3.24 \(\int \frac{1}{(d+e x)^2 (a+b \sinh ^{-1}(c x))} \, dx\)

Optimal. Leaf size=20 \[ \text{Unintegrable}\left (\frac{1}{(d+e x)^2 \left (a+b \sinh ^{-1}(c x)\right )},x\right ) \]

[Out]

Unintegrable[1/((d + e*x)^2*(a + b*ArcSinh[c*x])), x]

________________________________________________________________________________________

Rubi [A]  time = 0.0392964, 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 = {} \[ \int \frac{1}{(d+e x)^2 \left (a+b \sinh ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/((d + e*x)^2*(a + b*ArcSinh[c*x])),x]

[Out]

Defer[Int][1/((d + e*x)^2*(a + b*ArcSinh[c*x])), x]

Rubi steps

\begin{align*} \int \frac{1}{(d+e x)^2 \left (a+b \sinh ^{-1}(c x)\right )} \, dx &=\int \frac{1}{(d+e x)^2 \left (a+b \sinh ^{-1}(c x)\right )} \, dx\\ \end{align*}

Mathematica [A]  time = 0.376284, size = 0, normalized size = 0. \[ \int \frac{1}{(d+e x)^2 \left (a+b \sinh ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/((d + e*x)^2*(a + b*ArcSinh[c*x])),x]

[Out]

Integrate[1/((d + e*x)^2*(a + b*ArcSinh[c*x])), x]

________________________________________________________________________________________

Maple [A]  time = 0.23, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( ex+d \right ) ^{2} \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^2/(a+b*arcsinh(c*x)),x)

[Out]

int(1/(e*x+d)^2/(a+b*arcsinh(c*x)),x)

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (e x + d\right )}^{2}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(a+b*arcsinh(c*x)),x, algorithm="maxima")

[Out]

integrate(1/((e*x + d)^2*(b*arcsinh(c*x) + a)), x)

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{a e^{2} x^{2} + 2 \, a d e x + a d^{2} +{\left (b e^{2} x^{2} + 2 \, b d e x + b d^{2}\right )} \operatorname{arsinh}\left (c x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(a+b*arcsinh(c*x)),x, algorithm="fricas")

[Out]

integral(1/(a*e^2*x^2 + 2*a*d*e*x + a*d^2 + (b*e^2*x^2 + 2*b*d*e*x + b*d^2)*arcsinh(c*x)), x)

________________________________________________________________________________________

Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b \operatorname{asinh}{\left (c x \right )}\right ) \left (d + e x\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**2/(a+b*asinh(c*x)),x)

[Out]

Integral(1/((a + b*asinh(c*x))*(d + e*x)**2), x)

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (e x + d\right )}^{2}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(a+b*arcsinh(c*x)),x, algorithm="giac")

[Out]

integrate(1/((e*x + d)^2*(b*arcsinh(c*x) + a)), x)