Optimal. Leaf size=263 \[ -\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{e (d+e x)}+\frac {2 b c \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\sinh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e \sqrt {c^2 d^2+e^2}}-\frac {2 b c \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\sinh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e \sqrt {c^2 d^2+e^2}}+\frac {2 b^2 c \text {PolyLog}\left (2,-\frac {e e^{\sinh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e \sqrt {c^2 d^2+e^2}}-\frac {2 b^2 c \text {PolyLog}\left (2,-\frac {e e^{\sinh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e \sqrt {c^2 d^2+e^2}} \]
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Rubi [A]
time = 0.32, antiderivative size = 263, 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 = {5828, 5843,
3403, 2296, 2221, 2317, 2438} \begin {gather*} \frac {2 b c \left (a+b \sinh ^{-1}(c x)\right ) \log \left (\frac {e e^{\sinh ^{-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 \sinh ^{-1}(c x)\right ) \log \left (\frac {e e^{\sinh ^{-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 \sinh ^{-1}(c x)\right )^2}{e (d+e x)}+\frac {2 b^2 c \text {Li}_2\left (-\frac {e e^{\sinh ^{-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^{\sinh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e \sqrt {c^2 d^2+e^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 3403
Rule 5828
Rule 5843
Rubi steps
\begin {align*} \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{(d+e x)^2} \, dx &=-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{e (d+e x)}+\frac {(2 b c) \int \frac {a+b \sinh ^{-1}(c x)}{(d+e x) \sqrt {1+c^2 x^2}} \, dx}{e}\\ &=-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{e (d+e x)}+\frac {(2 b c) \text {Subst}\left (\int \frac {a+b x}{c d+e \sinh (x)} \, dx,x,\sinh ^{-1}(c x)\right )}{e}\\ &=-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{e (d+e x)}+\frac {(4 b c) \text {Subst}\left (\int \frac {e^x (a+b x)}{-e+2 c d e^x+e e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )}{e}\\ &=-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{e (d+e x)}+\frac {(4 b c) \text {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,\sinh ^{-1}(c x)\right )}{\sqrt {c^2 d^2+e^2}}-\frac {(4 b c) \text {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,\sinh ^{-1}(c x)\right )}{\sqrt {c^2 d^2+e^2}}\\ &=-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{e (d+e x)}+\frac {2 b c \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\sinh ^{-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 \sinh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\sinh ^{-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 ) \text {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,\sinh ^{-1}(c x)\right )}{e \sqrt {c^2 d^2+e^2}}+\frac {\left (2 b^2 c\right ) \text {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,\sinh ^{-1}(c x)\right )}{e \sqrt {c^2 d^2+e^2}}\\ &=-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{e (d+e x)}+\frac {2 b c \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\sinh ^{-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 \sinh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\sinh ^{-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 ) \text {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^{\sinh ^{-1}(c x)}\right )}{e \sqrt {c^2 d^2+e^2}}+\frac {\left (2 b^2 c\right ) \text {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^{\sinh ^{-1}(c x)}\right )}{e \sqrt {c^2 d^2+e^2}}\\ &=-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{e (d+e x)}+\frac {2 b c \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\sinh ^{-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 \sinh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\sinh ^{-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^{\sinh ^{-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^{\sinh ^{-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 [A]
time = 0.14, size = 191, normalized size = 0.73 \begin {gather*} \frac {-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{d+e x}+\frac {2 b c \left (\left (a+b \sinh ^{-1}(c x)\right ) \left (\log \left (1+\frac {e e^{\sinh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )-\log \left (1+\frac {e e^{\sinh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )\right )+b \text {PolyLog}\left (2,\frac {e e^{\sinh ^{-1}(c x)}}{-c d+\sqrt {c^2 d^2+e^2}}\right )-b \text {PolyLog}\left (2,-\frac {e e^{\sinh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )\right )}{\sqrt {c^2 d^2+e^2}}}{e} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 9.32, size = 549, normalized size = 2.09
method | result | size |
derivativedivides | \(\frac {-\frac {a^{2} c^{2}}{\left (c e x +c d \right ) e}-\frac {b^{2} c^{2} \arcsinh \left (c x \right )^{2}}{e \left (c e x +c d \right )}+\frac {2 b^{2} c^{2} \arcsinh \left (c x \right ) \ln \left (\frac {-c d -e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{-c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \sqrt {c^{2} d^{2}+e^{2}}}-\frac {2 b^{2} c^{2} \arcsinh \left (c x \right ) \ln \left (\frac {c d +e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \sqrt {c^{2} d^{2}+e^{2}}}+\frac {2 b^{2} c^{2} \dilog \left (\frac {-c d -e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{-c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \sqrt {c^{2} d^{2}+e^{2}}}-\frac {2 b^{2} c^{2} \dilog \left (\frac {c d +e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \sqrt {c^{2} d^{2}+e^{2}}}-\frac {2 a b \,c^{2} \arcsinh \left (c x \right )}{\left (c e x +c d \right ) e}-\frac {2 a b \,c^{2} \ln \left (\frac {\frac {2 c^{2} d^{2}+2 e^{2}}{e^{2}}-\frac {2 d c \left (c x +\frac {c d}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {\left (c x +\frac {c d}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {c d}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{c x +\frac {c d}{e}}\right )}{e^{2} \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}}{c}\) | \(549\) |
default | \(\frac {-\frac {a^{2} c^{2}}{\left (c e x +c d \right ) e}-\frac {b^{2} c^{2} \arcsinh \left (c x \right )^{2}}{e \left (c e x +c d \right )}+\frac {2 b^{2} c^{2} \arcsinh \left (c x \right ) \ln \left (\frac {-c d -e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{-c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \sqrt {c^{2} d^{2}+e^{2}}}-\frac {2 b^{2} c^{2} \arcsinh \left (c x \right ) \ln \left (\frac {c d +e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \sqrt {c^{2} d^{2}+e^{2}}}+\frac {2 b^{2} c^{2} \dilog \left (\frac {-c d -e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{-c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \sqrt {c^{2} d^{2}+e^{2}}}-\frac {2 b^{2} c^{2} \dilog \left (\frac {c d +e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \sqrt {c^{2} d^{2}+e^{2}}}-\frac {2 a b \,c^{2} \arcsinh \left (c x \right )}{\left (c e x +c d \right ) e}-\frac {2 a b \,c^{2} \ln \left (\frac {\frac {2 c^{2} d^{2}+2 e^{2}}{e^{2}}-\frac {2 d c \left (c x +\frac {c d}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {\left (c x +\frac {c d}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {c d}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{c x +\frac {c d}{e}}\right )}{e^{2} \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}}{c}\) | \(549\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\left (d + e x\right )^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (d+e\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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