Integrand size = 10, antiderivative size = 87 \[ \int \frac {e^{\arccos (a x)}}{x^2} \, dx=(1+i) a e^{(1+i) \arccos (a x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-\frac {i}{2},1,\frac {3}{2}-\frac {i}{2},-e^{2 i \arccos (a x)}\right )-(2+2 i) a e^{(1+i) \arccos (a x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-\frac {i}{2},2,\frac {3}{2}-\frac {i}{2},-e^{2 i \arccos (a x)}\right ) \] Output:
(1+I)*a*exp((1+I)*arccos(a*x))*hypergeom([1, 1/2-1/2*I],[3/2-1/2*I],-(a*x+ I*(-a^2*x^2+1)^(1/2))^2)-(2+2*I)*a*exp((1+I)*arccos(a*x))*hypergeom([2, 1/ 2-1/2*I],[3/2-1/2*I],-(a*x+I*(-a^2*x^2+1)^(1/2))^2)
Time = 0.05 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.63 \[ \int \frac {e^{\arccos (a x)}}{x^2} \, dx=-\frac {e^{\arccos (a x)}}{x}+(1-i) a e^{(1+i) \arccos (a x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-\frac {i}{2},1,\frac {3}{2}-\frac {i}{2},-e^{2 i \arccos (a x)}\right ) \] Input:
Integrate[E^ArcCos[a*x]/x^2,x]
Output:
-(E^ArcCos[a*x]/x) + (1 - I)*a*E^((1 + I)*ArcCos[a*x])*Hypergeometric2F1[1 /2 - I/2, 1, 3/2 - I/2, -E^((2*I)*ArcCos[a*x])]
Time = 0.31 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5336, 27, 4974, 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.
\(\displaystyle \int \frac {e^{\arccos (a x)}}{x^2} \, dx\) |
\(\Big \downarrow \) 5336 |
\(\displaystyle -\frac {\int \frac {e^{\arccos (a x)} \sqrt {1-a^2 x^2}}{x^2}d\arccos (a x)}{a}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -a \int \frac {e^{\arccos (a x)} \sqrt {1-a^2 x^2}}{a^2 x^2}d\arccos (a x)\) |
\(\Big \downarrow \) 4974 |
\(\displaystyle -a \int \left (\frac {4 i e^{(1+i) \arccos (a x)}}{\left (1+e^{2 i \arccos (a x)}\right )^2}-\frac {2 i e^{(1+i) \arccos (a x)}}{1+e^{2 i \arccos (a x)}}\right )d\arccos (a x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -a \left ((2+2 i) e^{(1+i) \arccos (a x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-\frac {i}{2},2,\frac {3}{2}-\frac {i}{2},-e^{2 i \arccos (a x)}\right )-(1+i) e^{(1+i) \arccos (a x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-\frac {i}{2},1,\frac {3}{2}-\frac {i}{2},-e^{2 i \arccos (a x)}\right )\right )\) |
Input:
Int[E^ArcCos[a*x]/x^2,x]
Output:
-(a*((-1 - I)*E^((1 + I)*ArcCos[a*x])*Hypergeometric2F1[1/2 - I/2, 1, 3/2 - I/2, -E^((2*I)*ArcCos[a*x])] + (2 + 2*I)*E^((1 + I)*ArcCos[a*x])*Hyperge ometric2F1[1/2 - I/2, 2, 3/2 - I/2, -E^((2*I)*ArcCos[a*x])]))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*(G_)[(d_.) + (e_.)*(x_)]^(m_.)*(H_)[( d_.) + (e_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigToExp[F^(c*(a + b*x)), G[d + e*x]^m*H[d + e*x]^n, x], x] /; FreeQ[{F, a, b, c, d, e}, x] && IGtQ[ m, 0] && IGtQ[n, 0] && TrigQ[G] && TrigQ[H]
Int[(u_.)*(f_)^(ArcCos[(a_.) + (b_.)*(x_)]^(n_.)*(c_.)), x_Symbol] :> Simp[ -b^(-1) Subst[Int[(u /. x -> -a/b + Cos[x]/b)*f^(c*x^n)*Sin[x], x], x, Ar cCos[a + b*x]], x] /; FreeQ[{a, b, c, f}, x] && IGtQ[n, 0]
\[\int \frac {{\mathrm e}^{\arccos \left (a x \right )}}{x^{2}}d x\]
Input:
int(exp(arccos(a*x))/x^2,x)
Output:
int(exp(arccos(a*x))/x^2,x)
\[ \int \frac {e^{\arccos (a x)}}{x^2} \, dx=\int { \frac {e^{\left (\arccos \left (a x\right )\right )}}{x^{2}} \,d x } \] Input:
integrate(exp(arccos(a*x))/x^2,x, algorithm="fricas")
Output:
integral(e^(arccos(a*x))/x^2, x)
\[ \int \frac {e^{\arccos (a x)}}{x^2} \, dx=\int \frac {e^{\operatorname {acos}{\left (a x \right )}}}{x^{2}}\, dx \] Input:
integrate(exp(acos(a*x))/x**2,x)
Output:
Integral(exp(acos(a*x))/x**2, x)
\[ \int \frac {e^{\arccos (a x)}}{x^2} \, dx=\int { \frac {e^{\left (\arccos \left (a x\right )\right )}}{x^{2}} \,d x } \] Input:
integrate(exp(arccos(a*x))/x^2,x, algorithm="maxima")
Output:
integrate(e^(arccos(a*x))/x^2, x)
\[ \int \frac {e^{\arccos (a x)}}{x^2} \, dx=\int { \frac {e^{\left (\arccos \left (a x\right )\right )}}{x^{2}} \,d x } \] Input:
integrate(exp(arccos(a*x))/x^2,x, algorithm="giac")
Output:
integrate(e^(arccos(a*x))/x^2, x)
Timed out. \[ \int \frac {e^{\arccos (a x)}}{x^2} \, dx=\int \frac {{\mathrm {e}}^{\mathrm {acos}\left (a\,x\right )}}{x^2} \,d x \] Input:
int(exp(acos(a*x))/x^2,x)
Output:
int(exp(acos(a*x))/x^2, x)
\[ \int \frac {e^{\arccos (a x)}}{x^2} \, dx=\int \frac {e^{\mathit {acos} \left (a x \right )}}{x^{2}}d x \] Input:
int(exp(acos(a*x))/x^2,x)
Output:
int(e**acos(a*x)/x**2,x)