\(\int \frac {\sin ^3(\frac {\sqrt {1-a x}}{\sqrt {1+a x}})}{1-a^2 x^2} \, dx\) [41]

Optimal result

Integrand size = 36, antiderivative size = 58 \[ \int \frac {\sin ^3\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=-\frac {3 \text {Si}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{4 a}+\frac {\text {Si}\left (\frac {3 \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{4 a} \] Output:

-3/4*Si((-a*x+1)^(1/2)/(a*x+1)^(1/2))/a+1/4*Si(3*(-a*x+1)^(1/2)/(a*x+1)^(1 
/2))/a
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.91 \[ \int \frac {\sin ^3\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\frac {-3 \text {Si}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )+\text {Si}\left (\frac {3 \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{4 a} \] Input:

Integrate[Sin[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^3/(1 - a^2*x^2),x]
 

Output:

(-3*SinIntegral[Sqrt[1 - a*x]/Sqrt[1 + a*x]] + SinIntegral[(3*Sqrt[1 - a*x 
])/Sqrt[1 + a*x]])/(4*a)
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {7232, 3042, 3793, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\sin ^3\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )}{1-a^2 x^2} \, dx\)

\(\Big \downarrow \) 7232

\(\displaystyle -\frac {\int \frac {\sqrt {a x+1} \sin ^3\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )}{\sqrt {1-a x}}d\frac {\sqrt {1-a x}}{\sqrt {a x+1}}}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int \frac {\sqrt {a x+1} \sin \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )^3}{\sqrt {1-a x}}d\frac {\sqrt {1-a x}}{\sqrt {a x+1}}}{a}\)

\(\Big \downarrow \) 3793

\(\displaystyle -\frac {\int \left (\frac {3 \sqrt {a x+1} \sin \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )}{4 \sqrt {1-a x}}-\frac {\sqrt {a x+1} \sin \left (\frac {3 \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{4 \sqrt {1-a x}}\right )d\frac {\sqrt {1-a x}}{\sqrt {a x+1}}}{a}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\frac {3}{4} \text {Si}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-\frac {1}{4} \text {Si}\left (\frac {3 \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}\)

Input:

Int[Sin[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^3/(1 - a^2*x^2),x]
 

Output:

-(((3*SinIntegral[Sqrt[1 - a*x]/Sqrt[1 + a*x]])/4 - SinIntegral[(3*Sqrt[1 
- a*x])/Sqrt[1 + a*x]]/4)/a)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 7232

Int[((a_.) + (b_.)*(F_)[((c_.)*Sqrt[(d_.) + (e_.)*(x_)])/Sqrt[(f_.) + (g_.) 
*(x_)]])^(n_.)/((A_.) + (C_.)*(x_)^2), x_Symbol] :> Simp[2*e*(g/(C*(e*f - d 
*g)))   Subst[Int[(a + b*F[c*x])^n/x, x], x, Sqrt[d + e*x]/Sqrt[f + g*x]], 
x] /; FreeQ[{a, b, c, d, e, f, g, A, C, F}, x] && EqQ[C*d*f - A*e*g, 0] && 
EqQ[e*f + d*g, 0] && IGtQ[n, 0]
 

Maple [F]

\[\int \frac {\sin \left (\frac {\sqrt {-a x +1}}{\sqrt {a x +1}}\right )^{3}}{-a^{2} x^{2}+1}d x\]

Input:

int(sin((-a*x+1)^(1/2)/(a*x+1)^(1/2))^3/(-a^2*x^2+1),x)
 

Output:

int(sin((-a*x+1)^(1/2)/(a*x+1)^(1/2))^3/(-a^2*x^2+1),x)
 

Fricas [F]

\[ \int \frac {\sin ^3\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\int { -\frac {\sin \left (\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )^{3}}{a^{2} x^{2} - 1} \,d x } \] Input:

integrate(sin((-a*x+1)^(1/2)/(a*x+1)^(1/2))^3/(-a^2*x^2+1),x, algorithm="f 
ricas")
 

Output:

integral((cos(sqrt(-a*x + 1)/sqrt(a*x + 1))^2 - 1)*sin(sqrt(-a*x + 1)/sqrt 
(a*x + 1))/(a^2*x^2 - 1), x)
 

Sympy [F]

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

integrate(sin((-a*x+1)**(1/2)/(a*x+1)**(1/2))**3/(-a**2*x**2+1),x)
 

Output:

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

Maxima [F]

\[ \int \frac {\sin ^3\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\int { -\frac {\sin \left (\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )^{3}}{a^{2} x^{2} - 1} \,d x } \] Input:

integrate(sin((-a*x+1)^(1/2)/(a*x+1)^(1/2))^3/(-a^2*x^2+1),x, algorithm="m 
axima")
 

Output:

-integrate(sin(sqrt(-a*x + 1)/sqrt(a*x + 1))^3/(a^2*x^2 - 1), x)
 

Giac [F]

\[ \int \frac {\sin ^3\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\int { -\frac {\sin \left (\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )^{3}}{a^{2} x^{2} - 1} \,d x } \] Input:

integrate(sin((-a*x+1)^(1/2)/(a*x+1)^(1/2))^3/(-a^2*x^2+1),x, algorithm="g 
iac")
 

Output:

integrate(-sin(sqrt(-a*x + 1)/sqrt(a*x + 1))^3/(a^2*x^2 - 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sin ^3\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=-\int \frac {{\sin \left (\frac {\sqrt {1-a\,x}}{\sqrt {a\,x+1}}\right )}^3}{a^2\,x^2-1} \,d x \] Input:

int(-sin((1 - a*x)^(1/2)/(a*x + 1)^(1/2))^3/(a^2*x^2 - 1),x)
 

Output:

-int(sin((1 - a*x)^(1/2)/(a*x + 1)^(1/2))^3/(a^2*x^2 - 1), x)
 

Reduce [F]

\[ \int \frac {\sin ^3\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=-\left (\int \frac {\sin \left (\frac {\sqrt {-a x +1}}{\sqrt {a x +1}}\right )^{3}}{a^{2} x^{2}-1}d x \right ) \] Input:

int(sin((-a*x+1)^(1/2)/(a*x+1)^(1/2))^3/(-a^2*x^2+1),x)
 

Output:

 - int(sin(sqrt( - a*x + 1)/sqrt(a*x + 1))**3/(a**2*x**2 - 1),x)