Optimal. Leaf size=427 \[ -\frac {i b c^2 \sqrt {d-c^2 d x^2} \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {i b c^2 \sqrt {d-c^2 d x^2} \text {Li}_2\left (i e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2}+\frac {c^2 \sqrt {d-c^2 d x^2} \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {i b^2 c^2 \sqrt {d-c^2 d x^2} \text {Li}_3\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {i b^2 c^2 \sqrt {d-c^2 d x^2} \text {Li}_3\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {b^2 c^2 \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\sqrt {c x-1} \sqrt {c x+1}\right )}{\sqrt {c x-1} \sqrt {c x+1}} \]
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Rubi [A] time = 0.88, antiderivative size = 427, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {5798, 5738, 5662, 92, 205, 5761, 4180, 2531, 2282, 6589} \[ -\frac {i b c^2 \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {i b c^2 \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {i b^2 c^2 \sqrt {d-c^2 d x^2} \text {PolyLog}\left (3,-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {i b^2 c^2 \sqrt {d-c^2 d x^2} \text {PolyLog}\left (3,i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2}+\frac {c^2 \sqrt {d-c^2 d x^2} \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {b^2 c^2 \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\sqrt {c x-1} \sqrt {c x+1}\right )}{\sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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Rule 92
Rule 205
Rule 2282
Rule 2531
Rule 4180
Rule 5662
Rule 5738
Rule 5761
Rule 5798
Rule 6589
Rubi steps
\begin {align*} \int \frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x^3} \, dx &=\frac {\sqrt {d-c^2 d x^2} \int \frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{x^3} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2}+\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x^2} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (c^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2}+\frac {\left (c^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^2 \text {sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b^2 c^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2}+\frac {c^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (i b c^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (i b c^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b^2 c^3 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{c+c x^2} \, dx,x,\sqrt {-1+c x} \sqrt {1+c x}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2}+\frac {c^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {b^2 c^2 \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {i b c^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {i b c^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (i b^2 c^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (i b^2 c^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2}+\frac {c^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {b^2 c^2 \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {i b c^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {i b c^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (i b^2 c^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (i b^2 c^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2}+\frac {c^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {b^2 c^2 \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {i b c^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {i b c^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {i b^2 c^2 \sqrt {d-c^2 d x^2} \text {Li}_3\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {i b^2 c^2 \sqrt {d-c^2 d x^2} \text {Li}_3\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A] time = 85.47, size = 547, normalized size = 1.28 \[ \frac {1}{2} a \left (-\frac {a \sqrt {d-c^2 d x^2}}{x^2}+a c^2 \sqrt {d} \log \left (\sqrt {d} \sqrt {d-c^2 d x^2}+d\right )-a c^2 \sqrt {d} \log (x)+\frac {2 b d (c x+1) \left (i c^2 x^2 \sqrt {\frac {c x-1}{c x+1}} \text {Li}_2\left (-i e^{-\cosh ^{-1}(c x)}\right )-i c^2 x^2 \sqrt {\frac {c x-1}{c x+1}} \text {Li}_2\left (i e^{-\cosh ^{-1}(c x)}\right )+i c^2 x^2 \sqrt {\frac {c x-1}{c x+1}} \cosh ^{-1}(c x) \log \left (1-i e^{-\cosh ^{-1}(c x)}\right )-i c^2 x^2 \sqrt {\frac {c x-1}{c x+1}} \cosh ^{-1}(c x) \log \left (1+i e^{-\cosh ^{-1}(c x)}\right )+c x \sqrt {\frac {c x-1}{c x+1}}+c x \cosh ^{-1}(c x)-\cosh ^{-1}(c x)\right )}{x^2 \sqrt {d-c^2 d x^2}}\right )+\frac {1}{2} b^2 c^2 \sqrt {d-c^2 d x^2} \left (-\frac {\cosh ^{-1}(c x)^2}{c^2 x^2}+\frac {2 \sqrt {\frac {c x-1}{c x+1}} \cosh ^{-1}(c x)}{c x-c^2 x^2}-\frac {i \left (2 \cosh ^{-1}(c x) \text {Li}_2\left (-i e^{-\cosh ^{-1}(c x)}\right )-2 \cosh ^{-1}(c x) \text {Li}_2\left (i e^{-\cosh ^{-1}(c x)}\right )+2 \text {Li}_3\left (-i e^{-\cosh ^{-1}(c x)}\right )-2 \text {Li}_3\left (i e^{-\cosh ^{-1}(c x)}\right )+\cosh ^{-1}(c x)^2 \log \left (1-i e^{-\cosh ^{-1}(c x)}\right )-\cosh ^{-1}(c x)^2 \log \left (1+i e^{-\cosh ^{-1}(c x)}\right )+4 i \tan ^{-1}\left (\tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )\right )}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-c^{2} d x^{2} + d} {\left (b^{2} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname {arcosh}\left (c x\right ) + a^{2}\right )}}{x^{3}}, x\right ) \]
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 [F] time = 0.87, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{2} \sqrt {-c^{2} d \,x^{2}+d}}{x^{3}}\, dx \]
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 \[ \frac {1}{2} \, {\left (c^{2} \sqrt {d} \log \left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {d}}{{\left | x \right |}} + \frac {2 \, d}{{\left | x \right |}}\right ) - \sqrt {-c^{2} d x^{2} + d} c^{2} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{d x^{2}}\right )} a^{2} + \int \frac {\sqrt {-c^{2} d x^{2} + d} b^{2} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )^{2}}{x^{3}} + \frac {2 \, \sqrt {-c^{2} d x^{2} + d} a b \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{x^{3}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,\sqrt {d-c^2\,d\,x^2}}{x^3} \,d x \]
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 \[ \int \frac {\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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