Optimal. Leaf size=191 \[ -\frac{1}{7} c^6 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{5} c^4 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-c^2 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )+d^3 x \left (a+b \cosh ^{-1}(c x)\right )+\frac{b d^3 (c x-1)^{7/2} (c x+1)^{7/2}}{49 c}-\frac{6 b d^3 (c x-1)^{5/2} (c x+1)^{5/2}}{175 c}+\frac{8 b d^3 (c x-1)^{3/2} (c x+1)^{3/2}}{105 c}-\frac{16 b d^3 \sqrt{c x-1} \sqrt{c x+1}}{35 c} \]
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Rubi [A] time = 0.26202, antiderivative size = 237, normalized size of antiderivative = 1.24, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {194, 5680, 12, 1610, 1799, 1850} \[ -\frac{1}{7} c^6 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{5} c^4 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-c^2 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )+d^3 x \left (a+b \cosh ^{-1}(c x)\right )+\frac{b d^3 \left (1-c^2 x^2\right )^4}{49 c \sqrt{c x-1} \sqrt{c x+1}}+\frac{6 b d^3 \left (1-c^2 x^2\right )^3}{175 c \sqrt{c x-1} \sqrt{c x+1}}+\frac{8 b d^3 \left (1-c^2 x^2\right )^2}{105 c \sqrt{c x-1} \sqrt{c x+1}}+\frac{16 b d^3 \left (1-c^2 x^2\right )}{35 c \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
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Rule 194
Rule 5680
Rule 12
Rule 1610
Rule 1799
Rule 1850
Rubi steps
\begin{align*} \int \left (d-c^2 d x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=d^3 x \left (a+b \cosh ^{-1}(c x)\right )-c^2 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{5} c^4 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{7} c^6 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac{d^3 x \left (35-35 c^2 x^2+21 c^4 x^4-5 c^6 x^6\right )}{35 \sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=d^3 x \left (a+b \cosh ^{-1}(c x)\right )-c^2 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{5} c^4 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{7} c^6 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{35} \left (b c d^3\right ) \int \frac{x \left (35-35 c^2 x^2+21 c^4 x^4-5 c^6 x^6\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=d^3 x \left (a+b \cosh ^{-1}(c x)\right )-c^2 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{5} c^4 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{7} c^6 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c d^3 \sqrt{-1+c^2 x^2}\right ) \int \frac{x \left (35-35 c^2 x^2+21 c^4 x^4-5 c^6 x^6\right )}{\sqrt{-1+c^2 x^2}} \, dx}{35 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=d^3 x \left (a+b \cosh ^{-1}(c x)\right )-c^2 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{5} c^4 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{7} c^6 d^3 x^7 \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{35-35 c^2 x+21 c^4 x^2-5 c^6 x^3}{\sqrt{-1+c^2 x}} \, dx,x,x^2\right )}{70 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=d^3 x \left (a+b \cosh ^{-1}(c x)\right )-c^2 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{5} c^4 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{7} c^6 d^3 x^7 \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 \left (\frac{16}{\sqrt{-1+c^2 x}}-8 \sqrt{-1+c^2 x}+6 \left (-1+c^2 x\right )^{3/2}-5 \left (-1+c^2 x\right )^{5/2}\right ) \, dx,x,x^2\right )}{70 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{16 b d^3 \left (1-c^2 x^2\right )}{35 c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{8 b d^3 \left (1-c^2 x^2\right )^2}{105 c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{6 b d^3 \left (1-c^2 x^2\right )^3}{175 c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b d^3 \left (1-c^2 x^2\right )^4}{49 c \sqrt{-1+c x} \sqrt{1+c x}}+d^3 x \left (a+b \cosh ^{-1}(c x)\right )-c^2 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{5} c^4 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{7} c^6 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )\\ \end{align*}
Mathematica [A] time = 0.258576, size = 123, normalized size = 0.64 \[ -\frac{d^3 \left (105 a c x \left (5 c^6 x^6-21 c^4 x^4+35 c^2 x^2-35\right )+b \sqrt{c x-1} \sqrt{c x+1} \left (-75 c^6 x^6+351 c^4 x^4-757 c^2 x^2+2161\right )+105 b c x \left (5 c^6 x^6-21 c^4 x^4+35 c^2 x^2-35\right ) \cosh ^{-1}(c x)\right )}{3675 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 132, normalized size = 0.7 \begin{align*}{\frac{1}{c} \left ( -{d}^{3}a \left ({\frac{{c}^{7}{x}^{7}}{7}}-{\frac{3\,{c}^{5}{x}^{5}}{5}}+{c}^{3}{x}^{3}-cx \right ) -{d}^{3}b \left ({\frac{{\rm arccosh} \left (cx\right ){c}^{7}{x}^{7}}{7}}-{\frac{3\,{\rm arccosh} \left (cx\right ){c}^{5}{x}^{5}}{5}}+{c}^{3}{x}^{3}{\rm arccosh} \left (cx\right )-cx{\rm arccosh} \left (cx\right )-{\frac{75\,{c}^{6}{x}^{6}-351\,{c}^{4}{x}^{4}+757\,{c}^{2}{x}^{2}-2161}{3675}\sqrt{cx-1}\sqrt{cx+1}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14356, size = 408, normalized size = 2.14 \begin{align*} -\frac{1}{7} \, a c^{6} d^{3} x^{7} + \frac{3}{5} \, a c^{4} d^{3} x^{5} - \frac{1}{245} \,{\left (35 \, x^{7} \operatorname{arcosh}\left (c x\right ) -{\left (\frac{5 \, \sqrt{c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac{6 \, \sqrt{c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac{8 \, \sqrt{c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac{16 \, \sqrt{c^{2} x^{2} - 1}}{c^{8}}\right )} c\right )} b c^{6} d^{3} + \frac{1}{25} \,{\left (15 \, x^{5} \operatorname{arcosh}\left (c x\right ) -{\left (\frac{3 \, \sqrt{c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac{4 \, \sqrt{c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac{8 \, \sqrt{c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b c^{4} d^{3} - a c^{2} d^{3} x^{3} - \frac{1}{3} \,{\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^{2} d^{3} + a d^{3} x + \frac{{\left (c x \operatorname{arcosh}\left (c x\right ) - \sqrt{c^{2} x^{2} - 1}\right )} b d^{3}}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.11889, size = 389, normalized size = 2.04 \begin{align*} -\frac{525 \, a c^{7} d^{3} x^{7} - 2205 \, a c^{5} d^{3} x^{5} + 3675 \, a c^{3} d^{3} x^{3} - 3675 \, a c d^{3} x + 105 \,{\left (5 \, b c^{7} d^{3} x^{7} - 21 \, b c^{5} d^{3} x^{5} + 35 \, b c^{3} d^{3} x^{3} - 35 \, b c d^{3} x\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (75 \, b c^{6} d^{3} x^{6} - 351 \, b c^{4} d^{3} x^{4} + 757 \, b c^{2} d^{3} x^{2} - 2161 \, b d^{3}\right )} \sqrt{c^{2} x^{2} - 1}}{3675 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 9.83384, size = 228, normalized size = 1.19 \begin{align*} \begin{cases} - \frac{a c^{6} d^{3} x^{7}}{7} + \frac{3 a c^{4} d^{3} x^{5}}{5} - a c^{2} d^{3} x^{3} + a d^{3} x - \frac{b c^{6} d^{3} x^{7} \operatorname{acosh}{\left (c x \right )}}{7} + \frac{b c^{5} d^{3} x^{6} \sqrt{c^{2} x^{2} - 1}}{49} + \frac{3 b c^{4} d^{3} x^{5} \operatorname{acosh}{\left (c x \right )}}{5} - \frac{117 b c^{3} d^{3} x^{4} \sqrt{c^{2} x^{2} - 1}}{1225} - b c^{2} d^{3} x^{3} \operatorname{acosh}{\left (c x \right )} + \frac{757 b c d^{3} x^{2} \sqrt{c^{2} x^{2} - 1}}{3675} + b d^{3} x \operatorname{acosh}{\left (c x \right )} - \frac{2161 b d^{3} \sqrt{c^{2} x^{2} - 1}}{3675 c} & \text{for}\: c \neq 0 \\d^{3} x \left (a + \frac{i \pi b}{2}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.60529, size = 417, normalized size = 2.18 \begin{align*} -\frac{1}{7} \, a c^{6} d^{3} x^{7} + \frac{3}{5} \, a c^{4} d^{3} x^{5} - \frac{1}{245} \,{\left (35 \, x^{7} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{5 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{7}{2}} + 21 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{5}{2}} + 35 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 35 \, \sqrt{c^{2} x^{2} - 1}}{c^{7}}\right )} b c^{6} d^{3} + \frac{1}{25} \,{\left (15 \, x^{5} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{3 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{5}{2}} + 10 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 15 \, \sqrt{c^{2} x^{2} - 1}}{c^{5}}\right )} b c^{4} d^{3} - a c^{2} d^{3} x^{3} - \frac{1}{3} \,{\left (3 \, x^{3} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{{\left (c^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 3 \, \sqrt{c^{2} x^{2} - 1}}{c^{3}}\right )} b c^{2} d^{3} +{\left (x \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{\sqrt{c^{2} x^{2} - 1}}{c}\right )} b d^{3} + a d^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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