Optimal. Leaf size=223 \[ \frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{12 a^5}-\frac{3 \sqrt{3 \pi } \text{Erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{8 a^5}+\frac{5 \sqrt{5 \pi } \text{Erf}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{24 a^5}+\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{12 a^5}-\frac{3 \sqrt{3 \pi } \text{Erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{8 a^5}+\frac{5 \sqrt{5 \pi } \text{Erfi}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{24 a^5}-\frac{2 x^4 \sqrt{a^2 x^2+1}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac{16 x^3}{3 a^2 \sqrt{\sinh ^{-1}(a x)}}-\frac{20 x^5}{3 \sqrt{\sinh ^{-1}(a x)}} \]
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Rubi [A] time = 0.583792, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 34, number of rules used = 8, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {5667, 5774, 5669, 5448, 3307, 2180, 2204, 2205} \[ \frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{12 a^5}-\frac{3 \sqrt{3 \pi } \text{Erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{8 a^5}+\frac{5 \sqrt{5 \pi } \text{Erf}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{24 a^5}+\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{12 a^5}-\frac{3 \sqrt{3 \pi } \text{Erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{8 a^5}+\frac{5 \sqrt{5 \pi } \text{Erfi}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{24 a^5}-\frac{2 x^4 \sqrt{a^2 x^2+1}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac{16 x^3}{3 a^2 \sqrt{\sinh ^{-1}(a x)}}-\frac{20 x^5}{3 \sqrt{\sinh ^{-1}(a x)}} \]
Antiderivative was successfully verified.
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Rule 5667
Rule 5774
Rule 5669
Rule 5448
Rule 3307
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \frac{x^4}{\sinh ^{-1}(a x)^{5/2}} \, dx &=-\frac{2 x^4 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}+\frac{8 \int \frac{x^3}{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}} \, dx}{3 a}+\frac{1}{3} (10 a) \int \frac{x^5}{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac{2 x^4 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac{16 x^3}{3 a^2 \sqrt{\sinh ^{-1}(a x)}}-\frac{20 x^5}{3 \sqrt{\sinh ^{-1}(a x)}}+\frac{100}{3} \int \frac{x^4}{\sqrt{\sinh ^{-1}(a x)}} \, dx+\frac{16 \int \frac{x^2}{\sqrt{\sinh ^{-1}(a x)}} \, dx}{a^2}\\ &=-\frac{2 x^4 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac{16 x^3}{3 a^2 \sqrt{\sinh ^{-1}(a x)}}-\frac{20 x^5}{3 \sqrt{\sinh ^{-1}(a x)}}+\frac{16 \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh ^2(x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{a^5}+\frac{100 \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh ^4(x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a^5}\\ &=-\frac{2 x^4 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac{16 x^3}{3 a^2 \sqrt{\sinh ^{-1}(a x)}}-\frac{20 x^5}{3 \sqrt{\sinh ^{-1}(a x)}}+\frac{16 \operatorname{Subst}\left (\int \left (-\frac{\cosh (x)}{4 \sqrt{x}}+\frac{\cosh (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a^5}+\frac{100 \operatorname{Subst}\left (\int \left (\frac{\cosh (x)}{8 \sqrt{x}}-\frac{3 \cosh (3 x)}{16 \sqrt{x}}+\frac{\cosh (5 x)}{16 \sqrt{x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{3 a^5}\\ &=-\frac{2 x^4 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac{16 x^3}{3 a^2 \sqrt{\sinh ^{-1}(a x)}}-\frac{20 x^5}{3 \sqrt{\sinh ^{-1}(a x)}}+\frac{25 \operatorname{Subst}\left (\int \frac{\cosh (5 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{12 a^5}-\frac{4 \operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{a^5}+\frac{4 \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{a^5}+\frac{25 \operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{6 a^5}-\frac{25 \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^5}\\ &=-\frac{2 x^4 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac{16 x^3}{3 a^2 \sqrt{\sinh ^{-1}(a x)}}-\frac{20 x^5}{3 \sqrt{\sinh ^{-1}(a x)}}+\frac{25 \operatorname{Subst}\left (\int \frac{e^{-5 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{24 a^5}+\frac{25 \operatorname{Subst}\left (\int \frac{e^{5 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{24 a^5}+\frac{2 \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{a^5}-\frac{2 \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{a^5}-\frac{2 \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{a^5}+\frac{2 \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{a^5}+\frac{25 \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{12 a^5}+\frac{25 \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{12 a^5}-\frac{25 \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^5}-\frac{25 \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^5}\\ &=-\frac{2 x^4 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac{16 x^3}{3 a^2 \sqrt{\sinh ^{-1}(a x)}}-\frac{20 x^5}{3 \sqrt{\sinh ^{-1}(a x)}}+\frac{25 \operatorname{Subst}\left (\int e^{-5 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{12 a^5}+\frac{25 \operatorname{Subst}\left (\int e^{5 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{12 a^5}+\frac{4 \operatorname{Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{a^5}-\frac{4 \operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{a^5}-\frac{4 \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{a^5}+\frac{4 \operatorname{Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{a^5}+\frac{25 \operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{6 a^5}+\frac{25 \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{6 a^5}-\frac{25 \operatorname{Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{4 a^5}-\frac{25 \operatorname{Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{4 a^5}\\ &=-\frac{2 x^4 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac{16 x^3}{3 a^2 \sqrt{\sinh ^{-1}(a x)}}-\frac{20 x^5}{3 \sqrt{\sinh ^{-1}(a x)}}+\frac{\sqrt{\pi } \text{erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{12 a^5}-\frac{3 \sqrt{3 \pi } \text{erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{8 a^5}+\frac{5 \sqrt{5 \pi } \text{erf}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{24 a^5}+\frac{\sqrt{\pi } \text{erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{12 a^5}-\frac{3 \sqrt{3 \pi } \text{erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{8 a^5}+\frac{5 \sqrt{5 \pi } \text{erfi}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{24 a^5}\\ \end{align*}
Mathematica [A] time = 0.333043, size = 343, normalized size = 1.54 \[ \frac{-\frac{10 \sqrt{5} \left (-\sinh ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-5 \sinh ^{-1}(a x)\right )+e^{5 \sinh ^{-1}(a x)} \left (10 \sinh ^{-1}(a x)+1\right )}{48 \sinh ^{-1}(a x)^{3/2}}+\frac{6 \sqrt{3} \left (-\sinh ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-3 \sinh ^{-1}(a x)\right )+e^{3 \sinh ^{-1}(a x)} \left (6 \sinh ^{-1}(a x)+1\right )}{16 \sinh ^{-1}(a x)^{3/2}}-\frac{2 \left (-\sinh ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-\sinh ^{-1}(a x)\right )+e^{\sinh ^{-1}(a x)} \left (2 \sinh ^{-1}(a x)+1\right )}{24 \sinh ^{-1}(a x)^{3/2}}-\frac{e^{-\sinh ^{-1}(a x)} \left (2 e^{\sinh ^{-1}(a x)} \sinh ^{-1}(a x)^{3/2} \text{Gamma}\left (\frac{1}{2},\sinh ^{-1}(a x)\right )-2 \sinh ^{-1}(a x)+1\right )}{24 \sinh ^{-1}(a x)^{3/2}}+\frac{e^{-3 \sinh ^{-1}(a x)} \left (6 \sqrt{3} e^{3 \sinh ^{-1}(a x)} \sinh ^{-1}(a x)^{3/2} \text{Gamma}\left (\frac{1}{2},3 \sinh ^{-1}(a x)\right )-6 \sinh ^{-1}(a x)+1\right )}{16 \sinh ^{-1}(a x)^{3/2}}-\frac{e^{-5 \sinh ^{-1}(a x)} \left (10 \sqrt{5} e^{5 \sinh ^{-1}(a x)} \sinh ^{-1}(a x)^{3/2} \text{Gamma}\left (\frac{1}{2},5 \sinh ^{-1}(a x)\right )-10 \sinh ^{-1}(a x)+1\right )}{48 \sinh ^{-1}(a x)^{3/2}}}{a^5} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.184, size = 0, normalized size = 0. \begin{align*} \int{{x}^{4} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\operatorname{arsinh}\left (a x\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\operatorname{asinh}^{\frac{5}{2}}{\left (a x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\operatorname{arsinh}\left (a x\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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