Optimal. Leaf size=123 \[ -\frac {1}{6} d \left (\frac {3 e}{c^2}+\frac {2 d^2}{e}\right ) \cosh ^{-1}(c x)-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (4 \left (4 c^2 d^2+e^2\right )+5 c^2 d e x\right )}{18 c^3}-\frac {\sqrt {c x-1} \sqrt {c x+1} (d+e x)^2}{9 c}+\frac {\cosh ^{-1}(c x) (d+e x)^3}{3 e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5802, 100, 147, 52} \[ -\frac {\sqrt {c x-1} \sqrt {c x+1} \left (4 \left (4 c^2 d^2+e^2\right )+5 c^2 d e x\right )}{18 c^3}-\frac {1}{6} d \left (\frac {3 e}{c^2}+\frac {2 d^2}{e}\right ) \cosh ^{-1}(c x)-\frac {\sqrt {c x-1} \sqrt {c x+1} (d+e x)^2}{9 c}+\frac {\cosh ^{-1}(c x) (d+e x)^3}{3 e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 52
Rule 100
Rule 147
Rule 5802
Rubi steps
\begin {align*} \int (d+e x)^2 \cosh ^{-1}(c x) \, dx &=\frac {(d+e x)^3 \cosh ^{-1}(c x)}{3 e}-\frac {c \int \frac {(d+e x)^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 e}\\ &=-\frac {\sqrt {-1+c x} \sqrt {1+c x} (d+e x)^2}{9 c}+\frac {(d+e x)^3 \cosh ^{-1}(c x)}{3 e}-\frac {\int \frac {(d+e x) \left (3 c^2 d^2+2 e^2+5 c^2 d e x\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{9 c e}\\ &=-\frac {\sqrt {-1+c x} \sqrt {1+c x} (d+e x)^2}{9 c}-\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (4 \left (4 c^2 d^2+e^2\right )+5 c^2 d e x\right )}{18 c^3}+\frac {(d+e x)^3 \cosh ^{-1}(c x)}{3 e}-\frac {1}{6} \left (d \left (\frac {2 c d^2}{e}+\frac {3 e}{c}\right )\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {\sqrt {-1+c x} \sqrt {1+c x} (d+e x)^2}{9 c}-\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (4 \left (4 c^2 d^2+e^2\right )+5 c^2 d e x\right )}{18 c^3}-\frac {1}{6} d \left (\frac {2 d^2}{e}+\frac {3 e}{c^2}\right ) \cosh ^{-1}(c x)+\frac {(d+e x)^3 \cosh ^{-1}(c x)}{3 e}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.21, size = 113, normalized size = 0.92 \[ -\frac {-6 c^3 x \cosh ^{-1}(c x) \left (3 d^2+3 d e x+e^2 x^2\right )+\sqrt {c x-1} \sqrt {c x+1} \left (c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )+4 e^2\right )+9 c d e \log \left (c x+\sqrt {c x-1} \sqrt {c x+1}\right )}{18 c^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.53, size = 106, normalized size = 0.86 \[ \frac {3 \, {\left (2 \, c^{3} e^{2} x^{3} + 6 \, c^{3} d e x^{2} + 6 \, c^{3} d^{2} x - 3 \, c d e\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (2 \, c^{2} e^{2} x^{2} + 9 \, c^{2} d e x + 18 \, c^{2} d^{2} + 4 \, e^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{18 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.35, size = 129, normalized size = 1.05 \[ \frac {1}{3} \, {\left (x e + d\right )}^{3} e^{\left (-1\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - \frac {1}{18} \, {\left (\sqrt {c^{2} x^{2} - 1} {\left (x {\left (\frac {2 \, x e^{3}}{c} + \frac {9 \, d e^{2}}{c}\right )} + \frac {2 \, {\left (9 \, c^{3} d^{2} e + 2 \, c e^{3}\right )}}{c^{4}}\right )} - \frac {3 \, {\left (2 \, c^{2} d^{3} + 3 \, d e^{2}\right )} \log \left ({\left | -x {\left | c \right |} + \sqrt {c^{2} x^{2} - 1} \right |}\right )}{c {\left | c \right |}}\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.01, size = 233, normalized size = 1.89 \[ \frac {e^{2} \mathrm {arccosh}\left (c x \right ) x^{3}}{3}+e \,\mathrm {arccosh}\left (c x \right ) x^{2} d +\mathrm {arccosh}\left (c x \right ) x \,d^{2}+\frac {\mathrm {arccosh}\left (c x \right ) d^{3}}{3 e}-\frac {e^{2} \sqrt {c x -1}\, \sqrt {c x +1}\, x^{2}}{9 c}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, d^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{3 e \sqrt {c^{2} x^{2}-1}}-\frac {e \sqrt {c x -1}\, \sqrt {c x +1}\, d x}{2 c}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, d^{2}}{c}-\frac {e \sqrt {c x -1}\, \sqrt {c x +1}\, d \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{2 c^{2} \sqrt {c^{2} x^{2}-1}}-\frac {2 e^{2} \sqrt {c x -1}\, \sqrt {c x +1}}{9 c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.49, size = 140, normalized size = 1.14 \[ -\frac {1}{18} \, {\left (\frac {2 \, \sqrt {c^{2} x^{2} - 1} e^{2} x^{2}}{c^{2}} + \frac {9 \, \sqrt {c^{2} x^{2} - 1} d e x}{c^{2}} + \frac {18 \, \sqrt {c^{2} x^{2} - 1} d^{2}}{c^{2}} + \frac {9 \, d e \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{3}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} e^{2}}{c^{4}}\right )} c + \frac {1}{3} \, {\left (e^{2} x^{3} + 3 \, d e x^{2} + 3 \, d^{2} x\right )} \operatorname {arcosh}\left (c x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \mathrm {acosh}\left (c\,x\right )\,{\left (d+e\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.58, size = 155, normalized size = 1.26 \[ \begin {cases} d^{2} x \operatorname {acosh}{\left (c x \right )} + d e x^{2} \operatorname {acosh}{\left (c x \right )} + \frac {e^{2} x^{3} \operatorname {acosh}{\left (c x \right )}}{3} - \frac {d^{2} \sqrt {c^{2} x^{2} - 1}}{c} - \frac {d e x \sqrt {c^{2} x^{2} - 1}}{2 c} - \frac {e^{2} x^{2} \sqrt {c^{2} x^{2} - 1}}{9 c} - \frac {d e \operatorname {acosh}{\left (c x \right )}}{2 c^{2}} - \frac {2 e^{2} \sqrt {c^{2} x^{2} - 1}}{9 c^{3}} & \text {for}\: c \neq 0 \\\frac {i \pi \left (d^{2} x + d e x^{2} + \frac {e^{2} x^{3}}{3}\right )}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________