Optimal. Leaf size=233 \[ -\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^6 d^3}+\frac{2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{c^6 d^2}+\frac{a+b \cosh ^{-1}(c x)}{c^6 d \sqrt{d-c^2 d x^2}}-\frac{b x^3 \sqrt{d-c^2 d x^2}}{9 c^3 d^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{5 b x \sqrt{d-c^2 d x^2}}{3 c^5 d^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b \sqrt{d-c^2 d x^2} \tanh ^{-1}(c x)}{c^6 d^2 \sqrt{c x-1} \sqrt{c x+1}} \]
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Rubi [A] time = 0.434414, antiderivative size = 262, normalized size of antiderivative = 1.12, number of steps used = 5, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5798, 98, 21, 100, 12, 74, 5733, 1153, 208} \[ \frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{4 x^2 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 d \sqrt{d-c^2 d x^2}}+\frac{8 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^6 d \sqrt{d-c^2 d x^2}}+\frac{b x^3 \sqrt{c x-1} \sqrt{c x+1}}{9 c^3 d \sqrt{d-c^2 d x^2}}+\frac{5 b x \sqrt{c x-1} \sqrt{c x+1}}{3 c^5 d \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}(c x)}{c^6 d \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 5798
Rule 98
Rule 21
Rule 100
Rule 12
Rule 74
Rule 5733
Rule 1153
Rule 208
Rubi steps
\begin{align*} \int \frac{x^5 \left (a+b \cosh ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^{3/2}} \, dx &=-\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x^5 \left (a+b \cosh ^{-1}(c x)\right )}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{d \sqrt{d-c^2 d x^2}}\\ &=\frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^6 d \sqrt{d-c^2 d x^2}}+\frac{4 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 d \sqrt{d-c^2 d x^2}}+\frac{\left (b c \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{8-4 c^2 x^2-c^4 x^4}{3 c^6-3 c^8 x^2} \, dx}{d \sqrt{d-c^2 d x^2}}\\ &=\frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^6 d \sqrt{d-c^2 d x^2}}+\frac{4 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 d \sqrt{d-c^2 d x^2}}+\frac{\left (b c \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \left (\frac{5}{3 c^6}+\frac{x^2}{3 c^4}+\frac{3}{3 c^6-3 c^8 x^2}\right ) \, dx}{d \sqrt{d-c^2 d x^2}}\\ &=\frac{5 b x \sqrt{-1+c x} \sqrt{1+c x}}{3 c^5 d \sqrt{d-c^2 d x^2}}+\frac{b x^3 \sqrt{-1+c x} \sqrt{1+c x}}{9 c^3 d \sqrt{d-c^2 d x^2}}+\frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^6 d \sqrt{d-c^2 d x^2}}+\frac{4 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 d \sqrt{d-c^2 d x^2}}+\frac{\left (3 b c \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{1}{3 c^6-3 c^8 x^2} \, dx}{d \sqrt{d-c^2 d x^2}}\\ &=\frac{5 b x \sqrt{-1+c x} \sqrt{1+c x}}{3 c^5 d \sqrt{d-c^2 d x^2}}+\frac{b x^3 \sqrt{-1+c x} \sqrt{1+c x}}{9 c^3 d \sqrt{d-c^2 d x^2}}+\frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^6 d \sqrt{d-c^2 d x^2}}+\frac{4 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 d \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{-1+c x} \sqrt{1+c x} \tanh ^{-1}(c x)}{c^6 d \sqrt{d-c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 0.101714, size = 145, normalized size = 0.62 \[ \frac{-3 a c^4 x^4-12 a c^2 x^2+24 a+b c^3 x^3 \sqrt{c x-1} \sqrt{c x+1}-3 b \left (c^4 x^4+4 c^2 x^2-8\right ) \cosh ^{-1}(c x)+15 b c x \sqrt{c x-1} \sqrt{c x+1}+9 b \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}(c x)}{9 c^6 d \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.285, size = 431, normalized size = 1.9 \begin{align*} -{\frac{{x}^{4}a}{3\,{c}^{2}d}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}}-{\frac{4\,a{x}^{2}}{3\,d{c}^{4}}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}}+{\frac{8\,a}{3\,d{c}^{6}}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}}-{\frac{8\,b{\rm arccosh} \left (cx\right )}{3\,{d}^{2}{c}^{6} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{b}{{d}^{2}{c}^{6} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx-1}\sqrt{cx+1}\ln \left ( cx+\sqrt{cx-1}\sqrt{cx+1}-1 \right ) }-{\frac{b{x}^{3}}{9\,{c}^{3}{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx-1}\sqrt{cx+1}}-{\frac{5\,bx}{3\,{c}^{5}{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx-1}\sqrt{cx+1}}+{\frac{b{x}^{4}{\rm arccosh} \left (cx\right )}{3\,{c}^{2}{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{4\,b{x}^{2}{\rm arccosh} \left (cx\right )}{3\,{d}^{2}{c}^{4} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{b}{{d}^{2}{c}^{6} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx-1}\sqrt{cx+1}\ln \left ( 1+cx+\sqrt{cx-1}\sqrt{cx+1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.68376, size = 1046, normalized size = 4.49 \begin{align*} \left [\frac{12 \,{\left (b c^{4} x^{4} + 4 \, b c^{2} x^{2} - 8 \, b\right )} \sqrt{-c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - 9 \,{\left (b c^{2} x^{2} - b\right )} \sqrt{-d} \log \left (-\frac{c^{6} d x^{6} + 5 \, c^{4} d x^{4} - 5 \, c^{2} d x^{2} - 4 \,{\left (c^{3} x^{3} + c x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} \sqrt{-d} - d}{c^{6} x^{6} - 3 \, c^{4} x^{4} + 3 \, c^{2} x^{2} - 1}\right ) - 4 \,{\left (b c^{3} x^{3} + 15 \, b c x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} + 12 \,{\left (a c^{4} x^{4} + 4 \, a c^{2} x^{2} - 8 \, a\right )} \sqrt{-c^{2} d x^{2} + d}}{36 \,{\left (c^{8} d^{2} x^{2} - c^{6} d^{2}\right )}}, -\frac{9 \,{\left (b c^{2} x^{2} - b\right )} \sqrt{d} \arctan \left (\frac{2 \, \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} c \sqrt{d} x}{c^{4} d x^{4} - d}\right ) - 6 \,{\left (b c^{4} x^{4} + 4 \, b c^{2} x^{2} - 8 \, b\right )} \sqrt{-c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) + 2 \,{\left (b c^{3} x^{3} + 15 \, b c x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} - 6 \,{\left (a c^{4} x^{4} + 4 \, a c^{2} x^{2} - 8 \, a\right )} \sqrt{-c^{2} d x^{2} + d}}{18 \,{\left (c^{8} d^{2} x^{2} - c^{6} d^{2}\right )}}\right ] \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*} \int \frac{x^{5} \left (a + b \operatorname{acosh}{\left (c x \right )}\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac{3}{2}}}\, dx \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 (b \operatorname{arcosh}\left (c x\right ) + a\right )} x^{5}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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