Integrand size = 34, antiderivative size = 26 \[ \int \frac {\cosh \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=-\frac {\text {Chi}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a} \] Output:
-Chi((-a*x+1)^(1/2)/(a*x+1)^(1/2))/a
Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=-\frac {\text {Chi}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a} \] Input:
Integrate[Cosh[Sqrt[1 - a*x]/Sqrt[1 + a*x]]/(1 - a^2*x^2),x]
Output:
-(CoshIntegral[Sqrt[1 - a*x]/Sqrt[1 + a*x]]/a)
Time = 0.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {7232, 3042, 3782}
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.
\(\displaystyle \int \frac {\cosh \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )}{1-a^2 x^2} \, dx\) |
\(\Big \downarrow \) 7232 |
\(\displaystyle -\frac {\int \frac {\sqrt {a x+1} \cosh \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )}{\sqrt {1-a x}}d\frac {\sqrt {1-a x}}{\sqrt {a x+1}}}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\int \frac {\sqrt {a x+1} \sin \left (\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}+\frac {\pi }{2}\right )}{\sqrt {1-a x}}d\frac {\sqrt {1-a x}}{\sqrt {a x+1}}}{a}\) |
\(\Big \downarrow \) 3782 |
\(\displaystyle -\frac {\text {Chi}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}\) |
Input:
Int[Cosh[Sqrt[1 - a*x]/Sqrt[1 + a*x]]/(1 - a^2*x^2),x]
Output:
-(CoshIntegral[Sqrt[1 - a*x]/Sqrt[1 + a*x]]/a)
Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo l] :> Simp[CoshIntegral[c*f*(fz/d) + f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz }, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]
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]
\[\int \frac {\cosh \left (\frac {\sqrt {-a x +1}}{\sqrt {a x +1}}\right )}{-a^{2} x^{2}+1}d x\]
Input:
int(cosh((-a*x+1)^(1/2)/(a*x+1)^(1/2))/(-a^2*x^2+1),x)
Output:
int(cosh((-a*x+1)^(1/2)/(a*x+1)^(1/2))/(-a^2*x^2+1),x)
\[ \int \frac {\cosh \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\int { -\frac {\cosh \left (\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )}{a^{2} x^{2} - 1} \,d x } \] Input:
integrate(cosh((-a*x+1)^(1/2)/(a*x+1)^(1/2))/(-a^2*x^2+1),x, algorithm="fr icas")
Output:
integral(-cosh(sqrt(-a*x + 1)/sqrt(a*x + 1))/(a^2*x^2 - 1), x)
\[ \int \frac {\cosh \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=- \int \frac {\cosh {\left (\frac {\sqrt {- a x + 1}}{\sqrt {a x + 1}} \right )}}{a^{2} x^{2} - 1}\, dx \] Input:
integrate(cosh((-a*x+1)**(1/2)/(a*x+1)**(1/2))/(-a**2*x**2+1),x)
Output:
-Integral(cosh(sqrt(-a*x + 1)/sqrt(a*x + 1))/(a**2*x**2 - 1), x)
\[ \int \frac {\cosh \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\int { -\frac {\cosh \left (\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )}{a^{2} x^{2} - 1} \,d x } \] Input:
integrate(cosh((-a*x+1)^(1/2)/(a*x+1)^(1/2))/(-a^2*x^2+1),x, algorithm="ma xima")
Output:
-integrate(cosh(sqrt(-a*x + 1)/sqrt(a*x + 1))/(a^2*x^2 - 1), x)
\[ \int \frac {\cosh \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\int { -\frac {\cosh \left (\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )}{a^{2} x^{2} - 1} \,d x } \] Input:
integrate(cosh((-a*x+1)^(1/2)/(a*x+1)^(1/2))/(-a^2*x^2+1),x, algorithm="gi ac")
Output:
integrate(-cosh(sqrt(-a*x + 1)/sqrt(a*x + 1))/(a^2*x^2 - 1), x)
Timed out. \[ \int \frac {\cosh \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=-\int \frac {\mathrm {cosh}\left (\frac {\sqrt {1-a\,x}}{\sqrt {a\,x+1}}\right )}{a^2\,x^2-1} \,d x \] Input:
int(-cosh((1 - a*x)^(1/2)/(a*x + 1)^(1/2))/(a^2*x^2 - 1),x)
Output:
-int(cosh((1 - a*x)^(1/2)/(a*x + 1)^(1/2))/(a^2*x^2 - 1), x)
\[ \int \frac {\cosh \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=-\left (\int \frac {\cosh \left (\frac {\sqrt {-a x +1}}{\sqrt {a x +1}}\right )}{a^{2} x^{2}-1}d x \right ) \] Input:
int(cosh((-a*x+1)^(1/2)/(a*x+1)^(1/2))/(-a^2*x^2+1),x)
Output:
- int(cosh(sqrt( - a*x + 1)/sqrt(a*x + 1))/(a**2*x**2 - 1),x)