Optimal. Leaf size=388 \[ \frac{6 c^2 \left (1-a^2 x^2\right )^3}{625 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{8 c^2 \left (1-a^2 x^2\right )^2}{375 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{16 c^2 \left (1-a^2 x^2\right )}{125 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{6}{125} a^4 c^2 x^5 \cosh ^{-1}(a x)-\frac{76}{225} a^2 c^2 x^3 \cosh ^{-1}(a x)+\frac{1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3+\frac{4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac{8}{135} a c^2 x^2 \sqrt{a x-1} \sqrt{a x+1}-\frac{488 c^2 \sqrt{a x-1} \sqrt{a x+1}}{135 a}+\frac{8}{15} c^2 x \cosh ^{-1}(a x)^3+\frac{298}{75} c^2 x \cosh ^{-1}(a x)-\frac{3 c^2 (a x-1)^{5/2} (a x+1)^{5/2} \cosh ^{-1}(a x)^2}{25 a}+\frac{4 c^2 (a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)^2}{15 a}-\frac{8 c^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{5 a} \]
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Rubi [A] time = 0.844015, antiderivative size = 388, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 11, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.55, Rules used = {5681, 5718, 194, 5680, 12, 520, 1247, 698, 460, 74, 5654} \[ \frac{6 c^2 \left (1-a^2 x^2\right )^3}{625 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{8 c^2 \left (1-a^2 x^2\right )^2}{375 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{16 c^2 \left (1-a^2 x^2\right )}{125 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{6}{125} a^4 c^2 x^5 \cosh ^{-1}(a x)-\frac{76}{225} a^2 c^2 x^3 \cosh ^{-1}(a x)+\frac{1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3+\frac{4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac{8}{135} a c^2 x^2 \sqrt{a x-1} \sqrt{a x+1}-\frac{488 c^2 \sqrt{a x-1} \sqrt{a x+1}}{135 a}+\frac{8}{15} c^2 x \cosh ^{-1}(a x)^3+\frac{298}{75} c^2 x \cosh ^{-1}(a x)-\frac{3 c^2 (a x-1)^{5/2} (a x+1)^{5/2} \cosh ^{-1}(a x)^2}{25 a}+\frac{4 c^2 (a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)^2}{15 a}-\frac{8 c^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{5 a} \]
Antiderivative was successfully verified.
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Rule 5681
Rule 5718
Rule 194
Rule 5680
Rule 12
Rule 520
Rule 1247
Rule 698
Rule 460
Rule 74
Rule 5654
Rubi steps
\begin{align*} \int \left (c-a^2 c x^2\right )^2 \cosh ^{-1}(a x)^3 \, dx &=\frac{1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3+\frac{1}{5} (4 c) \int \left (c-a^2 c x^2\right ) \cosh ^{-1}(a x)^3 \, dx-\frac{1}{5} \left (3 a c^2\right ) \int x (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2 \, dx\\ &=-\frac{3 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^2}{25 a}+\frac{4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac{1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3+\frac{1}{25} \left (6 c^2\right ) \int \left (-1+a^2 x^2\right )^2 \cosh ^{-1}(a x) \, dx+\frac{1}{15} \left (8 c^2\right ) \int \cosh ^{-1}(a x)^3 \, dx+\frac{1}{5} \left (4 a c^2\right ) \int x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \, dx\\ &=\frac{6}{25} c^2 x \cosh ^{-1}(a x)-\frac{4}{25} a^2 c^2 x^3 \cosh ^{-1}(a x)+\frac{6}{125} a^4 c^2 x^5 \cosh ^{-1}(a x)+\frac{4 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{15 a}-\frac{3 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^2}{25 a}+\frac{8}{15} c^2 x \cosh ^{-1}(a x)^3+\frac{4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac{1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3-\frac{1}{15} \left (8 c^2\right ) \int \left (-1+a^2 x^2\right ) \cosh ^{-1}(a x) \, dx-\frac{1}{25} \left (6 a c^2\right ) \int \frac{x \left (15-10 a^2 x^2+3 a^4 x^4\right )}{15 \sqrt{-1+a x} \sqrt{1+a x}} \, dx-\frac{1}{5} \left (8 a c^2\right ) \int \frac{x \cosh ^{-1}(a x)^2}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=\frac{58}{75} c^2 x \cosh ^{-1}(a x)-\frac{76}{225} a^2 c^2 x^3 \cosh ^{-1}(a x)+\frac{6}{125} a^4 c^2 x^5 \cosh ^{-1}(a x)-\frac{8 c^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{5 a}+\frac{4 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{15 a}-\frac{3 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^2}{25 a}+\frac{8}{15} c^2 x \cosh ^{-1}(a x)^3+\frac{4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac{1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3+\frac{1}{5} \left (16 c^2\right ) \int \cosh ^{-1}(a x) \, dx-\frac{1}{125} \left (2 a c^2\right ) \int \frac{x \left (15-10 a^2 x^2+3 a^4 x^4\right )}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx+\frac{1}{15} \left (8 a c^2\right ) \int \frac{x \left (-3+a^2 x^2\right )}{3 \sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=\frac{298}{75} c^2 x \cosh ^{-1}(a x)-\frac{76}{225} a^2 c^2 x^3 \cosh ^{-1}(a x)+\frac{6}{125} a^4 c^2 x^5 \cosh ^{-1}(a x)-\frac{8 c^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{5 a}+\frac{4 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{15 a}-\frac{3 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^2}{25 a}+\frac{8}{15} c^2 x \cosh ^{-1}(a x)^3+\frac{4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac{1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3+\frac{1}{45} \left (8 a c^2\right ) \int \frac{x \left (-3+a^2 x^2\right )}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx-\frac{1}{5} \left (16 a c^2\right ) \int \frac{x}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx-\frac{\left (2 a c^2 \sqrt{-1+a^2 x^2}\right ) \int \frac{x \left (15-10 a^2 x^2+3 a^4 x^4\right )}{\sqrt{-1+a^2 x^2}} \, dx}{125 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=-\frac{16 c^2 \sqrt{-1+a x} \sqrt{1+a x}}{5 a}+\frac{8}{135} a c^2 x^2 \sqrt{-1+a x} \sqrt{1+a x}+\frac{298}{75} c^2 x \cosh ^{-1}(a x)-\frac{76}{225} a^2 c^2 x^3 \cosh ^{-1}(a x)+\frac{6}{125} a^4 c^2 x^5 \cosh ^{-1}(a x)-\frac{8 c^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{5 a}+\frac{4 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{15 a}-\frac{3 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^2}{25 a}+\frac{8}{15} c^2 x \cosh ^{-1}(a x)^3+\frac{4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac{1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3-\frac{1}{135} \left (56 a c^2\right ) \int \frac{x}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx-\frac{\left (a c^2 \sqrt{-1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{15-10 a^2 x+3 a^4 x^2}{\sqrt{-1+a^2 x}} \, dx,x,x^2\right )}{125 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=-\frac{488 c^2 \sqrt{-1+a x} \sqrt{1+a x}}{135 a}+\frac{8}{135} a c^2 x^2 \sqrt{-1+a x} \sqrt{1+a x}+\frac{298}{75} c^2 x \cosh ^{-1}(a x)-\frac{76}{225} a^2 c^2 x^3 \cosh ^{-1}(a x)+\frac{6}{125} a^4 c^2 x^5 \cosh ^{-1}(a x)-\frac{8 c^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{5 a}+\frac{4 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{15 a}-\frac{3 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^2}{25 a}+\frac{8}{15} c^2 x \cosh ^{-1}(a x)^3+\frac{4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac{1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3-\frac{\left (a c^2 \sqrt{-1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{8}{\sqrt{-1+a^2 x}}-4 \sqrt{-1+a^2 x}+3 \left (-1+a^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )}{125 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=-\frac{488 c^2 \sqrt{-1+a x} \sqrt{1+a x}}{135 a}+\frac{8}{135} a c^2 x^2 \sqrt{-1+a x} \sqrt{1+a x}+\frac{16 c^2 \left (1-a^2 x^2\right )}{125 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{8 c^2 \left (1-a^2 x^2\right )^2}{375 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{6 c^2 \left (1-a^2 x^2\right )^3}{625 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{298}{75} c^2 x \cosh ^{-1}(a x)-\frac{76}{225} a^2 c^2 x^3 \cosh ^{-1}(a x)+\frac{6}{125} a^4 c^2 x^5 \cosh ^{-1}(a x)-\frac{8 c^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{5 a}+\frac{4 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{15 a}-\frac{3 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^2}{25 a}+\frac{8}{15} c^2 x \cosh ^{-1}(a x)^3+\frac{4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac{1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3\\ \end{align*}
Mathematica [A] time = 0.209834, size = 147, normalized size = 0.38 \[ \frac{c^2 \left (-2 \sqrt{a x-1} \sqrt{a x+1} \left (81 a^4 x^4-842 a^2 x^2+31841\right )+1125 a x \left (3 a^4 x^4-10 a^2 x^2+15\right ) \cosh ^{-1}(a x)^3-225 \sqrt{a x-1} \sqrt{a x+1} \left (9 a^4 x^4-38 a^2 x^2+149\right ) \cosh ^{-1}(a x)^2+30 a x \left (27 a^4 x^4-190 a^2 x^2+2235\right ) \cosh ^{-1}(a x)\right )}{16875 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 218, normalized size = 0.6 \begin{align*}{\frac{{c}^{2}}{16875\,a} \left ( 3375\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}{a}^{5}{x}^{5}-2025\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}\sqrt{ax-1}\sqrt{ax+1}{a}^{4}{x}^{4}-11250\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}{a}^{3}{x}^{3}+8550\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}\sqrt{ax-1}\sqrt{ax+1}{a}^{2}{x}^{2}+810\,{a}^{5}{x}^{5}{\rm arccosh} \left (ax\right )-162\,\sqrt{ax+1}\sqrt{ax-1}{x}^{4}{a}^{4}+16875\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}ax-33525\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}\sqrt{ax-1}\sqrt{ax+1}-5700\,{\rm arccosh} \left (ax\right ){a}^{3}{x}^{3}+1684\,\sqrt{ax+1}\sqrt{ax-1}{x}^{2}{a}^{2}+67050\,ax{\rm arccosh} \left (ax\right )-63682\,\sqrt{ax-1}\sqrt{ax+1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.2006, size = 284, normalized size = 0.73 \begin{align*} -\frac{1}{75} \,{\left (9 \, \sqrt{a^{2} x^{2} - 1} a^{2} c^{2} x^{4} - 38 \, \sqrt{a^{2} x^{2} - 1} c^{2} x^{2} + \frac{149 \, \sqrt{a^{2} x^{2} - 1} c^{2}}{a^{2}}\right )} a \operatorname{arcosh}\left (a x\right )^{2} + \frac{1}{15} \,{\left (3 \, a^{4} c^{2} x^{5} - 10 \, a^{2} c^{2} x^{3} + 15 \, c^{2} x\right )} \operatorname{arcosh}\left (a x\right )^{3} - \frac{2}{16875} \,{\left (81 \, \sqrt{a^{2} x^{2} - 1} a^{2} c^{2} x^{4} - 842 \, \sqrt{a^{2} x^{2} - 1} c^{2} x^{2} - \frac{15 \,{\left (27 \, a^{4} c^{2} x^{5} - 190 \, a^{2} c^{2} x^{3} + 2235 \, c^{2} x\right )} \operatorname{arcosh}\left (a x\right )}{a} + \frac{31841 \, \sqrt{a^{2} x^{2} - 1} c^{2}}{a^{2}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.17825, size = 467, normalized size = 1.2 \begin{align*} \frac{1125 \,{\left (3 \, a^{5} c^{2} x^{5} - 10 \, a^{3} c^{2} x^{3} + 15 \, a c^{2} x\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{3} - 225 \,{\left (9 \, a^{4} c^{2} x^{4} - 38 \, a^{2} c^{2} x^{2} + 149 \, c^{2}\right )} \sqrt{a^{2} x^{2} - 1} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2} + 30 \,{\left (27 \, a^{5} c^{2} x^{5} - 190 \, a^{3} c^{2} x^{3} + 2235 \, a c^{2} x\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) - 2 \,{\left (81 \, a^{4} c^{2} x^{4} - 842 \, a^{2} c^{2} x^{2} + 31841 \, c^{2}\right )} \sqrt{a^{2} x^{2} - 1}}{16875 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.76748, size = 274, normalized size = 0.71 \begin{align*} \begin{cases} \frac{a^{4} c^{2} x^{5} \operatorname{acosh}^{3}{\left (a x \right )}}{5} + \frac{6 a^{4} c^{2} x^{5} \operatorname{acosh}{\left (a x \right )}}{125} - \frac{3 a^{3} c^{2} x^{4} \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}^{2}{\left (a x \right )}}{25} - \frac{6 a^{3} c^{2} x^{4} \sqrt{a^{2} x^{2} - 1}}{625} - \frac{2 a^{2} c^{2} x^{3} \operatorname{acosh}^{3}{\left (a x \right )}}{3} - \frac{76 a^{2} c^{2} x^{3} \operatorname{acosh}{\left (a x \right )}}{225} + \frac{38 a c^{2} x^{2} \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}^{2}{\left (a x \right )}}{75} + \frac{1684 a c^{2} x^{2} \sqrt{a^{2} x^{2} - 1}}{16875} + c^{2} x \operatorname{acosh}^{3}{\left (a x \right )} + \frac{298 c^{2} x \operatorname{acosh}{\left (a x \right )}}{75} - \frac{149 c^{2} \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}^{2}{\left (a x \right )}}{75 a} - \frac{63682 c^{2} \sqrt{a^{2} x^{2} - 1}}{16875 a} & \text{for}\: a \neq 0 \\- \frac{i \pi ^{3} c^{2} x}{8} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.33745, size = 273, normalized size = 0.7 \begin{align*} \frac{1}{15} \,{\left (3 \, a^{4} c^{2} x^{5} - 10 \, a^{2} c^{2} x^{3} + 15 \, c^{2} x\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{3} + \frac{1}{16875} \,{\left (30 \,{\left (27 \, a^{4} x^{5} - 190 \, a^{2} x^{3} + 2235 \, x\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) - \frac{225 \,{\left (9 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{5}{2}} - 20 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 120 \, \sqrt{a^{2} x^{2} - 1}\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2}}{a} - \frac{2 \,{\left (81 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{5}{2}} - 680 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 31080 \, \sqrt{a^{2} x^{2} - 1}\right )}}{a}\right )} c^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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