Optimal. Leaf size=250 \[ -\frac {4 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d x \sqrt {d-c^2 d x^2}}-\frac {a+b \cosh ^{-1}(c x)}{3 d x^3 \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {d-c^2 d x^2}}{6 d^2 x^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 b c^3 \log (x) \sqrt {d-c^2 d x^2}}{3 d^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 \sqrt {d-c^2 d x^2} \log \left (1-c^2 x^2\right )}{2 d^2 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rubi [A] time = 0.46, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 6, 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, 1251, 893} \[ \frac {8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {4 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d x \sqrt {d-c^2 d x^2}}-\frac {a+b \cosh ^{-1}(c x)}{3 d x^3 \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{6 d x^2 \sqrt {d-c^2 d x^2}}-\frac {5 b c^3 \sqrt {c x-1} \sqrt {c x+1} \log (x)}{3 d \sqrt {d-c^2 d x^2}}-\frac {b c^3 \sqrt {c x-1} \sqrt {c x+1} \log \left (1-c^2 x^2\right )}{2 d \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 39
Rule 103
Rule 893
Rule 1251
Rule 5733
Rule 5798
Rubi steps
\begin {align*} \int \frac {a+b \cosh ^{-1}(c x)}{x^4 \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^4 (-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)}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {-1-4 c^2 x^2+8 c^4 x^4}{3 x^3 \left (1-c^2 x^2\right )} \, dx}{d \sqrt {d-c^2 d x^2}}\\ &=-\frac {a+b \cosh ^{-1}(c x)}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {-1-4 c^2 x^2+8 c^4 x^4}{x^3 \left (1-c^2 x^2\right )} \, dx}{3 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {a+b \cosh ^{-1}(c x)}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{3 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 {-1-4 c^2 x+8 c^4 x^2}{x^2 \left (1-c^2 x\right )} \, dx,x,x^2\right )}{6 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {a+b \cosh ^{-1}(c x)}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{3 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^2}-\frac {5 c^2}{x}-\frac {3 c^4}{-1+c^2 x}\right ) \, dx,x,x^2\right )}{6 d \sqrt {d-c^2 d x^2}}\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 d x^2 \sqrt {d-c^2 d x^2}}-\frac {a+b \cosh ^{-1}(c x)}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {5 b c^3 \sqrt {-1+c x} \sqrt {1+c x} \log (x)}{3 d \sqrt {d-c^2 d x^2}}-\frac {b c^3 \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 = 0.12, size = 161, normalized size = 0.64 \[ \frac {16 a c^4 x^4-8 a c^2 x^2-2 a-10 b c^3 x^3 \sqrt {c x-1} \sqrt {c x+1} \log (x)+2 b \left (8 c^4 x^4-4 c^2 x^2-1\right ) \cosh ^{-1}(c x)-3 b c^3 x^3 \sqrt {c x-1} \sqrt {c x+1} \log \left (1-c^2 x^2\right )+b c x \sqrt {c x-1} \sqrt {c x+1}}{6 d x^3 \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{c^{4} d^{2} x^{8} - 2 \, c^{2} d^{2} x^{6} + d^{2} x^{4}}, x\right ) \]
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 \[ \int \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.66, size = 1050, normalized size = 4.20 \[ -\frac {a}{3 d \,x^{3} \sqrt {-c^{2} d \,x^{2}+d}}-\frac {4 a \,c^{2}}{3 d x \sqrt {-c^{2} d \,x^{2}+d}}+\frac {8 a \,c^{4} x}{3 d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {16 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^{3}}{3 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {32 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{5} \left (c x +1\right ) \left (c x -1\right ) c^{8}}{3 d^{2} \left (8 c^{4} x^{4}-7 c^{2} x^{2}-1\right )}-\frac {32 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{7} c^{10}}{3 d^{2} \left (8 c^{4} x^{4}-7 c^{2} x^{2}-1\right )}-\frac {16 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{3} \left (c x +1\right ) \left (c x -1\right ) c^{6}}{3 d^{2} \left (8 c^{4} x^{4}-7 c^{2} x^{2}-1\right )}+\frac {16 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{5} c^{8}}{d^{2} \left (8 c^{4} x^{4}-7 c^{2} x^{2}-1\right )}+\frac {64 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{2} \mathrm {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5}}{3 d^{2} \left (8 c^{4} x^{4}-7 c^{2} x^{2}-1\right )}-\frac {64 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{3} \mathrm {arccosh}\left (c x \right ) c^{6}}{3 d^{2} \left (8 c^{4} x^{4}-7 c^{2} x^{2}-1\right )}-\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x \left (c x +1\right ) \left (c x -1\right ) c^{4}}{3 d^{2} \left (8 c^{4} x^{4}-7 c^{2} x^{2}-1\right )}-\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{3} c^{6}}{d^{2} \left (8 c^{4} x^{4}-7 c^{2} x^{2}-1\right )}+\frac {8 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c^{3}}{3 d^{2} \left (8 c^{4} x^{4}-7 c^{2} x^{2}-1\right )}+\frac {8 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x \,\mathrm {arccosh}\left (c x \right ) c^{4}}{d^{2} \left (8 c^{4} x^{4}-7 c^{2} x^{2}-1\right )}-\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x +1}\, \sqrt {c x -1}\, c^{3}}{3 d^{2} \left (8 c^{4} x^{4}-7 c^{2} x^{2}-1\right )}-\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x \,c^{4}}{3 d^{2} \left (8 c^{4} x^{4}-7 c^{2} x^{2}-1\right )}+\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) c^{2}}{d^{2} \left (8 c^{4} x^{4}-7 c^{2} x^{2}-1\right ) x}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x +1}\, \sqrt {c x -1}\, c}{6 d^{2} \left (8 c^{4} x^{4}-7 c^{2} x^{2}-1\right ) x^{2}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )}{3 d^{2} \left (8 c^{4} x^{4}-7 c^{2} x^{2}-1\right ) x^{3}}+\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 )^{2}-1\right ) c^{3}}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {5 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) c^{3}}{3 d^{2} \left (c^{2} x^{2}-1\right )} \]
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 \[ \frac {1}{3} \, {\left (\frac {8 \, c^{4} x}{\sqrt {-c^{2} d x^{2} + d} d} - \frac {4 \, c^{2}}{\sqrt {-c^{2} d x^{2} + d} d x} - \frac {1}{\sqrt {-c^{2} d x^{2} + d} d x^{3}}\right )} a + b \int \frac {\log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{4}}\,{d x} \]
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 \[ \int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^4\,{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{x^{4} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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