Optimal. Leaf size=263 \[ \frac{e^4 \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \cosh ^{-1}(c+d x)}{b}\right )}{8 b^2 d}+\frac{9 e^4 \cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{16 b^2 d}+\frac{5 e^4 \cosh \left (\frac{5 a}{b}\right ) \text{Chi}\left (\frac{5 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{16 b^2 d}-\frac{e^4 \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \cosh ^{-1}(c+d x)}{b}\right )}{8 b^2 d}-\frac{9 e^4 \sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{16 b^2 d}-\frac{5 e^4 \sinh \left (\frac{5 a}{b}\right ) \text{Shi}\left (\frac{5 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{16 b^2 d}-\frac{e^4 \sqrt{c+d x-1} (c+d x)^4 \sqrt{c+d x+1}}{b d \left (a+b \cosh ^{-1}(c+d x)\right )} \]
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Rubi [A] time = 0.393699, antiderivative size = 259, 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 = {5866, 12, 5666, 3303, 3298, 3301} \[ \frac{e^4 \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )}{8 b^2 d}+\frac{9 e^4 \cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c+d x)\right )}{16 b^2 d}+\frac{5 e^4 \cosh \left (\frac{5 a}{b}\right ) \text{Chi}\left (\frac{5 a}{b}+5 \cosh ^{-1}(c+d x)\right )}{16 b^2 d}-\frac{e^4 \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )}{8 b^2 d}-\frac{9 e^4 \sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c+d x)\right )}{16 b^2 d}-\frac{5 e^4 \sinh \left (\frac{5 a}{b}\right ) \text{Shi}\left (\frac{5 a}{b}+5 \cosh ^{-1}(c+d x)\right )}{16 b^2 d}-\frac{e^4 \sqrt{c+d x-1} (c+d x)^4 \sqrt{c+d x+1}}{b d \left (a+b \cosh ^{-1}(c+d x)\right )} \]
Antiderivative was successfully verified.
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Rule 5866
Rule 12
Rule 5666
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{(c e+d e x)^4}{\left (a+b \cosh ^{-1}(c+d x)\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{e^4 x^4}{\left (a+b \cosh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^4 \operatorname{Subst}\left (\int \frac{x^4}{\left (a+b \cosh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{d}\\ &=-\frac{e^4 \sqrt{-1+c+d x} (c+d x)^4 \sqrt{1+c+d x}}{b d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac{e^4 \operatorname{Subst}\left (\int \left (-\frac{\cosh (x)}{8 (a+b x)}-\frac{9 \cosh (3 x)}{16 (a+b x)}-\frac{5 \cosh (5 x)}{16 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{b d}\\ &=-\frac{e^4 \sqrt{-1+c+d x} (c+d x)^4 \sqrt{1+c+d x}}{b d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac{e^4 \operatorname{Subst}\left (\int \frac{\cosh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{8 b d}+\frac{\left (5 e^4\right ) \operatorname{Subst}\left (\int \frac{\cosh (5 x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{16 b d}+\frac{\left (9 e^4\right ) \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{16 b d}\\ &=-\frac{e^4 \sqrt{-1+c+d x} (c+d x)^4 \sqrt{1+c+d x}}{b d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac{\left (e^4 \cosh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-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{\cosh \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-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{\cosh \left (\frac{5 a}{b}+5 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{16 b d}-\frac{\left (e^4 \sinh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-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{\sinh \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-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{\sinh \left (\frac{5 a}{b}+5 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{16 b d}\\ &=-\frac{e^4 \sqrt{-1+c+d x} (c+d x)^4 \sqrt{1+c+d x}}{b d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac{e^4 \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )}{8 b^2 d}+\frac{9 e^4 \cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c+d x)\right )}{16 b^2 d}+\frac{5 e^4 \cosh \left (\frac{5 a}{b}\right ) \text{Chi}\left (\frac{5 a}{b}+5 \cosh ^{-1}(c+d x)\right )}{16 b^2 d}-\frac{e^4 \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )}{8 b^2 d}-\frac{9 e^4 \sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c+d x)\right )}{16 b^2 d}-\frac{5 e^4 \sinh \left (\frac{5 a}{b}\right ) \text{Shi}\left (\frac{5 a}{b}+5 \cosh ^{-1}(c+d x)\right )}{16 b^2 d}\\ \end{align*}
Mathematica [A] time = 2.0537, size = 293, normalized size = 1.11 \[ \frac{e^4 \left (-16 \left (3 \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )+\cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (3 \left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )\right )-3 \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )-\sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (3 \left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )\right )\right )+5 \left (10 \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )+5 \cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (3 \left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )\right )+\cosh \left (\frac{5 a}{b}\right ) \text{Chi}\left (5 \left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )\right )-10 \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )-5 \sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (3 \left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )\right )-\sinh \left (\frac{5 a}{b}\right ) \text{Shi}\left (5 \left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )\right )\right )-\frac{16 b \sqrt{\frac{c+d x-1}{c+d x+1}} (c+d x+1) (c+d x)^4}{a+b \cosh ^{-1}(c+d x)}\right )}{16 b^2 d} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.204, size = 665, normalized size = 2.5 \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*} \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{arcosh}\left (d x + c\right )^{2} + 2 \, a b \operatorname{arcosh}\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] time = 0., size = 0, normalized size = 0. \begin{align*} e^{4} \left (\int \frac{c^{4}}{a^{2} + 2 a b \operatorname{acosh}{\left (c + d x \right )} + b^{2} \operatorname{acosh}^{2}{\left (c + d x \right )}}\, dx + \int \frac{d^{4} x^{4}}{a^{2} + 2 a b \operatorname{acosh}{\left (c + d x \right )} + b^{2} \operatorname{acosh}^{2}{\left (c + d x \right )}}\, dx + \int \frac{4 c d^{3} x^{3}}{a^{2} + 2 a b \operatorname{acosh}{\left (c + d x \right )} + b^{2} \operatorname{acosh}^{2}{\left (c + d x \right )}}\, dx + \int \frac{6 c^{2} d^{2} x^{2}}{a^{2} + 2 a b \operatorname{acosh}{\left (c + d x \right )} + b^{2} \operatorname{acosh}^{2}{\left (c + d x \right )}}\, dx + \int \frac{4 c^{3} d x}{a^{2} + 2 a b \operatorname{acosh}{\left (c + d x \right )} + b^{2} \operatorname{acosh}^{2}{\left (c + d x \right )}}\, dx\right ) \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{arcosh}\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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