Optimal. Leaf size=204 \[ -\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b c \sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{2} x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{b c x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{4} b^2 x \sqrt{d-c^2 d x^2}+\frac{b^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{4 c \sqrt{c x-1} \sqrt{c x+1}} \]
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Rubi [A] time = 0.348636, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {5713, 5683, 5676, 5662, 90, 52} \[ -\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b c \sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{2} x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{b c x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{4} b^2 x \sqrt{d-c^2 d x^2}+\frac{b^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{4 c \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
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Rule 5713
Rule 5683
Rule 5676
Rule 5662
Rule 90
Rule 52
Rubi steps
\begin{align*} \int \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 \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}{2} x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{\sqrt{d-c^2 d x^2} \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b c \sqrt{d-c^2 d x^2}\right ) \int x \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{1}{2} x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b^2 c^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{4} b^2 x \sqrt{d-c^2 d x^2}-\frac{b c x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{1}{2} x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{4 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{4} b^2 x \sqrt{d-c^2 d x^2}+\frac{b^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{4 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{1}{2} x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b c \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 1.03956, size = 235, normalized size = 1.15 \[ \frac{1}{24} \left (12 a^2 x \sqrt{d-c^2 d x^2}-\frac{12 a^2 \sqrt{d} \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )}{c}-\frac{6 a b \sqrt{d-c^2 d x^2} \left (2 \cosh ^{-1}(c x)^2+\cosh \left (2 \cosh ^{-1}(c x)\right )-2 \cosh ^{-1}(c x) \sinh \left (2 \cosh ^{-1}(c x)\right )\right )}{c \sqrt{\frac{c x-1}{c x+1}} (c x+1)}+\frac{b^2 \sqrt{d-c^2 d x^2} \left (-4 \cosh ^{-1}(c x)^3-6 \cosh \left (2 \cosh ^{-1}(c x)\right ) \cosh ^{-1}(c x)+\left (6 \cosh ^{-1}(c x)^2+3\right ) \sinh \left (2 \cosh ^{-1}(c x)\right )\right )}{c \sqrt{\frac{c x-1}{c x+1}} (c x+1)}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.224, size = 528, normalized size = 2.6 \begin{align*}{\frac{x{a}^{2}}{2}\sqrt{-{c}^{2}d{x}^{2}+d}}+{\frac{{a}^{2}d}{2}\arctan \left ({x\sqrt{{c}^{2}d}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}} \right ){\frac{1}{\sqrt{{c}^{2}d}}}}-{\frac{{b}^{2} \left ({\rm arccosh} \left (cx\right ) \right ) ^{3}}{6\,c}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}-{\frac{{b}^{2}c{\rm arccosh} \left (cx\right ){x}^{2}}{2}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}+{\frac{{b}^{2}{c}^{2} \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}{x}^{3}}{ \left ( 2\,cx+2 \right ) \left ( cx-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{{b}^{2} \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}x}{ \left ( 2\,cx+2 \right ) \left ( cx-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{{b}^{2}{c}^{2}{x}^{3}}{ \left ( 4\,cx+4 \right ) \left ( cx-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{{b}^{2}x}{ \left ( 4\,cx+4 \right ) \left ( cx-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{{b}^{2}{\rm arccosh} \left (cx\right )}{4\,c}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}-{\frac{ab \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}}{2\,c}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}+{\frac{ab{c}^{2}{\rm arccosh} \left (cx\right ){x}^{3}}{ \left ( cx+1 \right ) \left ( cx-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{abc{x}^{2}}{2}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}-{\frac{ab{\rm arccosh} \left (cx\right )x}{ \left ( cx+1 \right ) \left ( cx-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{ab}{4\,c}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{-c^{2} d x^{2} + d}{\left (b^{2} \operatorname{arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname{arcosh}\left (c x\right ) + a^{2}\right )}, x\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 \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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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