Optimal. Leaf size=352 \[ \frac {c^2 \log (d+e x)}{e \left (c^2 d^2-e^2\right )}-\frac {c \sqrt {-\frac {1-c x}{c x+1}} (c x+1) \cosh ^{-1}(c x)}{\left (c^2 d^2-e^2\right ) (d+e x)}+\frac {c^3 d \text {Li}_2\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {c^3 d \text {Li}_2\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}+\frac {c^3 d \cosh ^{-1}(c x) \log \left (\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}+1\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {c^3 d \cosh ^{-1}(c x) \log \left (\frac {e e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}+1\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {\cosh ^{-1}(c x)^2}{2 e (d+e x)^2} \]
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Rubi [A] time = 0.69, antiderivative size = 352, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {5802, 5832, 3324, 3320, 2264, 2190, 2279, 2391, 2668, 31} \[ \frac {c^3 d \text {PolyLog}\left (2,-\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {c^3 d \text {PolyLog}\left (2,-\frac {e e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}+\frac {c^2 \log (d+e x)}{e \left (c^2 d^2-e^2\right )}-\frac {c \sqrt {-\frac {1-c x}{c x+1}} (c x+1) \cosh ^{-1}(c x)}{\left (c^2 d^2-e^2\right ) (d+e x)}+\frac {c^3 d \cosh ^{-1}(c x) \log \left (\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}+1\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {c^3 d \cosh ^{-1}(c x) \log \left (\frac {e e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}+1\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {\cosh ^{-1}(c x)^2}{2 e (d+e x)^2} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2190
Rule 2264
Rule 2279
Rule 2391
Rule 2668
Rule 3320
Rule 3324
Rule 5802
Rule 5832
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(c x)^2}{(d+e x)^3} \, dx &=-\frac {\cosh ^{-1}(c x)^2}{2 e (d+e x)^2}+\frac {c \int \frac {\cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x} (d+e x)^2} \, dx}{e}\\ &=-\frac {\cosh ^{-1}(c x)^2}{2 e (d+e x)^2}+\frac {c^2 \operatorname {Subst}\left (\int \frac {x}{(c d+e \cosh (x))^2} \, dx,x,\cosh ^{-1}(c x)\right )}{e}\\ &=-\frac {c \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \cosh ^{-1}(c x)}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\cosh ^{-1}(c x)^2}{2 e (d+e x)^2}+\frac {c^2 \operatorname {Subst}\left (\int \frac {\sinh (x)}{c d+e \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{c^2 d^2-e^2}+\frac {\left (c^3 d\right ) \operatorname {Subst}\left (\int \frac {x}{c d+e \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{e \left (c^2 d^2-e^2\right )}\\ &=-\frac {c \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \cosh ^{-1}(c x)}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\cosh ^{-1}(c x)^2}{2 e (d+e x)^2}+\frac {c^2 \operatorname {Subst}\left (\int \frac {1}{c d+x} \, dx,x,c e x\right )}{e \left (c^2 d^2-e^2\right )}+\frac {\left (2 c^3 d\right ) \operatorname {Subst}\left (\int \frac {e^x x}{e+2 c d e^x+e e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )}{e \left (c^2 d^2-e^2\right )}\\ &=-\frac {c \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \cosh ^{-1}(c x)}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\cosh ^{-1}(c x)^2}{2 e (d+e x)^2}+\frac {c^2 \log (d+e x)}{e \left (c^2 d^2-e^2\right )}+\frac {\left (2 c^3 d\right ) \operatorname {Subst}\left (\int \frac {e^x x}{2 c d-2 \sqrt {c^2 d^2-e^2}+2 e e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{\left (c^2 d^2-e^2\right )^{3/2}}-\frac {\left (2 c^3 d\right ) \operatorname {Subst}\left (\int \frac {e^x x}{2 c d+2 \sqrt {c^2 d^2-e^2}+2 e e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{\left (c^2 d^2-e^2\right )^{3/2}}\\ &=-\frac {c \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \cosh ^{-1}(c x)}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\cosh ^{-1}(c x)^2}{2 e (d+e x)^2}+\frac {c^3 d \cosh ^{-1}(c x) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {c^3 d \cosh ^{-1}(c x) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}+\frac {c^2 \log (d+e x)}{e \left (c^2 d^2-e^2\right )}-\frac {\left (c^3 d\right ) \operatorname {Subst}\left (\int \log \left (1+\frac {2 e e^x}{2 c d-2 \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}+\frac {\left (c^3 d\right ) \operatorname {Subst}\left (\int \log \left (1+\frac {2 e e^x}{2 c d+2 \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}\\ &=-\frac {c \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \cosh ^{-1}(c x)}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\cosh ^{-1}(c x)^2}{2 e (d+e x)^2}+\frac {c^3 d \cosh ^{-1}(c x) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {c^3 d \cosh ^{-1}(c x) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}+\frac {c^2 \log (d+e x)}{e \left (c^2 d^2-e^2\right )}-\frac {\left (c^3 d\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 e x}{2 c d-2 \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}+\frac {\left (c^3 d\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 e x}{2 c d+2 \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}\\ &=-\frac {c \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \cosh ^{-1}(c x)}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\cosh ^{-1}(c x)^2}{2 e (d+e x)^2}+\frac {c^3 d \cosh ^{-1}(c x) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {c^3 d \cosh ^{-1}(c x) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}+\frac {c^2 \log (d+e x)}{e \left (c^2 d^2-e^2\right )}+\frac {c^3 d \text {Li}_2\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {c^3 d \text {Li}_2\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}\\ \end {align*}
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Mathematica [C] time = 4.53, size = 936, normalized size = 2.66 \[ c^2 \left (-\frac {\cosh ^{-1}(c x)^2}{2 e (c d+c e x)^2}-\frac {\sqrt {\frac {c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)}{(c d-e) (c d+e) (c d+c e x)}+\frac {\log \left (\frac {e x}{d}+1\right )}{c^2 d^2 e-e^3}+\frac {c d \left (2 \cosh ^{-1}(c x) \tan ^{-1}\left (\frac {(c d+e) \coth \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {e^2-c^2 d^2}}\right )-2 i \cos ^{-1}\left (-\frac {c d}{e}\right ) \tan ^{-1}\left (\frac {(e-c d) \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {e^2-c^2 d^2}}\right )+\left (\cos ^{-1}\left (-\frac {c d}{e}\right )+2 \left (\tan ^{-1}\left (\frac {(c d+e) \coth \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {e^2-c^2 d^2}}\right )+\tan ^{-1}\left (\frac {(e-c d) \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {e^2-c^2 d^2}}\right )\right )\right ) \log \left (\frac {\sqrt {e^2-c^2 d^2} e^{-\frac {1}{2} \cosh ^{-1}(c x)}}{\sqrt {2} \sqrt {e} \sqrt {c d+c e x}}\right )+\left (\cos ^{-1}\left (-\frac {c d}{e}\right )-2 \left (\tan ^{-1}\left (\frac {(c d+e) \coth \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {e^2-c^2 d^2}}\right )+\tan ^{-1}\left (\frac {(e-c d) \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {e^2-c^2 d^2}}\right )\right )\right ) \log \left (\frac {\sqrt {e^2-c^2 d^2} e^{\frac {1}{2} \cosh ^{-1}(c x)}}{\sqrt {2} \sqrt {e} \sqrt {c d+c e x}}\right )-\left (\cos ^{-1}\left (-\frac {c d}{e}\right )+2 \tan ^{-1}\left (\frac {(e-c d) \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {e^2-c^2 d^2}}\right )\right ) \log \left (\frac {(c d+e) \left (c d-e+i \sqrt {e^2-c^2 d^2}\right ) \left (\tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )-1\right )}{e \left (c d+e+i \sqrt {e^2-c^2 d^2} \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}\right )-\left (\cos ^{-1}\left (-\frac {c d}{e}\right )-2 \tan ^{-1}\left (\frac {(e-c d) \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {e^2-c^2 d^2}}\right )\right ) \log \left (\frac {(c d+e) \left (-c d+e+i \sqrt {e^2-c^2 d^2}\right ) \left (\tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )+1\right )}{e \left (c d+e+i \sqrt {e^2-c^2 d^2} \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}\right )+i \left (\text {Li}_2\left (\frac {\left (c d-i \sqrt {e^2-c^2 d^2}\right ) \left (c d+e-i \sqrt {e^2-c^2 d^2} \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}{e \left (c d+e+i \sqrt {e^2-c^2 d^2} \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}\right )-\text {Li}_2\left (\frac {\left (c d+i \sqrt {e^2-c^2 d^2}\right ) \left (c d+e-i \sqrt {e^2-c^2 d^2} \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}{e \left (c d+e+i \sqrt {e^2-c^2 d^2} \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}\right )\right )\right )}{e \left (e^2-c^2 d^2\right )^{3/2}}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arcosh}\left (c x\right )^{2}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.52, size = 766, normalized size = 2.18 \[ -\frac {c^{3} \mathrm {arccosh}\left (c x \right ) e \sqrt {c x +1}\, \sqrt {c x -1}\, x}{\left (c x e +c d \right )^{2} \left (c^{2} d^{2}-e^{2}\right )}+\frac {c^{4} \mathrm {arccosh}\left (c x \right ) e \,x^{2}}{\left (c x e +c d \right )^{2} \left (c^{2} d^{2}-e^{2}\right )}-\frac {c^{4} \mathrm {arccosh}\left (c x \right )^{2} d^{2}}{2 e \left (c x e +c d \right )^{2} \left (c^{2} d^{2}-e^{2}\right )}-\frac {c^{3} \mathrm {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, d}{\left (c x e +c d \right )^{2} \left (c^{2} d^{2}-e^{2}\right )}+\frac {2 c^{4} \mathrm {arccosh}\left (c x \right ) d x}{\left (c x e +c d \right )^{2} \left (c^{2} d^{2}-e^{2}\right )}+\frac {c^{4} \mathrm {arccosh}\left (c x \right ) d^{2}}{e \left (c x e +c d \right )^{2} \left (c^{2} d^{2}-e^{2}\right )}+\frac {c^{2} \mathrm {arccosh}\left (c x \right )^{2} e}{2 \left (c x e +c d \right )^{2} \left (c^{2} d^{2}-e^{2}\right )}+\frac {c^{3} \mathrm {arccosh}\left (c x \right ) \ln \left (\frac {-\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) e -c d +\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right ) d}{\left (c^{2} d^{2}-e^{2}\right )^{\frac {3}{2}} e}-\frac {c^{3} \mathrm {arccosh}\left (c x \right ) \ln \left (\frac {\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) e +c d +\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right ) d}{\left (c^{2} d^{2}-e^{2}\right )^{\frac {3}{2}} e}+\frac {c^{3} \dilog \left (\frac {-\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) e -c d +\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right ) d}{\left (c^{2} d^{2}-e^{2}\right )^{\frac {3}{2}} e}-\frac {c^{3} \dilog \left (\frac {\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) e +c d +\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right ) d}{\left (c^{2} d^{2}-e^{2}\right )^{\frac {3}{2}} e}+\frac {c^{2} \ln \left (2 c d \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2} e +e \right )}{\left (c^{2} d^{2}-e^{2}\right ) e}-\frac {2 c^{2} \ln \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{\left (c^{2} d^{2}-e^{2}\right ) e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {acosh}\left (c\,x\right )}^2}{{\left (d+e\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acosh}^{2}{\left (c x \right )}}{\left (d + e x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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