Optimal. Leaf size=285 \[ -\frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{30 a^5}+\frac{9 \sqrt{3 \pi } \text{Erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{20 a^5}-\frac{5 \sqrt{5 \pi } \text{Erf}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{12 a^5}+\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{30 a^5}-\frac{9 \sqrt{3 \pi } \text{Erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{20 a^5}+\frac{5 \sqrt{5 \pi } \text{Erfi}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{12 a^5}-\frac{40 x^4 \sqrt{a^2 x^2+1}}{3 a \sqrt{\sinh ^{-1}(a x)}}-\frac{2 x^4 \sqrt{a^2 x^2+1}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac{16 x^3}{15 a^2 \sinh ^{-1}(a x)^{3/2}}-\frac{32 x^2 \sqrt{a^2 x^2+1}}{5 a^3 \sqrt{\sinh ^{-1}(a x)}}-\frac{4 x^5}{3 \sinh ^{-1}(a x)^{3/2}} \]
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Rubi [A] time = 0.543711, antiderivative size = 285, normalized size of antiderivative = 1., number of steps used = 32, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {5667, 5774, 5665, 3308, 2180, 2204, 2205} \[ -\frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{30 a^5}+\frac{9 \sqrt{3 \pi } \text{Erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{20 a^5}-\frac{5 \sqrt{5 \pi } \text{Erf}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{12 a^5}+\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{30 a^5}-\frac{9 \sqrt{3 \pi } \text{Erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{20 a^5}+\frac{5 \sqrt{5 \pi } \text{Erfi}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{12 a^5}-\frac{40 x^4 \sqrt{a^2 x^2+1}}{3 a \sqrt{\sinh ^{-1}(a x)}}-\frac{2 x^4 \sqrt{a^2 x^2+1}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac{16 x^3}{15 a^2 \sinh ^{-1}(a x)^{3/2}}-\frac{32 x^2 \sqrt{a^2 x^2+1}}{5 a^3 \sqrt{\sinh ^{-1}(a x)}}-\frac{4 x^5}{3 \sinh ^{-1}(a x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 5667
Rule 5774
Rule 5665
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \frac{x^4}{\sinh ^{-1}(a x)^{7/2}} \, dx &=-\frac{2 x^4 \sqrt{1+a^2 x^2}}{5 a \sinh ^{-1}(a x)^{5/2}}+\frac{8 \int \frac{x^3}{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{5/2}} \, dx}{5 a}+(2 a) \int \frac{x^5}{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{5/2}} \, dx\\ &=-\frac{2 x^4 \sqrt{1+a^2 x^2}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac{16 x^3}{15 a^2 \sinh ^{-1}(a x)^{3/2}}-\frac{4 x^5}{3 \sinh ^{-1}(a x)^{3/2}}+\frac{20}{3} \int \frac{x^4}{\sinh ^{-1}(a x)^{3/2}} \, dx+\frac{16 \int \frac{x^2}{\sinh ^{-1}(a x)^{3/2}} \, dx}{5 a^2}\\ &=-\frac{2 x^4 \sqrt{1+a^2 x^2}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac{16 x^3}{15 a^2 \sinh ^{-1}(a x)^{3/2}}-\frac{4 x^5}{3 \sinh ^{-1}(a x)^{3/2}}-\frac{32 x^2 \sqrt{1+a^2 x^2}}{5 a^3 \sqrt{\sinh ^{-1}(a x)}}-\frac{40 x^4 \sqrt{1+a^2 x^2}}{3 a \sqrt{\sinh ^{-1}(a x)}}+\frac{32 \operatorname{Subst}\left (\int \left (-\frac{\sinh (x)}{4 \sqrt{x}}+\frac{3 \sinh (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{5 a^5}+\frac{40 \operatorname{Subst}\left (\int \left (\frac{\sinh (x)}{8 \sqrt{x}}-\frac{9 \sinh (3 x)}{16 \sqrt{x}}+\frac{5 \sinh (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}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac{16 x^3}{15 a^2 \sinh ^{-1}(a x)^{3/2}}-\frac{4 x^5}{3 \sinh ^{-1}(a x)^{3/2}}-\frac{32 x^2 \sqrt{1+a^2 x^2}}{5 a^3 \sqrt{\sinh ^{-1}(a x)}}-\frac{40 x^4 \sqrt{1+a^2 x^2}}{3 a \sqrt{\sinh ^{-1}(a x)}}-\frac{8 \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{5 a^5}+\frac{5 \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a^5}+\frac{25 \operatorname{Subst}\left (\int \frac{\sinh (5 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{6 a^5}+\frac{24 \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{5 a^5}-\frac{15 \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^5}\\ &=-\frac{2 x^4 \sqrt{1+a^2 x^2}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac{16 x^3}{15 a^2 \sinh ^{-1}(a x)^{3/2}}-\frac{4 x^5}{3 \sinh ^{-1}(a x)^{3/2}}-\frac{32 x^2 \sqrt{1+a^2 x^2}}{5 a^3 \sqrt{\sinh ^{-1}(a x)}}-\frac{40 x^4 \sqrt{1+a^2 x^2}}{3 a \sqrt{\sinh ^{-1}(a x)}}+\frac{4 \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{5 a^5}-\frac{4 \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{5 a^5}-\frac{5 \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{6 a^5}+\frac{5 \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{6 a^5}-\frac{25 \operatorname{Subst}\left (\int \frac{e^{-5 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{12 a^5}+\frac{25 \operatorname{Subst}\left (\int \frac{e^{5 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{12 a^5}-\frac{12 \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{5 a^5}+\frac{12 \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{5 a^5}+\frac{15 \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^5}-\frac{15 \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^5}\\ &=-\frac{2 x^4 \sqrt{1+a^2 x^2}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac{16 x^3}{15 a^2 \sinh ^{-1}(a x)^{3/2}}-\frac{4 x^5}{3 \sinh ^{-1}(a x)^{3/2}}-\frac{32 x^2 \sqrt{1+a^2 x^2}}{5 a^3 \sqrt{\sinh ^{-1}(a x)}}-\frac{40 x^4 \sqrt{1+a^2 x^2}}{3 a \sqrt{\sinh ^{-1}(a x)}}+\frac{8 \operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{5 a^5}-\frac{8 \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{5 a^5}-\frac{5 \operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{3 a^5}+\frac{5 \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{3 a^5}-\frac{25 \operatorname{Subst}\left (\int e^{-5 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{6 a^5}+\frac{25 \operatorname{Subst}\left (\int e^{5 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{6 a^5}-\frac{24 \operatorname{Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{5 a^5}+\frac{24 \operatorname{Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{5 a^5}+\frac{15 \operatorname{Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{2 a^5}-\frac{15 \operatorname{Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{2 a^5}\\ &=-\frac{2 x^4 \sqrt{1+a^2 x^2}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac{16 x^3}{15 a^2 \sinh ^{-1}(a x)^{3/2}}-\frac{4 x^5}{3 \sinh ^{-1}(a x)^{3/2}}-\frac{32 x^2 \sqrt{1+a^2 x^2}}{5 a^3 \sqrt{\sinh ^{-1}(a x)}}-\frac{40 x^4 \sqrt{1+a^2 x^2}}{3 a \sqrt{\sinh ^{-1}(a x)}}-\frac{\sqrt{\pi } \text{erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{30 a^5}+\frac{9 \sqrt{3 \pi } \text{erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{20 a^5}-\frac{5 \sqrt{5 \pi } \text{erf}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{12 a^5}+\frac{\sqrt{\pi } \text{erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{30 a^5}-\frac{9 \sqrt{3 \pi } \text{erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{20 a^5}+\frac{5 \sqrt{5 \pi } \text{erfi}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{12 a^5}\\ \end{align*}
Mathematica [A] time = 0.673252, size = 334, normalized size = 1.17 \[ \frac{100 \sqrt{5} \left (-\sinh ^{-1}(a x)\right )^{5/2} \text{Gamma}\left (\frac{1}{2},-5 \sinh ^{-1}(a x)\right )-108 \sqrt{3} \left (-\sinh ^{-1}(a x)\right )^{5/2} \text{Gamma}\left (\frac{1}{2},-3 \sinh ^{-1}(a x)\right )+8 \left (-\sinh ^{-1}(a x)\right )^{5/2} \text{Gamma}\left (\frac{1}{2},-\sinh ^{-1}(a x)\right )+e^{-\sinh ^{-1}(a x)} \left (8 e^{\sinh ^{-1}(a x)} \sinh ^{-1}(a x)^{5/2} \text{Gamma}\left (\frac{1}{2},\sinh ^{-1}(a x)\right )-8 \sinh ^{-1}(a x)^2+4 \sinh ^{-1}(a x)-6\right )+9 e^{-3 \sinh ^{-1}(a x)} \left (-12 \sqrt{3} e^{3 \sinh ^{-1}(a x)} \sinh ^{-1}(a x)^{5/2} \text{Gamma}\left (\frac{1}{2},3 \sinh ^{-1}(a x)\right )+12 \sinh ^{-1}(a x)^2-2 \sinh ^{-1}(a x)+1\right )+e^{-5 \sinh ^{-1}(a x)} \left (100 \sqrt{5} e^{5 \sinh ^{-1}(a x)} \sinh ^{-1}(a x)^{5/2} \text{Gamma}\left (\frac{1}{2},5 \sinh ^{-1}(a x)\right )-100 \sinh ^{-1}(a x)^2+10 \sinh ^{-1}(a x)-3\right )-2 e^{\sinh ^{-1}(a x)} \left (4 \sinh ^{-1}(a x)^2+2 \sinh ^{-1}(a x)+3\right )+9 e^{3 \sinh ^{-1}(a x)} \left (12 \sinh ^{-1}(a x)^2+2 \sinh ^{-1}(a x)+1\right )-e^{5 \sinh ^{-1}(a x)} \left (100 \sinh ^{-1}(a x)^2+10 \sinh ^{-1}(a x)+3\right )}{240 a^5 \sinh ^{-1}(a x)^{5/2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.18, size = 0, normalized size = 0. \begin{align*} \int{{x}^{4} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{-{\frac{7}{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{7}{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(-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.
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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{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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