Optimal. Leaf size=24 \[ \text{Unintegrable}\left (\frac{1}{\left ((a+b x)^2+1\right )^{3/2} \sinh ^{-1}(a+b x)},x\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0821379, 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 = {} \[ \int \frac{1}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \sinh ^{-1}(a+b x)} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin{align*} \int \frac{1}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \sinh ^{-1}(a+b x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right )^{3/2} \sinh ^{-1}(x)} \, dx,x,a+b x\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.551142, size = 0, normalized size = 0. \[ \int \frac{1}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \sinh ^{-1}(a+b x)} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.113, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{\it Arcsinh} \left ( bx+a \right ) } \left ({b}^{2}{x}^{2}+2\,xab+{a}^{2}+1 \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac{3}{2}} \operatorname{arsinh}\left (b x + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{{\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 2 \,{\left (3 \, a^{2} + 1\right )} b^{2} x^{2} + a^{4} + 4 \,{\left (a^{3} + a\right )} b x + 2 \, a^{2} + 1\right )} \operatorname{arsinh}\left (b x + a\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a^{2} + 2 a b x + b^{2} x^{2} + 1\right )^{\frac{3}{2}} \operatorname{asinh}{\left (a + b x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac{3}{2}} \operatorname{arsinh}\left (b x + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]