Optimal. Leaf size=180 \[ -\frac {1}{5} c^6 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-3 c^2 d^3 x \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}+\frac {1}{25} b c d^3 (c x-1)^{5/2} (c x+1)^{5/2}-\frac {1}{5} b c d^3 (c x-1)^{3/2} (c x+1)^{3/2}+\frac {11}{5} b c d^3 \sqrt {c x-1} \sqrt {c x+1}+b c d^3 \tan ^{-1}\left (\sqrt {c x-1} \sqrt {c x+1}\right ) \]
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Rubi [A] time = 0.36, antiderivative size = 239, normalized size of antiderivative = 1.33, 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, 1610, 1799, 1620, 63, 205} \[ -\frac {1}{5} c^6 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-3 c^2 d^3 x \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}-\frac {b c d^3 \left (1-c^2 x^2\right )^3}{25 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d^3 \left (1-c^2 x^2\right )^2}{5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {11 b c d^3 \left (1-c^2 x^2\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c d^3 \sqrt {c^2 x^2-1} \tan ^{-1}\left (\sqrt {c^2 x^2-1}\right )}{\sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 63
Rule 205
Rule 270
Rule 1610
Rule 1620
Rule 1799
Rule 5731
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{x^2} \, dx &=-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}-3 c^2 d^3 x \left (a+b \cosh ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{5} c^6 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac {d^3 \left (-5-15 c^2 x^2+5 c^4 x^4-c^6 x^6\right )}{5 x \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}-3 c^2 d^3 x \left (a+b \cosh ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{5} c^6 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{5} \left (b c d^3\right ) \int \frac {-5-15 c^2 x^2+5 c^4 x^4-c^6 x^6}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}-3 c^2 d^3 x \left (a+b \cosh ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{5} c^6 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c d^3 \sqrt {-1+c^2 x^2}\right ) \int \frac {-5-15 c^2 x^2+5 c^4 x^4-c^6 x^6}{x \sqrt {-1+c^2 x^2}} \, dx}{5 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}-3 c^2 d^3 x \left (a+b \cosh ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{5} c^6 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c d^3 \sqrt {-1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {-5-15 c^2 x+5 c^4 x^2-c^6 x^3}{x \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{10 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}-3 c^2 d^3 x \left (a+b \cosh ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{5} c^6 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c d^3 \sqrt {-1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \left (-\frac {11 c^2}{\sqrt {-1+c^2 x}}-\frac {5}{x \sqrt {-1+c^2 x}}+3 c^2 \sqrt {-1+c^2 x}-c^2 \left (-1+c^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )}{10 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {11 b c d^3 \left (1-c^2 x^2\right )}{5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^3 \left (1-c^2 x^2\right )^2}{5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^3 \left (1-c^2 x^2\right )^3}{25 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}-3 c^2 d^3 x \left (a+b \cosh ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{5} c^6 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {\left (b c d^3 \sqrt {-1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {11 b c d^3 \left (1-c^2 x^2\right )}{5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^3 \left (1-c^2 x^2\right )^2}{5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^3 \left (1-c^2 x^2\right )^3}{25 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}-3 c^2 d^3 x \left (a+b \cosh ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{5} c^6 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {\left (b d^3 \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 {11 b c d^3 \left (1-c^2 x^2\right )}{5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^3 \left (1-c^2 x^2\right )^2}{5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^3 \left (1-c^2 x^2\right )^3}{25 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}-3 c^2 d^3 x \left (a+b \cosh ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{5} c^6 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {b c d^3 \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 = 0.31, size = 136, normalized size = 0.76 \[ \frac {1}{25} d^3 \left (-5 a c^6 x^5+25 a c^4 x^3-75 a c^2 x-\frac {25 a}{x}+b c \sqrt {c x-1} \sqrt {c x+1} \left (c^4 x^4-7 c^2 x^2+61\right )-\frac {5 b \left (c^6 x^6-5 c^4 x^4+15 c^2 x^2+5\right ) \cosh ^{-1}(c x)}{x}-25 b c \tan ^{-1}\left (\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 249, normalized size = 1.38 \[ -\frac {5 \, a c^{6} d^{3} x^{6} - 25 \, a c^{4} d^{3} x^{4} + 75 \, a c^{2} d^{3} x^{2} - 50 \, b c d^{3} x \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - 5 \, {\left (b c^{6} - 5 \, b c^{4} + 15 \, b c^{2} + 5 \, b\right )} d^{3} x \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 25 \, a d^{3} + 5 \, {\left (b c^{6} d^{3} x^{6} - 5 \, b c^{4} d^{3} x^{4} + 15 \, b c^{2} d^{3} x^{2} - {\left (b c^{6} - 5 \, b c^{4} + 15 \, b c^{2} + 5 \, b\right )} d^{3} x + 5 \, b d^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (b c^{5} d^{3} x^{5} - 7 \, b c^{3} d^{3} x^{3} + 61 \, b c d^{3} x\right )} \sqrt {c^{2} x^{2} - 1}}{25 \, x} \]
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 = 219, normalized size = 1.22 \[ -\frac {d^{3} a \,c^{6} x^{5}}{5}+d^{3} a \,c^{4} x^{3}-3 d^{3} a \,c^{2} x -\frac {d^{3} a}{x}-\frac {d^{3} b \,\mathrm {arccosh}\left (c x \right ) c^{6} x^{5}}{5}+d^{3} b \,\mathrm {arccosh}\left (c x \right ) c^{4} x^{3}-3 d^{3} b \,\mathrm {arccosh}\left (c x \right ) c^{2} x -\frac {d^{3} b \,\mathrm {arccosh}\left (c x \right )}{x}+\frac {d^{3} b \sqrt {c x -1}\, \sqrt {c x +1}\, c^{5} x^{4}}{25}-\frac {7 d^{3} b \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{2}}{25}+\frac {61 b c \,d^{3} \sqrt {c x -1}\, \sqrt {c x +1}}{25}-\frac {c \,d^{3} b \sqrt {c x -1}\, \sqrt {c x +1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{\sqrt {c^{2} x^{2}-1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 231, normalized size = 1.28 \[ -\frac {1}{5} \, a c^{6} d^{3} x^{5} - \frac {1}{75} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b c^{6} d^{3} + a c^{4} d^{3} x^{3} + \frac {1}{3} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b c^{4} d^{3} - 3 \, a c^{2} d^{3} x - 3 \, {\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b c d^{3} - {\left (c \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arcosh}\left (c x\right )}{x}\right )} b d^{3} - \frac {a d^{3}}{x} \]
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 \[ \int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^3}{x^2} \,d x \]
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 \[ - d^{3} \left (\int 3 a c^{2}\, dx + \int \left (- \frac {a}{x^{2}}\right )\, dx + \int \left (- 3 a c^{4} x^{2}\right )\, dx + \int a c^{6} x^{4}\, dx + \int 3 b c^{2} \operatorname {acosh}{\left (c x \right )}\, dx + \int \left (- \frac {b \operatorname {acosh}{\left (c x \right )}}{x^{2}}\right )\, dx + \int \left (- 3 b c^{4} x^{2} \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int b c^{6} x^{4} \operatorname {acosh}{\left (c x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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