Optimal. Leaf size=142 \[ c^4 d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac{2 c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-b c^3 d^2 \sqrt{c x-1} \sqrt{c x+1}-\frac{11}{6} b c^3 d^2 \tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right )+\frac{b c d^2 \sqrt{c x-1} \sqrt{c x+1}}{6 x^2} \]
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Rubi [A] time = 0.234374, antiderivative size = 186, normalized size of antiderivative = 1.31, number of steps used = 8, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {270, 5731, 12, 520, 1251, 897, 1157, 388, 205} \[ c^4 d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac{2 c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac{b c^3 d^2 \left (1-c^2 x^2\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d^2 \left (1-c^2 x^2\right )}{6 x^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{11 b c^3 d^2 \sqrt{c^2 x^2-1} \tan ^{-1}\left (\sqrt{c^2 x^2-1}\right )}{6 \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
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Rule 270
Rule 5731
Rule 12
Rule 520
Rule 1251
Rule 897
Rule 1157
Rule 388
Rule 205
Rubi steps
\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{x^4} \, dx &=-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac{2 c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}+c^4 d^2 x \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac{d^2 \left (-1+6 c^2 x^2+3 c^4 x^4\right )}{3 x^3 \sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac{2 c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}+c^4 d^2 x \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{3} \left (b c d^2\right ) \int \frac{-1+6 c^2 x^2+3 c^4 x^4}{x^3 \sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac{2 c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}+c^4 d^2 x \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c d^2 \sqrt{-1+c^2 x^2}\right ) \int \frac{-1+6 c^2 x^2+3 c^4 x^4}{x^3 \sqrt{-1+c^2 x^2}} \, dx}{3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac{2 c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}+c^4 d^2 x \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c d^2 \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{-1+6 c^2 x+3 c^4 x^2}{x^2 \sqrt{-1+c^2 x}} \, dx,x,x^2\right )}{6 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac{2 c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}+c^4 d^2 x \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b d^2 \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{8+12 x^2+3 x^4}{\left (\frac{1}{c^2}+\frac{x^2}{c^2}\right )^2} \, dx,x,\sqrt{-1+c^2 x^2}\right )}{3 c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d^2 \left (1-c^2 x^2\right )}{6 x^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac{2 c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}+c^4 d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac{\left (b c d^2 \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{-17-6 x^2}{\frac{1}{c^2}+\frac{x^2}{c^2}} \, dx,x,\sqrt{-1+c^2 x^2}\right )}{6 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b c^3 d^2 \left (1-c^2 x^2\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c d^2 \left (1-c^2 x^2\right )}{6 x^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac{2 c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}+c^4 d^2 x \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (11 b c d^2 \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{c^2}+\frac{x^2}{c^2}} \, dx,x,\sqrt{-1+c^2 x^2}\right )}{6 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b c^3 d^2 \left (1-c^2 x^2\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c d^2 \left (1-c^2 x^2\right )}{6 x^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac{2 c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}+c^4 d^2 x \left (a+b \cosh ^{-1}(c x)\right )-\frac{11 b c^3 d^2 \sqrt{-1+c^2 x^2} \tan ^{-1}\left (\sqrt{-1+c^2 x^2}\right )}{6 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 0.160193, size = 135, normalized size = 0.95 \[ \frac{d^2 \left (6 a c^4 x^4+12 a c^2 x^2-2 a-6 b c^3 x^3 \sqrt{c x-1} \sqrt{c x+1}+11 b c^3 x^3 \tan ^{-1}\left (\frac{1}{\sqrt{c x-1} \sqrt{c x+1}}\right )+2 b \left (3 c^4 x^4+6 c^2 x^2-1\right ) \cosh ^{-1}(c x)+b c x \sqrt{c x-1} \sqrt{c x+1}\right )}{6 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 167, normalized size = 1.2 \begin{align*}{c}^{4}{d}^{2}ax+2\,{\frac{{c}^{2}{d}^{2}a}{x}}-{\frac{{d}^{2}a}{3\,{x}^{3}}}+{c}^{4}{d}^{2}b{\rm arccosh} \left (cx\right )x+2\,{\frac{b{c}^{2}{d}^{2}{\rm arccosh} \left (cx\right )}{x}}-{\frac{b{d}^{2}{\rm arccosh} \left (cx\right )}{3\,{x}^{3}}}-b{c}^{3}{d}^{2}\sqrt{cx-1}\sqrt{cx+1}+{\frac{11\,b{c}^{3}{d}^{2}}{6}\sqrt{cx-1}\sqrt{cx+1}\arctan \left ({\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}}+{\frac{{d}^{2}bc}{6\,{x}^{2}}\sqrt{cx-1}\sqrt{cx+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.74458, size = 190, normalized size = 1.34 \begin{align*} a c^{4} d^{2} x +{\left (c x \operatorname{arcosh}\left (c x\right ) - \sqrt{c^{2} x^{2} - 1}\right )} b c^{3} d^{2} + 2 \,{\left (c \arcsin \left (\frac{1}{\sqrt{c^{2}}{\left | x \right |}}\right ) + \frac{\operatorname{arcosh}\left (c x\right )}{x}\right )} b c^{2} d^{2} - \frac{1}{6} \,{\left ({\left (c^{2} \arcsin \left (\frac{1}{\sqrt{c^{2}}{\left | x \right |}}\right ) - \frac{\sqrt{c^{2} x^{2} - 1}}{x^{2}}\right )} c + \frac{2 \, \operatorname{arcosh}\left (c x\right )}{x^{3}}\right )} b d^{2} + \frac{2 \, a c^{2} d^{2}}{x} - \frac{a d^{2}}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.96439, size = 452, normalized size = 3.18 \begin{align*} \frac{6 \, a c^{4} d^{2} x^{4} - 22 \, b c^{3} d^{2} x^{3} \arctan \left (-c x + \sqrt{c^{2} x^{2} - 1}\right ) + 12 \, a c^{2} d^{2} x^{2} - 2 \,{\left (3 \, b c^{4} + 6 \, b c^{2} - b\right )} d^{2} x^{3} \log \left (-c x + \sqrt{c^{2} x^{2} - 1}\right ) - 2 \, a d^{2} + 2 \,{\left (3 \, b c^{4} d^{2} x^{4} + 6 \, b c^{2} d^{2} x^{2} -{\left (3 \, b c^{4} + 6 \, b c^{2} - b\right )} d^{2} x^{3} - b d^{2}\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (6 \, b c^{3} d^{2} x^{3} - b c d^{2} x\right )} \sqrt{c^{2} x^{2} - 1}}{6 \, x^{3}} \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*} d^{2} \left (\int a c^{4}\, dx + \int \frac{a}{x^{4}}\, dx + \int - \frac{2 a c^{2}}{x^{2}}\, dx + \int b c^{4} \operatorname{acosh}{\left (c x \right )}\, dx + \int \frac{b \operatorname{acosh}{\left (c x \right )}}{x^{4}}\, dx + \int - \frac{2 b c^{2} \operatorname{acosh}{\left (c x \right )}}{x^{2}}\, dx\right ) \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 )}^{2}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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