Optimal. Leaf size=195 \[ -\frac{1}{3} c^6 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )+3 c^4 d^3 x \left (a+b \cosh ^{-1}(c x)\right )+\frac{3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac{1}{9} b c^3 d^3 (c x-1)^{3/2} (c x+1)^{3/2}-\frac{8}{3} b c^3 d^3 \sqrt{c x-1} \sqrt{c x+1}-\frac{17}{6} b c^3 d^3 \tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right )+\frac{b c d^3 \sqrt{c x-1} \sqrt{c x+1}}{6 x^2} \]
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Rubi [A] time = 0.389749, antiderivative size = 252, normalized size of antiderivative = 1.29, number of steps used = 9, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {270, 5731, 12, 1610, 1799, 1621, 897, 1153, 205} \[ -\frac{1}{3} c^6 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )+3 c^4 d^3 x \left (a+b \cosh ^{-1}(c x)\right )+\frac{3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac{b c^3 d^3 \left (1-c^2 x^2\right )^2}{9 \sqrt{c x-1} \sqrt{c x+1}}+\frac{8 b c^3 d^3 \left (1-c^2 x^2\right )}{3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d^3 \left (1-c^2 x^2\right )}{6 x^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{17 b c^3 d^3 \sqrt{c^2 x^2-1} \tan ^{-1}\left (\sqrt{c^2 x^2-1}\right )}{6 \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
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Rule 270
Rule 5731
Rule 12
Rule 1610
Rule 1799
Rule 1621
Rule 897
Rule 1153
Rule 205
Rubi steps
\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{x^4} \, dx &=-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac{3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}+3 c^4 d^3 x \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{3} c^6 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac{d^3 \left (-1+9 c^2 x^2+9 c^4 x^4-c^6 x^6\right )}{3 x^3 \sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac{3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}+3 c^4 d^3 x \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{3} c^6 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{3} \left (b c d^3\right ) \int \frac{-1+9 c^2 x^2+9 c^4 x^4-c^6 x^6}{x^3 \sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac{3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}+3 c^4 d^3 x \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{3} c^6 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c d^3 \sqrt{-1+c^2 x^2}\right ) \int \frac{-1+9 c^2 x^2+9 c^4 x^4-c^6 x^6}{x^3 \sqrt{-1+c^2 x^2}} \, dx}{3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac{3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}+3 c^4 d^3 x \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{3} c^6 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c d^3 \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{-1+9 c^2 x+9 c^4 x^2-c^6 x^3}{x^2 \sqrt{-1+c^2 x}} \, dx,x,x^2\right )}{6 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d^3 \left (1-c^2 x^2\right )}{6 x^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac{3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}+3 c^4 d^3 x \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{3} c^6 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c d^3 \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\frac{17 c^2}{2}+9 c^4 x-c^6 x^2}{x \sqrt{-1+c^2 x}} \, dx,x,x^2\right )}{6 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d^3 \left (1-c^2 x^2\right )}{6 x^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac{3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}+3 c^4 d^3 x \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{3} c^6 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b d^3 \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\frac{33 c^2}{2}+7 c^2 x^2-c^2 x^4}{\frac{1}{c^2}+\frac{x^2}{c^2}} \, dx,x,\sqrt{-1+c^2 x^2}\right )}{3 c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d^3 \left (1-c^2 x^2\right )}{6 x^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac{3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}+3 c^4 d^3 x \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{3} c^6 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b d^3 \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (8 c^4-c^4 x^2+\frac{17 c^2}{2 \left (\frac{1}{c^2}+\frac{x^2}{c^2}\right )}\right ) \, dx,x,\sqrt{-1+c^2 x^2}\right )}{3 c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{8 b c^3 d^3 \left (1-c^2 x^2\right )}{3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c d^3 \left (1-c^2 x^2\right )}{6 x^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d^3 \left (1-c^2 x^2\right )^2}{9 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac{3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}+3 c^4 d^3 x \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{3} c^6 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (17 b c d^3 \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{8 b c^3 d^3 \left (1-c^2 x^2\right )}{3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c d^3 \left (1-c^2 x^2\right )}{6 x^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d^3 \left (1-c^2 x^2\right )^2}{9 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac{3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}+3 c^4 d^3 x \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{3} c^6 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{17 b c^3 d^3 \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 = 0.291029, size = 142, normalized size = 0.73 \[ \frac{d^3 \left (-6 a c^6 x^6+54 a c^4 x^4+54 a c^2 x^2-6 a+b c x \sqrt{c x-1} \sqrt{c x+1} \left (2 c^4 x^4-50 c^2 x^2+3\right )+51 b c^3 x^3 \tan ^{-1}\left (\frac{1}{\sqrt{c x-1} \sqrt{c x+1}}\right )-6 b \left (c^6 x^6-9 c^4 x^4-9 c^2 x^2+1\right ) \cosh ^{-1}(c x)\right )}{18 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 223, normalized size = 1.1 \begin{align*} -{\frac{{c}^{6}{d}^{3}a{x}^{3}}{3}}+3\,{c}^{4}{d}^{3}ax+3\,{\frac{{c}^{2}{d}^{3}a}{x}}-{\frac{{d}^{3}a}{3\,{x}^{3}}}-{\frac{{c}^{6}{d}^{3}b{\rm arccosh} \left (cx\right ){x}^{3}}{3}}+3\,{c}^{4}{d}^{3}b{\rm arccosh} \left (cx\right )x+3\,{\frac{b{c}^{2}{d}^{3}{\rm arccosh} \left (cx\right )}{x}}-{\frac{b{d}^{3}{\rm arccosh} \left (cx\right )}{3\,{x}^{3}}}+{\frac{{c}^{5}{d}^{3}b{x}^{2}}{9}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{25\,{d}^{3}b{c}^{3}}{9}\sqrt{cx-1}\sqrt{cx+1}}+{\frac{17\,{d}^{3}b{c}^{3}}{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}^{3}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 = 2.18894, size = 286, normalized size = 1.47 \begin{align*} -\frac{1}{3} \, a c^{6} d^{3} x^{3} - \frac{1}{9} \,{\left (3 \, x^{3} \operatorname{arcosh}\left (c x\right ) - c{\left (\frac{\sqrt{c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac{2 \, \sqrt{c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b c^{6} d^{3} + 3 \, a c^{4} d^{3} x + 3 \,{\left (c x \operatorname{arcosh}\left (c x\right ) - \sqrt{c^{2} x^{2} - 1}\right )} b c^{3} d^{3} + 3 \,{\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^{3} - \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^{3} + \frac{3 \, a c^{2} d^{3}}{x} - \frac{a d^{3}}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.40329, size = 554, normalized size = 2.84 \begin{align*} -\frac{6 \, a c^{6} d^{3} x^{6} - 54 \, a c^{4} d^{3} x^{4} + 102 \, b c^{3} d^{3} x^{3} \arctan \left (-c x + \sqrt{c^{2} x^{2} - 1}\right ) - 54 \, a c^{2} d^{3} x^{2} - 6 \,{\left (b c^{6} - 9 \, b c^{4} - 9 \, b c^{2} + b\right )} d^{3} x^{3} \log \left (-c x + \sqrt{c^{2} x^{2} - 1}\right ) + 6 \, a d^{3} + 6 \,{\left (b c^{6} d^{3} x^{6} - 9 \, b c^{4} d^{3} x^{4} - 9 \, b c^{2} d^{3} x^{2} -{\left (b c^{6} - 9 \, b c^{4} - 9 \, b c^{2} + b\right )} d^{3} x^{3} + b d^{3}\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (2 \, b c^{5} d^{3} x^{5} - 50 \, b c^{3} d^{3} x^{3} + 3 \, b c d^{3} x\right )} \sqrt{c^{2} x^{2} - 1}}{18 \, 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^{3} \left (\int - 3 a c^{4}\, dx + \int - \frac{a}{x^{4}}\, dx + \int \frac{3 a c^{2}}{x^{2}}\, dx + \int a c^{6} x^{2}\, dx + \int - 3 b c^{4} \operatorname{acosh}{\left (c x \right )}\, dx + \int - \frac{b \operatorname{acosh}{\left (c x \right )}}{x^{4}}\, dx + \int \frac{3 b c^{2} \operatorname{acosh}{\left (c x \right )}}{x^{2}}\, dx + \int b c^{6} x^{2} \operatorname{acosh}{\left (c x \right )}\, 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 )}^{3}{\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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