Optimal. Leaf size=142 \[ c^4 d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {2 c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-b c^3 d^2 \sqrt {c x-1} \sqrt {c x+1}-\frac {11}{6} b c^3 d^2 \tan ^{-1}\left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {b c d^2 \sqrt {c x-1} \sqrt {c x+1}}{6 x^2} \]
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Rubi [A] time = 0.23, antiderivative size = 186, normalized size of antiderivative = 1.31, number of steps used = 8, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {270, 5731, 12, 520, 1251, 897, 1157, 388, 205} \[ c^4 d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {2 c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac {b c^3 d^2 \left (1-c^2 x^2\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d^2 \left (1-c^2 x^2\right )}{6 x^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {11 b c^3 d^2 \sqrt {c^2 x^2-1} \tan ^{-1}\left (\sqrt {c^2 x^2-1}\right )}{6 \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 270
Rule 388
Rule 520
Rule 897
Rule 1157
Rule 1251
Rule 5731
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{x^4} \, dx &=-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac {2 c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}+c^4 d^2 x \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac {d^2 \left (-1+6 c^2 x^2+3 c^4 x^4\right )}{3 x^3 \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac {2 c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}+c^4 d^2 x \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{3} \left (b c d^2\right ) \int \frac {-1+6 c^2 x^2+3 c^4 x^4}{x^3 \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac {2 c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}+c^4 d^2 x \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c d^2 \sqrt {-1+c^2 x^2}\right ) \int \frac {-1+6 c^2 x^2+3 c^4 x^4}{x^3 \sqrt {-1+c^2 x^2}} \, dx}{3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac {2 c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}+c^4 d^2 x \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c d^2 \sqrt {-1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {-1+6 c^2 x+3 c^4 x^2}{x^2 \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{6 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac {2 c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}+c^4 d^2 x \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b d^2 \sqrt {-1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {8+12 x^2+3 x^4}{\left (\frac {1}{c^2}+\frac {x^2}{c^2}\right )^2} \, dx,x,\sqrt {-1+c^2 x^2}\right )}{3 c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d^2 \left (1-c^2 x^2\right )}{6 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac {2 c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}+c^4 d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {\left (b c d^2 \sqrt {-1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {-17-6 x^2}{\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {-1+c^2 x^2}\right )}{6 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b c^3 d^2 \left (1-c^2 x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \left (1-c^2 x^2\right )}{6 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac {2 c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}+c^4 d^2 x \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (11 b c d^2 \sqrt {-1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {-1+c^2 x^2}\right )}{6 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b c^3 d^2 \left (1-c^2 x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \left (1-c^2 x^2\right )}{6 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac {2 c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}+c^4 d^2 x \left (a+b \cosh ^{-1}(c x)\right )-\frac {11 b c^3 d^2 \sqrt {-1+c^2 x^2} \tan ^{-1}\left (\sqrt {-1+c^2 x^2}\right )}{6 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 135, normalized size = 0.95 \[ \frac {d^2 \left (6 a c^4 x^4+12 a c^2 x^2-2 a-6 b c^3 x^3 \sqrt {c x-1} \sqrt {c x+1}+11 b c^3 x^3 \tan ^{-1}\left (\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )+2 b \left (3 c^4 x^4+6 c^2 x^2-1\right ) \cosh ^{-1}(c x)+b c x \sqrt {c x-1} \sqrt {c x+1}\right )}{6 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 213, normalized size = 1.50 \[ \frac {6 \, a c^{4} d^{2} x^{4} - 22 \, b c^{3} d^{2} x^{3} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 12 \, a c^{2} d^{2} x^{2} - 2 \, {\left (3 \, b c^{4} + 6 \, b c^{2} - b\right )} d^{2} x^{3} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - 2 \, a d^{2} + 2 \, {\left (3 \, b c^{4} d^{2} x^{4} + 6 \, b c^{2} d^{2} x^{2} - {\left (3 \, b c^{4} + 6 \, b c^{2} - b\right )} d^{2} x^{3} - b d^{2}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (6 \, b c^{3} d^{2} x^{3} - b c d^{2} x\right )} \sqrt {c^{2} x^{2} - 1}}{6 \, x^{3}} \]
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 [A] time = 0.02, size = 167, normalized size = 1.18 \[ c^{4} d^{2} a x +\frac {2 c^{2} d^{2} a}{x}-\frac {d^{2} a}{3 x^{3}}+c^{4} d^{2} b \,\mathrm {arccosh}\left (c x \right ) x +\frac {2 c^{2} d^{2} b \,\mathrm {arccosh}\left (c x \right )}{x}-\frac {d^{2} b \,\mathrm {arccosh}\left (c x \right )}{3 x^{3}}-b \,c^{3} d^{2} \sqrt {c x -1}\, \sqrt {c x +1}+\frac {11 c^{3} d^{2} b \sqrt {c x -1}\, \sqrt {c x +1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{6 \sqrt {c^{2} x^{2}-1}}+\frac {b c \,d^{2} \sqrt {c x -1}\, \sqrt {c x +1}}{6 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 137, normalized size = 0.96 \[ a c^{4} d^{2} x + {\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b c^{3} d^{2} + 2 \, {\left (c \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arcosh}\left (c x\right )}{x}\right )} b c^{2} d^{2} - \frac {1}{6} \, {\left ({\left (c^{2} \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) - \frac {\sqrt {c^{2} x^{2} - 1}}{x^{2}}\right )} c + \frac {2 \, \operatorname {arcosh}\left (c x\right )}{x^{3}}\right )} b d^{2} + \frac {2 \, a c^{2} d^{2}}{x} - \frac {a d^{2}}{3 \, x^{3}} \]
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 \[ \int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^2}{x^4} \,d x \]
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 \[ d^{2} \left (\int a c^{4}\, dx + \int \frac {a}{x^{4}}\, dx + \int \left (- \frac {2 a c^{2}}{x^{2}}\right )\, dx + \int b c^{4} \operatorname {acosh}{\left (c x \right )}\, dx + \int \frac {b \operatorname {acosh}{\left (c x \right )}}{x^{4}}\, dx + \int \left (- \frac {2 b c^{2} \operatorname {acosh}{\left (c x \right )}}{x^{2}}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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