Optimal. Leaf size=186 \[ -\frac{2 b c x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 d}+\frac{2}{27} b^2 x^2 \sqrt{d-c^2 d x^2}-\frac{14 b^2 \sqrt{d-c^2 d x^2}}{27 c^2} \]
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Rubi [A] time = 0.370248, antiderivative size = 194, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {5798, 5718, 5680, 12, 460, 74} \[ -\frac{2 b c x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{(1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2}+\frac{2}{27} b^2 x^2 \sqrt{d-c^2 d x^2}-\frac{14 b^2 \sqrt{d-c^2 d x^2}}{27 c^2} \]
Antiderivative was successfully verified.
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Rule 5798
Rule 5718
Rule 5680
Rule 12
Rule 460
Rule 74
Rubi steps
\begin{align*} \int x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx &=\frac{\sqrt{d-c^2 d x^2} \int x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{(1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2}-\frac{\left (2 b \sqrt{d-c^2 d x^2}\right ) \int \left (-1+c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{3 c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{2 b x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{(1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2}+\frac{\left (2 b^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x \left (-3+c^2 x^2\right )}{3 \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{2 b x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{(1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2}+\frac{\left (2 b^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x \left (-3+c^2 x^2\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{9 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{2}{27} b^2 x^2 \sqrt{d-c^2 d x^2}+\frac{2 b x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{(1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2}-\frac{\left (14 b^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{27 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{14 b^2 \sqrt{d-c^2 d x^2}}{27 c^2}+\frac{2}{27} b^2 x^2 \sqrt{d-c^2 d x^2}+\frac{2 b x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{(1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2}\\ \end{align*}
Mathematica [A] time = 0.347932, size = 181, normalized size = 0.97 \[ \frac{\sqrt{d-c^2 d x^2} \left (9 a^2 \left (c^2 x^2-1\right )^2-6 a b c x \sqrt{c x-1} \sqrt{c x+1} \left (c^2 x^2-3\right )+6 b \cosh ^{-1}(c x) \left (3 a \left (c^2 x^2-1\right )^2+b c x \sqrt{c x-1} \sqrt{c x+1} \left (3-c^2 x^2\right )\right )+2 b^2 \left (c^4 x^4-8 c^2 x^2+7\right )+9 b^2 \left (c^2 x^2-1\right )^2 \cosh ^{-1}(c x)^2\right )}{27 c^2 \left (c^2 x^2-1\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.366, size = 726, normalized size = 3.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13214, size = 275, normalized size = 1.48 \begin{align*} \frac{2}{27} \, b^{2}{\left (\frac{\sqrt{c^{2} x^{2} - 1} \sqrt{-d} d x^{2} - \frac{7 \, \sqrt{c^{2} x^{2} - 1} \sqrt{-d} d}{c^{2}}}{d} - \frac{3 \,{\left (c^{2} \sqrt{-d} d x^{3} - 3 \, \sqrt{-d} d x\right )} \operatorname{arcosh}\left (c x\right )}{c d}\right )} - \frac{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} b^{2} \operatorname{arcosh}\left (c x\right )^{2}}{3 \, c^{2} d} - \frac{2 \,{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} a b \operatorname{arcosh}\left (c x\right )}{3 \, c^{2} d} - \frac{2 \,{\left (c^{2} \sqrt{-d} d x^{3} - 3 \, \sqrt{-d} d x\right )} a b}{9 \, c d} - \frac{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} a^{2}}{3 \, c^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.27491, size = 591, normalized size = 3.18 \begin{align*} \frac{9 \,{\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \sqrt{-c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right )^{2} - 6 \,{\left (a b c^{3} x^{3} - 3 \, a b c x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} - 6 \,{\left ({\left (b^{2} c^{3} x^{3} - 3 \, b^{2} c x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} - 3 \,{\left (a b c^{4} x^{4} - 2 \, a b c^{2} x^{2} + a b\right )} \sqrt{-c^{2} d x^{2} + d}\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) +{\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{4} x^{4} - 2 \,{\left (9 \, a^{2} + 8 \, b^{2}\right )} c^{2} x^{2} + 9 \, a^{2} + 14 \, b^{2}\right )} \sqrt{-c^{2} d x^{2} + d}}{27 \,{\left (c^{4} x^{2} - c^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sqrt{- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname{acosh}{\left (c x \right )}\right )^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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