Optimal. Leaf size=269 \[ \frac {b \left (44 c^4 d^2+44 c^2 d e+15 e^2\right ) x \left (1-c^2 x^2\right )}{288 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b \left (2 c^2 d+e\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )}{144 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{36 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d+e x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{6 e}-\frac {b \left (2 c^2 d+e\right ) \left (8 c^4 d^2+8 c^2 d e+5 e^2\right ) \sqrt {-1+c^2 x^2} \tanh ^{-1}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{96 c^6 e \sqrt {-1+c x} \sqrt {1+c x}} \]
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Rubi [A]
time = 0.17, antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {5957, 916, 427,
542, 396, 223, 212} \begin {gather*} \frac {\left (d+e x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{6 e}+\frac {b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{36 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 b x \left (1-c^2 x^2\right ) \left (2 c^2 d+e\right ) \left (d+e x^2\right )}{144 c^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b \sqrt {c^2 x^2-1} \left (2 c^2 d+e\right ) \left (8 c^4 d^2+8 c^2 d e+5 e^2\right ) \tanh ^{-1}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{96 c^6 e \sqrt {c x-1} \sqrt {c x+1}}+\frac {b x \left (1-c^2 x^2\right ) \left (44 c^4 d^2+44 c^2 d e+15 e^2\right )}{288 c^5 \sqrt {c x-1} \sqrt {c x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 396
Rule 427
Rule 542
Rule 916
Rule 5957
Rubi steps
\begin {align*} \int x \left (d+e x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {\left (d+e x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{6 e}-\frac {(b c) \int \frac {\left (d+e x^2\right )^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{6 e}\\ &=\frac {\left (d+e x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{6 e}-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \int \frac {\left (d+e x^2\right )^3}{\sqrt {-1+c^2 x^2}} \, dx}{6 e \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{36 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d+e x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{6 e}-\frac {\left (b \sqrt {-1+c^2 x^2}\right ) \int \frac {\left (d+e x^2\right ) \left (d \left (6 c^2 d+e\right )+5 e \left (2 c^2 d+e\right ) x^2\right )}{\sqrt {-1+c^2 x^2}} \, dx}{36 c e \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {5 b \left (2 c^2 d+e\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )}{144 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{36 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d+e x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{6 e}-\frac {\left (b \sqrt {-1+c^2 x^2}\right ) \int \frac {d \left (24 c^4 d^2+14 c^2 d e+5 e^2\right )+e \left (44 c^4 d^2+44 c^2 d e+15 e^2\right ) x^2}{\sqrt {-1+c^2 x^2}} \, dx}{144 c^3 e \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b \left (44 c^4 d^2+44 c^2 d e+15 e^2\right ) x \left (1-c^2 x^2\right )}{288 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b \left (2 c^2 d+e\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )}{144 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{36 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d+e x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{6 e}-\frac {\left (b \left (2 c^2 d+e\right ) \left (8 c^4 d^2+8 c^2 d e+5 e^2\right ) \sqrt {-1+c^2 x^2}\right ) \int \frac {1}{\sqrt {-1+c^2 x^2}} \, dx}{96 c^5 e \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b \left (44 c^4 d^2+44 c^2 d e+15 e^2\right ) x \left (1-c^2 x^2\right )}{288 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b \left (2 c^2 d+e\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )}{144 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{36 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d+e x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{6 e}-\frac {\left (b \left (2 c^2 d+e\right ) \left (8 c^4 d^2+8 c^2 d e+5 e^2\right ) \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\frac {x}{\sqrt {-1+c^2 x^2}}\right )}{96 c^5 e \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b \left (44 c^4 d^2+44 c^2 d e+15 e^2\right ) x \left (1-c^2 x^2\right )}{288 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b \left (2 c^2 d+e\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )}{144 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{36 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d+e x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{6 e}-\frac {b \left (2 c^2 d+e\right ) \left (8 c^4 d^2+8 c^2 d e+5 e^2\right ) \sqrt {-1+c^2 x^2} \tanh ^{-1}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{96 c^6 e \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 189, normalized size = 0.70 \begin {gather*} \frac {c x \left (48 a c^5 x \left (3 d^2+3 d e x^2+e^2 x^4\right )-b \sqrt {-1+c x} \sqrt {1+c x} \left (15 e^2+2 c^2 e \left (27 d+5 e x^2\right )+4 c^4 \left (18 d^2+9 d e x^2+2 e^2 x^4\right )\right )\right )+48 b c^6 x^2 \left (3 d^2+3 d e x^2+e^2 x^4\right ) \cosh ^{-1}(c x)-3 b \left (24 c^4 d^2+18 c^2 d e+5 e^2\right ) \log \left (c x+\sqrt {-1+c x} \sqrt {1+c x}\right )}{288 c^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.80, size = 435, normalized size = 1.62
method | result | size |
derivativedivides | \(\frac {\frac {\left (c^{2} e \,x^{2}+c^{2} d \right )^{3} a}{6 c^{4} e}+\frac {b \,c^{2} \mathrm {arccosh}\left (c x \right ) d^{3}}{6 e}+\frac {d^{2} b \,\mathrm {arccosh}\left (c x \right ) c^{2} x^{2}}{2}+\frac {b \,c^{2} e \,\mathrm {arccosh}\left (c x \right ) d \,x^{4}}{2}+\frac {b \,c^{2} e^{2} \mathrm {arccosh}\left (c x \right ) x^{6}}{6}-\frac {b \,c^{2} \sqrt {c x -1}\, \sqrt {c x +1}\, d^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{6 e \sqrt {c^{2} x^{2}-1}}-\frac {b c \,d^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{4}-\frac {b c e \sqrt {c x -1}\, \sqrt {c x +1}\, d \,x^{3}}{8}-\frac {b c \,e^{2} \sqrt {c x -1}\, \sqrt {c x +1}\, x^{5}}{36}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right ) d^{2}}{4 \sqrt {c^{2} x^{2}-1}}-\frac {3 b d e x \sqrt {c x -1}\, \sqrt {c x +1}}{16 c}-\frac {5 b \,e^{2} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}}{144 c}-\frac {3 b e \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right ) d}{16 c^{2} \sqrt {c^{2} x^{2}-1}}-\frac {5 b \,e^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{96 c^{3}}-\frac {5 b \,e^{2} \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{96 c^{4} \sqrt {c^{2} x^{2}-1}}}{c^{2}}\) | \(435\) |
default | \(\frac {\frac {\left (c^{2} e \,x^{2}+c^{2} d \right )^{3} a}{6 c^{4} e}+\frac {b \,c^{2} \mathrm {arccosh}\left (c x \right ) d^{3}}{6 e}+\frac {d^{2} b \,\mathrm {arccosh}\left (c x \right ) c^{2} x^{2}}{2}+\frac {b \,c^{2} e \,\mathrm {arccosh}\left (c x \right ) d \,x^{4}}{2}+\frac {b \,c^{2} e^{2} \mathrm {arccosh}\left (c x \right ) x^{6}}{6}-\frac {b \,c^{2} \sqrt {c x -1}\, \sqrt {c x +1}\, d^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{6 e \sqrt {c^{2} x^{2}-1}}-\frac {b c \,d^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{4}-\frac {b c e \sqrt {c x -1}\, \sqrt {c x +1}\, d \,x^{3}}{8}-\frac {b c \,e^{2} \sqrt {c x -1}\, \sqrt {c x +1}\, x^{5}}{36}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right ) d^{2}}{4 \sqrt {c^{2} x^{2}-1}}-\frac {3 b d e x \sqrt {c x -1}\, \sqrt {c x +1}}{16 c}-\frac {5 b \,e^{2} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}}{144 c}-\frac {3 b e \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right ) d}{16 c^{2} \sqrt {c^{2} x^{2}-1}}-\frac {5 b \,e^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{96 c^{3}}-\frac {5 b \,e^{2} \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{96 c^{4} \sqrt {c^{2} x^{2}-1}}}{c^{2}}\) | \(435\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 273, normalized size = 1.01 \begin {gather*} \frac {1}{6} \, a x^{6} e^{2} + \frac {1}{2} \, a d x^{4} e + \frac {1}{2} \, a d^{2} x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x}{c^{2}} + \frac {\log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{3}}\right )}\right )} b d^{2} + \frac {1}{16} \, {\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 e + \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^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 373, normalized size = 1.39 \begin {gather*} \frac {48 \, a c^{6} x^{6} \cosh \left (1\right )^{2} + 48 \, a c^{6} x^{6} \sinh \left (1\right )^{2} + 144 \, a c^{6} d x^{4} \cosh \left (1\right ) + 144 \, a c^{6} d^{2} x^{2} + 3 \, {\left (48 \, b c^{6} d^{2} x^{2} - 24 \, b c^{4} d^{2} + {\left (16 \, b c^{6} x^{6} - 5 \, b\right )} \cosh \left (1\right )^{2} + {\left (16 \, b c^{6} x^{6} - 5 \, b\right )} \sinh \left (1\right )^{2} + 6 \, {\left (8 \, b c^{6} d x^{4} - 3 \, b c^{2} d\right )} \cosh \left (1\right ) + 2 \, {\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 )\right )} \sinh \left (1\right )\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 48 \, {\left (2 \, a c^{6} x^{6} \cosh \left (1\right ) + 3 \, a c^{6} d x^{4}\right )} \sinh \left (1\right ) - {\left (72 \, b c^{5} d^{2} x + {\left (8 \, b c^{5} x^{5} + 10 \, b c^{3} x^{3} + 15 \, b c x\right )} \cosh \left (1\right )^{2} + {\left (8 \, b c^{5} x^{5} + 10 \, b c^{3} x^{3} + 15 \, b c x\right )} \sinh \left (1\right )^{2} + 18 \, {\left (2 \, b c^{5} d x^{3} + 3 \, b c^{3} d x\right )} \cosh \left (1\right ) + 2 \, {\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 )\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 = 0.53, size = 306, normalized size = 1.14 \begin {gather*} \begin {cases} \frac {a d^{2} x^{2}}{2} + \frac {a d e x^{4}}{2} + \frac {a e^{2} x^{6}}{6} + \frac {b d^{2} x^{2} \operatorname {acosh}{\left (c x \right )}}{2} + \frac {b d e x^{4} \operatorname {acosh}{\left (c x \right )}}{2} + \frac {b e^{2} x^{6} \operatorname {acosh}{\left (c x \right )}}{6} - \frac {b d^{2} x \sqrt {c^{2} x^{2} - 1}}{4 c} - \frac {b d e x^{3} \sqrt {c^{2} x^{2} - 1}}{8 c} - \frac {b e^{2} x^{5} \sqrt {c^{2} x^{2} - 1}}{36 c} - \frac {b d^{2} \operatorname {acosh}{\left (c x \right )}}{4 c^{2}} - \frac {3 b d e x \sqrt {c^{2} x^{2} - 1}}{16 c^{3}} - \frac {5 b e^{2} x^{3} \sqrt {c^{2} x^{2} - 1}}{144 c^{3}} - \frac {3 b d e \operatorname {acosh}{\left (c x \right )}}{16 c^{4}} - \frac {5 b e^{2} x \sqrt {c^{2} x^{2} - 1}}{96 c^{5}} - \frac {5 b e^{2} \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^{2} x^{2}}{2} + \frac {d e x^{4}}{2} + \frac {e^{2} 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: RuntimeError} \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 x\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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