Optimal. Leaf size=452 \[ -\frac{b^2 c^3 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right )}{d \sqrt{c^2 d x^2+d}}-\frac{5 b^2 c^3 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )}{3 d \sqrt{c^2 d x^2+d}}+\frac{8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt{c^2 d x^2+d}}+\frac{8 c^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt{c^2 d x^2+d}}+\frac{4 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x \sqrt{c^2 d x^2+d}}-\frac{b c \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2 \sqrt{c^2 d x^2+d}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{c^2 d x^2+d}}-\frac{16 b c^3 \sqrt{c^2 x^2+1} \log \left (e^{2 \sinh ^{-1}(c x)}+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 d \sqrt{c^2 d x^2+d}}+\frac{20 b c^3 \sqrt{c^2 x^2+1} \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 d \sqrt{c^2 d x^2+d}}-\frac{b^2 c^2 \left (c^2 x^2+1\right )}{3 d x \sqrt{c^2 d x^2+d}} \]
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Rubi [A] time = 0.840918, antiderivative size = 452, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 11, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.393, Rules used = {5747, 5687, 5714, 3718, 2190, 2279, 2391, 5720, 5461, 4182, 264} \[ -\frac{b^2 c^3 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right )}{d \sqrt{c^2 d x^2+d}}-\frac{5 b^2 c^3 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )}{3 d \sqrt{c^2 d x^2+d}}+\frac{8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt{c^2 d x^2+d}}+\frac{8 c^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt{c^2 d x^2+d}}+\frac{4 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x \sqrt{c^2 d x^2+d}}-\frac{b c \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2 \sqrt{c^2 d x^2+d}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{c^2 d x^2+d}}-\frac{16 b c^3 \sqrt{c^2 x^2+1} \log \left (e^{2 \sinh ^{-1}(c x)}+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 d \sqrt{c^2 d x^2+d}}+\frac{20 b c^3 \sqrt{c^2 x^2+1} \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 d \sqrt{c^2 d x^2+d}}-\frac{b^2 c^2 \left (c^2 x^2+1\right )}{3 d x \sqrt{c^2 d x^2+d}} \]
Antiderivative was successfully verified.
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Rule 5747
Rule 5687
Rule 5714
Rule 3718
Rule 2190
Rule 2279
Rule 2391
Rule 5720
Rule 5461
Rule 4182
Rule 264
Rubi steps
\begin{align*} \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{x^4 \left (d+c^2 d x^2\right )^{3/2}} \, dx &=-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{d+c^2 d x^2}}-\frac{1}{3} \left (4 c^2\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{x^2 \left (d+c^2 d x^2\right )^{3/2}} \, dx+\frac{\left (2 b c \sqrt{1+c^2 x^2}\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x^3 \left (1+c^2 x^2\right )} \, dx}{3 d \sqrt{d+c^2 d x^2}}\\ &=-\frac{b c \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2 \sqrt{d+c^2 d x^2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{d+c^2 d x^2}}+\frac{4 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x \sqrt{d+c^2 d x^2}}+\frac{1}{3} \left (8 c^4\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx+\frac{\left (b^2 c^2 \sqrt{1+c^2 x^2}\right ) \int \frac{1}{x^2 \sqrt{1+c^2 x^2}} \, dx}{3 d \sqrt{d+c^2 d x^2}}-\frac{\left (2 b c^3 \sqrt{1+c^2 x^2}\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x \left (1+c^2 x^2\right )} \, dx}{3 d \sqrt{d+c^2 d x^2}}-\frac{\left (8 b c^3 \sqrt{1+c^2 x^2}\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x \left (1+c^2 x^2\right )} \, dx}{3 d \sqrt{d+c^2 d x^2}}\\ &=-\frac{b^2 c^2 \left (1+c^2 x^2\right )}{3 d x \sqrt{d+c^2 d x^2}}-\frac{b c \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2 \sqrt{d+c^2 d x^2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{d+c^2 d x^2}}+\frac{4 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x \sqrt{d+c^2 d x^2}}+\frac{8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt{d+c^2 d x^2}}-\frac{\left (2 b c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \text{sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d \sqrt{d+c^2 d x^2}}-\frac{\left (8 b c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \text{sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d \sqrt{d+c^2 d x^2}}-\frac{\left (16 b c^5 \sqrt{1+c^2 x^2}\right ) \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 d \sqrt{d+c^2 d x^2}}\\ &=-\frac{b^2 c^2 \left (1+c^2 x^2\right )}{3 d x \sqrt{d+c^2 d x^2}}-\frac{b c \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2 \sqrt{d+c^2 d x^2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{d+c^2 d x^2}}+\frac{4 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x \sqrt{d+c^2 d x^2}}+\frac{8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt{d+c^2 d x^2}}-\frac{\left (4 b c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(2 x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d \sqrt{d+c^2 d x^2}}-\frac{\left (16 b c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(2 x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d \sqrt{d+c^2 d x^2}}-\frac{\left (16 b c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d \sqrt{d+c^2 d x^2}}\\ &=-\frac{b^2 c^2 \left (1+c^2 x^2\right )}{3 d x \sqrt{d+c^2 d x^2}}-\frac{b c \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2 \sqrt{d+c^2 d x^2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{d+c^2 d x^2}}+\frac{4 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x \sqrt{d+c^2 d x^2}}+\frac{8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt{d+c^2 d x^2}}+\frac{8 c^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt{d+c^2 d x^2}}+\frac{20 b c^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{3 d \sqrt{d+c^2 d x^2}}-\frac{\left (32 b c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )}{3 d \sqrt{d+c^2 d x^2}}+\frac{\left (2 b^2 c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d \sqrt{d+c^2 d x^2}}-\frac{\left (2 b^2 c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d \sqrt{d+c^2 d x^2}}+\frac{\left (8 b^2 c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d \sqrt{d+c^2 d x^2}}-\frac{\left (8 b^2 c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d \sqrt{d+c^2 d x^2}}\\ &=-\frac{b^2 c^2 \left (1+c^2 x^2\right )}{3 d x \sqrt{d+c^2 d x^2}}-\frac{b c \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2 \sqrt{d+c^2 d x^2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{d+c^2 d x^2}}+\frac{4 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x \sqrt{d+c^2 d x^2}}+\frac{8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt{d+c^2 d x^2}}+\frac{8 c^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt{d+c^2 d x^2}}+\frac{20 b c^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{3 d \sqrt{d+c^2 d x^2}}-\frac{16 b c^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{3 d \sqrt{d+c^2 d x^2}}+\frac{\left (b^2 c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{3 d \sqrt{d+c^2 d x^2}}-\frac{\left (b^2 c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{3 d \sqrt{d+c^2 d x^2}}+\frac{\left (4 b^2 c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{3 d \sqrt{d+c^2 d x^2}}-\frac{\left (4 b^2 c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{3 d \sqrt{d+c^2 d x^2}}+\frac{\left (16 b^2 c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d \sqrt{d+c^2 d x^2}}\\ &=-\frac{b^2 c^2 \left (1+c^2 x^2\right )}{3 d x \sqrt{d+c^2 d x^2}}-\frac{b c \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2 \sqrt{d+c^2 d x^2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{d+c^2 d x^2}}+\frac{4 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x \sqrt{d+c^2 d x^2}}+\frac{8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt{d+c^2 d x^2}}+\frac{8 c^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt{d+c^2 d x^2}}+\frac{20 b c^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{3 d \sqrt{d+c^2 d x^2}}-\frac{16 b c^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{3 d \sqrt{d+c^2 d x^2}}+\frac{5 b^2 c^3 \sqrt{1+c^2 x^2} \text{Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{3 d \sqrt{d+c^2 d x^2}}-\frac{5 b^2 c^3 \sqrt{1+c^2 x^2} \text{Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{3 d \sqrt{d+c^2 d x^2}}+\frac{\left (8 b^2 c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{3 d \sqrt{d+c^2 d x^2}}\\ &=-\frac{b^2 c^2 \left (1+c^2 x^2\right )}{3 d x \sqrt{d+c^2 d x^2}}-\frac{b c \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2 \sqrt{d+c^2 d x^2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{d+c^2 d x^2}}+\frac{4 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x \sqrt{d+c^2 d x^2}}+\frac{8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt{d+c^2 d x^2}}+\frac{8 c^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt{d+c^2 d x^2}}+\frac{20 b c^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{3 d \sqrt{d+c^2 d x^2}}-\frac{16 b c^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{3 d \sqrt{d+c^2 d x^2}}-\frac{b^2 c^3 \sqrt{1+c^2 x^2} \text{Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{d \sqrt{d+c^2 d x^2}}-\frac{5 b^2 c^3 \sqrt{1+c^2 x^2} \text{Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{3 d \sqrt{d+c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 0.848985, size = 438, normalized size = 0.97 \[ \frac{3 b^2 c^3 x^3 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-e^{-2 \sinh ^{-1}(c x)}\right )+5 b^2 c^3 x^3 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )+8 a^2 c^4 x^4+4 a^2 c^2 x^2-a^2-a b c x \sqrt{c^2 x^2+1}-10 a b c^3 x^3 \sqrt{c^2 x^2+1} \log (c x)-3 a b c^3 x^3 \sqrt{c^2 x^2+1} \log \left (c^2 x^2+1\right )+16 a b c^4 x^4 \sinh ^{-1}(c x)+8 a b c^2 x^2 \sinh ^{-1}(c x)-2 a b \sinh ^{-1}(c x)-b^2 c^4 x^4-b^2 c^2 x^2+8 b^2 c^4 x^4 \sinh ^{-1}(c x)^2-8 b^2 c^3 x^3 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)^2+4 b^2 c^2 x^2 \sinh ^{-1}(c x)^2-b^2 c x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)-10 b^2 c^3 x^3 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )-6 b^2 c^3 x^3 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left (e^{-2 \sinh ^{-1}(c x)}+1\right )-b^2 \sinh ^{-1}(c x)^2}{3 d x^3 \sqrt{c^2 d x^2+d}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.314, size = 2609, normalized size = 5.8 \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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{\sqrt{c^{2} d x^{2} + d}{\left (b^{2} \operatorname{arsinh}\left (c x\right )^{2} + 2 \, a b \operatorname{arsinh}\left (c x\right ) + a^{2}\right )}}{c^{4} d^{2} x^{8} + 2 \, c^{2} d^{2} x^{6} + d^{2} x^{4}}, 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 (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac{3}{2}} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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