Optimal. Leaf size=106 \[ -\frac {3 b d \sqrt {-1+c x} \sqrt {1+c x}}{4 c}-\frac {b \sqrt {-1+c x} \sqrt {1+c x} (d+e x)}{4 c}-\frac {b \left (2 d^2+\frac {e^2}{c^2}\right ) \cosh ^{-1}(c x)}{4 e}+\frac {(d+e x)^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 e} \]
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Rubi [A]
time = 0.03, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5963, 92, 81,
54} \begin {gather*} \frac {(d+e x)^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 e}-\frac {b \left (\frac {e^2}{c^2}+2 d^2\right ) \cosh ^{-1}(c x)}{4 e}-\frac {b \sqrt {c x-1} \sqrt {c x+1} (d+e x)}{4 c}-\frac {3 b d \sqrt {c x-1} \sqrt {c x+1}}{4 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 54
Rule 81
Rule 92
Rule 5963
Rubi steps
\begin {align*} \int (d+e x) \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {(d+e x)^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 e}-\frac {(b c) \int \frac {(d+e x)^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 e}\\ &=-\frac {b \sqrt {-1+c x} \sqrt {1+c x} (d+e x)}{4 c}+\frac {(d+e x)^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 e}-\frac {b \int \frac {2 c^2 d^2+e^2+3 c^2 d e x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{4 c e}\\ &=-\frac {3 b d \sqrt {-1+c x} \sqrt {1+c x}}{4 c}-\frac {b \sqrt {-1+c x} \sqrt {1+c x} (d+e x)}{4 c}+\frac {(d+e x)^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 e}-\frac {\left (b \left (2 c^2 d^2+e^2\right )\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{4 c e}\\ &=-\frac {3 b d \sqrt {-1+c x} \sqrt {1+c x}}{4 c}-\frac {b \sqrt {-1+c x} \sqrt {1+c x} (d+e x)}{4 c}-\frac {b \left (2 d^2+\frac {e^2}{c^2}\right ) \cosh ^{-1}(c x)}{4 e}+\frac {(d+e x)^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 e}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 117, normalized size = 1.10 \begin {gather*} a d x+\frac {1}{2} a e x^2-\frac {b d \sqrt {-1+c x} \sqrt {1+c x}}{c}-\frac {b e x \sqrt {-1+c x} \sqrt {1+c x}}{4 c}+b d x \cosh ^{-1}(c x)+\frac {1}{2} b e x^2 \cosh ^{-1}(c x)-\frac {b e \tanh ^{-1}\left (\frac {\sqrt {-1+c x}}{\sqrt {1+c x}}\right )}{2 c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.10, size = 133, normalized size = 1.25
method | result | size |
derivativedivides | \(\frac {\frac {a \left (c^{2} d x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+b \,\mathrm {arccosh}\left (c x \right ) d c x +\frac {b c \,\mathrm {arccosh}\left (c x \right ) e \,x^{2}}{2}-b \sqrt {c x -1}\, \sqrt {c x +1}\, d -\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, e x}{4}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{4 c \sqrt {c^{2} x^{2}-1}}}{c}\) | \(133\) |
default | \(\frac {\frac {a \left (c^{2} d x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+b \,\mathrm {arccosh}\left (c x \right ) d c x +\frac {b c \,\mathrm {arccosh}\left (c x \right ) e \,x^{2}}{2}-b \sqrt {c x -1}\, \sqrt {c x +1}\, d -\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, e x}{4}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{4 c \sqrt {c^{2} x^{2}-1}}}{c}\) | \(133\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 101, normalized size = 0.95 \begin {gather*} \frac {1}{2} \, a x^{2} e + a d x + \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 e + \frac {{\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b d}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 125, normalized size = 1.18 \begin {gather*} \frac {2 \, a c^{2} x^{2} \cosh \left (1\right ) + 2 \, a c^{2} x^{2} \sinh \left (1\right ) + 4 \, a c^{2} d x + {\left (4 \, b c^{2} d x + {\left (2 \, b c^{2} x^{2} - b\right )} \cosh \left (1\right ) + {\left (2 \, b c^{2} x^{2} - b\right )} \sinh \left (1\right )\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - \sqrt {c^{2} x^{2} - 1} {\left (b c x \cosh \left (1\right ) + b c x \sinh \left (1\right ) + 4 \, b c d\right )}}{4 \, c^{2}} \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 = 0.11, size = 105, normalized size = 0.99 \begin {gather*} \begin {cases} a d x + \frac {a e x^{2}}{2} + b d x \operatorname {acosh}{\left (c x \right )} + \frac {b e x^{2} \operatorname {acosh}{\left (c x \right )}}{2} - \frac {b d \sqrt {c^{2} x^{2} - 1}}{c} - \frac {b e x \sqrt {c^{2} x^{2} - 1}}{4 c} - \frac {b e \operatorname {acosh}{\left (c x \right )}}{4 c^{2}} & \text {for}\: c \neq 0 \\\left (a + \frac {i \pi b}{2}\right ) \left (d x + \frac {e x^{2}}{2}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 124, normalized size = 1.17 \begin {gather*} \frac {1}{2} \, a e x^{2} + {\left (x \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - \frac {\sqrt {c^{2} x^{2} - 1}}{c}\right )} b d + \frac {1}{4} \, {\left (2 \, x^{2} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x}{c^{2}} - \frac {\log \left ({\left | -x {\left | c \right |} + \sqrt {c^{2} x^{2} - 1} \right |}\right )}{c^{2} {\left | c \right |}}\right )}\right )} b e + a d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.03, size = 83, normalized size = 0.78 \begin {gather*} \frac {a\,x\,\left (2\,d+e\,x\right )}{2}+b\,d\,x\,\mathrm {acosh}\left (c\,x\right )+b\,e\,x\,\mathrm {acosh}\left (c\,x\right )\,\left (\frac {x}{2}-\frac {1}{4\,c^2\,x}\right )-\frac {b\,d\,\sqrt {c\,x-1}\,\sqrt {c\,x+1}}{c}-\frac {b\,e\,x\,\sqrt {c\,x-1}\,\sqrt {c\,x+1}}{4\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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