Optimal. Leaf size=158 \[ \frac {2 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}-\frac {a+b \cosh ^{-1}(c x)}{d x \sqrt {d-c^2 d x^2}}+\frac {b c \log (x) \sqrt {d-c^2 d x^2}}{d^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c \sqrt {d-c^2 d x^2} \log \left (1-c^2 x^2\right )}{2 d^2 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rubi [A] time = 0.40, antiderivative size = 159, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {5798, 103, 12, 39, 5733, 446, 72} \[ \frac {2 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}-\frac {a+b \cosh ^{-1}(c x)}{d x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \log (x)}{d \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \log \left (1-c^2 x^2\right )}{2 d \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 39
Rule 72
Rule 103
Rule 446
Rule 5733
Rule 5798
Rubi steps
\begin {align*} \int \frac {a+b \cosh ^{-1}(c x)}{x^2 \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^2 (-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)}{d x \sqrt {d-c^2 d x^2}}+\frac {2 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {-1+2 c^2 x^2}{x \left (1-c^2 x^2\right )} \, dx}{d \sqrt {d-c^2 d x^2}}\\ &=-\frac {a+b \cosh ^{-1}(c x)}{d x \sqrt {d-c^2 d x^2}}+\frac {2 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {-1+2 c^2 x}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )}{2 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {a+b \cosh ^{-1}(c x)}{d x \sqrt {d-c^2 d x^2}}+\frac {2 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{x}-\frac {c^2}{-1+c^2 x}\right ) \, dx,x,x^2\right )}{2 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {a+b \cosh ^{-1}(c x)}{d x \sqrt {d-c^2 d x^2}}+\frac {2 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {-1+c x} \sqrt {1+c x} \log (x)}{d \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {-1+c x} \sqrt {1+c x} \log \left (1-c^2 x^2\right )}{2 d \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 114, normalized size = 0.72 \[ \frac {4 a c^2 x^2-2 a-b c x \sqrt {c x-1} \sqrt {c x+1} \log \left (1-c^2 x^2\right )+2 b \left (2 c^2 x^2-1\right ) \cosh ^{-1}(c x)-2 b c x \sqrt {c x-1} \sqrt {c x+1} \log (x)}{2 d x \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{c^{4} d^{2} x^{6} - 2 \, c^{2} d^{2} x^{4} + d^{2} x^{2}}, x\right ) \]
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 \[ \int \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 242, normalized size = 1.53 \[ -\frac {a}{d x \sqrt {-c^{2} d \,x^{2}+d}}+\frac {2 a \,c^{2} x}{d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \mathrm {arccosh}\left (c x \right ) c}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) x \,c^{2}}{\left (c^{2} x^{2}-1\right ) d^{2}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )}{x \left (c^{2} x^{2}-1\right ) d^{2}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{4}-1\right ) c}{d^{2} \left (c^{2} x^{2}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.74, size = 144, normalized size = 0.91 \[ \frac {1}{2} \, b c {\left (\frac {\sqrt {-d} \log \left (c x + 1\right )}{d^{2}} + \frac {\sqrt {-d} \log \left (c x - 1\right )}{d^{2}} + \frac {2 \, \sqrt {-d} \log \relax (x)}{d^{2}}\right )} + {\left (\frac {2 \, c^{2} x}{\sqrt {-c^{2} d x^{2} + d} d} - \frac {1}{\sqrt {-c^{2} d x^{2} + d} d x}\right )} b \operatorname {arcosh}\left (c x\right ) + {\left (\frac {2 \, c^{2} x}{\sqrt {-c^{2} d x^{2} + d} d} - \frac {1}{\sqrt {-c^{2} d x^{2} + d} d x}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{x^{2} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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