Optimal. Leaf size=161 \[ \frac {1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \cosh ^{-1}(c x)\right )-\frac {b \left (9 c^2 d+5 e\right ) \cosh ^{-1}(c x)}{96 c^6}-\frac {b x \sqrt {c x-1} \sqrt {c x+1} \left (9 c^2 d+5 e\right )}{96 c^5}-\frac {b x^3 \sqrt {c x-1} \sqrt {c x+1} \left (9 c^2 d+5 e\right )}{144 c^3}-\frac {b e x^5 \sqrt {c x-1} \sqrt {c x+1}}{36 c} \]
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Rubi [A] time = 0.14, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {5786, 460, 100, 12, 90, 52} \[ \frac {1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \cosh ^{-1}(c x)\right )-\frac {b x^3 \sqrt {c x-1} \sqrt {c x+1} \left (9 c^2 d+5 e\right )}{144 c^3}-\frac {b x \sqrt {c x-1} \sqrt {c x+1} \left (9 c^2 d+5 e\right )}{96 c^5}-\frac {b \left (9 c^2 d+5 e\right ) \cosh ^{-1}(c x)}{96 c^6}-\frac {b e x^5 \sqrt {c x-1} \sqrt {c x+1}}{36 c} \]
Antiderivative was successfully verified.
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Rule 12
Rule 52
Rule 90
Rule 100
Rule 460
Rule 5786
Rubi steps
\begin {align*} \int x^3 \left (d+e x^2\right ) \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{24} (b c) \int \frac {x^4 \left (6 d+4 e x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {b e x^5 \sqrt {-1+c x} \sqrt {1+c x}}{36 c}+\frac {1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{36} \left (b c \left (9 d+\frac {5 e}{c^2}\right )\right ) \int \frac {x^4}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {b \left (9 c^2 d+5 e\right ) x^3 \sqrt {-1+c x} \sqrt {1+c x}}{144 c^3}-\frac {b e x^5 \sqrt {-1+c x} \sqrt {1+c x}}{36 c}+\frac {1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b \left (9 c^2 d+5 e\right )\right ) \int \frac {3 x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{144 c^3}\\ &=-\frac {b \left (9 c^2 d+5 e\right ) x^3 \sqrt {-1+c x} \sqrt {1+c x}}{144 c^3}-\frac {b e x^5 \sqrt {-1+c x} \sqrt {1+c x}}{36 c}+\frac {1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b \left (9 c^2 d+5 e\right )\right ) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{48 c^3}\\ &=-\frac {b \left (9 c^2 d+5 e\right ) x \sqrt {-1+c x} \sqrt {1+c x}}{96 c^5}-\frac {b \left (9 c^2 d+5 e\right ) x^3 \sqrt {-1+c x} \sqrt {1+c x}}{144 c^3}-\frac {b e x^5 \sqrt {-1+c x} \sqrt {1+c x}}{36 c}+\frac {1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b \left (9 c^2 d+5 e\right )\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{96 c^5}\\ &=-\frac {b \left (9 c^2 d+5 e\right ) x \sqrt {-1+c x} \sqrt {1+c x}}{96 c^5}-\frac {b \left (9 c^2 d+5 e\right ) x^3 \sqrt {-1+c x} \sqrt {1+c x}}{144 c^3}-\frac {b e x^5 \sqrt {-1+c x} \sqrt {1+c x}}{36 c}-\frac {b \left (9 c^2 d+5 e\right ) \cosh ^{-1}(c x)}{96 c^6}+\frac {1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \cosh ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A] time = 0.20, size = 140, normalized size = 0.87 \[ \frac {24 a c^6 x^4 \left (3 d+2 e x^2\right )+24 b c^6 x^4 \cosh ^{-1}(c x) \left (3 d+2 e x^2\right )-6 b \left (9 c^2 d+5 e\right ) \tanh ^{-1}\left (\sqrt {\frac {c x-1}{c x+1}}\right )-b c x \sqrt {c x-1} \sqrt {c x+1} \left (2 c^4 \left (9 d x^2+4 e x^4\right )+c^2 \left (27 d+10 e x^2\right )+15 e\right )}{288 c^6} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.61, size = 136, normalized size = 0.84 \[ \frac {48 \, a c^{6} e x^{6} + 72 \, a c^{6} d x^{4} + 3 \, {\left (16 \, b c^{6} e x^{6} + 24 \, b c^{6} d x^{4} - 9 \, b c^{2} d - 5 \, b e\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (8 \, b c^{5} e x^{5} + 2 \, {\left (9 \, b c^{5} d + 5 \, b c^{3} e\right )} x^{3} + 3 \, {\left (9 \, b c^{3} d + 5 \, b c e\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{288 \, c^{6}} \]
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 = 250, normalized size = 1.55 \[ \frac {a e \,x^{6}}{6}+\frac {a \,x^{4} d}{4}+\frac {b \,\mathrm {arccosh}\left (c x \right ) e \,x^{6}}{6}+\frac {b \,\mathrm {arccosh}\left (c x \right ) x^{4} d}{4}-\frac {b e \,x^{5} \sqrt {c x -1}\, \sqrt {c x +1}}{36 c}-\frac {b d \,x^{3} \sqrt {c x -1}\, \sqrt {c x +1}}{16 c}-\frac {5 b \sqrt {c x -1}\, \sqrt {c x +1}\, e \,x^{3}}{144 c^{3}}-\frac {3 b d x \sqrt {c x -1}\, \sqrt {c x +1}}{32 c^{3}}-\frac {3 b \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right ) d}{32 c^{4} \sqrt {c^{2} x^{2}-1}}-\frac {5 b \sqrt {c x -1}\, \sqrt {c x +1}\, e x}{96 c^{5}}-\frac {5 b \sqrt {c x -1}\, \sqrt {c x +1}\, e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{96 c^{6} \sqrt {c^{2} x^{2}-1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 196, normalized size = 1.22 \[ \frac {1}{6} \, a e x^{6} + \frac {1}{4} \, a d x^{4} + \frac {1}{32} \, {\left (8 \, x^{4} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {c^{2} x^{2} - 1} x}{c^{4}} + \frac {3 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{5}}\right )} c\right )} b d + \frac {1}{288} \, {\left (48 \, x^{6} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {c^{2} x^{2} - 1} x}{c^{6}} + \frac {15 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{7}}\right )} c\right )} b e \]
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 \[ \int x^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\left (e\,x^2+d\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.83, size = 212, normalized size = 1.32 \[ \begin {cases} \frac {a d x^{4}}{4} + \frac {a e x^{6}}{6} + \frac {b d x^{4} \operatorname {acosh}{\left (c x \right )}}{4} + \frac {b e x^{6} \operatorname {acosh}{\left (c x \right )}}{6} - \frac {b d x^{3} \sqrt {c^{2} x^{2} - 1}}{16 c} - \frac {b e x^{5} \sqrt {c^{2} x^{2} - 1}}{36 c} - \frac {3 b d x \sqrt {c^{2} x^{2} - 1}}{32 c^{3}} - \frac {5 b e x^{3} \sqrt {c^{2} x^{2} - 1}}{144 c^{3}} - \frac {3 b d \operatorname {acosh}{\left (c x \right )}}{32 c^{4}} - \frac {5 b e x \sqrt {c^{2} x^{2} - 1}}{96 c^{5}} - \frac {5 b e \operatorname {acosh}{\left (c x \right )}}{96 c^{6}} & \text {for}\: c \neq 0 \\\left (a + \frac {i \pi b}{2}\right ) \left (\frac {d x^{4}}{4} + \frac {e x^{6}}{6}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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