Optimal. Leaf size=292 \[ -\frac{4 a b x \sqrt{c x-1} \sqrt{c x+1}}{3 c^3 \sqrt{d-c^2 d x^2}}-\frac{2 b x^3 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{9 c \sqrt{d-c^2 d x^2}}-\frac{x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 d}-\frac{2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d}-\frac{2 b^2 x^2 (1-c x) (c x+1)}{27 c^2 \sqrt{d-c^2 d x^2}}-\frac{40 b^2 (1-c x) (c x+1)}{27 c^4 \sqrt{d-c^2 d x^2}}-\frac{4 b^2 x \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{3 c^3 \sqrt{d-c^2 d x^2}} \]
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Rubi [A] time = 0.769446, antiderivative size = 308, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {5798, 5759, 5718, 5654, 74, 5662, 100, 12} \[ -\frac{4 a b x \sqrt{c x-1} \sqrt{c x+1}}{3 c^3 \sqrt{d-c^2 d x^2}}-\frac{2 b x^3 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{9 c \sqrt{d-c^2 d x^2}}-\frac{x^2 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 \sqrt{d-c^2 d x^2}}-\frac{2 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 \sqrt{d-c^2 d x^2}}-\frac{2 b^2 x^2 (1-c x) (c x+1)}{27 c^2 \sqrt{d-c^2 d x^2}}-\frac{40 b^2 (1-c x) (c x+1)}{27 c^4 \sqrt{d-c^2 d x^2}}-\frac{4 b^2 x \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{3 c^3 \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 5798
Rule 5759
Rule 5718
Rule 5654
Rule 74
Rule 5662
Rule 100
Rule 12
Rubi steps
\begin{align*} \int \frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{d-c^2 d x^2}} \, dx &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{d-c^2 d x^2}}\\ &=-\frac{x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 \sqrt{d-c^2 d x^2}}+\frac{\left (2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{3 c^2 \sqrt{d-c^2 d x^2}}-\frac{\left (2 b \sqrt{-1+c x} \sqrt{1+c x}\right ) \int x^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{3 c \sqrt{d-c^2 d x^2}}\\ &=-\frac{2 b x^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{9 c \sqrt{d-c^2 d x^2}}-\frac{2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 \sqrt{d-c^2 d x^2}}-\frac{x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 \sqrt{d-c^2 d x^2}}+\frac{\left (2 b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x^3}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{9 \sqrt{d-c^2 d x^2}}-\frac{\left (4 b \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{3 c^3 \sqrt{d-c^2 d x^2}}\\ &=-\frac{4 a b x \sqrt{-1+c x} \sqrt{1+c x}}{3 c^3 \sqrt{d-c^2 d x^2}}-\frac{2 b^2 x^2 (1-c x) (1+c x)}{27 c^2 \sqrt{d-c^2 d x^2}}-\frac{2 b x^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{9 c \sqrt{d-c^2 d x^2}}-\frac{2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 \sqrt{d-c^2 d x^2}}-\frac{x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 \sqrt{d-c^2 d x^2}}-\frac{\left (4 b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \cosh ^{-1}(c x) \, dx}{3 c^3 \sqrt{d-c^2 d x^2}}+\frac{\left (2 b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{2 x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{27 c^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{4 a b x \sqrt{-1+c x} \sqrt{1+c x}}{3 c^3 \sqrt{d-c^2 d x^2}}-\frac{2 b^2 x^2 (1-c x) (1+c x)}{27 c^2 \sqrt{d-c^2 d x^2}}-\frac{4 b^2 x \sqrt{-1+c x} \sqrt{1+c x} \cosh ^{-1}(c x)}{3 c^3 \sqrt{d-c^2 d x^2}}-\frac{2 b x^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{9 c \sqrt{d-c^2 d x^2}}-\frac{2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 \sqrt{d-c^2 d x^2}}-\frac{x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 \sqrt{d-c^2 d x^2}}+\frac{\left (4 b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{27 c^2 \sqrt{d-c^2 d x^2}}+\frac{\left (4 b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{3 c^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{4 a b x \sqrt{-1+c x} \sqrt{1+c x}}{3 c^3 \sqrt{d-c^2 d x^2}}-\frac{40 b^2 (1-c x) (1+c x)}{27 c^4 \sqrt{d-c^2 d x^2}}-\frac{2 b^2 x^2 (1-c x) (1+c x)}{27 c^2 \sqrt{d-c^2 d x^2}}-\frac{4 b^2 x \sqrt{-1+c x} \sqrt{1+c x} \cosh ^{-1}(c x)}{3 c^3 \sqrt{d-c^2 d x^2}}-\frac{2 b x^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{9 c \sqrt{d-c^2 d x^2}}-\frac{2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 \sqrt{d-c^2 d x^2}}-\frac{x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 \sqrt{d-c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 0.444763, size = 201, normalized size = 0.69 \[ \frac{\sqrt{d-c^2 d x^2} \left (-9 a^2 \left (c^4 x^4+c^2 x^2-2\right )+6 a b c x \sqrt{c x-1} \sqrt{c x+1} \left (c^2 x^2+6\right )+6 b \cosh ^{-1}(c x) \left (b c x \sqrt{c x-1} \sqrt{c x+1} \left (c^2 x^2+6\right )-3 a \left (c^4 x^4+c^2 x^2-2\right )\right )-2 b^2 \left (c^4 x^4+19 c^2 x^2-20\right )-9 b^2 \left (c^4 x^4+c^2 x^2-2\right ) \cosh ^{-1}(c x)^2\right )}{27 c^4 d (c x-1) (c x+1)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.375, size = 752, normalized size = 2.6 \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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.20677, size = 598, normalized size = 2.05 \begin{align*} -\frac{9 \,{\left (b^{2} c^{4} x^{4} + b^{2} c^{2} x^{2} - 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} + 6 \, 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} + 6 \, 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} + a b c^{2} x^{2} - 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} +{\left (9 \, a^{2} + 38 \, b^{2}\right )} c^{2} x^{2} - 18 \, a^{2} - 40 \, b^{2}\right )} \sqrt{-c^{2} d x^{2} + d}}{27 \,{\left (c^{6} d x^{2} - c^{4} d\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 \frac{x^{3} \left (a + b \operatorname{acosh}{\left (c x \right )}\right )^{2}}{\sqrt{- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{2} x^{3}}{\sqrt{-c^{2} d x^{2} + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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