Optimal. Leaf size=166 \[ -\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^2}+\frac{35 b d^3 \cosh ^{-1}(c x)}{1024 c^2}+\frac{b d^3 x (c x-1)^{7/2} (c x+1)^{7/2}}{64 c}-\frac{7 b d^3 x (c x-1)^{5/2} (c x+1)^{5/2}}{384 c}+\frac{35 b d^3 x (c x-1)^{3/2} (c x+1)^{3/2}}{1536 c}-\frac{35 b d^3 x \sqrt{c x-1} \sqrt{c x+1}}{1024 c} \]
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Rubi [A] time = 0.0787097, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {5716, 38, 52} \[ -\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^2}+\frac{35 b d^3 \cosh ^{-1}(c x)}{1024 c^2}+\frac{b d^3 x (c x-1)^{7/2} (c x+1)^{7/2}}{64 c}-\frac{7 b d^3 x (c x-1)^{5/2} (c x+1)^{5/2}}{384 c}+\frac{35 b d^3 x (c x-1)^{3/2} (c x+1)^{3/2}}{1536 c}-\frac{35 b d^3 x \sqrt{c x-1} \sqrt{c x+1}}{1024 c} \]
Antiderivative was successfully verified.
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Rule 5716
Rule 38
Rule 52
Rubi steps
\begin{align*} \int x \left (d-c^2 d x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^2}+\frac{\left (b d^3\right ) \int (-1+c x)^{7/2} (1+c x)^{7/2} \, dx}{8 c}\\ &=\frac{b d^3 x (-1+c x)^{7/2} (1+c x)^{7/2}}{64 c}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^2}-\frac{\left (7 b d^3\right ) \int (-1+c x)^{5/2} (1+c x)^{5/2} \, dx}{64 c}\\ &=-\frac{7 b d^3 x (-1+c x)^{5/2} (1+c x)^{5/2}}{384 c}+\frac{b d^3 x (-1+c x)^{7/2} (1+c x)^{7/2}}{64 c}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^2}+\frac{\left (35 b d^3\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \, dx}{384 c}\\ &=\frac{35 b d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}}{1536 c}-\frac{7 b d^3 x (-1+c x)^{5/2} (1+c x)^{5/2}}{384 c}+\frac{b d^3 x (-1+c x)^{7/2} (1+c x)^{7/2}}{64 c}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^2}-\frac{\left (35 b d^3\right ) \int \sqrt{-1+c x} \sqrt{1+c x} \, dx}{512 c}\\ &=-\frac{35 b d^3 x \sqrt{-1+c x} \sqrt{1+c x}}{1024 c}+\frac{35 b d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}}{1536 c}-\frac{7 b d^3 x (-1+c x)^{5/2} (1+c x)^{5/2}}{384 c}+\frac{b d^3 x (-1+c x)^{7/2} (1+c x)^{7/2}}{64 c}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^2}+\frac{\left (35 b d^3\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{1024 c}\\ &=-\frac{35 b d^3 x \sqrt{-1+c x} \sqrt{1+c x}}{1024 c}+\frac{35 b d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}}{1536 c}-\frac{7 b d^3 x (-1+c x)^{5/2} (1+c x)^{5/2}}{384 c}+\frac{b d^3 x (-1+c x)^{7/2} (1+c x)^{7/2}}{64 c}+\frac{35 b d^3 \cosh ^{-1}(c x)}{1024 c^2}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^2}\\ \end{align*}
Mathematica [A] time = 0.365045, size = 150, normalized size = 0.9 \[ -\frac{d^3 \left (c x \left (384 a c x \left (c^6 x^6-4 c^4 x^4+6 c^2 x^2-4\right )+b \sqrt{c x-1} \sqrt{c x+1} \left (-48 c^6 x^6+200 c^4 x^4-326 c^2 x^2+279\right )\right )+384 b c^2 x^2 \left (c^6 x^6-4 c^4 x^4+6 c^2 x^2-4\right ) \cosh ^{-1}(c x)+558 b \tanh ^{-1}\left (\sqrt{\frac{c x-1}{c x+1}}\right )\right )}{3072 c^2} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.014, size = 258, normalized size = 1.6 \begin{align*} -{\frac{{c}^{6}{d}^{3}a{x}^{8}}{8}}+{\frac{{c}^{4}{d}^{3}a{x}^{6}}{2}}-{\frac{3\,{c}^{2}{d}^{3}a{x}^{4}}{4}}+{\frac{{d}^{3}a{x}^{2}}{2}}-{\frac{{c}^{6}{d}^{3}b{\rm arccosh} \left (cx\right ){x}^{8}}{8}}+{\frac{{c}^{4}{d}^{3}b{\rm arccosh} \left (cx\right ){x}^{6}}{2}}-{\frac{3\,{c}^{2}{d}^{3}b{\rm arccosh} \left (cx\right ){x}^{4}}{4}}+{\frac{{d}^{3}b{\rm arccosh} \left (cx\right ){x}^{2}}{2}}+{\frac{{d}^{3}b{c}^{5}{x}^{7}}{64}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{25\,{d}^{3}b{c}^{3}{x}^{5}}{384}\sqrt{cx-1}\sqrt{cx+1}}+{\frac{163\,{d}^{3}bc{x}^{3}}{1536}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{93\,{d}^{3}bx}{1024\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{93\,{d}^{3}b}{1024\,{c}^{2}}\sqrt{cx-1}\sqrt{cx+1}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.19907, size = 620, normalized size = 3.73 \begin{align*} -\frac{1}{8} \, a c^{6} d^{3} x^{8} + \frac{1}{2} \, a c^{4} d^{3} x^{6} - \frac{1}{3072} \,{\left (384 \, x^{8} \operatorname{arcosh}\left (c x\right ) -{\left (\frac{48 \, \sqrt{c^{2} x^{2} - 1} x^{7}}{c^{2}} + \frac{56 \, \sqrt{c^{2} x^{2} - 1} x^{5}}{c^{4}} + \frac{70 \, \sqrt{c^{2} x^{2} - 1} x^{3}}{c^{6}} + \frac{105 \, \sqrt{c^{2} x^{2} - 1} x}{c^{8}} + \frac{105 \, \log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{\sqrt{c^{2}} c^{8}}\right )} c\right )} b c^{6} d^{3} - \frac{3}{4} \, a c^{2} d^{3} x^{4} + \frac{1}{96} \,{\left (48 \, x^{6} \operatorname{arcosh}\left (c x\right ) -{\left (\frac{8 \, \sqrt{c^{2} x^{2} - 1} x^{5}}{c^{2}} + \frac{10 \, \sqrt{c^{2} x^{2} - 1} x^{3}}{c^{4}} + \frac{15 \, \sqrt{c^{2} x^{2} - 1} x}{c^{6}} + \frac{15 \, \log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{\sqrt{c^{2}} c^{6}}\right )} c\right )} b c^{4} d^{3} - \frac{3}{32} \,{\left (8 \, x^{4} \operatorname{arcosh}\left (c x\right ) -{\left (\frac{2 \, \sqrt{c^{2} x^{2} - 1} x^{3}}{c^{2}} + \frac{3 \, \sqrt{c^{2} x^{2} - 1} x}{c^{4}} + \frac{3 \, \log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{\sqrt{c^{2}} c^{4}}\right )} c\right )} b c^{2} d^{3} + \frac{1}{2} \, a d^{3} x^{2} + \frac{1}{4} \,{\left (2 \, x^{2} \operatorname{arcosh}\left (c x\right ) - c{\left (\frac{\sqrt{c^{2} x^{2} - 1} x}{c^{2}} + \frac{\log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{\sqrt{c^{2}} c^{2}}\right )}\right )} b d^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.91986, size = 425, normalized size = 2.56 \begin{align*} -\frac{384 \, a c^{8} d^{3} x^{8} - 1536 \, a c^{6} d^{3} x^{6} + 2304 \, a c^{4} d^{3} x^{4} - 1536 \, a c^{2} d^{3} x^{2} + 3 \,{\left (128 \, b c^{8} d^{3} x^{8} - 512 \, b c^{6} d^{3} x^{6} + 768 \, b c^{4} d^{3} x^{4} - 512 \, b c^{2} d^{3} x^{2} + 93 \, b d^{3}\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (48 \, b c^{7} d^{3} x^{7} - 200 \, b c^{5} d^{3} x^{5} + 326 \, b c^{3} d^{3} x^{3} - 279 \, b c d^{3} x\right )} \sqrt{c^{2} x^{2} - 1}}{3072 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 17.5542, size = 260, normalized size = 1.57 \begin{align*} \begin{cases} - \frac{a c^{6} d^{3} x^{8}}{8} + \frac{a c^{4} d^{3} x^{6}}{2} - \frac{3 a c^{2} d^{3} x^{4}}{4} + \frac{a d^{3} x^{2}}{2} - \frac{b c^{6} d^{3} x^{8} \operatorname{acosh}{\left (c x \right )}}{8} + \frac{b c^{5} d^{3} x^{7} \sqrt{c^{2} x^{2} - 1}}{64} + \frac{b c^{4} d^{3} x^{6} \operatorname{acosh}{\left (c x \right )}}{2} - \frac{25 b c^{3} d^{3} x^{5} \sqrt{c^{2} x^{2} - 1}}{384} - \frac{3 b c^{2} d^{3} x^{4} \operatorname{acosh}{\left (c x \right )}}{4} + \frac{163 b c d^{3} x^{3} \sqrt{c^{2} x^{2} - 1}}{1536} + \frac{b d^{3} x^{2} \operatorname{acosh}{\left (c x \right )}}{2} - \frac{93 b d^{3} x \sqrt{c^{2} x^{2} - 1}}{1024 c} - \frac{93 b d^{3} \operatorname{acosh}{\left (c x \right )}}{1024 c^{2}} & \text{for}\: c \neq 0 \\\frac{d^{3} x^{2} \left (a + \frac{i \pi b}{2}\right )}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.82256, size = 574, normalized size = 3.46 \begin{align*} -\frac{1}{8} \, a c^{6} d^{3} x^{8} + \frac{1}{2} \, a c^{4} d^{3} x^{6} - \frac{1}{3072} \,{\left (384 \, x^{8} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (\sqrt{c^{2} x^{2} - 1}{\left (2 \,{\left (4 \, x^{2}{\left (\frac{6 \, x^{2}}{c^{2}} + \frac{7}{c^{4}}\right )} + \frac{35}{c^{6}}\right )} x^{2} + \frac{105}{c^{8}}\right )} x - \frac{105 \, \log \left ({\left | -x{\left | c \right |} + \sqrt{c^{2} x^{2} - 1} \right |}\right )}{c^{8}{\left | c \right |}}\right )} c\right )} b c^{6} d^{3} - \frac{3}{4} \, a c^{2} d^{3} x^{4} + \frac{1}{96} \,{\left (48 \, x^{6} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (\sqrt{c^{2} x^{2} - 1}{\left (2 \, x^{2}{\left (\frac{4 \, x^{2}}{c^{2}} + \frac{5}{c^{4}}\right )} + \frac{15}{c^{6}}\right )} x - \frac{15 \, \log \left ({\left | -x{\left | c \right |} + \sqrt{c^{2} x^{2} - 1} \right |}\right )}{c^{6}{\left | c \right |}}\right )} c\right )} b c^{4} d^{3} - \frac{3}{32} \,{\left (8 \, x^{4} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (\sqrt{c^{2} x^{2} - 1} x{\left (\frac{2 \, x^{2}}{c^{2}} + \frac{3}{c^{4}}\right )} - \frac{3 \, \log \left ({\left | -x{\left | c \right |} + \sqrt{c^{2} x^{2} - 1} \right |}\right )}{c^{4}{\left | c \right |}}\right )} c\right )} b c^{2} d^{3} + \frac{1}{2} \, a d^{3} x^{2} + \frac{1}{4} \,{\left (2 \, x^{2} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - c{\left (\frac{\sqrt{c^{2} x^{2} - 1} x}{c^{2}} - \frac{\log \left ({\left | -x{\left | c \right |} + \sqrt{c^{2} x^{2} - 1} \right |}\right )}{c^{2}{\left | c \right |}}\right )}\right )} b d^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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