Optimal. Leaf size=279 \[ \frac {2 b c \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}+1\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {2 b c \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {e e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}+1\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{e (d+e x)}+\frac {2 b^2 c \text {Li}_2\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {2 b^2 c \text {Li}_2\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}} \]
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Rubi [A] time = 0.61, antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {5802, 5832, 3320, 2264, 2190, 2279, 2391} \[ \frac {2 b^2 c \text {PolyLog}\left (2,-\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {2 b^2 c \text {PolyLog}\left (2,-\frac {e e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e \sqrt {c^2 d^2-e^2}}+\frac {2 b c \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}+1\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {2 b c \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {e e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}+1\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{e (d+e x)} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2264
Rule 2279
Rule 2391
Rule 3320
Rule 5802
Rule 5832
Rubi steps
\begin {align*} \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{(d+e x)^2} \, dx &=-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{e (d+e x)}+\frac {(2 b c) \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x} (d+e x)} \, dx}{e}\\ &=-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{e (d+e x)}+\frac {(2 b c) \operatorname {Subst}\left (\int \frac {a+b x}{c d+e \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{e}\\ &=-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{e (d+e x)}+\frac {(4 b c) \operatorname {Subst}\left (\int \frac {e^x (a+b x)}{e+2 c d e^x+e e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )}{e}\\ &=-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{e (d+e x)}+\frac {(4 b c) \operatorname {Subst}\left (\int \frac {e^x (a+b x)}{2 c d-2 \sqrt {c^2 d^2-e^2}+2 e e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {c^2 d^2-e^2}}-\frac {(4 b c) \operatorname {Subst}\left (\int \frac {e^x (a+b x)}{2 c d+2 \sqrt {c^2 d^2-e^2}+2 e e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {c^2 d^2-e^2}}\\ &=-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{e (d+e x)}+\frac {2 b c \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {2 b c \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {\left (2 b^2 c\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 \sqrt {c^2 d^2-e^2}}+\frac {\left (2 b^2 c\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 \sqrt {c^2 d^2-e^2}}\\ &=-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{e (d+e x)}+\frac {2 b c \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {2 b c \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {\left (2 b^2 c\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 \sqrt {c^2 d^2-e^2}}+\frac {\left (2 b^2 c\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 \sqrt {c^2 d^2-e^2}}\\ &=-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{e (d+e x)}+\frac {2 b c \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {2 b c \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}+\frac {2 b^2 c \text {Li}_2\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {2 b^2 c \text {Li}_2\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}\\ \end {align*}
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Mathematica [C] time = 4.60, size = 961, normalized size = 3.44 \[ -\frac {a^2}{e (d+e x)}+2 b c \left (\frac {2 \tanh ^{-1}\left (\frac {\sqrt {(c d-e) e} \sqrt {\frac {c x-1}{c x+1}}}{\sqrt {e (c d+e)}}\right )}{\sqrt {(c d-e) e} \sqrt {e (c d+e)}}-\frac {\cosh ^{-1}(c x)}{e (c d+c e x)}\right ) a-\frac {b^2 c \left (\frac {\cosh ^{-1}(c x)^2}{c d+c e x}+\frac {2 \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 )}{\sqrt {e^2-c^2 d^2}}\right )}{e} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.98, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname {arcosh}\left (c x\right ) + a^{2}}{e^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 536, normalized size = 1.92 \[ -\frac {c \,a^{2}}{\left (c x e +c d \right ) e}-\frac {c \,b^{2} \mathrm {arccosh}\left (c x \right )^{2}}{e \left (c x e +c d \right )}+\frac {2 c \,b^{2} \mathrm {arccosh}\left (c x \right ) \ln \left (\frac {-\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) e -c d +\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}-\frac {2 c \,b^{2} \mathrm {arccosh}\left (c x \right ) \ln \left (\frac {\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) e +c d +\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}+\frac {2 c \,b^{2} \dilog \left (\frac {-\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) e -c d +\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}-\frac {2 c \,b^{2} \dilog \left (\frac {\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) e +c d +\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}-\frac {2 c a b \,\mathrm {arccosh}\left (c x \right )}{\left (c x e +c d \right ) e}-\frac {2 c a b \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{c x e +c d}\right )}{e^{2} \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{{\left (d+e\,x\right )}^2} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{\left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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