Optimal. Leaf size=135 \[ \frac {1}{3} c^4 d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )-2 c^2 d^2 x \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}-\frac {1}{9} b c d^2 (c x-1)^{3/2} (c x+1)^{3/2}+\frac {5}{3} b c d^2 \sqrt {c x-1} \sqrt {c x+1}+b c d^2 \tan ^{-1}\left (\sqrt {c x-1} \sqrt {c x+1}\right ) \]
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Rubi [A] time = 0.23, antiderivative size = 182, normalized size of antiderivative = 1.35, number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {270, 5731, 12, 520, 1251, 897, 1153, 205} \[ \frac {1}{3} c^4 d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )-2 c^2 d^2 x \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}-\frac {b c d^2 \left (1-c^2 x^2\right )^2}{9 \sqrt {c x-1} \sqrt {c x+1}}-\frac {5 b c d^2 \left (1-c^2 x^2\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c d^2 \sqrt {c^2 x^2-1} \tan ^{-1}\left (\sqrt {c^2 x^2-1}\right )}{\sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 270
Rule 520
Rule 897
Rule 1153
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^2} \, dx &=-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}-2 c^2 d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} c^4 d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac {d^2 \left (-3-6 c^2 x^2+c^4 x^4\right )}{3 x \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}-2 c^2 d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} c^4 d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{3} \left (b c d^2\right ) \int \frac {-3-6 c^2 x^2+c^4 x^4}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}-2 c^2 d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} c^4 d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c d^2 \sqrt {-1+c^2 x^2}\right ) \int \frac {-3-6 c^2 x^2+c^4 x^4}{x \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 )}{x}-2 c^2 d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} c^4 d^2 x^3 \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 {-3-6 c^2 x+c^4 x^2}{x \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 )}{x}-2 c^2 d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} c^4 d^2 x^3 \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 {-8-4 x^2+x^4}{\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {-1+c^2 x^2}\right )}{3 c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}-2 c^2 d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} c^4 d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b d^2 \sqrt {-1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \left (-5 c^2+c^2 x^2-\frac {3}{\frac {1}{c^2}+\frac {x^2}{c^2}}\right ) \, dx,x,\sqrt {-1+c^2 x^2}\right )}{3 c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {5 b c d^2 \left (1-c^2 x^2\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \left (1-c^2 x^2\right )^2}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}-2 c^2 d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} c^4 d^2 x^3 \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 {1}{\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {-1+c^2 x^2}\right )}{c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {5 b c d^2 \left (1-c^2 x^2\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \left (1-c^2 x^2\right )^2}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}-2 c^2 d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} c^4 d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {b c d^2 \sqrt {-1+c^2 x^2} \tan ^{-1}\left (\sqrt {-1+c^2 x^2}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 131, normalized size = 0.97 \[ \frac {d^2 \left (3 a c^4 x^4-18 a c^2 x^2-9 a-b c^3 x^3 \sqrt {c x-1} \sqrt {c x+1}+3 b \left (c^4 x^4-6 c^2 x^2-3\right ) \cosh ^{-1}(c x)+16 b c x \sqrt {c x-1} \sqrt {c x+1}-9 b c x \tan ^{-1}\left (\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{9 x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 201, normalized size = 1.49 \[ \frac {3 \, a c^{4} d^{2} x^{4} - 18 \, a c^{2} d^{2} x^{2} + 18 \, b c d^{2} x \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - 3 \, {\left (b c^{4} - 6 \, b c^{2} - 3 \, b\right )} d^{2} x \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - 9 \, a d^{2} + 3 \, {\left (b c^{4} d^{2} x^{4} - 6 \, b c^{2} d^{2} x^{2} - {\left (b c^{4} - 6 \, b c^{2} - 3 \, b\right )} d^{2} x - 3 \, b d^{2}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (b c^{3} d^{2} x^{3} - 16 \, b c d^{2} x\right )} \sqrt {c^{2} x^{2} - 1}}{9 \, x} \]
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 = 1.24 \[ \frac {d^{2} a \,c^{4} x^{3}}{3}-2 d^{2} a \,c^{2} x -\frac {d^{2} a}{x}+\frac {d^{2} b \,\mathrm {arccosh}\left (c x \right ) c^{4} x^{3}}{3}-2 d^{2} b \,\mathrm {arccosh}\left (c x \right ) c^{2} x -\frac {d^{2} b \,\mathrm {arccosh}\left (c x \right )}{x}-\frac {d^{2} b \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{2}}{9}+\frac {16 b c \,d^{2} \sqrt {c x -1}\, \sqrt {c x +1}}{9}-\frac {c \,d^{2} b \sqrt {c x -1}\, \sqrt {c x +1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{\sqrt {c^{2} x^{2}-1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 143, normalized size = 1.06 \[ \frac {1}{3} \, a c^{4} d^{2} x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b c^{4} d^{2} - 2 \, a c^{2} d^{2} x - 2 \, {\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b c d^{2} - {\left (c \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arcosh}\left (c x\right )}{x}\right )} b d^{2} - \frac {a d^{2}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^2}{x^2} \,d x \]
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 \[ d^{2} \left (\int \left (- 2 a c^{2}\right )\, dx + \int \frac {a}{x^{2}}\, dx + \int a c^{4} x^{2}\, dx + \int \left (- 2 b c^{2} \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int \frac {b \operatorname {acosh}{\left (c x \right )}}{x^{2}}\, dx + \int b c^{4} x^{2} \operatorname {acosh}{\left (c x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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