Optimal. Leaf size=161 \[ -\frac {b \left (9 c^2 d+5 e\right ) x \sqrt {-1+c x} \sqrt {1+c x}}{96 c^5}-\frac {b \left (9 c^2 d+5 e\right ) x^3 \sqrt {-1+c x} \sqrt {1+c x}}{144 c^3}-\frac {b e x^5 \sqrt {-1+c x} \sqrt {1+c x}}{36 c}-\frac {b \left (9 c^2 d+5 e\right ) \cosh ^{-1}(c x)}{96 c^6}+\frac {1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \cosh ^{-1}(c x)\right ) \]
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Rubi [A]
time = 0.10, 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 = {5956, 471, 102,
12, 92, 54} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 54
Rule 92
Rule 102
Rule 471
Rule 5956
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 = 0.12, size = 140, normalized size = 0.87 \begin {gather*} \frac {24 a c^6 x^4 \left (3 d+2 e x^2\right )-b c x \sqrt {-1+c x} \sqrt {1+c x} \left (15 e+c^2 \left (27 d+10 e x^2\right )+2 c^4 \left (9 d x^2+4 e x^4\right )\right )+24 b c^6 x^4 \left (3 d+2 e x^2\right ) \cosh ^{-1}(c x)-6 b \left (9 c^2 d+5 e\right ) \tanh ^{-1}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )}{288 c^6} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(331\) vs.
\(2(137)=274\).
time = 3.07, size = 332, normalized size = 2.06
method | result | size |
derivativedivides | \(\frac {\frac {a \left (\frac {1}{4} c^{6} d \,x^{4}+\frac {1}{6} c^{6} e \,x^{6}\right )}{c^{2}}-\frac {b \,c^{4} \mathrm {arccosh}\left (c x \right ) d^{3}}{12 e^{2}}+\frac {b \,\mathrm {arccosh}\left (c x \right ) d \,c^{4} x^{4}}{4}+\frac {b \,c^{4} e \,\mathrm {arccosh}\left (c x \right ) x^{6}}{6}+\frac {b \,c^{4} \sqrt {c x -1}\, \sqrt {c x +1}\, d^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{12 e^{2} \sqrt {c^{2} x^{2}-1}}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, d \,c^{3} x^{3}}{16}-\frac {b \,c^{3} e \sqrt {c x -1}\, \sqrt {c x +1}\, x^{5}}{36}-\frac {3 b c d x \sqrt {c x -1}\, \sqrt {c x +1}}{32}-\frac {5 b c e \sqrt {c x -1}\, \sqrt {c x +1}\, x^{3}}{144}-\frac {3 b \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right ) d}{32 \sqrt {c^{2} x^{2}-1}}-\frac {5 b e x \sqrt {c x -1}\, \sqrt {c x +1}}{96 c}-\frac {5 b e \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{96 c^{2} \sqrt {c^{2} x^{2}-1}}}{c^{4}}\) | \(332\) |
default | \(\frac {\frac {a \left (\frac {1}{4} c^{6} d \,x^{4}+\frac {1}{6} c^{6} e \,x^{6}\right )}{c^{2}}-\frac {b \,c^{4} \mathrm {arccosh}\left (c x \right ) d^{3}}{12 e^{2}}+\frac {b \,\mathrm {arccosh}\left (c x \right ) d \,c^{4} x^{4}}{4}+\frac {b \,c^{4} e \,\mathrm {arccosh}\left (c x \right ) x^{6}}{6}+\frac {b \,c^{4} \sqrt {c x -1}\, \sqrt {c x +1}\, d^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{12 e^{2} \sqrt {c^{2} x^{2}-1}}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, d \,c^{3} x^{3}}{16}-\frac {b \,c^{3} e \sqrt {c x -1}\, \sqrt {c x +1}\, x^{5}}{36}-\frac {3 b c d x \sqrt {c x -1}\, \sqrt {c x +1}}{32}-\frac {5 b c e \sqrt {c x -1}\, \sqrt {c x +1}\, x^{3}}{144}-\frac {3 b \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right ) d}{32 \sqrt {c^{2} x^{2}-1}}-\frac {5 b e x \sqrt {c x -1}\, \sqrt {c x +1}}{96 c}-\frac {5 b e \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{96 c^{2} \sqrt {c^{2} x^{2}-1}}}{c^{4}}\) | \(332\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 198, normalized size = 1.23 \begin {gather*} \frac {1}{6} \, a x^{6} e + \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 192, normalized size = 1.19 \begin {gather*} \frac {48 \, a c^{6} x^{6} \cosh \left (1\right ) + 48 \, a c^{6} x^{6} \sinh \left (1\right ) + 72 \, a c^{6} d x^{4} + 3 \, {\left (24 \, b c^{6} d x^{4} - 9 \, b c^{2} d + {\left (16 \, b c^{6} x^{6} - 5 \, b\right )} \cosh \left (1\right ) + {\left (16 \, b c^{6} x^{6} - 5 \, b\right )} \sinh \left (1\right )\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (18 \, b c^{5} d x^{3} + 27 \, b c^{3} d x + {\left (8 \, b c^{5} x^{5} + 10 \, b c^{3} x^{3} + 15 \, b c x\right )} \cosh \left (1\right ) + {\left (8 \, b c^{5} x^{5} + 10 \, b c^{3} x^{3} + 15 \, b c x\right )} \sinh \left (1\right )\right )} \sqrt {c^{2} x^{2} - 1}}{288 \, c^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.51, size = 212, normalized size = 1.32 \begin {gather*} \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} \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 = -0.01 \begin {gather*} \int x^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\left (e\,x^2+d\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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