Optimal. Leaf size=281 \[ \frac{\sinh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3}-\frac{\sinh \left (\frac{4 a}{b}\right ) \text{Chi}\left (\frac{4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{8 b^2 c^3}-\frac{3 \sinh \left (\frac{6 a}{b}\right ) \text{Chi}\left (\frac{6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3}-\frac{\sinh \left (\frac{8 a}{b}\right ) \text{Chi}\left (\frac{8 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3}-\frac{\cosh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3}+\frac{\cosh \left (\frac{4 a}{b}\right ) \text{Shi}\left (\frac{4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{8 b^2 c^3}+\frac{3 \cosh \left (\frac{6 a}{b}\right ) \text{Shi}\left (\frac{6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3}+\frac{\cosh \left (\frac{8 a}{b}\right ) \text{Shi}\left (\frac{8 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3}-\frac{x^2 \left (c^2 x^2+1\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]
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Rubi [A] time = 1.07198, antiderivative size = 281, normalized size of antiderivative = 1., 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} \[ \frac{\sinh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 a}{b}+2 \sinh ^{-1}(c x)\right )}{16 b^2 c^3}-\frac{\sinh \left (\frac{4 a}{b}\right ) \text{Chi}\left (\frac{4 a}{b}+4 \sinh ^{-1}(c x)\right )}{8 b^2 c^3}-\frac{3 \sinh \left (\frac{6 a}{b}\right ) \text{Chi}\left (\frac{6 a}{b}+6 \sinh ^{-1}(c x)\right )}{16 b^2 c^3}-\frac{\sinh \left (\frac{8 a}{b}\right ) \text{Chi}\left (\frac{8 a}{b}+8 \sinh ^{-1}(c x)\right )}{16 b^2 c^3}-\frac{\cosh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \sinh ^{-1}(c x)\right )}{16 b^2 c^3}+\frac{\cosh \left (\frac{4 a}{b}\right ) \text{Shi}\left (\frac{4 a}{b}+4 \sinh ^{-1}(c x)\right )}{8 b^2 c^3}+\frac{3 \cosh \left (\frac{6 a}{b}\right ) \text{Shi}\left (\frac{6 a}{b}+6 \sinh ^{-1}(c x)\right )}{16 b^2 c^3}+\frac{\cosh \left (\frac{8 a}{b}\right ) \text{Shi}\left (\frac{8 a}{b}+8 \sinh ^{-1}(c x)\right )}{16 b^2 c^3}-\frac{x^2 \left (c^2 x^2+1\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Rule 5777
Rule 5779
Rule 5448
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{x^2 \left (1+c^2 x^2\right )^{5/2}}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx &=-\frac{x^2 \left (1+c^2 x^2\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac{2 \int \frac{x \left (1+c^2 x^2\right )^2}{a+b \sinh ^{-1}(c x)} \, dx}{b c}+\frac{(8 c) \int \frac{x^3 \left (1+c^2 x^2\right )^2}{a+b \sinh ^{-1}(c x)} \, dx}{b}\\ &=-\frac{x^2 \left (1+c^2 x^2\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac{2 \operatorname{Subst}\left (\int \frac{\cosh ^5(x) \sinh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^3}+\frac{8 \operatorname{Subst}\left (\int \frac{\cosh ^5(x) \sinh ^3(x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^3}\\ &=-\frac{x^2 \left (1+c^2 x^2\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac{2 \operatorname{Subst}\left (\int \left (\frac{5 \sinh (2 x)}{32 (a+b x)}+\frac{\sinh (4 x)}{8 (a+b x)}+\frac{\sinh (6 x)}{32 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b c^3}+\frac{8 \operatorname{Subst}\left (\int \left (-\frac{3 \sinh (2 x)}{64 (a+b x)}-\frac{\sinh (4 x)}{64 (a+b x)}+\frac{\sinh (6 x)}{64 (a+b x)}+\frac{\sinh (8 x)}{128 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b c^3}\\ &=-\frac{x^2 \left (1+c^2 x^2\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{\sinh (6 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^3}+\frac{\operatorname{Subst}\left (\int \frac{\sinh (8 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^3}-\frac{\operatorname{Subst}\left (\int \frac{\sinh (4 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^3}+\frac{\operatorname{Subst}\left (\int \frac{\sinh (6 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^3}+\frac{\operatorname{Subst}\left (\int \frac{\sinh (4 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b c^3}+\frac{5 \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^3}-\frac{3 \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^3}\\ &=-\frac{x^2 \left (1+c^2 x^2\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac{\left (5 \cosh \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^3}-\frac{\left (3 \cosh \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^3}-\frac{\cosh \left (\frac{4 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{4 a}{b}+4 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^3}+\frac{\cosh \left (\frac{4 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{4 a}{b}+4 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b c^3}+\frac{\cosh \left (\frac{6 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{6 a}{b}+6 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^3}+\frac{\cosh \left (\frac{6 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{6 a}{b}+6 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^3}+\frac{\cosh \left (\frac{8 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{8 a}{b}+8 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^3}-\frac{\left (5 \sinh \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^3}+\frac{\left (3 \sinh \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^3}+\frac{\sinh \left (\frac{4 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{4 a}{b}+4 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^3}-\frac{\sinh \left (\frac{4 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{4 a}{b}+4 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b c^3}-\frac{\sinh \left (\frac{6 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{6 a}{b}+6 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^3}-\frac{\sinh \left (\frac{6 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{6 a}{b}+6 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^3}-\frac{\sinh \left (\frac{8 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{8 a}{b}+8 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^3}\\ &=-\frac{x^2 \left (1+c^2 x^2\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac{\text{Chi}\left (\frac{2 a}{b}+2 \sinh ^{-1}(c x)\right ) \sinh \left (\frac{2 a}{b}\right )}{16 b^2 c^3}-\frac{\text{Chi}\left (\frac{4 a}{b}+4 \sinh ^{-1}(c x)\right ) \sinh \left (\frac{4 a}{b}\right )}{8 b^2 c^3}-\frac{3 \text{Chi}\left (\frac{6 a}{b}+6 \sinh ^{-1}(c x)\right ) \sinh \left (\frac{6 a}{b}\right )}{16 b^2 c^3}-\frac{\text{Chi}\left (\frac{8 a}{b}+8 \sinh ^{-1}(c x)\right ) \sinh \left (\frac{8 a}{b}\right )}{16 b^2 c^3}-\frac{\cosh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \sinh ^{-1}(c x)\right )}{16 b^2 c^3}+\frac{\cosh \left (\frac{4 a}{b}\right ) \text{Shi}\left (\frac{4 a}{b}+4 \sinh ^{-1}(c x)\right )}{8 b^2 c^3}+\frac{3 \cosh \left (\frac{6 a}{b}\right ) \text{Shi}\left (\frac{6 a}{b}+6 \sinh ^{-1}(c x)\right )}{16 b^2 c^3}+\frac{\cosh \left (\frac{8 a}{b}\right ) \text{Shi}\left (\frac{8 a}{b}+8 \sinh ^{-1}(c x)\right )}{16 b^2 c^3}\\ \end{align*}
Mathematica [A] time = 1.20148, size = 413, normalized size = 1.47 \[ -\frac{-\sinh \left (\frac{2 a}{b}\right ) \left (a+b \sinh ^{-1}(c x)\right ) \text{Chi}\left (2 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )+2 \sinh \left (\frac{4 a}{b}\right ) \left (a+b \sinh ^{-1}(c x)\right ) \text{Chi}\left (4 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )+3 a \sinh \left (\frac{6 a}{b}\right ) \text{Chi}\left (6 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )+3 b \sinh \left (\frac{6 a}{b}\right ) \sinh ^{-1}(c x) \text{Chi}\left (6 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )+a \sinh \left (\frac{8 a}{b}\right ) \text{Chi}\left (8 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )+b \sinh \left (\frac{8 a}{b}\right ) \sinh ^{-1}(c x) \text{Chi}\left (8 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )+a \cosh \left (\frac{2 a}{b}\right ) \text{Shi}\left (2 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )+b \cosh \left (\frac{2 a}{b}\right ) \sinh ^{-1}(c x) \text{Shi}\left (2 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )-2 a \cosh \left (\frac{4 a}{b}\right ) \text{Shi}\left (4 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )-2 b \cosh \left (\frac{4 a}{b}\right ) \sinh ^{-1}(c x) \text{Shi}\left (4 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )-3 a \cosh \left (\frac{6 a}{b}\right ) \text{Shi}\left (6 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )-3 b \cosh \left (\frac{6 a}{b}\right ) \sinh ^{-1}(c x) \text{Shi}\left (6 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )-a \cosh \left (\frac{8 a}{b}\right ) \text{Shi}\left (8 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )-b \cosh \left (\frac{8 a}{b}\right ) \sinh ^{-1}(c x) \text{Shi}\left (8 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )+16 b c^8 x^8+48 b c^6 x^6+48 b c^4 x^4+16 b c^2 x^2}{16 b^2 c^3 \left (a+b \sinh ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.416, size = 1044, normalized size = 3.7 \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^{8} + 3 \, c^{4} x^{6} + 3 \, c^{2} x^{4} + x^{2}\right )}{\left (c^{2} x^{2} + 1\right )} +{\left (c^{7} x^{9} + 3 \, c^{5} x^{7} + 3 \, c^{3} x^{5} + c x^{3}\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 (8 \, c^{7} x^{8} + 17 \, c^{5} x^{6} + 10 \, c^{3} x^{4} + c x^{2}\right )}{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} + 2 \,{\left (8 \, c^{8} x^{9} + 22 \, c^{6} x^{7} + 21 \, c^{4} x^{5} + 8 \, c^{2} x^{3} + x\right )}{\left (c^{2} x^{2} + 1\right )} +{\left (8 \, c^{9} x^{10} + 27 \, c^{7} x^{8} + 33 \, c^{5} x^{6} + 17 \, c^{3} x^{4} + 3 \, c x^{2}\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^{6} + 2 \, c^{2} x^{4} + x^{2}\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^{2}}{{\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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