Optimal. Leaf size=197 \[ -\frac{3}{2} c^2 d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{3 c d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b \sqrt{c x-1} \sqrt{c x+1}}-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{x}+\frac{b c^3 d x^2 \sqrt{d-c^2 d x^2}}{4 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c d \log (x) \sqrt{d-c^2 d x^2}}{\sqrt{c x-1} \sqrt{c x+1}} \]
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Rubi [A] time = 0.536116, antiderivative size = 209, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {5798, 5740, 5683, 5676, 30, 14} \[ -\frac{3}{2} c^2 d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{3 c d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b \sqrt{c x-1} \sqrt{c x+1}}-\frac{d (1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x}+\frac{b c^3 d x^2 \sqrt{d-c^2 d x^2}}{4 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c d \log (x) \sqrt{d-c^2 d x^2}}{\sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
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Rule 5798
Rule 5740
Rule 5683
Rule 5676
Rule 30
Rule 14
Rubi steps
\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{x^2} \, dx &=-\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \int \frac{(-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{x^2} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x}-\frac{\left (b c d \sqrt{d-c^2 d x^2}\right ) \int \frac{-1+c^2 x^2}{x} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 c^2 d \sqrt{d-c^2 d x^2}\right ) \int \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{3}{2} c^2 d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x}-\frac{\left (b c d \sqrt{d-c^2 d x^2}\right ) \int \left (-\frac{1}{x}+c^2 x\right ) \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (3 c^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (3 b c^3 d \sqrt{d-c^2 d x^2}\right ) \int x \, dx}{2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b c^3 d x^2 \sqrt{d-c^2 d x^2}}{4 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{3}{2} c^2 d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x}+\frac{3 c d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c d \sqrt{d-c^2 d x^2} \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 1.01072, size = 223, normalized size = 1.13 \[ \frac{1}{8} \left (12 a c d^{3/2} \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )-\frac{4 a d \left (c^2 x^2+2\right ) \sqrt{d-c^2 d x^2}}{x}+4 b c d \sqrt{d-c^2 d x^2} \left (\frac{2 \log (c x)+\cosh ^{-1}(c x)^2}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}-\frac{2 \cosh ^{-1}(c x)}{c x}\right )+\frac{b c d \sqrt{d-c^2 d x^2} \left (2 \cosh ^{-1}(c x)^2+\cosh \left (2 \cosh ^{-1}(c x)\right )-2 \cosh ^{-1}(c x) \sinh \left (2 \cosh ^{-1}(c x)\right )\right )}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.204, size = 427, normalized size = 2.2 \begin{align*} -{\frac{a}{dx} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{5}{2}}}}-a{c}^{2}x \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{3}{2}}}-{\frac{3\,da{c}^{2}x}{2}\sqrt{-{c}^{2}d{x}^{2}+d}}-{\frac{3\,a{c}^{2}{d}^{2}}{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{3\,b \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}cd}{4}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}-{\frac{b{c}^{4}d{\rm arccosh} \left (cx\right ){x}^{3}}{ \left ( 2\,cx+2 \right ) \left ( cx-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{b{c}^{3}d{x}^{2}}{4}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}-{bcd{\rm arccosh} \left (cx\right )\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}-{\frac{db{\rm arccosh} \left (cx\right ){c}^{2}x}{ \left ( 2\,cx+2 \right ) \left ( cx-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{bcd}{8}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}+{\frac{bd{\rm arccosh} \left (cx\right )}{ \left ( cx+1 \right ) \left ( cx-1 \right ) x}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{bcd\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\ln \left ( \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{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 (-\frac{{\left (a c^{2} d x^{2} - a d +{\left (b c^{2} d x^{2} - b d\right )} \operatorname{arcosh}\left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{x^{2}}, 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 \frac{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac{3}{2}} \left (a + b \operatorname{acosh}{\left (c x \right )}\right )}{x^{2}}\, 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 (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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