Optimal. Leaf size=267 \[ -\frac {3}{32} b c^3 d^3 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {7}{16} b c^3 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac {b c d^3 (-1+c x)^{5/2} (1+c x)^{5/2}}{2 x}+\frac {3}{32} b c^2 d^3 \cosh ^{-1}(c x)-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}-\frac {3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}-3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{-2 \cosh ^{-1}(c x)}\right )+\frac {3}{2} b c^2 d^3 \text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right ) \]
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Rubi [A]
time = 0.25, antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps
used = 18, number of rules used = 11, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {5920, 99, 12,
38, 54, 5919, 5882, 3799, 2221, 2317, 2438} \begin {gather*} -\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}-3 c^2 d^3 \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {7}{16} b c^3 d^3 x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{32} b c^3 d^3 x \sqrt {c x-1} \sqrt {c x+1}+\frac {3}{2} b c^2 d^3 \text {Li}_2\left (-e^{-2 \cosh ^{-1}(c x)}\right )+\frac {3}{32} b c^2 d^3 \cosh ^{-1}(c x)+\frac {b c d^3 (c x-1)^{5/2} (c x+1)^{5/2}}{2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 38
Rule 54
Rule 99
Rule 2221
Rule 2317
Rule 2438
Rule 3799
Rule 5882
Rule 5919
Rule 5920
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{x^3} \, dx &=-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}-\left (3 c^2 d\right ) \int \frac {\left (d-c^2 d x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{x} \, dx-\frac {1}{2} \left (b c d^3\right ) \int \frac {(-1+c x)^{5/2} (1+c x)^{5/2}}{x^2} \, dx\\ &=\frac {b c d^3 (-1+c x)^{5/2} (1+c x)^{5/2}}{2 x}-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}-\left (3 c^2 d^2\right ) \int \frac {\left (d-c^2 d x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{x} \, dx-\frac {1}{2} \left (b c d^3\right ) \int 5 c^2 (-1+c x)^{3/2} (1+c x)^{3/2} \, dx+\frac {1}{4} \left (3 b c^3 d^3\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \, dx\\ &=\frac {3}{16} b c^3 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac {b c d^3 (-1+c x)^{5/2} (1+c x)^{5/2}}{2 x}-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}-\left (3 c^2 d^3\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x} \, dx-\frac {1}{16} \left (9 b c^3 d^3\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx-\frac {1}{2} \left (3 b c^3 d^3\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx-\frac {1}{2} \left (5 b c^3 d^3\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \, dx\\ &=-\frac {33}{32} b c^3 d^3 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {7}{16} b c^3 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac {b c d^3 (-1+c x)^{5/2} (1+c x)^{5/2}}{2 x}-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}-\left (3 c^2 d^3\right ) \text {Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\cosh ^{-1}(c x)\right )+\frac {1}{32} \left (9 b c^3 d^3\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx+\frac {1}{4} \left (3 b c^3 d^3\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx+\frac {1}{8} \left (15 b c^3 d^3\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx\\ &=-\frac {3}{32} b c^3 d^3 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {7}{16} b c^3 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac {b c d^3 (-1+c x)^{5/2} (1+c x)^{5/2}}{2 x}+\frac {33}{32} b c^2 d^3 \cosh ^{-1}(c x)-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac {3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}-\left (6 c^2 d^3\right ) \text {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )-\frac {1}{16} \left (15 b c^3 d^3\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {3}{32} b c^3 d^3 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {7}{16} b c^3 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac {b c d^3 (-1+c x)^{5/2} (1+c x)^{5/2}}{2 x}+\frac {3}{32} b c^2 d^3 \cosh ^{-1}(c x)-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac {3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}-3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )+\left (3 b c^2 d^3\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )\\ &=-\frac {3}{32} b c^3 d^3 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {7}{16} b c^3 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac {b c d^3 (-1+c x)^{5/2} (1+c x)^{5/2}}{2 x}+\frac {3}{32} b c^2 d^3 \cosh ^{-1}(c x)-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac {3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}-3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )+\frac {1}{2} \left (3 b c^2 d^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )\\ &=-\frac {3}{32} b c^3 d^3 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {7}{16} b c^3 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac {b c d^3 (-1+c x)^{5/2} (1+c x)^{5/2}}{2 x}+\frac {3}{32} b c^2 d^3 \cosh ^{-1}(c x)-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac {3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}-3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )-\frac {3}{2} b c^2 d^3 \text {Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A]
time = 0.37, size = 303, normalized size = 1.13 \begin {gather*} \frac {d^3 \left (-16 a+48 a c^4 x^4-8 a c^6 x^6+3 b c^3 x^3 \sqrt {\frac {-1+c x}{1+c x}}+3 b c^4 x^4 \sqrt {\frac {-1+c x}{1+c x}}+2 b c^5 x^5 \sqrt {\frac {-1+c x}{1+c x}}+2 b c^6 x^6 \sqrt {\frac {-1+c x}{1+c x}}+16 b c x \sqrt {-1+c x} \sqrt {1+c x}-24 b c^3 x^3 \sqrt {-1+c x} \sqrt {1+c x}-48 b c^2 x^2 \cosh ^{-1}(c x)^2-42 b c^2 x^2 \tanh ^{-1}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )-8 b \cosh ^{-1}(c x) \left (2-6 c^4 x^4+c^6 x^6+12 c^2 x^2 \log \left (1+e^{-2 \cosh ^{-1}(c x)}\right )\right )-96 a c^2 x^2 \log (x)+48 b c^2 x^2 \text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )\right )}{32 x^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 10.75, size = 267, normalized size = 1.00
method | result | size |
derivativedivides | \(c^{2} \left (-\frac {a \,d^{3} c^{4} x^{4}}{4}+\frac {3 a \,d^{3} c^{2} x^{2}}{2}-\frac {a \,d^{3}}{2 c^{2} x^{2}}-3 a \,d^{3} \ln \left (c x \right )-3 d^{3} b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {d^{3} b \,\mathrm {arccosh}\left (c x \right ) c^{4} x^{4}}{4}+\frac {3 d^{3} b \,\mathrm {arccosh}\left (c x \right ) c^{2} x^{2}}{2}-\frac {d^{3} b}{2}-\frac {21 b \,d^{3} \mathrm {arccosh}\left (c x \right )}{32}+\frac {d^{3} b \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}}{16}-\frac {21 b c \,d^{3} x \sqrt {c x -1}\, \sqrt {c x +1}}{32}-\frac {3 d^{3} b \polylog \left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}+\frac {3 d^{3} b \mathrm {arccosh}\left (c x \right )^{2}}{2}+\frac {d^{3} b \sqrt {c x +1}\, \sqrt {c x -1}}{2 c x}-\frac {d^{3} b \,\mathrm {arccosh}\left (c x \right )}{2 c^{2} x^{2}}\right )\) | \(267\) |
default | \(c^{2} \left (-\frac {a \,d^{3} c^{4} x^{4}}{4}+\frac {3 a \,d^{3} c^{2} x^{2}}{2}-\frac {a \,d^{3}}{2 c^{2} x^{2}}-3 a \,d^{3} \ln \left (c x \right )-3 d^{3} b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {d^{3} b \,\mathrm {arccosh}\left (c x \right ) c^{4} x^{4}}{4}+\frac {3 d^{3} b \,\mathrm {arccosh}\left (c x \right ) c^{2} x^{2}}{2}-\frac {d^{3} b}{2}-\frac {21 b \,d^{3} \mathrm {arccosh}\left (c x \right )}{32}+\frac {d^{3} b \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}}{16}-\frac {21 b c \,d^{3} x \sqrt {c x -1}\, \sqrt {c x +1}}{32}-\frac {3 d^{3} b \polylog \left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}+\frac {3 d^{3} b \mathrm {arccosh}\left (c x \right )^{2}}{2}+\frac {d^{3} b \sqrt {c x +1}\, \sqrt {c x -1}}{2 c x}-\frac {d^{3} b \,\mathrm {arccosh}\left (c x \right )}{2 c^{2} x^{2}}\right )\) | \(267\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} - d^{3} \left (\int \left (- \frac {a}{x^{3}}\right )\, dx + \int \frac {3 a c^{2}}{x}\, dx + \int \left (- 3 a c^{4} x\right )\, dx + \int a c^{6} x^{3}\, dx + \int \left (- \frac {b \operatorname {acosh}{\left (c x \right )}}{x^{3}}\right )\, dx + \int \frac {3 b c^{2} \operatorname {acosh}{\left (c x \right )}}{x}\, dx + \int \left (- 3 b c^{4} x \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int b c^{6} x^{3} \operatorname {acosh}{\left (c x \right )}\, dx\right ) \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^3}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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