Optimal. Leaf size=297 \[ -\frac{i b^2 \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^4}+\frac{i b^2 \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^4}+\frac{i b^3 \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}-\frac{i b^3 \text{PolyLog}\left (3,i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}+\frac{b^2 \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^4 (c+d x)}+\frac{b \sqrt{c+d x-1} \sqrt{c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^4 (c+d x)^2}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^3}+\frac{b \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^4}-\frac{b^3 \tan ^{-1}\left (\sqrt{c+d x-1} \sqrt{c+d x+1}\right )}{d e^4} \]
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Rubi [A] time = 0.593879, antiderivative size = 297, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 11, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.478, Rules used = {5866, 12, 5662, 5748, 5761, 4180, 2531, 2282, 6589, 92, 203} \[ -\frac{i b^2 \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^4}+\frac{i b^2 \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^4}+\frac{i b^3 \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}-\frac{i b^3 \text{PolyLog}\left (3,i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}+\frac{b^2 \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^4 (c+d x)}+\frac{b \sqrt{c+d x-1} \sqrt{c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^4 (c+d x)^2}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^3}+\frac{b \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^4}-\frac{b^3 \tan ^{-1}\left (\sqrt{c+d x-1} \sqrt{c+d x+1}\right )}{d e^4} \]
Antiderivative was successfully verified.
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Rule 5866
Rule 12
Rule 5662
Rule 5748
Rule 5761
Rule 4180
Rule 2531
Rule 2282
Rule 6589
Rule 92
Rule 203
Rubi steps
\begin{align*} \int \frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{(c e+d e x)^4} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \cosh ^{-1}(x)\right )^3}{e^4 x^4} \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \cosh ^{-1}(x)\right )^3}{x^4} \, dx,x,c+d x\right )}{d e^4}\\ &=-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^3}+\frac{b \operatorname{Subst}\left (\int \frac{\left (a+b \cosh ^{-1}(x)\right )^2}{\sqrt{-1+x} x^3 \sqrt{1+x}} \, dx,x,c+d x\right )}{d e^4}\\ &=\frac{b \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^4 (c+d x)^2}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^3}+\frac{b \operatorname{Subst}\left (\int \frac{\left (a+b \cosh ^{-1}(x)\right )^2}{\sqrt{-1+x} x \sqrt{1+x}} \, dx,x,c+d x\right )}{2 d e^4}-\frac{b^2 \operatorname{Subst}\left (\int \frac{a+b \cosh ^{-1}(x)}{x^2} \, dx,x,c+d x\right )}{d e^4}\\ &=\frac{b^2 \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^4 (c+d x)}+\frac{b \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^4 (c+d x)^2}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^3}+\frac{b \operatorname{Subst}\left (\int (a+b x)^2 \text{sech}(x) \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 d e^4}-\frac{b^3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x} x \sqrt{1+x}} \, dx,x,c+d x\right )}{d e^4}\\ &=\frac{b^2 \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^4 (c+d x)}+\frac{b \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^4 (c+d x)^2}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^3}+\frac{b \left (a+b \cosh ^{-1}(c+d x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}-\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^4}+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^4}-\frac{b^3 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{-1+c+d x} \sqrt{1+c+d x}\right )}{d e^4}\\ &=\frac{b^2 \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^4 (c+d x)}+\frac{b \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^4 (c+d x)^2}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^3}+\frac{b \left (a+b \cosh ^{-1}(c+d x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}-\frac{b^3 \tan ^{-1}\left (\sqrt{-1+c+d x} \sqrt{1+c+d x}\right )}{d e^4}-\frac{i b^2 \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Li}_2\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}+\frac{i b^2 \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Li}_2\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}+\frac{\left (i b^3\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^4}-\frac{\left (i b^3\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^x\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^4}\\ &=\frac{b^2 \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^4 (c+d x)}+\frac{b \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^4 (c+d x)^2}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^3}+\frac{b \left (a+b \cosh ^{-1}(c+d x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}-\frac{b^3 \tan ^{-1}\left (\sqrt{-1+c+d x} \sqrt{1+c+d x}\right )}{d e^4}-\frac{i b^2 \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Li}_2\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}+\frac{i b^2 \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Li}_2\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}+\frac{\left (i b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}-\frac{\left (i b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}\\ &=\frac{b^2 \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^4 (c+d x)}+\frac{b \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^4 (c+d x)^2}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^3}+\frac{b \left (a+b \cosh ^{-1}(c+d x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}-\frac{b^3 \tan ^{-1}\left (\sqrt{-1+c+d x} \sqrt{1+c+d x}\right )}{d e^4}-\frac{i b^2 \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Li}_2\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}+\frac{i b^2 \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Li}_2\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}+\frac{i b^3 \text{Li}_3\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}-\frac{i b^3 \text{Li}_3\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}\\ \end{align*}
Mathematica [A] time = 2.16661, size = 504, normalized size = 1.7 \[ \frac{6 a b^2 \left (-i \text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(c+d x)}\right )+i \text{PolyLog}\left (2,i e^{-\cosh ^{-1}(c+d x)}\right )+\frac{1}{c+d x}-\frac{\cosh ^{-1}(c+d x)^2}{(c+d x)^3}+\frac{\sqrt{\frac{c+d x-1}{c+d x+1}} (c+d x+1) \cosh ^{-1}(c+d x)}{(c+d x)^2}-i \cosh ^{-1}(c+d x) \log \left (1-i e^{-\cosh ^{-1}(c+d x)}\right )+i \cosh ^{-1}(c+d x) \log \left (1+i e^{-\cosh ^{-1}(c+d x)}\right )\right )+b^3 \left (-3 i \left (2 \cosh ^{-1}(c+d x) \text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(c+d x)}\right )-2 \cosh ^{-1}(c+d x) \text{PolyLog}\left (2,i e^{-\cosh ^{-1}(c+d x)}\right )+2 \text{PolyLog}\left (3,-i e^{-\cosh ^{-1}(c+d x)}\right )-2 \text{PolyLog}\left (3,i e^{-\cosh ^{-1}(c+d x)}\right )+\cosh ^{-1}(c+d x)^2 \log \left (1-i e^{-\cosh ^{-1}(c+d x)}\right )-\cosh ^{-1}(c+d x)^2 \log \left (1+i e^{-\cosh ^{-1}(c+d x)}\right )-4 i \tan ^{-1}\left (\tanh \left (\frac{1}{2} \cosh ^{-1}(c+d x)\right )\right )\right )-\frac{2 \cosh ^{-1}(c+d x)^3}{(c+d x)^3}+\frac{3 \sqrt{\frac{c+d x-1}{c+d x+1}} (c+d x+1) \cosh ^{-1}(c+d x)^2}{(c+d x)^2}+\frac{6 \cosh ^{-1}(c+d x)}{c+d x}\right )+\frac{3 a^2 b \sqrt{c+d x-1} \sqrt{c+d x+1}}{(c+d x)^2}-3 a^2 b \tan ^{-1}\left (\frac{1}{\sqrt{c+d x-1} \sqrt{c+d x+1}}\right )-\frac{6 a^2 b \cosh ^{-1}(c+d x)}{(c+d x)^3}-\frac{2 a^3}{(c+d x)^3}}{6 d e^4} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.184, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\rm arccosh} \left (dx+c\right ) \right ) ^{3}}{ \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^{3} \operatorname{arcosh}\left (d x + c\right )^{3} + 3 \, a b^{2} \operatorname{arcosh}\left (d x + c\right )^{2} + 3 \, a^{2} b \operatorname{arcosh}\left (d x + c\right ) + a^{3}}{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^{3}}{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^{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{3 a 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{3 a^{2} 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 )}^{3}}{{\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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