3.456 \(\int \frac{e^{\sin ^{-1}(a+b x)}}{x^2} \, dx\)

Optimal. Leaf size=21 \[ b^2 \text{CannotIntegrate}\left (\frac{e^{\sin ^{-1}(a+b x)}}{b^2 x^2},x\right ) \]

[Out]

b^2*CannotIntegrate[E^ArcSin[a + b*x]/(b^2*x^2), x]

________________________________________________________________________________________

Rubi [A]  time = 0.263681, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{e^{\sin ^{-1}(a+b x)}}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[E^ArcSin[a + b*x]/x^2,x]

[Out]

b*Defer[Subst][Defer[Int][(E^x*Cos[x])/(a - Sin[x])^2, x], x, ArcSin[a + b*x]]

Rubi steps

\begin{align*} \int \frac{e^{\sin ^{-1}(a+b x)}}{x^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{e^x \cos (x)}{\left (-\frac{a}{b}+\frac{\sin (x)}{b}\right )^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{b^2 e^x \cos (x)}{(a-\sin (x))^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=b \operatorname{Subst}\left (\int \frac{e^x \cos (x)}{(a-\sin (x))^2} \, dx,x,\sin ^{-1}(a+b x)\right )\\ \end{align*}

Mathematica [A]  time = 0.244126, size = 0, normalized size = 0. \[ \int \frac{e^{\sin ^{-1}(a+b x)}}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[E^ArcSin[a + b*x]/x^2,x]

[Out]

Integrate[E^ArcSin[a + b*x]/x^2, x]

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{\arcsin \left ( bx+a \right ) }}}{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(arcsin(b*x+a))/x^2,x)

[Out]

int(exp(arcsin(b*x+a))/x^2,x)

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (\arcsin \left (b x + a\right )\right )}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arcsin(b*x+a))/x^2,x, algorithm="maxima")

[Out]

integrate(e^(arcsin(b*x + a))/x^2, x)

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{e^{\left (\arcsin \left (b x + a\right )\right )}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arcsin(b*x+a))/x^2,x, algorithm="fricas")

[Out]

integral(e^(arcsin(b*x + a))/x^2, x)

________________________________________________________________________________________

Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\operatorname{asin}{\left (a + b x \right )}}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(asin(b*x+a))/x**2,x)

[Out]

Integral(exp(asin(a + b*x))/x**2, x)

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (\arcsin \left (b x + a\right )\right )}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arcsin(b*x+a))/x^2,x, algorithm="giac")

[Out]

integrate(e^(arcsin(b*x + a))/x^2, x)