Optimal. Leaf size=182 \[ \frac{2 \sqrt{2 \pi } \sqrt{a^2 c x^2+c} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{3 a \sqrt{a^2 x^2+1}}+\frac{2 \sqrt{2 \pi } \sqrt{a^2 c x^2+c} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{3 a \sqrt{a^2 x^2+1}}-\frac{8 x \sqrt{a^2 c x^2+c}}{3 \sqrt{\sinh ^{-1}(a x)}}-\frac{2 \sqrt{a^2 x^2+1} \sqrt{a^2 c x^2+c}}{3 a \sinh ^{-1}(a x)^{3/2}} \]
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Rubi [A] time = 0.114611, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {5696, 5665, 3307, 2180, 2204, 2205} \[ \frac{2 \sqrt{2 \pi } \sqrt{a^2 c x^2+c} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{3 a \sqrt{a^2 x^2+1}}+\frac{2 \sqrt{2 \pi } \sqrt{a^2 c x^2+c} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{3 a \sqrt{a^2 x^2+1}}-\frac{8 x \sqrt{a^2 c x^2+c}}{3 \sqrt{\sinh ^{-1}(a x)}}-\frac{2 \sqrt{a^2 x^2+1} \sqrt{a^2 c x^2+c}}{3 a \sinh ^{-1}(a x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 5696
Rule 5665
Rule 3307
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \frac{\sqrt{c+a^2 c x^2}}{\sinh ^{-1}(a x)^{5/2}} \, dx &=-\frac{2 \sqrt{1+a^2 x^2} \sqrt{c+a^2 c x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}+\frac{\left (4 a \sqrt{c+a^2 c x^2}\right ) \int \frac{x}{\sinh ^{-1}(a x)^{3/2}} \, dx}{3 \sqrt{1+a^2 x^2}}\\ &=-\frac{2 \sqrt{1+a^2 x^2} \sqrt{c+a^2 c x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac{8 x \sqrt{c+a^2 c x^2}}{3 \sqrt{\sinh ^{-1}(a x)}}+\frac{\left (8 \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a \sqrt{1+a^2 x^2}}\\ &=-\frac{2 \sqrt{1+a^2 x^2} \sqrt{c+a^2 c x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac{8 x \sqrt{c+a^2 c x^2}}{3 \sqrt{\sinh ^{-1}(a x)}}+\frac{\left (4 \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a \sqrt{1+a^2 x^2}}+\frac{\left (4 \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a \sqrt{1+a^2 x^2}}\\ &=-\frac{2 \sqrt{1+a^2 x^2} \sqrt{c+a^2 c x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac{8 x \sqrt{c+a^2 c x^2}}{3 \sqrt{\sinh ^{-1}(a x)}}+\frac{\left (8 \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{3 a \sqrt{1+a^2 x^2}}+\frac{\left (8 \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{3 a \sqrt{1+a^2 x^2}}\\ &=-\frac{2 \sqrt{1+a^2 x^2} \sqrt{c+a^2 c x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac{8 x \sqrt{c+a^2 c x^2}}{3 \sqrt{\sinh ^{-1}(a x)}}+\frac{2 \sqrt{2 \pi } \sqrt{c+a^2 c x^2} \text{erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{3 a \sqrt{1+a^2 x^2}}+\frac{2 \sqrt{2 \pi } \sqrt{c+a^2 c x^2} \text{erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{3 a \sqrt{1+a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.145474, size = 122, normalized size = 0.67 \[ -\frac{2 \sqrt{a^2 c x^2+c} \left (\sqrt{2} \left (-\sinh ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-2 \sinh ^{-1}(a x)\right )+\sqrt{2} \sinh ^{-1}(a x)^{3/2} \text{Gamma}\left (\frac{1}{2},2 \sinh ^{-1}(a x)\right )+a^2 x^2+4 a x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)+1\right )}{3 a \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.21, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{{a}^{2}c{x}^{2}+c} \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{\sqrt{a^{2} c x^{2} + c}}{\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(-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{\sqrt{a^{2} c x^{2} + c}}{\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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