Optimal. Leaf size=135 \[ -\frac {b d x \sqrt {-1+c x} \sqrt {1+c x}}{24 c^3}-\frac {b d x^3 \sqrt {-1+c x} \sqrt {1+c x}}{36 c}+\frac {1}{36} b c d x^5 \sqrt {-1+c x} \sqrt {1+c x}-\frac {b d \cosh ^{-1}(c x)}{24 c^4}+\frac {1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{6} c^2 d x^6 \left (a+b \cosh ^{-1}(c x)\right ) \]
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Rubi [A]
time = 0.09, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {14, 5921, 12,
471, 102, 92, 54} \begin {gather*} -\frac {1}{6} c^2 d x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {b d \cosh ^{-1}(c x)}{24 c^4}-\frac {b d x \sqrt {c x-1} \sqrt {c x+1}}{24 c^3}+\frac {1}{36} b c d x^5 \sqrt {c x-1} \sqrt {c x+1}-\frac {b d x^3 \sqrt {c x-1} \sqrt {c x+1}}{36 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 54
Rule 92
Rule 102
Rule 471
Rule 5921
Rubi steps
\begin {align*} \int x^3 \left (d-c^2 d 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} c^2 d x^6 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac {d x^4 \left (3-2 c^2 x^2\right )}{12 \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{6} c^2 d x^6 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{12} (b c d) \int \frac {x^4 \left (3-2 c^2 x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {1}{36} b c d x^5 \sqrt {-1+c x} \sqrt {1+c x}+\frac {1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{6} c^2 d x^6 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{9} (b c d) \int \frac {x^4}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {b d x^3 \sqrt {-1+c x} \sqrt {1+c x}}{36 c}+\frac {1}{36} b c d x^5 \sqrt {-1+c x} \sqrt {1+c x}+\frac {1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{6} c^2 d x^6 \left (a+b \cosh ^{-1}(c x)\right )-\frac {(b d) \int \frac {3 x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{36 c}\\ &=-\frac {b d x^3 \sqrt {-1+c x} \sqrt {1+c x}}{36 c}+\frac {1}{36} b c d x^5 \sqrt {-1+c x} \sqrt {1+c x}+\frac {1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{6} c^2 d x^6 \left (a+b \cosh ^{-1}(c x)\right )-\frac {(b d) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{12 c}\\ &=-\frac {b d x \sqrt {-1+c x} \sqrt {1+c x}}{24 c^3}-\frac {b d x^3 \sqrt {-1+c x} \sqrt {1+c x}}{36 c}+\frac {1}{36} b c d x^5 \sqrt {-1+c x} \sqrt {1+c x}+\frac {1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{6} c^2 d x^6 \left (a+b \cosh ^{-1}(c x)\right )-\frac {(b d) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{24 c^3}\\ &=-\frac {b d x \sqrt {-1+c x} \sqrt {1+c x}}{24 c^3}-\frac {b d x^3 \sqrt {-1+c x} \sqrt {1+c x}}{36 c}+\frac {1}{36} b c d x^5 \sqrt {-1+c x} \sqrt {1+c x}-\frac {b d \cosh ^{-1}(c x)}{24 c^4}+\frac {1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{6} c^2 d x^6 \left (a+b \cosh ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 166, normalized size = 1.23 \begin {gather*} \frac {1}{4} a d x^4-\frac {1}{6} a c^2 d x^6-\frac {b d x \sqrt {-1+c x} \sqrt {1+c x}}{24 c^3}-\frac {b d x^3 \sqrt {-1+c x} \sqrt {1+c x}}{36 c}+\frac {1}{36} b c d x^5 \sqrt {-1+c x} \sqrt {1+c x}+\frac {1}{4} b d x^4 \cosh ^{-1}(c x)-\frac {1}{6} b c^2 d x^6 \cosh ^{-1}(c x)-\frac {b d \tanh ^{-1}\left (\frac {\sqrt {-1+c x}}{\sqrt {1+c x}}\right )}{12 c^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.66, size = 186, normalized size = 1.38
method | result | size |
derivativedivides | \(\frac {-d a \left (\frac {\left (c^{2} x^{2}-1\right )^{3}}{6}+\frac {\left (c^{2} x^{2}-1\right )^{2}}{4}\right )-\frac {b d \,\mathrm {arccosh}\left (c x \right ) c^{6} x^{6}}{6}+\frac {b d \,\mathrm {arccosh}\left (c x \right ) c^{4} x^{4}}{4}-\frac {b d \,\mathrm {arccosh}\left (c x \right )}{12}+\frac {b d \sqrt {c x -1}\, \sqrt {c x +1}\, c^{5} x^{5}}{36}-\frac {b d \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}}{36}-\frac {b c d x \sqrt {c x -1}\, \sqrt {c x +1}}{24}+\frac {b d \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{24 \sqrt {c^{2} x^{2}-1}}}{c^{4}}\) | \(186\) |
default | \(\frac {-d a \left (\frac {\left (c^{2} x^{2}-1\right )^{3}}{6}+\frac {\left (c^{2} x^{2}-1\right )^{2}}{4}\right )-\frac {b d \,\mathrm {arccosh}\left (c x \right ) c^{6} x^{6}}{6}+\frac {b d \,\mathrm {arccosh}\left (c x \right ) c^{4} x^{4}}{4}-\frac {b d \,\mathrm {arccosh}\left (c x \right )}{12}+\frac {b d \sqrt {c x -1}\, \sqrt {c x +1}\, c^{5} x^{5}}{36}-\frac {b d \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}}{36}-\frac {b c d x \sqrt {c x -1}\, \sqrt {c x +1}}{24}+\frac {b d \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{24 \sqrt {c^{2} x^{2}-1}}}{c^{4}}\) | \(186\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 202, normalized size = 1.50 \begin {gather*} -\frac {1}{6} \, a c^{2} d x^{6} + \frac {1}{4} \, a d x^{4} - \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 c^{2} d + \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 108, normalized size = 0.80 \begin {gather*} -\frac {12 \, a c^{6} d x^{6} - 18 \, a c^{4} d x^{4} + 3 \, {\left (4 \, b c^{6} d x^{6} - 6 \, b c^{4} d x^{4} + b d\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (2 \, b c^{5} d x^{5} - 2 \, b c^{3} d x^{3} - 3 \, b c d x\right )} \sqrt {c^{2} x^{2} - 1}}{72 \, c^{4}} \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 = 144, normalized size = 1.07 \begin {gather*} \begin {cases} - \frac {a c^{2} d x^{6}}{6} + \frac {a d x^{4}}{4} - \frac {b c^{2} d x^{6} \operatorname {acosh}{\left (c x \right )}}{6} + \frac {b c d x^{5} \sqrt {c^{2} x^{2} - 1}}{36} + \frac {b d x^{4} \operatorname {acosh}{\left (c x \right )}}{4} - \frac {b d x^{3} \sqrt {c^{2} x^{2} - 1}}{36 c} - \frac {b d x \sqrt {c^{2} x^{2} - 1}}{24 c^{3}} - \frac {b d \operatorname {acosh}{\left (c x \right )}}{24 c^{4}} & \text {for}\: c \neq 0 \\\frac {d x^{4} \left (a + \frac {i \pi b}{2}\right )}{4} & \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 (d-c^2\,d\,x^2\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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