Optimal. Leaf size=112 \[ -\frac{8 \sqrt{a^2 x^2+1}}{15 a \sqrt{\sinh ^{-1}(a x)}}-\frac{2 \sqrt{a^2 x^2+1}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac{4 \sqrt{\pi } \text{Erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{15 a}+\frac{4 \sqrt{\pi } \text{Erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{15 a}-\frac{4 x}{15 \sinh ^{-1}(a x)^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.189101, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.875, Rules used = {5655, 5774, 5779, 3308, 2180, 2204, 2205} \[ -\frac{8 \sqrt{a^2 x^2+1}}{15 a \sqrt{\sinh ^{-1}(a x)}}-\frac{2 \sqrt{a^2 x^2+1}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac{4 \sqrt{\pi } \text{Erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{15 a}+\frac{4 \sqrt{\pi } \text{Erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{15 a}-\frac{4 x}{15 \sinh ^{-1}(a x)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5655
Rule 5774
Rule 5779
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \frac{1}{\sinh ^{-1}(a x)^{7/2}} \, dx &=-\frac{2 \sqrt{1+a^2 x^2}}{5 a \sinh ^{-1}(a x)^{5/2}}+\frac{1}{5} (2 a) \int \frac{x}{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{5/2}} \, dx\\ &=-\frac{2 \sqrt{1+a^2 x^2}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac{4 x}{15 \sinh ^{-1}(a x)^{3/2}}+\frac{4}{15} \int \frac{1}{\sinh ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac{2 \sqrt{1+a^2 x^2}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac{4 x}{15 \sinh ^{-1}(a x)^{3/2}}-\frac{8 \sqrt{1+a^2 x^2}}{15 a \sqrt{\sinh ^{-1}(a x)}}+\frac{1}{15} (8 a) \int \frac{x}{\sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}} \, dx\\ &=-\frac{2 \sqrt{1+a^2 x^2}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac{4 x}{15 \sinh ^{-1}(a x)^{3/2}}-\frac{8 \sqrt{1+a^2 x^2}}{15 a \sqrt{\sinh ^{-1}(a x)}}+\frac{8 \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{15 a}\\ &=-\frac{2 \sqrt{1+a^2 x^2}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac{4 x}{15 \sinh ^{-1}(a x)^{3/2}}-\frac{8 \sqrt{1+a^2 x^2}}{15 a \sqrt{\sinh ^{-1}(a x)}}-\frac{4 \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{15 a}+\frac{4 \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{15 a}\\ &=-\frac{2 \sqrt{1+a^2 x^2}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac{4 x}{15 \sinh ^{-1}(a x)^{3/2}}-\frac{8 \sqrt{1+a^2 x^2}}{15 a \sqrt{\sinh ^{-1}(a x)}}-\frac{8 \operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{15 a}+\frac{8 \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{15 a}\\ &=-\frac{2 \sqrt{1+a^2 x^2}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac{4 x}{15 \sinh ^{-1}(a x)^{3/2}}-\frac{8 \sqrt{1+a^2 x^2}}{15 a \sqrt{\sinh ^{-1}(a x)}}-\frac{4 \sqrt{\pi } \text{erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{15 a}+\frac{4 \sqrt{\pi } \text{erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{15 a}\\ \end{align*}
Mathematica [A] time = 0.18052, size = 111, normalized size = 0.99 \[ \frac{8 \left (-\sinh ^{-1}(a x)\right )^{5/2} \text{Gamma}\left (\frac{1}{2},-\sinh ^{-1}(a x)\right )+e^{-\sinh ^{-1}(a x)} \left (8 e^{\sinh ^{-1}(a x)} \sinh ^{-1}(a x)^{5/2} \text{Gamma}\left (\frac{1}{2},\sinh ^{-1}(a x)\right )-8 \sinh ^{-1}(a x)^2+4 \sinh ^{-1}(a x)-6\right )-2 e^{\sinh ^{-1}(a x)} \left (4 \sinh ^{-1}(a x)^2+2 \sinh ^{-1}(a x)+3\right )}{30 a \sinh ^{-1}(a x)^{5/2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.082, size = 105, normalized size = 0.9 \begin{align*} -{\frac{2}{15\,\sqrt{\pi }a \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}} \left ( 2\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}\pi \,{\it Erf} \left ( \sqrt{{\it Arcsinh} \left ( ax \right ) } \right ) -2\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}\pi \,{\it erfi} \left ( \sqrt{{\it Arcsinh} \left ( ax \right ) } \right ) +4\,\sqrt{{a}^{2}{x}^{2}+1} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{5/2}\sqrt{\pi }+2\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3/2}\sqrt{\pi }xa+3\,\sqrt{{\it Arcsinh} \left ( ax \right ) }\sqrt{\pi }\sqrt{{a}^{2}{x}^{2}+1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\operatorname{arsinh}\left (a x\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\operatorname{arsinh}\left (a x\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]