Optimal. Leaf size=432 \[ \frac{4 i b^3 \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^4}-\frac{4 i b^3 \text{PolyLog}\left (3,i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^4}-\frac{2 i b^2 \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^4}+\frac{2 i b^2 \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^4}+\frac{4 i b^4 \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}-\frac{4 i b^4 \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}-\frac{4 i b^4 \text{PolyLog}\left (4,-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}+\frac{4 i b^4 \text{PolyLog}\left (4,i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}+\frac{2 b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^4 (c+d x)}-\frac{8 b^3 \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^4}+\frac{2 b \sqrt{c+d x-1} \sqrt{c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^2}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4}{3 d e^4 (c+d x)^3}+\frac{4 b \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d e^4} \]
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Rubi [A] time = 0.841757, antiderivative size = 432, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 12, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.522, Rules used = {5866, 12, 5662, 5748, 5761, 4180, 2531, 6609, 2282, 6589, 2279, 2391} \[ \frac{4 i b^3 \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^4}-\frac{4 i b^3 \text{PolyLog}\left (3,i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^4}-\frac{2 i b^2 \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^4}+\frac{2 i b^2 \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^4}+\frac{4 i b^4 \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}-\frac{4 i b^4 \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}-\frac{4 i b^4 \text{PolyLog}\left (4,-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}+\frac{4 i b^4 \text{PolyLog}\left (4,i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}+\frac{2 b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^4 (c+d x)}-\frac{8 b^3 \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^4}+\frac{2 b \sqrt{c+d x-1} \sqrt{c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^2}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4}{3 d e^4 (c+d x)^3}+\frac{4 b \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d e^4} \]
Antiderivative was successfully verified.
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Rule 5866
Rule 12
Rule 5662
Rule 5748
Rule 5761
Rule 4180
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4}{(c e+d e x)^4} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \cosh ^{-1}(x)\right )^4}{e^4 x^4} \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \cosh ^{-1}(x)\right )^4}{x^4} \, dx,x,c+d x\right )}{d e^4}\\ &=-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4}{3 d e^4 (c+d x)^3}+\frac{(4 b) \operatorname{Subst}\left (\int \frac{\left (a+b \cosh ^{-1}(x)\right )^3}{\sqrt{-1+x} x^3 \sqrt{1+x}} \, dx,x,c+d x\right )}{3 d e^4}\\ &=\frac{2 b \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^2}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4}{3 d e^4 (c+d x)^3}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{\left (a+b \cosh ^{-1}(x)\right )^3}{\sqrt{-1+x} x \sqrt{1+x}} \, dx,x,c+d x\right )}{3 d e^4}-\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \cosh ^{-1}(x)\right )^2}{x^2} \, dx,x,c+d x\right )}{d e^4}\\ &=\frac{2 b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^4 (c+d x)}+\frac{2 b \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^2}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4}{3 d e^4 (c+d x)^3}+\frac{(2 b) \operatorname{Subst}\left (\int (a+b x)^3 \text{sech}(x) \, dx,x,\cosh ^{-1}(c+d x)\right )}{3 d e^4}-\frac{\left (4 b^3\right ) \operatorname{Subst}\left (\int \frac{a+b \cosh ^{-1}(x)}{\sqrt{-1+x} x \sqrt{1+x}} \, dx,x,c+d x\right )}{d e^4}\\ &=\frac{2 b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^4 (c+d x)}+\frac{2 b \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^2}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4}{3 d e^4 (c+d x)^3}+\frac{4 b \left (a+b \cosh ^{-1}(c+d x)\right )^3 \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right )}{3 d e^4}-\frac{\left (2 i b^2\right ) \operatorname{Subst}\left (\int (a+b x)^2 \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^4}+\frac{\left (2 i b^2\right ) \operatorname{Subst}\left (\int (a+b x)^2 \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^4}-\frac{\left (4 b^3\right ) \operatorname{Subst}\left (\int (a+b x) \text{sech}(x) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^4}\\ &=\frac{2 b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^4 (c+d x)}+\frac{2 b \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^2}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4}{3 d e^4 (c+d x)^3}-\frac{8 b^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}+\frac{4 b \left (a+b \cosh ^{-1}(c+d x)\right )^3 \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right )}{3 d e^4}-\frac{2 i b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text{Li}_2\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}+\frac{2 i b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text{Li}_2\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}+\frac{\left (4 i b^3\right ) \operatorname{Subst}\left (\int (a+b x) \text{Li}_2\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^4}-\frac{\left (4 i b^3\right ) \operatorname{Subst}\left (\int (a+b x) \text{Li}_2\left (i e^x\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^4}+\frac{\left (4 i b^4\right ) \operatorname{Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^4}-\frac{\left (4 i b^4\right ) \operatorname{Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^4}\\ &=\frac{2 b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^4 (c+d x)}+\frac{2 b \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^2}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4}{3 d e^4 (c+d x)^3}-\frac{8 b^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}+\frac{4 b \left (a+b \cosh ^{-1}(c+d x)\right )^3 \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right )}{3 d e^4}-\frac{2 i b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text{Li}_2\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}+\frac{2 i b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text{Li}_2\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}+\frac{4 i b^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Li}_3\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}-\frac{4 i b^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Li}_3\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}+\frac{\left (4 i b^4\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}-\frac{\left (4 i b^4\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}-\frac{\left (4 i b^4\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^4}+\frac{\left (4 i b^4\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (i e^x\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^4}\\ &=\frac{2 b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^4 (c+d x)}+\frac{2 b \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^2}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4}{3 d e^4 (c+d x)^3}-\frac{8 b^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}+\frac{4 b \left (a+b \cosh ^{-1}(c+d x)\right )^3 \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right )}{3 d e^4}+\frac{4 i b^4 \text{Li}_2\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}-\frac{2 i b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text{Li}_2\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}-\frac{4 i b^4 \text{Li}_2\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}+\frac{2 i b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text{Li}_2\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}+\frac{4 i b^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Li}_3\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}-\frac{4 i b^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Li}_3\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}-\frac{\left (4 i b^4\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-i x)}{x} \, dx,x,e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}+\frac{\left (4 i b^4\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(i x)}{x} \, dx,x,e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}\\ &=\frac{2 b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^4 (c+d x)}+\frac{2 b \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^2}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4}{3 d e^4 (c+d x)^3}-\frac{8 b^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}+\frac{4 b \left (a+b \cosh ^{-1}(c+d x)\right )^3 \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right )}{3 d e^4}+\frac{4 i b^4 \text{Li}_2\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}-\frac{2 i b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text{Li}_2\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}-\frac{4 i b^4 \text{Li}_2\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}+\frac{2 i b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text{Li}_2\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}+\frac{4 i b^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Li}_3\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}-\frac{4 i b^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Li}_3\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}-\frac{4 i b^4 \text{Li}_4\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}+\frac{4 i b^4 \text{Li}_4\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}\\ \end{align*}
Mathematica [B] time = 8.80474, size = 1213, normalized size = 2.81 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.187, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\rm arccosh} \left (dx+c\right ) \right ) ^{4}}{ \left ( dex+ce \right ) ^{4}}}\, dx \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{b^{4} \operatorname{arcosh}\left (d x + c\right )^{4} + 4 \, a b^{3} \operatorname{arcosh}\left (d x + c\right )^{3} + 6 \, a^{2} b^{2} \operatorname{arcosh}\left (d x + c\right )^{2} + 4 \, a^{3} b \operatorname{arcosh}\left (d x + c\right ) + a^{4}}{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}}, 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*} \frac{\int \frac{a^{4}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac{b^{4} \operatorname{acosh}^{4}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac{4 a b^{3} \operatorname{acosh}^{3}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac{6 a^{2} b^{2} \operatorname{acosh}^{2}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac{4 a^{3} b \operatorname{acosh}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx}{e^{4}} \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{arcosh}\left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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