Optimal. Leaf size=256 \[ -\frac{e^4 \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{8 b^2 d}+\frac{9 e^4 \sinh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{16 b^2 d}-\frac{5 e^4 \sinh \left (\frac{5 a}{b}\right ) \text{Chi}\left (\frac{5 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{16 b^2 d}+\frac{e^4 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{8 b^2 d}-\frac{9 e^4 \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{16 b^2 d}+\frac{5 e^4 \cosh \left (\frac{5 a}{b}\right ) \text{Shi}\left (\frac{5 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{16 b^2 d}-\frac{e^4 (c+d x)^4 \sqrt{(c+d x)^2+1}}{b d \left (a+b \sinh ^{-1}(c+d x)\right )} \]
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Rubi [A] time = 0.409048, antiderivative size = 252, normalized size of antiderivative = 0.98, number of steps used = 13, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {5865, 12, 5665, 3303, 3298, 3301} \[ -\frac{e^4 \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\sinh ^{-1}(c+d x)\right )}{8 b^2 d}+\frac{9 e^4 \sinh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 a}{b}+3 \sinh ^{-1}(c+d x)\right )}{16 b^2 d}-\frac{5 e^4 \sinh \left (\frac{5 a}{b}\right ) \text{Chi}\left (\frac{5 a}{b}+5 \sinh ^{-1}(c+d x)\right )}{16 b^2 d}+\frac{e^4 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\sinh ^{-1}(c+d x)\right )}{8 b^2 d}-\frac{9 e^4 \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \sinh ^{-1}(c+d x)\right )}{16 b^2 d}+\frac{5 e^4 \cosh \left (\frac{5 a}{b}\right ) \text{Shi}\left (\frac{5 a}{b}+5 \sinh ^{-1}(c+d x)\right )}{16 b^2 d}-\frac{e^4 (c+d x)^4 \sqrt{(c+d x)^2+1}}{b d \left (a+b \sinh ^{-1}(c+d x)\right )} \]
Antiderivative was successfully verified.
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Rule 5865
Rule 12
Rule 5665
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{(c e+d e x)^4}{\left (a+b \sinh ^{-1}(c+d x)\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{e^4 x^4}{\left (a+b \sinh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^4 \operatorname{Subst}\left (\int \frac{x^4}{\left (a+b \sinh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{d}\\ &=-\frac{e^4 (c+d x)^4 \sqrt{1+(c+d x)^2}}{b d \left (a+b \sinh ^{-1}(c+d x)\right )}+\frac{e^4 \operatorname{Subst}\left (\int \left (\frac{\sinh (x)}{8 (a+b x)}-\frac{9 \sinh (3 x)}{16 (a+b x)}+\frac{5 \sinh (5 x)}{16 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{b d}\\ &=-\frac{e^4 (c+d x)^4 \sqrt{1+(c+d x)^2}}{b d \left (a+b \sinh ^{-1}(c+d x)\right )}+\frac{e^4 \operatorname{Subst}\left (\int \frac{\sinh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{8 b d}+\frac{\left (5 e^4\right ) \operatorname{Subst}\left (\int \frac{\sinh (5 x)}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{16 b d}-\frac{\left (9 e^4\right ) \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{16 b d}\\ &=-\frac{e^4 (c+d x)^4 \sqrt{1+(c+d x)^2}}{b d \left (a+b \sinh ^{-1}(c+d x)\right )}+\frac{\left (e^4 \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+d x)\right )}{8 b d}-\frac{\left (9 e^4 \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+d x)\right )}{16 b d}+\frac{\left (5 e^4 \cosh \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+d x)\right )}{16 b d}-\frac{\left (e^4 \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+d x)\right )}{8 b d}+\frac{\left (9 e^4 \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+d x)\right )}{16 b d}-\frac{\left (5 e^4 \sinh \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+d x)\right )}{16 b d}\\ &=-\frac{e^4 (c+d x)^4 \sqrt{1+(c+d x)^2}}{b d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac{e^4 \text{Chi}\left (\frac{a}{b}+\sinh ^{-1}(c+d x)\right ) \sinh \left (\frac{a}{b}\right )}{8 b^2 d}+\frac{9 e^4 \text{Chi}\left (\frac{3 a}{b}+3 \sinh ^{-1}(c+d x)\right ) \sinh \left (\frac{3 a}{b}\right )}{16 b^2 d}-\frac{5 e^4 \text{Chi}\left (\frac{5 a}{b}+5 \sinh ^{-1}(c+d x)\right ) \sinh \left (\frac{5 a}{b}\right )}{16 b^2 d}+\frac{e^4 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\sinh ^{-1}(c+d x)\right )}{8 b^2 d}-\frac{9 e^4 \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \sinh ^{-1}(c+d x)\right )}{16 b^2 d}+\frac{5 e^4 \cosh \left (\frac{5 a}{b}\right ) \text{Shi}\left (\frac{5 a}{b}+5 \sinh ^{-1}(c+d x)\right )}{16 b^2 d}\\ \end{align*}
Mathematica [A] time = 1.08561, size = 281, normalized size = 1.1 \[ \frac{e^4 \left (16 \left (3 \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\sinh ^{-1}(c+d x)\right )-\sinh \left (\frac{3 a}{b}\right ) \text{Chi}\left (3 \left (\frac{a}{b}+\sinh ^{-1}(c+d x)\right )\right )-3 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\sinh ^{-1}(c+d x)\right )+\cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (3 \left (\frac{a}{b}+\sinh ^{-1}(c+d x)\right )\right )\right )-5 \left (10 \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\sinh ^{-1}(c+d x)\right )-5 \sinh \left (\frac{3 a}{b}\right ) \text{Chi}\left (3 \left (\frac{a}{b}+\sinh ^{-1}(c+d x)\right )\right )+\sinh \left (\frac{5 a}{b}\right ) \text{Chi}\left (5 \left (\frac{a}{b}+\sinh ^{-1}(c+d x)\right )\right )-10 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\sinh ^{-1}(c+d x)\right )+5 \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (3 \left (\frac{a}{b}+\sinh ^{-1}(c+d x)\right )\right )-\cosh \left (\frac{5 a}{b}\right ) \text{Shi}\left (5 \left (\frac{a}{b}+\sinh ^{-1}(c+d x)\right )\right )\right )-\frac{16 b \sqrt{(c+d x)^2+1} (c+d x)^4}{a+b \sinh ^{-1}(c+d x)}\right )}{16 b^2 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.207, size = 602, normalized size = 2.4 \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*} \text{result too large to display} \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{d^{4} e^{4} x^{4} + 4 \, c d^{3} e^{4} x^{3} + 6 \, c^{2} d^{2} e^{4} x^{2} + 4 \, c^{3} d e^{4} x + c^{4} e^{4}}{b^{2} \operatorname{arsinh}\left (d x + c\right )^{2} + 2 \, a b \operatorname{arsinh}\left (d x + c\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 (d e x + c e\right )}^{4}}{{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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