Optimal. Leaf size=136 \[ \frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {b \sqrt {c x-1}}{4 c^4 d^3 \sqrt {c x+1}}+\frac {b}{4 c^4 d^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b \cosh ^{-1}(c x)}{4 c^4 d^3}+\frac {b x^3}{12 c d^3 (c x-1)^{3/2} (c x+1)^{3/2}} \]
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Rubi [A] time = 0.10, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {5722, 98, 21, 89, 12, 78, 52} \[ \frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {b \sqrt {c x-1}}{4 c^4 d^3 \sqrt {c x+1}}+\frac {b}{4 c^4 d^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b \cosh ^{-1}(c x)}{4 c^4 d^3}+\frac {b x^3}{12 c d^3 (c x-1)^{3/2} (c x+1)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 21
Rule 52
Rule 78
Rule 89
Rule 98
Rule 5722
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^3} \, dx &=\frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {(b c) \int \frac {x^4}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{4 d^3}\\ &=\frac {b x^3}{12 c d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {b \int \frac {x^2 (-3-3 c x)}{(-1+c x)^{3/2} (1+c x)^{5/2}} \, dx}{12 c d^3}\\ &=\frac {b x^3}{12 c d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b \int \frac {x^2}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{4 c d^3}\\ &=\frac {b x^3}{12 c d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b}{4 c^4 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b \int \frac {c^2 x}{\sqrt {-1+c x} (1+c x)^{3/2}} \, dx}{4 c^4 d^3}\\ &=\frac {b x^3}{12 c d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b}{4 c^4 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b \int \frac {x}{\sqrt {-1+c x} (1+c x)^{3/2}} \, dx}{4 c^2 d^3}\\ &=\frac {b x^3}{12 c d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b}{4 c^4 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b \sqrt {-1+c x}}{4 c^4 d^3 \sqrt {1+c x}}+\frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{4 c^3 d^3}\\ &=\frac {b x^3}{12 c d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b}{4 c^4 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b \sqrt {-1+c x}}{4 c^4 d^3 \sqrt {1+c x}}-\frac {b \cosh ^{-1}(c x)}{4 c^4 d^3}+\frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 83, normalized size = 0.61 \[ \frac {a \left (6 c^2 x^2-3\right )+b c x \sqrt {c x-1} \sqrt {c x+1} \left (4 c^2 x^2-3\right )+3 b \left (2 c^2 x^2-1\right ) \cosh ^{-1}(c x)}{12 c^4 d^3 \left (c^2 x^2-1\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 101, normalized size = 0.74 \[ \frac {3 \, a c^{4} x^{4} + 3 \, {\left (2 \, b c^{2} x^{2} - b\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (4 \, b c^{3} x^{3} - 3 \, b c x\right )} \sqrt {c^{2} x^{2} - 1}}{12 \, {\left (c^{8} d^{3} x^{4} - 2 \, c^{6} d^{3} x^{2} + c^{4} d^{3}\right )}} \]
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 = 136, normalized size = 1.00 \[ \frac {-\frac {a \left (-\frac {1}{16 \left (c x -1\right )^{2}}-\frac {3}{16 \left (c x -1\right )}-\frac {1}{16 \left (c x +1\right )^{2}}+\frac {3}{16 \left (c x +1\right )}\right )}{d^{3}}-\frac {b \left (-\frac {\mathrm {arccosh}\left (c x \right )}{16 \left (c x -1\right )^{2}}-\frac {3 \,\mathrm {arccosh}\left (c x \right )}{16 \left (c x -1\right )}-\frac {\mathrm {arccosh}\left (c x \right )}{16 \left (c x +1\right )^{2}}+\frac {3 \,\mathrm {arccosh}\left (c x \right )}{16 \left (c x +1\right )}-\frac {c x \left (4 c^{2} x^{2}-3\right )}{12 \left (c x +1\right )^{\frac {3}{2}} \left (c x -1\right )^{\frac {3}{2}}}\right )}{d^{3}}}{c^{4}} \]
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 \[ \frac {1}{16} \, b {\left (\frac {4 \, c^{2} x^{2} + 4 \, {\left (2 \, c^{2} x^{2} - 1\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) - 3}{c^{8} d^{3} x^{4} - 2 \, c^{6} d^{3} x^{2} + c^{4} d^{3}} + 16 \, \int \frac {2 \, c^{2} x^{2} - 1}{4 \, {\left (c^{10} d^{3} x^{7} - 3 \, c^{8} d^{3} x^{5} + 3 \, c^{6} d^{3} x^{3} - c^{4} d^{3} x + {\left (c^{9} d^{3} x^{6} - 3 \, c^{7} d^{3} x^{4} + 3 \, c^{5} d^{3} x^{2} - c^{3} d^{3}\right )} e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (c x - 1\right )\right )}\right )}}\,{d x}\right )} + \frac {{\left (2 \, c^{2} x^{2} - 1\right )} a}{4 \, {\left (c^{8} d^{3} x^{4} - 2 \, c^{6} d^{3} x^{2} + c^{4} d^{3}\right )}} \]
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 \[ \int \frac {x^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^3} \,d x \]
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 \[ - \frac {\int \frac {a x^{3}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac {b x^{3} \operatorname {acosh}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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