\(\int \frac {\sinh (\frac {\sqrt {1-a x}}{\sqrt {1+a x}})}{1-a^2 x^2} \, dx\) [263]

Optimal result

Integrand size = 34, antiderivative size = 26 \[ \int \frac {\sinh \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=-\frac {\text {Shi}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a} \] Output:

-Shi((-a*x+1)^(1/2)/(a*x+1)^(1/2))/a
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\sinh \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=-\frac {\text {Shi}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a} \] Input:

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

Output:

-(SinhIntegral[Sqrt[1 - a*x]/Sqrt[1 + a*x]]/a)
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {7232, 3042, 26, 3779}

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.

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

\(\Big \downarrow \) 7232

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3779

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

Input:

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

Output:

-(SinhIntegral[Sqrt[1 - a*x]/Sqrt[1 + a*x]]/a)
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 3042

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

rule 3779

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> Simp[I*(SinhIntegral[c*f*(fz/d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f 
, fz}, x] && EqQ[d*e - c*f*fz*I, 0]
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

integrate(sinh((-a*x+1)^(1/2)/(a*x+1)^(1/2))/(-a^2*x^2+1),x, algorithm="fr 
icas")
 

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(sinh((-a*x+1)^(1/2)/(a*x+1)^(1/2))/(-a^2*x^2+1),x, algorithm="ma 
xima")
 

Output:

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

Giac [F]

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

integrate(sinh((-a*x+1)^(1/2)/(a*x+1)^(1/2))/(-a^2*x^2+1),x, algorithm="gi 
ac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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