Optimal. Leaf size=273 \[ \frac {2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \text {ArcTan}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text {PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {d-c^2 d x^2}}+\frac {2 i b^2 \sqrt {-1+c x} \sqrt {1+c x} \text {PolyLog}\left (3,-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {d-c^2 d x^2}}-\frac {2 i b^2 \sqrt {-1+c x} \sqrt {1+c x} \text {PolyLog}\left (3,i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {d-c^2 d x^2}} \]
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Rubi [A]
time = 0.17, antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {5946, 4265,
2611, 2320, 6724} \begin {gather*} \frac {2 \sqrt {c x-1} \sqrt {c x+1} \text {ArcTan}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (i e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {d-c^2 d x^2}}+\frac {2 i b^2 \sqrt {c x-1} \sqrt {c x+1} \text {Li}_3\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {d-c^2 d x^2}}-\frac {2 i b^2 \sqrt {c x-1} \sqrt {c x+1} \text {Li}_3\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {d-c^2 d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2320
Rule 2611
Rule 4265
Rule 5946
Rule 6724
Rubi steps
\begin {align*} \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{x \sqrt {d-c^2 d x^2}} \, dx &=\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {d-c^2 d x^2}}\\ &=\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int (a+b x)^2 \text {sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {d-c^2 d x^2}}\\ &=\frac {2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt {d-c^2 d x^2}}-\frac {\left (2 i b \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int (a+b x) \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {d-c^2 d x^2}}+\frac {\left (2 i b \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int (a+b x) \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {d-c^2 d x^2}}\\ &=\frac {2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {d-c^2 d x^2}}+\frac {\left (2 i b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \text {Li}_2\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {d-c^2 d x^2}}-\frac {\left (2 i b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \text {Li}_2\left (i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {d-c^2 d x^2}}\\ &=\frac {2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {d-c^2 d x^2}}+\frac {\left (2 i b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{\sqrt {d-c^2 d x^2}}-\frac {\left (2 i b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{\sqrt {d-c^2 d x^2}}\\ &=\frac {2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {d-c^2 d x^2}}+\frac {2 i b^2 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_3\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {d-c^2 d x^2}}-\frac {2 i b^2 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_3\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.38, size = 315, normalized size = 1.15 \begin {gather*} \frac {a^2 \log (c x)}{\sqrt {d}}-\frac {a^2 \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )}{\sqrt {d}}-\frac {2 i a b \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (\cosh ^{-1}(c x) \left (\log \left (1-i e^{-\cosh ^{-1}(c x)}\right )-\log \left (1+i e^{-\cosh ^{-1}(c x)}\right )\right )+\text {PolyLog}\left (2,-i e^{-\cosh ^{-1}(c x)}\right )-\text {PolyLog}\left (2,i e^{-\cosh ^{-1}(c x)}\right )\right )}{\sqrt {d-c^2 d x^2}}+\frac {i b^2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (-\cosh ^{-1}(c x)^2 \left (\log \left (1-i e^{-\cosh ^{-1}(c x)}\right )-\log \left (1+i e^{-\cosh ^{-1}(c x)}\right )\right )-2 \cosh ^{-1}(c x) \left (\text {PolyLog}\left (2,-i e^{-\cosh ^{-1}(c x)}\right )-\text {PolyLog}\left (2,i e^{-\cosh ^{-1}(c x)}\right )\right )-2 \text {PolyLog}\left (3,-i e^{-\cosh ^{-1}(c x)}\right )+2 \text {PolyLog}\left (3,i e^{-\cosh ^{-1}(c x)}\right )\right )}{\sqrt {d-c^2 d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{2}}{x \sqrt {-c^{2} d \,x^{2}+d}}\, dx\]
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 )^{2}}{x \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, 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 )}^2}{x\,\sqrt {d-c^2\,d\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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