Optimal. Leaf size=183 \[ -\frac {\left (8 c^4 d^4+24 c^2 d^2 e^2+3 e^4\right ) \cosh ^{-1}(c x)}{32 c^4 e}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (e x \left (26 c^2 d^2+9 e^2\right )+4 d \left (19 c^2 d^2+16 e^2\right )\right )}{96 c^3}-\frac {\sqrt {c x-1} \sqrt {c x+1} (d+e x)^3}{16 c}-\frac {7 d \sqrt {c x-1} \sqrt {c x+1} (d+e x)^2}{48 c}+\frac {\cosh ^{-1}(c x) (d+e x)^4}{4 e} \]
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Rubi [A] time = 0.15, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5802, 100, 153, 147, 52} \[ -\frac {\sqrt {c x-1} \sqrt {c x+1} \left (e x \left (26 c^2 d^2+9 e^2\right )+4 d \left (19 c^2 d^2+16 e^2\right )\right )}{96 c^3}-\frac {\left (24 c^2 d^2 e^2+8 c^4 d^4+3 e^4\right ) \cosh ^{-1}(c x)}{32 c^4 e}-\frac {\sqrt {c x-1} \sqrt {c x+1} (d+e x)^3}{16 c}-\frac {7 d \sqrt {c x-1} \sqrt {c x+1} (d+e x)^2}{48 c}+\frac {\cosh ^{-1}(c x) (d+e x)^4}{4 e} \]
Antiderivative was successfully verified.
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Rule 52
Rule 100
Rule 147
Rule 153
Rule 5802
Rubi steps
\begin {align*} \int (d+e x)^3 \cosh ^{-1}(c x) \, dx &=\frac {(d+e x)^4 \cosh ^{-1}(c x)}{4 e}-\frac {c \int \frac {(d+e x)^4}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{4 e}\\ &=-\frac {\sqrt {-1+c x} \sqrt {1+c x} (d+e x)^3}{16 c}+\frac {(d+e x)^4 \cosh ^{-1}(c x)}{4 e}-\frac {\int \frac {(d+e x)^2 \left (4 c^2 d^2+3 e^2+7 c^2 d e x\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{16 c e}\\ &=-\frac {7 d \sqrt {-1+c x} \sqrt {1+c x} (d+e x)^2}{48 c}-\frac {\sqrt {-1+c x} \sqrt {1+c x} (d+e x)^3}{16 c}+\frac {(d+e x)^4 \cosh ^{-1}(c x)}{4 e}-\frac {\int \frac {(d+e x) \left (c^2 d \left (12 c^2 d^2+23 e^2\right )+c^2 e \left (26 c^2 d^2+9 e^2\right ) x\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{48 c^3 e}\\ &=-\frac {7 d \sqrt {-1+c x} \sqrt {1+c x} (d+e x)^2}{48 c}-\frac {\sqrt {-1+c x} \sqrt {1+c x} (d+e x)^3}{16 c}-\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (4 d \left (19 c^2 d^2+16 e^2\right )+e \left (26 c^2 d^2+9 e^2\right ) x\right )}{96 c^3}+\frac {(d+e x)^4 \cosh ^{-1}(c x)}{4 e}-\frac {\left (8 c^4 d^4+24 c^2 d^2 e^2+3 e^4\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{32 c^3 e}\\ &=-\frac {7 d \sqrt {-1+c x} \sqrt {1+c x} (d+e x)^2}{48 c}-\frac {\sqrt {-1+c x} \sqrt {1+c x} (d+e x)^3}{16 c}-\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (4 d \left (19 c^2 d^2+16 e^2\right )+e \left (26 c^2 d^2+9 e^2\right ) x\right )}{96 c^3}-\frac {\left (8 c^4 d^4+24 c^2 d^2 e^2+3 e^4\right ) \cosh ^{-1}(c x)}{32 c^4 e}+\frac {(d+e x)^4 \cosh ^{-1}(c x)}{4 e}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 153, normalized size = 0.84 \[ -\frac {-24 c^4 x \cosh ^{-1}(c x) \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+9 \left (8 c^2 d^2 e+e^3\right ) \log \left (c x+\sqrt {c x-1} \sqrt {c x+1}\right )+c \sqrt {c x-1} \sqrt {c x+1} \left (c^2 \left (96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )+e^2 (64 d+9 e x)\right )}{96 c^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 153, normalized size = 0.84 \[ \frac {3 \, {\left (8 \, c^{4} e^{3} x^{4} + 32 \, c^{4} d e^{2} x^{3} + 48 \, c^{4} d^{2} e x^{2} + 32 \, c^{4} d^{3} x - 24 \, c^{2} d^{2} e - 3 \, e^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (6 \, c^{3} e^{3} x^{3} + 32 \, c^{3} d e^{2} x^{2} + 96 \, c^{3} d^{3} + 64 \, c d e^{2} + 9 \, {\left (8 \, c^{3} d^{2} e + c e^{3}\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{96 \, c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 167, normalized size = 0.91 \[ \frac {1}{4} \, {\left (x e + d\right )}^{4} e^{\left (-1\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - \frac {1}{96} \, {\left (\sqrt {c^{2} x^{2} - 1} {\left ({\left (2 \, x {\left (\frac {3 \, x e^{4}}{c} + \frac {16 \, d e^{3}}{c}\right )} + \frac {9 \, {\left (8 \, c^{5} d^{2} e^{2} + c^{3} e^{4}\right )}}{c^{6}}\right )} x + \frac {32 \, {\left (3 \, c^{5} d^{3} e + 2 \, c^{3} d e^{3}\right )}}{c^{6}}\right )} - \frac {3 \, {\left (8 \, c^{4} d^{4} + 24 \, c^{2} d^{2} e^{2} + 3 \, e^{4}\right )} \log \left ({\left | -x {\left | c \right |} + \sqrt {c^{2} x^{2} - 1} \right |}\right )}{c^{3} {\left | c \right |}}\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 351, normalized size = 1.92 \[ \frac {e^{3} \mathrm {arccosh}\left (c x \right ) x^{4}}{4}+e^{2} \mathrm {arccosh}\left (c x \right ) x^{3} d +\frac {3 e \,\mathrm {arccosh}\left (c x \right ) x^{2} d^{2}}{2}+\mathrm {arccosh}\left (c x \right ) x \,d^{3}+\frac {\mathrm {arccosh}\left (c x \right ) d^{4}}{4 e}-\frac {e^{3} \sqrt {c x -1}\, \sqrt {c x +1}\, x^{3}}{16 c}-\frac {e^{2} \sqrt {c x -1}\, \sqrt {c x +1}\, x^{2} d}{3 c}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, d^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{4 e \sqrt {c^{2} x^{2}-1}}-\frac {3 e \sqrt {c x -1}\, \sqrt {c x +1}\, d^{2} x}{4 c}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, d^{3}}{c}-\frac {3 e \sqrt {c x -1}\, \sqrt {c x +1}\, d^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{4 c^{2} \sqrt {c^{2} x^{2}-1}}-\frac {3 e^{3} \sqrt {c x -1}\, \sqrt {c x +1}\, x}{32 c^{3}}-\frac {2 e^{2} \sqrt {c x -1}\, \sqrt {c x +1}\, d}{3 c^{3}}-\frac {3 e^{3} \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{32 c^{4} \sqrt {c^{2} x^{2}-1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 229, normalized size = 1.25 \[ -\frac {1}{96} \, {\left (\frac {6 \, \sqrt {c^{2} x^{2} - 1} e^{3} x^{3}}{c^{2}} + \frac {32 \, \sqrt {c^{2} x^{2} - 1} d e^{2} x^{2}}{c^{2}} + \frac {72 \, \sqrt {c^{2} x^{2} - 1} d^{2} e x}{c^{2}} + \frac {96 \, \sqrt {c^{2} x^{2} - 1} d^{3}}{c^{2}} + \frac {72 \, d^{2} e \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{3}} + \frac {9 \, \sqrt {c^{2} x^{2} - 1} e^{3} x}{c^{4}} + \frac {64 \, \sqrt {c^{2} x^{2} - 1} d e^{2}}{c^{4}} + \frac {9 \, e^{3} \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{5}}\right )} c + \frac {1}{4} \, {\left (e^{3} x^{4} + 4 \, d e^{2} x^{3} + 6 \, d^{2} e x^{2} + 4 \, d^{3} x\right )} \operatorname {arcosh}\left (c x\right ) \]
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 \[ \int \mathrm {acosh}\left (c\,x\right )\,{\left (d+e\,x\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.28, size = 258, normalized size = 1.41 \[ \begin {cases} d^{3} x \operatorname {acosh}{\left (c x \right )} + \frac {3 d^{2} e x^{2} \operatorname {acosh}{\left (c x \right )}}{2} + d e^{2} x^{3} \operatorname {acosh}{\left (c x \right )} + \frac {e^{3} x^{4} \operatorname {acosh}{\left (c x \right )}}{4} - \frac {d^{3} \sqrt {c^{2} x^{2} - 1}}{c} - \frac {3 d^{2} e x \sqrt {c^{2} x^{2} - 1}}{4 c} - \frac {d e^{2} x^{2} \sqrt {c^{2} x^{2} - 1}}{3 c} - \frac {e^{3} x^{3} \sqrt {c^{2} x^{2} - 1}}{16 c} - \frac {3 d^{2} e \operatorname {acosh}{\left (c x \right )}}{4 c^{2}} - \frac {2 d e^{2} \sqrt {c^{2} x^{2} - 1}}{3 c^{3}} - \frac {3 e^{3} x \sqrt {c^{2} x^{2} - 1}}{32 c^{3}} - \frac {3 e^{3} \operatorname {acosh}{\left (c x \right )}}{32 c^{4}} & \text {for}\: c \neq 0 \\\frac {i \pi \left (d^{3} x + \frac {3 d^{2} e x^{2}}{2} + d e^{2} x^{3} + \frac {e^{3} x^{4}}{4}\right )}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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