + 2*b^2)*ArcTanh[Cos[x]])/(2*a
^3) + (b*Cot[x])/a^2 - (Cot[x]*Csc[x])/(2*a)

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Rubi [A]  time = 0.271398, antiderivative size = 84, 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 = {2802, 3055, 3001, 3770, 2660, 618, 204} \[ -\frac{2 b^3 \tan ^{-1}\left (\frac{a \tan \left (\frac{x}{2}\right )+b}{\sqrt{a^2-b^2}}\right )}{a^3 \sqrt{a^2-b^2}}-\frac{\left (a^2+2 b^2\right ) \tanh ^{-1}(\cos (x))}{2 a^3}+\frac{b \cot (x)}{a^2}-\frac{\cot (x) \csc (x)}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]^3/(a + b*Sin[x]),x]

[Out]

(-2*b^3*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^3*Sqrt[a^2 - b^2]) - ((a^2 + 2*b^2)*ArcTanh[Cos[x]])/(2*a
^3) + (b*Cot[x])/a^2 - (Cot[x]*Csc[x])/(2*a)

Rule 2802

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -S
imp[(b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 -
b^2)), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n
*Simp[a*(b*c - a*d)*(m + 1) + b^2*d*(m + n + 2) - (b^2*c + b*(b*c - a*d)*(m + 1))*Sin[e + f*x] - b^2*d*(m + n
+ 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] &
& NeQ[c^2 - d^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*n] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !
(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) || EqQ[a, 0])))

Rule 3055

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((A*b^2 - a*b*B + a^2*C)*Cos[e +
f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)), x] + Dis
t[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b
*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*
c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /;
FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && Lt
Q[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&
!IntegerQ[m]) || EqQ[a, 0])))

Rule 3001

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)])), x_Symbol] :> Dist[(A*b - a*B)/(b*c - a*d), Int[1/(a + b*Sin[e + f*x]), x], x] + Dist[(B*c - A
*d)/(b*c - a*d), Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0]
&& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 3770

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

Rule 2660

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(2*e)/d, Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.47448, size = 144, normalized size = 1.71 \[ \frac{-\frac{16 b^3 \tan ^{-1}\left (\frac{a \tan \left (\frac{x}{2}\right )+b}{\sqrt{a^2-b^2}}\right )}{\sqrt{a^2-b^2}}-a^2 \csc ^2\left (\frac{x}{2}\right )+a^2 \sec ^2\left (\frac{x}{2}\right )+4 a^2 \log \left (\sin \left (\frac{x}{2}\right )\right )-4 a^2 \log \left (\cos \left (\frac{x}{2}\right )\right )-4 a b \tan \left (\frac{x}{2}\right )+4 a b \cot \left (\frac{x}{2}\right )+8 b^2 \log \left (\sin \left (\frac{x}{2}\right )\right )-8 b^2 \log \left (\cos \left (\frac{x}{2}\right )\right )}{8 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]^3/(a + b*Sin[x]),x]

[Out]

((-16*b^3*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + 4*a*b*Cot[x/2] - a^2*Csc[x/2]^2 - 4*a^2*
Log[Cos[x/2]] - 8*b^2*Log[Cos[x/2]] + 4*a^2*Log[Sin[x/2]] + 8*b^2*Log[Sin[x/2]] + a^2*Sec[x/2]^2 - 4*a*b*Tan[x
/2])/(8*a^3)

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Maple [A]  time = 0.05, size = 112, normalized size = 1.3 \begin{align*}{\frac{1}{8\,a} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}}-{\frac{b}{2\,{a}^{2}}\tan \left ({\frac{x}{2}} \right ) }-2\,{\frac{{b}^{3}}{{a}^{3}\sqrt{{a}^{2}-{b}^{2}}}\arctan \left ( 1/2\,{\frac{2\,a\tan \left ( x/2 \right ) +2\,b}{\sqrt{{a}^{2}-{b}^{2}}}} \right ) }-{\frac{1}{8\,a} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-2}}+{\frac{1}{2\,a}\ln \left ( \tan \left ({\frac{x}{2}} \right ) \right ) }+{\frac{{b}^{2}}{{a}^{3}}\ln \left ( \tan \left ({\frac{x}{2}} \right ) \right ) }+{\frac{b}{2\,{a}^{2}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8/a*tan(1/2*x)^2-1/2/a^2*tan(1/2*x)*b-2/a^3*b^3/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*x)+2*b)/(a^2-b^2)^(1
/2))-1/8/a/tan(1/2*x)^2+1/2/a*ln(tan(1/2*x))+1/a^3*ln(tan(1/2*x))*b^2+1/2*b/a^2/tan(1/2*x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.95383, size = 1131, normalized size = 13.46 \begin{align*} \left [\frac{4 \,{\left (a^{3} b - a b^{3}\right )} \cos \left (x\right ) \sin \left (x\right ) + 2 \,{\left (b^{3} \cos \left (x\right )^{2} - b^{3}\right )} \sqrt{-a^{2} + b^{2}} \log \left (-\frac{{\left (2 \, a^{2} - b^{2}\right )} \cos \left (x\right )^{2} - 2 \, a b \sin \left (x\right ) - a^{2} - b^{2} - 2 \,{\left (a \cos \left (x\right ) \sin \left (x\right ) + b \cos \left (x\right )\right )} \sqrt{-a^{2} + b^{2}}}{b^{2} \cos \left (x\right )^{2} - 2 \, a b \sin \left (x\right ) - a^{2} - b^{2}}\right ) - 2 \,{\left (a^{4} - a^{2} b^{2}\right )} \cos \left (x\right ) -{\left (a^{4} + a^{2} b^{2} - 2 \, b^{4} -{\left (a^{4} + a^{2} b^{2} - 2 \, b^{4}\right )} \cos \left (x\right )^{2}\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) +{\left (a^{4} + a^{2} b^{2} - 2 \, b^{4} -{\left (a^{4} + a^{2} b^{2} - 2 \, b^{4}\right )} \cos \left (x\right )^{2}\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right )}{4 \,{\left (a^{5} - a^{3} b^{2} -{\left (a^{5} - a^{3} b^{2}\right )} \cos \left (x\right )^{2}\right )}}, \frac{4 \,{\left (a^{3} b - a b^{3}\right )} \cos \left (x\right ) \sin \left (x\right ) - 4 \,{\left (b^{3} \cos \left (x\right )^{2} - b^{3}\right )} \sqrt{a^{2} - b^{2}} \arctan \left (-\frac{a \sin \left (x\right ) + b}{\sqrt{a^{2} - b^{2}} \cos \left (x\right )}\right ) - 2 \,{\left (a^{4} - a^{2} b^{2}\right )} \cos \left (x\right ) -{\left (a^{4} + a^{2} b^{2} - 2 \, b^{4} -{\left (a^{4} + a^{2} b^{2} - 2 \, b^{4}\right )} \cos \left (x\right )^{2}\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) +{\left (a^{4} + a^{2} b^{2} - 2 \, b^{4} -{\left (a^{4} + a^{2} b^{2} - 2 \, b^{4}\right )} \cos \left (x\right )^{2}\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right )}{4 \,{\left (a^{5} - a^{3} b^{2} -{\left (a^{5} - a^{3} b^{2}\right )} \cos \left (x\right )^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(4*(a^3*b - a*b^3)*cos(x)*sin(x) + 2*(b^3*cos(x)^2 - b^3)*sqrt(-a^2 + b^2)*log(-((2*a^2 - b^2)*cos(x)^2 -
2*a*b*sin(x) - a^2 - b^2 - 2*(a*cos(x)*sin(x) + b*cos(x))*sqrt(-a^2 + b^2))/(b^2*cos(x)^2 - 2*a*b*sin(x) - a^
2 - b^2)) - 2*(a^4 - a^2*b^2)*cos(x) - (a^4 + a^2*b^2 - 2*b^4 - (a^4 + a^2*b^2 - 2*b^4)*cos(x)^2)*log(1/2*cos(
x) + 1/2) + (a^4 + a^2*b^2 - 2*b^4 - (a^4 + a^2*b^2 - 2*b^4)*cos(x)^2)*log(-1/2*cos(x) + 1/2))/(a^5 - a^3*b^2
- (a^5 - a^3*b^2)*cos(x)^2), 1/4*(4*(a^3*b - a*b^3)*cos(x)*sin(x) - 4*(b^3*cos(x)^2 - b^3)*sqrt(a^2 - b^2)*arc
tan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*cos(x))) - 2*(a^4 - a^2*b^2)*cos(x) - (a^4 + a^2*b^2 - 2*b^4 - (a^4 + a^2
*b^2 - 2*b^4)*cos(x)^2)*log(1/2*cos(x) + 1/2) + (a^4 + a^2*b^2 - 2*b^4 - (a^4 + a^2*b^2 - 2*b^4)*cos(x)^2)*log
(-1/2*cos(x) + 1/2))/(a^5 - a^3*b^2 - (a^5 - a^3*b^2)*cos(x)^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(csc(x)**3/(a + b*sin(x)), x)

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Giac [A]  time = 1.24654, size = 190, normalized size = 2.26 \begin{align*} -\frac{2 \,{\left (\pi \left \lfloor \frac{x}{2 \, \pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (a\right ) + \arctan \left (\frac{a \tan \left (\frac{1}{2} \, x\right ) + b}{\sqrt{a^{2} - b^{2}}}\right )\right )} b^{3}}{\sqrt{a^{2} - b^{2}} a^{3}} + \frac{a \tan \left (\frac{1}{2} \, x\right )^{2} - 4 \, b \tan \left (\frac{1}{2} \, x\right )}{8 \, a^{2}} + \frac{{\left (a^{2} + 2 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) \right |}\right )}{2 \, a^{3}} - \frac{6 \, a^{2} \tan \left (\frac{1}{2} \, x\right )^{2} + 12 \, b^{2} \tan \left (\frac{1}{2} \, x\right )^{2} - 4 \, a b \tan \left (\frac{1}{2} \, x\right ) + a^{2}}{8 \, a^{3} \tan \left (\frac{1}{2} \, x\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(pi*floor(1/2*x/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*x) + b)/sqrt(a^2 - b^2)))*b^3/(sqrt(a^2 - b^2)*a^3) +
1/8*(a*tan(1/2*x)^2 - 4*b*tan(1/2*x))/a^2 + 1/2*(a^2 + 2*b^2)*log(abs(tan(1/2*x)))/a^3 - 1/8*(6*a^2*tan(1/2*x)
^2 + 12*b^2*tan(1/2*x)^2 - 4*a*b*tan(1/2*x) + a^2)/(a^3*tan(1/2*x)^2)