Optimal. Leaf size=277 \[ -\frac{3 \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{64 b^2 c^4}-\frac{9 \cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^4}+\frac{5 \cosh \left (\frac{5 a}{b}\right ) \text{Chi}\left (\frac{5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^4}+\frac{7 \cosh \left (\frac{7 a}{b}\right ) \text{Chi}\left (\frac{7 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^4}+\frac{3 \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{64 b^2 c^4}+\frac{9 \sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^4}-\frac{5 \sinh \left (\frac{5 a}{b}\right ) \text{Shi}\left (\frac{5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^4}-\frac{7 \sinh \left (\frac{7 a}{b}\right ) \text{Shi}\left (\frac{7 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^4}-\frac{x^3 \left (c^2 x^2+1\right )^2}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]
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Rubi [A] time = 1.00638, antiderivative size = 273, normalized size of antiderivative = 0.99, number of steps used = 28, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {5777, 5779, 5448, 3303, 3298, 3301} \[ -\frac{3 \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )}{64 b^2 c^4}-\frac{9 \cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 a}{b}+3 \sinh ^{-1}(c x)\right )}{64 b^2 c^4}+\frac{5 \cosh \left (\frac{5 a}{b}\right ) \text{Chi}\left (\frac{5 a}{b}+5 \sinh ^{-1}(c x)\right )}{64 b^2 c^4}+\frac{7 \cosh \left (\frac{7 a}{b}\right ) \text{Chi}\left (\frac{7 a}{b}+7 \sinh ^{-1}(c x)\right )}{64 b^2 c^4}+\frac{3 \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )}{64 b^2 c^4}+\frac{9 \sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \sinh ^{-1}(c x)\right )}{64 b^2 c^4}-\frac{5 \sinh \left (\frac{5 a}{b}\right ) \text{Shi}\left (\frac{5 a}{b}+5 \sinh ^{-1}(c x)\right )}{64 b^2 c^4}-\frac{7 \sinh \left (\frac{7 a}{b}\right ) \text{Shi}\left (\frac{7 a}{b}+7 \sinh ^{-1}(c x)\right )}{64 b^2 c^4}-\frac{x^3 \left (c^2 x^2+1\right )^2}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Rule 5777
Rule 5779
Rule 5448
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{x^3 \left (1+c^2 x^2\right )^{3/2}}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx &=-\frac{x^3 \left (1+c^2 x^2\right )^2}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac{3 \int \frac{x^2 \left (1+c^2 x^2\right )}{a+b \sinh ^{-1}(c x)} \, dx}{b c}+\frac{(7 c) \int \frac{x^4 \left (1+c^2 x^2\right )}{a+b \sinh ^{-1}(c x)} \, dx}{b}\\ &=-\frac{x^3 \left (1+c^2 x^2\right )^2}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac{3 \operatorname{Subst}\left (\int \frac{\cosh ^3(x) \sinh ^2(x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^4}+\frac{7 \operatorname{Subst}\left (\int \frac{\cosh ^3(x) \sinh ^4(x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^4}\\ &=-\frac{x^3 \left (1+c^2 x^2\right )^2}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac{3 \operatorname{Subst}\left (\int \left (-\frac{\cosh (x)}{8 (a+b x)}+\frac{\cosh (3 x)}{16 (a+b x)}+\frac{\cosh (5 x)}{16 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b c^4}+\frac{7 \operatorname{Subst}\left (\int \left (\frac{3 \cosh (x)}{64 (a+b x)}-\frac{3 \cosh (3 x)}{64 (a+b x)}-\frac{\cosh (5 x)}{64 (a+b x)}+\frac{\cosh (7 x)}{64 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b c^4}\\ &=-\frac{x^3 \left (1+c^2 x^2\right )^2}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac{7 \operatorname{Subst}\left (\int \frac{\cosh (5 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^4}+\frac{7 \operatorname{Subst}\left (\int \frac{\cosh (7 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^4}+\frac{3 \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^4}+\frac{3 \operatorname{Subst}\left (\int \frac{\cosh (5 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^4}+\frac{21 \operatorname{Subst}\left (\int \frac{\cosh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^4}-\frac{21 \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^4}-\frac{3 \operatorname{Subst}\left (\int \frac{\cosh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^4}\\ &=-\frac{x^3 \left (1+c^2 x^2\right )^2}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac{\left (21 \cosh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^4}-\frac{\left (3 \cosh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^4}+\frac{\left (3 \cosh \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^4}-\frac{\left (21 \cosh \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^4}-\frac{\left (7 \cosh \left (\frac{5 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{5 a}{b}+5 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^4}+\frac{\left (3 \cosh \left (\frac{5 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{5 a}{b}+5 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^4}+\frac{\left (7 \cosh \left (\frac{7 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{7 a}{b}+7 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^4}-\frac{\left (21 \sinh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^4}+\frac{\left (3 \sinh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^4}-\frac{\left (3 \sinh \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^4}+\frac{\left (21 \sinh \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^4}+\frac{\left (7 \sinh \left (\frac{5 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{5 a}{b}+5 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^4}-\frac{\left (3 \sinh \left (\frac{5 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{5 a}{b}+5 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^4}-\frac{\left (7 \sinh \left (\frac{7 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{7 a}{b}+7 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^4}\\ &=-\frac{x^3 \left (1+c^2 x^2\right )^2}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac{3 \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )}{64 b^2 c^4}-\frac{9 \cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 a}{b}+3 \sinh ^{-1}(c x)\right )}{64 b^2 c^4}+\frac{5 \cosh \left (\frac{5 a}{b}\right ) \text{Chi}\left (\frac{5 a}{b}+5 \sinh ^{-1}(c x)\right )}{64 b^2 c^4}+\frac{7 \cosh \left (\frac{7 a}{b}\right ) \text{Chi}\left (\frac{7 a}{b}+7 \sinh ^{-1}(c x)\right )}{64 b^2 c^4}+\frac{3 \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )}{64 b^2 c^4}+\frac{9 \sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \sinh ^{-1}(c x)\right )}{64 b^2 c^4}-\frac{5 \sinh \left (\frac{5 a}{b}\right ) \text{Shi}\left (\frac{5 a}{b}+5 \sinh ^{-1}(c x)\right )}{64 b^2 c^4}-\frac{7 \sinh \left (\frac{7 a}{b}\right ) \text{Shi}\left (\frac{7 a}{b}+7 \sinh ^{-1}(c x)\right )}{64 b^2 c^4}\\ \end{align*}
Mathematica [A] time = 0.800154, size = 399, normalized size = 1.44 \[ \frac{-3 \cosh \left (\frac{a}{b}\right ) \left (a+b \sinh ^{-1}(c x)\right ) \text{Chi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )-9 \cosh \left (\frac{3 a}{b}\right ) \left (a+b \sinh ^{-1}(c x)\right ) \text{Chi}\left (3 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )+5 a \cosh \left (\frac{5 a}{b}\right ) \text{Chi}\left (5 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )+5 b \cosh \left (\frac{5 a}{b}\right ) \sinh ^{-1}(c x) \text{Chi}\left (5 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )+7 a \cosh \left (\frac{7 a}{b}\right ) \text{Chi}\left (7 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )+7 b \cosh \left (\frac{7 a}{b}\right ) \sinh ^{-1}(c x) \text{Chi}\left (7 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )+3 a \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )+3 b \sinh \left (\frac{a}{b}\right ) \sinh ^{-1}(c x) \text{Shi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )+9 a \sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (3 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )+9 b \sinh \left (\frac{3 a}{b}\right ) \sinh ^{-1}(c x) \text{Shi}\left (3 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )-5 a \sinh \left (\frac{5 a}{b}\right ) \text{Shi}\left (5 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )-5 b \sinh \left (\frac{5 a}{b}\right ) \sinh ^{-1}(c x) \text{Shi}\left (5 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )-7 a \sinh \left (\frac{7 a}{b}\right ) \text{Shi}\left (7 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )-7 b \sinh \left (\frac{7 a}{b}\right ) \sinh ^{-1}(c x) \text{Shi}\left (7 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )-64 b c^7 x^7-128 b c^5 x^5-64 b c^3 x^3}{64 b^2 c^4 \left (a+b \sinh ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.369, size = 958, normalized size = 3.5 \begin{align*} \text{result too large to display} \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*} -\frac{{\left (c^{4} x^{7} + 2 \, c^{2} x^{5} + x^{3}\right )}{\left (c^{2} x^{2} + 1\right )} +{\left (c^{5} x^{8} + 2 \, c^{3} x^{6} + c x^{4}\right )} \sqrt{c^{2} x^{2} + 1}}{a b c^{3} x^{2} + \sqrt{c^{2} x^{2} + 1} a b c^{2} x + a b c +{\left (b^{2} c^{3} x^{2} + \sqrt{c^{2} x^{2} + 1} b^{2} c^{2} x + b^{2} c\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )} + \int \frac{{\left (7 \, c^{5} x^{7} + 9 \, c^{3} x^{5} + 2 \, c x^{3}\right )}{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} +{\left (14 \, c^{6} x^{8} + 27 \, c^{4} x^{6} + 16 \, c^{2} x^{4} + 3 \, x^{2}\right )}{\left (c^{2} x^{2} + 1\right )} +{\left (7 \, c^{7} x^{9} + 18 \, c^{5} x^{7} + 15 \, c^{3} x^{5} + 4 \, c x^{3}\right )} \sqrt{c^{2} x^{2} + 1}}{a b c^{5} x^{4} +{\left (c^{2} x^{2} + 1\right )} a b c^{3} x^{2} + 2 \, a b c^{3} x^{2} + a b c +{\left (b^{2} c^{5} x^{4} +{\left (c^{2} x^{2} + 1\right )} b^{2} c^{3} x^{2} + 2 \, b^{2} c^{3} x^{2} + b^{2} c + 2 \,{\left (b^{2} c^{4} x^{3} + b^{2} c^{2} x\right )} \sqrt{c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) + 2 \,{\left (a b c^{4} x^{3} + a b c^{2} x\right )} \sqrt{c^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (c^{2} x^{5} + x^{3}\right )} \sqrt{c^{2} x^{2} + 1}}{b^{2} \operatorname{arsinh}\left (c x\right )^{2} + 2 \, a b \operatorname{arsinh}\left (c x\right ) + a^{2}}, x\right ) \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{{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} x^{3}}{{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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