Optimal. Leaf size=248 \[ \frac {8 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {4 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \cosh ^{-1}(c x)}{d x \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {d-c^2 d x^2}}{6 d^3 \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )}+\frac {b c \log (x) \sqrt {d-c^2 d x^2}}{d^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 b c \sqrt {d-c^2 d x^2} \log \left (1-c^2 x^2\right )}{6 d^3 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rubi [A] time = 0.44, antiderivative size = 279, normalized size of antiderivative = 1.12, number of steps used = 6, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {5798, 103, 12, 40, 39, 5733, 1251, 893} \[ \frac {8 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {4 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 (1-c x) (c x+1) \sqrt {d-c^2 d x^2}}-\frac {a+b \cosh ^{-1}(c x)}{d^2 x (1-c x) (c x+1) \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{6 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \log (x)}{d^2 \sqrt {d-c^2 d x^2}}-\frac {5 b c \sqrt {c x-1} \sqrt {c x+1} \log \left (1-c^2 x^2\right )}{6 d^2 \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 39
Rule 40
Rule 103
Rule 893
Rule 1251
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 )^{5/2}} \, 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)^{5/2} (1+c x)^{5/2}} \, dx}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {8 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {a+b \cosh ^{-1}(c x)}{d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {4 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}-\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {3-12 c^2 x^2+8 c^4 x^4}{3 x \left (1-c^2 x^2\right )^2} \, dx}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {8 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {a+b \cosh ^{-1}(c x)}{d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {4 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}-\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {3-12 c^2 x^2+8 c^4 x^4}{x \left (1-c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {8 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {a+b \cosh ^{-1}(c x)}{d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {4 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 (1-c x) (1+c x) \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 {3-12 c^2 x+8 c^4 x^2}{x \left (1-c^2 x\right )^2} \, dx,x,x^2\right )}{6 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {8 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {a+b \cosh ^{-1}(c x)}{d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {4 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 (1-c x) (1+c x) \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 {3}{x}-\frac {c^2}{\left (-1+c^2 x\right )^2}+\frac {5 c^2}{-1+c^2 x}\right ) \, dx,x,x^2\right )}{6 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {8 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {a+b \cosh ^{-1}(c x)}{d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {4 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {-1+c x} \sqrt {1+c x} \log (x)}{d^2 \sqrt {d-c^2 d x^2}}-\frac {5 b c \sqrt {-1+c x} \sqrt {1+c x} \log \left (1-c^2 x^2\right )}{6 d^2 \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.38, size = 147, normalized size = 0.59 \[ \frac {\sqrt {c x-1} \sqrt {c x+1} \left (\frac {4 c^2 x \left (2 c^2 x^2-3\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 (c x-1)^{3/2} (c x+1)^{3/2}}+\frac {a+b \cosh ^{-1}(c x)}{x (c x-1)^{3/2} (c x+1)^{3/2}}-\frac {1}{6} b c \left (\frac {1}{c^2 x^2-1}+5 \log \left (1-c^2 x^2\right )+6 \log (x)\right )\right )}{d^2 \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{c^{6} d^{3} x^{8} - 3 \, c^{4} d^{3} x^{6} + 3 \, c^{2} d^{3} x^{4} - d^{3} x^{2}}, x\right ) \]
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 \[ \int \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.46, size = 1350, normalized size = 5.44 \[ \text {result too large to display} \]
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 \[ \frac {1}{3} \, a {\left (\frac {8 \, c^{2} x}{\sqrt {-c^{2} d x^{2} + d} d^{2}} + \frac {4 \, c^{2} x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d} - \frac {3}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x}\right )} + b \int \frac {\log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}\,{d x} \]
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 \[ \int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]
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 \[ \int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{x^{2} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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