Optimal. Leaf size=329 \[ \frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d x \sqrt {d-c^2 d x^2}}+\frac {3 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 d \sqrt {d-c^2 d x^2}}-\frac {a+b \cosh ^{-1}(c x)}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {3 c^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text {ArcTan}\left (e^{\cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {b c^2 \sqrt {-1+c x} \sqrt {1+c x} \tanh ^{-1}(c x)}{d \sqrt {d-c^2 d x^2}}-\frac {3 i b c^2 \sqrt {-1+c x} \sqrt {1+c x} \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right )}{2 d \sqrt {d-c^2 d x^2}}+\frac {3 i b c^2 \sqrt {-1+c x} \sqrt {1+c x} \text {PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right )}{2 d \sqrt {d-c^2 d x^2}} \]
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Rubi [A]
time = 0.29, antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5932, 5936,
5946, 4265, 2317, 2438, 35, 213, 84} \begin {gather*} \frac {3 c^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 )}{d \sqrt {d-c^2 d x^2}}+\frac {3 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 d \sqrt {d-c^2 d x^2}}-\frac {a+b \cosh ^{-1}(c x)}{2 d x^2 \sqrt {d-c^2 d x^2}}-\frac {3 i b c^2 \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{2 d \sqrt {d-c^2 d x^2}}+\frac {3 i b c^2 \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{2 d \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 d x \sqrt {d-c^2 d x^2}}+\frac {b c^2 \sqrt {c x-1} \sqrt {c x+1} \tanh ^{-1}(c x)}{d \sqrt {d-c^2 d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 35
Rule 84
Rule 213
Rule 2317
Rule 2438
Rule 4265
Rule 5932
Rule 5936
Rule 5946
Rubi steps
\begin {align*} \int \frac {a+b \cosh ^{-1}(c x)}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx &=-\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x^3 (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{d \sqrt {d-c^2 d x^2}}\\ &=-\frac {a+b \cosh ^{-1}(c x)}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{x^2 \left (-1+c^2 x^2\right )} \, dx}{2 d \sqrt {d-c^2 d x^2}}-\frac {\left (3 c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 d \sqrt {d-c^2 d x^2}}\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d x \sqrt {d-c^2 d x^2}}+\frac {3 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 d \sqrt {d-c^2 d x^2}}-\frac {a+b \cosh ^{-1}(c x)}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {\left (3 c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 d \sqrt {d-c^2 d x^2}}+\frac {\left (b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{-1+c^2 x^2} \, dx}{2 d \sqrt {d-c^2 d x^2}}-\frac {\left (3 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{-1+c^2 x^2} \, dx}{2 d \sqrt {d-c^2 d x^2}}\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d x \sqrt {d-c^2 d x^2}}+\frac {3 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 d \sqrt {d-c^2 d x^2}}-\frac {a+b \cosh ^{-1}(c x)}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {b c^2 \sqrt {-1+c x} \sqrt {1+c x} \tanh ^{-1}(c x)}{d \sqrt {d-c^2 d x^2}}+\frac {\left (3 c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int (a+b x) \text {sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{2 d \sqrt {d-c^2 d x^2}}\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d x \sqrt {d-c^2 d x^2}}+\frac {3 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 d \sqrt {d-c^2 d x^2}}-\frac {a+b \cosh ^{-1}(c x)}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {3 c^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {b c^2 \sqrt {-1+c x} \sqrt {1+c x} \tanh ^{-1}(c x)}{d \sqrt {d-c^2 d x^2}}-\frac {\left (3 i b c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 d \sqrt {d-c^2 d x^2}}+\frac {\left (3 i b c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 d \sqrt {d-c^2 d x^2}}\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d x \sqrt {d-c^2 d x^2}}+\frac {3 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 d \sqrt {d-c^2 d x^2}}-\frac {a+b \cosh ^{-1}(c x)}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {3 c^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {b c^2 \sqrt {-1+c x} \sqrt {1+c x} \tanh ^{-1}(c x)}{d \sqrt {d-c^2 d x^2}}-\frac {\left (3 i b c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 d \sqrt {d-c^2 d x^2}}+\frac {\left (3 i b c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 d \sqrt {d-c^2 d x^2}}\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d x \sqrt {d-c^2 d x^2}}+\frac {3 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 d \sqrt {d-c^2 d x^2}}-\frac {a+b \cosh ^{-1}(c x)}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {3 c^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {b c^2 \sqrt {-1+c x} \sqrt {1+c x} \tanh ^{-1}(c x)}{d \sqrt {d-c^2 d x^2}}-\frac {3 i b c^2 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{2 d \sqrt {d-c^2 d x^2}}+\frac {3 i b c^2 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{2 d \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A]
time = 2.86, size = 405, normalized size = 1.23 \begin {gather*} \frac {1}{2} \left (-\frac {a \left (-1+3 c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{d^2 x^2 \left (-1+c^2 x^2\right )}+\frac {3 a c^2 \log (x)}{d^{3/2}}-\frac {3 a c^2 \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )}{d^{3/2}}-\frac {b c^2 \left (-\frac {\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}{c x}+\left (-1+\frac {1}{c^2 x^2}\right ) \cosh ^{-1}(c x)-2 \cosh ^{-1}(c x) \cosh ^2\left (\frac {1}{2} \cosh ^{-1}(c x)\right )+3 i \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \cosh ^{-1}(c x) \log \left (1-i e^{-\cosh ^{-1}(c x)}\right )-3 i \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \cosh ^{-1}(c x) \log \left (1+i e^{-\cosh ^{-1}(c x)}\right )+2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )+3 i \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {PolyLog}\left (2,-i e^{-\cosh ^{-1}(c x)}\right )-3 i \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {PolyLog}\left (2,i e^{-\cosh ^{-1}(c x)}\right )+2 \cosh ^{-1}(c x) \sinh ^2\left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}{d \sqrt {d-c^2 d x^2}}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 647 vs. \(2 (323 ) = 646\).
time = 4.92, size = 648, normalized size = 1.97
method | result | size |
default | \(-\frac {a}{2 d \,x^{2} \sqrt {-c^{2} d \,x^{2}+d}}+\frac {3 a \,c^{2}}{2 d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {3 a \,c^{2} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{2 d^{\frac {3}{2}}}-\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) c^{2}}{2 d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x +1}\, \sqrt {c x -1}\, c}{2 d^{2} \left (c^{2} x^{2}-1\right ) x}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )}{2 d^{2} \left (c^{2} x^{2}-1\right ) x^{2}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}-1\right ) c^{2}}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c^{2}}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {3 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \mathrm {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d^{2} \left (c^{2} x^{2}-1\right )}-\frac {3 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \mathrm {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {3 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \dilog \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d^{2} \left (c^{2} x^{2}-1\right )}-\frac {3 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \dilog \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d^{2} \left (c^{2} x^{2}-1\right )}\) | \(648\) |
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 {a + b \operatorname {acosh}{\left (c x \right )}}{x^{3} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{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 {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^3\,{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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