Optimal. Leaf size=275 \[ \frac{5 \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{64 b^2 c^2}+\frac{27 \cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2}+\frac{25 \cosh \left (\frac{5 a}{b}\right ) \text{Chi}\left (\frac{5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2}+\frac{7 \cosh \left (\frac{7 a}{b}\right ) \text{Chi}\left (\frac{7 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2}-\frac{5 \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{64 b^2 c^2}-\frac{27 \sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2}-\frac{25 \sinh \left (\frac{5 a}{b}\right ) \text{Shi}\left (\frac{5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2}-\frac{7 \sinh \left (\frac{7 a}{b}\right ) \text{Shi}\left (\frac{7 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2}-\frac{x \left (c^2 x^2+1\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]
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Rubi [A] time = 0.969514, antiderivative size = 271, normalized size of antiderivative = 0.99, number of steps used = 28, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {5777, 5699, 3312, 3303, 3298, 3301, 5779, 5448} \[ \frac{5 \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )}{64 b^2 c^2}+\frac{27 \cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 a}{b}+3 \sinh ^{-1}(c x)\right )}{64 b^2 c^2}+\frac{25 \cosh \left (\frac{5 a}{b}\right ) \text{Chi}\left (\frac{5 a}{b}+5 \sinh ^{-1}(c x)\right )}{64 b^2 c^2}+\frac{7 \cosh \left (\frac{7 a}{b}\right ) \text{Chi}\left (\frac{7 a}{b}+7 \sinh ^{-1}(c x)\right )}{64 b^2 c^2}-\frac{5 \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )}{64 b^2 c^2}-\frac{27 \sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \sinh ^{-1}(c x)\right )}{64 b^2 c^2}-\frac{25 \sinh \left (\frac{5 a}{b}\right ) \text{Shi}\left (\frac{5 a}{b}+5 \sinh ^{-1}(c x)\right )}{64 b^2 c^2}-\frac{7 \sinh \left (\frac{7 a}{b}\right ) \text{Shi}\left (\frac{7 a}{b}+7 \sinh ^{-1}(c x)\right )}{64 b^2 c^2}-\frac{x \left (c^2 x^2+1\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Rule 5777
Rule 5699
Rule 3312
Rule 3303
Rule 3298
Rule 3301
Rule 5779
Rule 5448
Rubi steps
\begin{align*} \int \frac{x \left (1+c^2 x^2\right )^{5/2}}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx &=-\frac{x \left (1+c^2 x^2\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac{\int \frac{\left (1+c^2 x^2\right )^2}{a+b \sinh ^{-1}(c x)} \, dx}{b c}+\frac{(7 c) \int \frac{x^2 \left (1+c^2 x^2\right )^2}{a+b \sinh ^{-1}(c x)} \, dx}{b}\\ &=-\frac{x \left (1+c^2 x^2\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{\cosh ^5(x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^2}+\frac{7 \operatorname{Subst}\left (\int \frac{\cosh ^5(x) \sinh ^2(x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^2}\\ &=-\frac{x \left (1+c^2 x^2\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac{\operatorname{Subst}\left (\int \left (\frac{5 \cosh (x)}{8 (a+b x)}+\frac{5 \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^2}+\frac{7 \operatorname{Subst}\left (\int \left (-\frac{5 \cosh (x)}{64 (a+b x)}+\frac{\cosh (3 x)}{64 (a+b x)}+\frac{3 \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^2}\\ &=-\frac{x \left (1+c^2 x^2\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (5 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^2}+\frac{7 \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^2}+\frac{7 \operatorname{Subst}\left (\int \frac{\cosh (7 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^2}+\frac{5 \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^2}+\frac{21 \operatorname{Subst}\left (\int \frac{\cosh (5 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^2}-\frac{35 \operatorname{Subst}\left (\int \frac{\cosh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^2}+\frac{5 \operatorname{Subst}\left (\int \frac{\cosh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^2}\\ &=-\frac{x \left (1+c^2 x^2\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac{\left (35 \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^2}+\frac{\left (5 \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^2}+\frac{\left (7 \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^2}+\frac{\left (5 \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^2}+\frac{\cosh \left (\frac{5 a}{b}\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^2}+\frac{\left (21 \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^2}+\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^2}+\frac{\left (35 \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^2}-\frac{\left (5 \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^2}-\frac{\left (7 \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^2}-\frac{\left (5 \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^2}-\frac{\sinh \left (\frac{5 a}{b}\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^2}-\frac{\left (21 \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^2}-\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^2}\\ &=-\frac{x \left (1+c^2 x^2\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac{5 \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )}{64 b^2 c^2}+\frac{27 \cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 a}{b}+3 \sinh ^{-1}(c x)\right )}{64 b^2 c^2}+\frac{25 \cosh \left (\frac{5 a}{b}\right ) \text{Chi}\left (\frac{5 a}{b}+5 \sinh ^{-1}(c x)\right )}{64 b^2 c^2}+\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^2}-\frac{5 \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )}{64 b^2 c^2}-\frac{27 \sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \sinh ^{-1}(c x)\right )}{64 b^2 c^2}-\frac{25 \sinh \left (\frac{5 a}{b}\right ) \text{Shi}\left (\frac{5 a}{b}+5 \sinh ^{-1}(c x)\right )}{64 b^2 c^2}-\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^2}\\ \end{align*}
Mathematica [A] time = 1.00757, size = 404, normalized size = 1.47 \[ -\frac{-5 \cosh \left (\frac{a}{b}\right ) \left (a+b \sinh ^{-1}(c x)\right ) \text{Chi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )-27 \cosh \left (\frac{3 a}{b}\right ) \left (a+b \sinh ^{-1}(c x)\right ) \text{Chi}\left (3 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )-25 a \cosh \left (\frac{5 a}{b}\right ) \text{Chi}\left (5 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )-25 b \cosh \left (\frac{5 a}{b}\right ) \sinh ^{-1}(c x) \text{Chi}\left (5 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )-7 a \cosh \left (\frac{7 a}{b}\right ) \text{Chi}\left (7 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )-7 b \cosh \left (\frac{7 a}{b}\right ) \sinh ^{-1}(c x) \text{Chi}\left (7 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )+5 a \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )+5 b \sinh \left (\frac{a}{b}\right ) \sinh ^{-1}(c x) \text{Shi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )+27 a \sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (3 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )+27 b \sinh \left (\frac{3 a}{b}\right ) \sinh ^{-1}(c x) \text{Shi}\left (3 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )+25 a \sinh \left (\frac{5 a}{b}\right ) \text{Shi}\left (5 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )+25 b \sinh \left (\frac{5 a}{b}\right ) \sinh ^{-1}(c x) \text{Shi}\left (5 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )+7 a \sinh \left (\frac{7 a}{b}\right ) \text{Shi}\left (7 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )+7 b \sinh \left (\frac{7 a}{b}\right ) \sinh ^{-1}(c x) \text{Shi}\left (7 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )+64 b c^7 x^7+192 b c^5 x^5+192 b c^3 x^3+64 b c x}{64 b^2 c^2 \left (a+b \sinh ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.346, 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^{6} x^{7} + 3 \, c^{4} x^{5} + 3 \, c^{2} x^{3} + x\right )}{\left (c^{2} x^{2} + 1\right )} +{\left (c^{7} x^{8} + 3 \, c^{5} x^{6} + 3 \, c^{3} x^{4} + c x^{2}\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{7 \,{\left (c^{7} x^{7} + 2 \, c^{5} x^{5} + c^{3} x^{3}\right )}{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} +{\left (14 \, c^{8} x^{8} + 37 \, c^{6} x^{6} + 33 \, c^{4} x^{4} + 11 \, c^{2} x^{2} + 1\right )}{\left (c^{2} x^{2} + 1\right )} +{\left (7 \, c^{9} x^{9} + 23 \, c^{7} x^{7} + 27 \, c^{5} x^{5} + 13 \, c^{3} x^{3} + 2 \, c x\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^{4} x^{5} + 2 \, c^{2} x^{3} + x\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{5}{2}} x}{{\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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