Optimal. Leaf size=359 \[ \frac{2 d e \cosh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{b^2 c^2}+\frac{e^2 \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{4 b^2 c^3}-\frac{3 e^2 \sinh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{4 b^2 c^3}-\frac{2 d e \sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{b^2 c^2}-\frac{e^2 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{4 b^2 c^3}+\frac{3 e^2 \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{4 b^2 c^3}-\frac{d^2 \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{b^2 c}+\frac{d^2 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{b^2 c}-\frac{d^2 \sqrt{c^2 x^2+1}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac{2 d e x \sqrt{c^2 x^2+1}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac{e^2 x^2 \sqrt{c^2 x^2+1}}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]
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Rubi [A] time = 0.688232, antiderivative size = 351, normalized size of antiderivative = 0.98, number of steps used = 19, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {5803, 5655, 5779, 3303, 3298, 3301, 5665} \[ \frac{2 d e \cosh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 a}{b}+2 \sinh ^{-1}(c x)\right )}{b^2 c^2}+\frac{e^2 \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )}{4 b^2 c^3}-\frac{3 e^2 \sinh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 a}{b}+3 \sinh ^{-1}(c x)\right )}{4 b^2 c^3}-\frac{2 d e \sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \sinh ^{-1}(c x)\right )}{b^2 c^2}-\frac{e^2 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )}{4 b^2 c^3}+\frac{3 e^2 \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \sinh ^{-1}(c x)\right )}{4 b^2 c^3}-\frac{d^2 \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )}{b^2 c}+\frac{d^2 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )}{b^2 c}-\frac{d^2 \sqrt{c^2 x^2+1}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac{2 d e x \sqrt{c^2 x^2+1}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac{e^2 x^2 \sqrt{c^2 x^2+1}}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Rule 5803
Rule 5655
Rule 5779
Rule 3303
Rule 3298
Rule 3301
Rule 5665
Rubi steps
\begin{align*} \int \frac{(d+e x)^2}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx &=\int \left (\frac{d^2}{\left (a+b \sinh ^{-1}(c x)\right )^2}+\frac{2 d e x}{\left (a+b \sinh ^{-1}(c x)\right )^2}+\frac{e^2 x^2}{\left (a+b \sinh ^{-1}(c x)\right )^2}\right ) \, dx\\ &=d^2 \int \frac{1}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx+(2 d e) \int \frac{x}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx+e^2 \int \frac{x^2}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx\\ &=-\frac{d^2 \sqrt{1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac{2 d e x \sqrt{1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac{e^2 x^2 \sqrt{1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac{\left (c d^2\right ) \int \frac{x}{\sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )} \, dx}{b}+\frac{(2 d e) \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^2}+\frac{e^2 \operatorname{Subst}\left (\int \left (-\frac{\sinh (x)}{4 (a+b x)}+\frac{3 \sinh (3 x)}{4 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b c^3}\\ &=-\frac{d^2 \sqrt{1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac{2 d e x \sqrt{1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac{e^2 x^2 \sqrt{1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac{d^2 \operatorname{Subst}\left (\int \frac{\sinh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{b c}-\frac{e^2 \operatorname{Subst}\left (\int \frac{\sinh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b c^3}+\frac{\left (3 e^2\right ) \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b c^3}+\frac{\left (2 d e \cosh \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 )}{b c^2}-\frac{\left (2 d e \sinh \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 )}{b c^2}\\ &=-\frac{d^2 \sqrt{1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac{2 d e x \sqrt{1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac{e^2 x^2 \sqrt{1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac{2 d e \cosh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 a}{b}+2 \sinh ^{-1}(c x)\right )}{b^2 c^2}-\frac{2 d e \sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \sinh ^{-1}(c x)\right )}{b^2 c^2}+\frac{\left (d^2 \cosh \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 )}{b c}-\frac{\left (e^2 \cosh \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 )}{4 b c^3}+\frac{\left (3 e^2 \cosh \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 )}{4 b c^3}-\frac{\left (d^2 \sinh \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 )}{b c}+\frac{\left (e^2 \sinh \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 )}{4 b c^3}-\frac{\left (3 e^2 \sinh \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 )}{4 b c^3}\\ &=-\frac{d^2 \sqrt{1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac{2 d e x \sqrt{1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac{e^2 x^2 \sqrt{1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac{2 d e \cosh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 a}{b}+2 \sinh ^{-1}(c x)\right )}{b^2 c^2}-\frac{d^2 \text{Chi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right ) \sinh \left (\frac{a}{b}\right )}{b^2 c}+\frac{e^2 \text{Chi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right ) \sinh \left (\frac{a}{b}\right )}{4 b^2 c^3}-\frac{3 e^2 \text{Chi}\left (\frac{3 a}{b}+3 \sinh ^{-1}(c x)\right ) \sinh \left (\frac{3 a}{b}\right )}{4 b^2 c^3}+\frac{d^2 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )}{b^2 c}-\frac{e^2 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )}{4 b^2 c^3}-\frac{2 d e \sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \sinh ^{-1}(c x)\right )}{b^2 c^2}+\frac{3 e^2 \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \sinh ^{-1}(c x)\right )}{4 b^2 c^3}\\ \end{align*}
Mathematica [A] time = 1.58072, size = 288, normalized size = 0.8 \[ -\frac{\sinh \left (\frac{a}{b}\right ) \left (4 c^2 d^2-e^2\right ) \text{Chi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )-4 c^2 d^2 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )+\frac{4 b c^2 d^2 \sqrt{c^2 x^2+1}}{a+b \sinh ^{-1}(c x)}+\frac{8 b c^2 d e x \sqrt{c^2 x^2+1}}{a+b \sinh ^{-1}(c x)}+\frac{4 b c^2 e^2 x^2 \sqrt{c^2 x^2+1}}{a+b \sinh ^{-1}(c x)}-8 c d e \cosh \left (\frac{2 a}{b}\right ) \text{Chi}\left (2 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )+3 e^2 \sinh \left (\frac{3 a}{b}\right ) \text{Chi}\left (3 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )+8 c d e \sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (2 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )+e^2 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )-3 e^2 \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (3 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )}{4 b^2 c^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.2, size = 616, normalized size = 1.7 \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{c^{3} e^{2} x^{5} + 2 \, c^{3} d e x^{4} + 2 \, c d e x^{2} + c d^{2} x +{\left (c^{3} d^{2} + c e^{2}\right )} x^{3} +{\left (c^{2} e^{2} x^{4} + 2 \, c^{2} d e x^{3} + 2 \, d e x +{\left (c^{2} d^{2} + e^{2}\right )} x^{2} + d^{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{3 \, c^{5} e^{2} x^{6} + 4 \, c^{5} d e x^{5} + 8 \, c^{3} d e x^{3} +{\left (c^{5} d^{2} + 6 \, c^{3} e^{2}\right )} x^{4} + 4 \, c d e x + c d^{2} +{\left (2 \, c^{3} d^{2} + 3 \, c e^{2}\right )} x^{2} +{\left (3 \, c^{3} e^{2} x^{4} + 4 \, c^{3} d e x^{3} - c d^{2} +{\left (c^{3} d^{2} + c e^{2}\right )} x^{2}\right )}{\left (c^{2} x^{2} + 1\right )} +{\left (6 \, c^{4} e^{2} x^{5} + 8 \, c^{4} d e x^{4} + 8 \, c^{2} d e x^{2} +{\left (2 \, c^{4} d^{2} + 7 \, c^{2} e^{2}\right )} x^{3} + 2 \, d e +{\left (c^{2} d^{2} + 2 \, e^{2}\right )} 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{e^{2} x^{2} + 2 \, d e x + d^{2}}{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e x\right )^{2}}{\left (a + b \operatorname{asinh}{\left (c x \right )}\right )^{2}}\, dx \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 (e x + d\right )}^{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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