Optimal. Leaf size=251 \[ \frac {b c d \sqrt {-1+c x} \sqrt {1+c x}}{2 x}-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}-\frac {i b e \sqrt {1-c^2 x^2} \text {ArcSin}(c x)^2}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e \sqrt {1-c^2 x^2} \text {ArcSin}(c x) \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac {b e \sqrt {1-c^2 x^2} \text {ArcSin}(c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {i b e \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}} \]
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Rubi [A]
time = 0.41, antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 11, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {14, 5958,
6874, 97, 2365, 2363, 4721, 3798, 2221, 2317, 2438} \begin {gather*} -\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+e \log (x) \left (a+b \cosh ^{-1}(c x)\right )-\frac {i b e \sqrt {1-c^2 x^2} \text {Li}_2\left (e^{2 i \text {ArcSin}(c x)}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {i b e \sqrt {1-c^2 x^2} \text {ArcSin}(c x)^2}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b e \sqrt {1-c^2 x^2} \text {ArcSin}(c x) \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {b e \sqrt {1-c^2 x^2} \log (x) \text {ArcSin}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {b c d \sqrt {c x-1} \sqrt {c x+1}}{2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 97
Rule 2221
Rule 2317
Rule 2363
Rule 2365
Rule 2438
Rule 3798
Rule 4721
Rule 5958
Rule 6874
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{x^3} \, dx &=-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-(b c) \int \frac {-\frac {d}{2 x^2}+e \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-(b c) \int \left (-\frac {d}{2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {e \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}\right ) \, dx\\ &=-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)+\frac {1}{2} (b c d) \int \frac {1}{x^2 \sqrt {-1+c x} \sqrt {1+c x}} \, dx-(b c e) \int \frac {\log (x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {b c d \sqrt {-1+c x} \sqrt {1+c x}}{2 x}-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac {\left (b c e \sqrt {1-c^2 x^2}\right ) \int \frac {\log (x)}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b c d \sqrt {-1+c x} \sqrt {1+c x}}{2 x}-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac {b e \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b e \sqrt {1-c^2 x^2}\right ) \int \frac {\sin ^{-1}(c x)}{x} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b c d \sqrt {-1+c x} \sqrt {1+c x}}{2 x}-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac {b e \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b e \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int x \cot (x) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b c d \sqrt {-1+c x} \sqrt {1+c x}}{2 x}-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}-\frac {i b e \sqrt {1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt {-1+c x} \sqrt {1+c x}}+e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac {b e \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 i b e \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b c d \sqrt {-1+c x} \sqrt {1+c x}}{2 x}-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}-\frac {i b e \sqrt {1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac {b e \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b e \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b c d \sqrt {-1+c x} \sqrt {1+c x}}{2 x}-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}-\frac {i b e \sqrt {1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac {b e \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (i b e \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b c d \sqrt {-1+c x} \sqrt {1+c x}}{2 x}-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}-\frac {i b e \sqrt {1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac {b e \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {i b e \sqrt {1-c^2 x^2} \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 101, normalized size = 0.40 \begin {gather*} \frac {-a d+b c d x \sqrt {-1+c x} \sqrt {1+c x}+b e x^2 \cosh ^{-1}(c x)^2-b \cosh ^{-1}(c x) \left (d-2 e x^2 \log \left (1+e^{-2 \cosh ^{-1}(c x)}\right )\right )+2 a e x^2 \log (x)-b e x^2 \text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 6.22, size = 147, normalized size = 0.59
method | result | size |
derivativedivides | \(c^{2} \left (-\frac {a d}{2 c^{2} x^{2}}+\frac {a e \ln \left (c x \right )}{c^{2}}-\frac {b e \mathrm {arccosh}\left (c x \right )^{2}}{2 c^{2}}+\frac {b d \sqrt {c x +1}\, \sqrt {c x -1}}{2 c x}-\frac {b d}{2}-\frac {b \,\mathrm {arccosh}\left (c x \right ) d}{2 c^{2} x^{2}}+\frac {b e \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{c^{2}}+\frac {b e \polylog \left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2 c^{2}}\right )\) | \(147\) |
default | \(c^{2} \left (-\frac {a d}{2 c^{2} x^{2}}+\frac {a e \ln \left (c x \right )}{c^{2}}-\frac {b e \mathrm {arccosh}\left (c x \right )^{2}}{2 c^{2}}+\frac {b d \sqrt {c x +1}\, \sqrt {c x -1}}{2 c x}-\frac {b d}{2}-\frac {b \,\mathrm {arccosh}\left (c x \right ) d}{2 c^{2} x^{2}}+\frac {b e \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{c^{2}}+\frac {b e \polylog \left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2 c^{2}}\right )\) | \(147\) |
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 {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x^{3}}\, 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 {acosh}\left (c\,x\right )\right )\,\left (e\,x^2+d\right )}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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