Optimal. Leaf size=314 \[ -\frac{2 c^2 \left (d-c^2 d x^2\right )^{7/2} \left (a+b \cosh ^{-1}(c x)\right )}{63 d x^7}-\frac{\left (d-c^2 d x^2\right )^{7/2} \left (a+b \cosh ^{-1}(c x)\right )}{9 d x^9}-\frac{b c^7 d^2 \sqrt{d-c^2 d x^2}}{21 x^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^5 d^2 \sqrt{d-c^2 d x^2}}{42 x^4 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^3 d^2 \sqrt{d-c^2 d x^2}}{189 x^6 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d^2 \left (1-c^2 x^2\right )^4 \sqrt{d-c^2 d x^2}}{72 x^8 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 b c^9 d^2 \log (x) \sqrt{d-c^2 d x^2}}{63 \sqrt{c x-1} \sqrt{c x+1}} \]
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Rubi [A] time = 0.525799, antiderivative size = 448, normalized size of antiderivative = 1.43, number of steps used = 7, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5798, 97, 12, 103, 95, 5733, 446, 78, 43} \[ \frac{2 c^8 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x}+\frac{c^6 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x^3}-\frac{c^4 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 x^5}+\frac{5 c^2 d^2 (1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x^7}-\frac{d^2 (1-c x)^2 (c x+1)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 x^9}-\frac{b c^7 d^2 \sqrt{d-c^2 d x^2}}{21 x^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^5 d^2 \sqrt{d-c^2 d x^2}}{42 x^4 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^3 d^2 \sqrt{d-c^2 d x^2}}{189 x^6 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d^2 \left (1-c^2 x^2\right )^4 \sqrt{d-c^2 d x^2}}{72 x^8 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 b c^9 d^2 \log (x) \sqrt{d-c^2 d x^2}}{63 \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
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Rule 5798
Rule 97
Rule 12
Rule 103
Rule 95
Rule 5733
Rule 446
Rule 78
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{x^{10}} \, dx &=\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{(-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{x^{10}} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{c^4 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 x^5}+\frac{c^6 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x^3}+\frac{2 c^8 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x}+\frac{5 c^2 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x^7}-\frac{d^2 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 x^9}-\frac{\left (b c d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (-7-2 c^2 x^2\right ) \left (1-c^2 x^2\right )^3}{63 x^9} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{c^4 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 x^5}+\frac{c^6 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x^3}+\frac{2 c^8 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x}+\frac{5 c^2 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x^7}-\frac{d^2 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 x^9}-\frac{\left (b c d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (-7-2 c^2 x^2\right ) \left (1-c^2 x^2\right )^3}{x^9} \, dx}{63 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{c^4 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 x^5}+\frac{c^6 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x^3}+\frac{2 c^8 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x}+\frac{5 c^2 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x^7}-\frac{d^2 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 x^9}-\frac{\left (b c d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\left (-7-2 c^2 x\right ) \left (1-c^2 x\right )^3}{x^5} \, dx,x,x^2\right )}{126 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d^2 \left (1-c^2 x^2\right )^4 \sqrt{d-c^2 d x^2}}{72 x^8 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{c^4 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 x^5}+\frac{c^6 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x^3}+\frac{2 c^8 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x}+\frac{5 c^2 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x^7}-\frac{d^2 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 x^9}+\frac{\left (b c^3 d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\left (1-c^2 x\right )^3}{x^4} \, dx,x,x^2\right )}{63 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d^2 \left (1-c^2 x^2\right )^4 \sqrt{d-c^2 d x^2}}{72 x^8 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{c^4 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 x^5}+\frac{c^6 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x^3}+\frac{2 c^8 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x}+\frac{5 c^2 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x^7}-\frac{d^2 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 x^9}+\frac{\left (b c^3 d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{x^4}-\frac{3 c^2}{x^3}+\frac{3 c^4}{x^2}-\frac{c^6}{x}\right ) \, dx,x,x^2\right )}{63 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c^3 d^2 \sqrt{d-c^2 d x^2}}{189 x^6 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^5 d^2 \sqrt{d-c^2 d x^2}}{42 x^4 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^7 d^2 \sqrt{d-c^2 d x^2}}{21 x^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c d^2 \left (1-c^2 x^2\right )^4 \sqrt{d-c^2 d x^2}}{72 x^8 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{c^4 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 x^5}+\frac{c^6 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x^3}+\frac{2 c^8 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x}+\frac{5 c^2 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x^7}-\frac{d^2 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 x^9}-\frac{2 b c^9 d^2 \sqrt{d-c^2 d x^2} \log (x)}{63 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 0.162677, size = 147, normalized size = 0.47 \[ \frac{d^2 \sqrt{d-c^2 d x^2} \left (48 c^2 x^2 (c x-1)^{7/2} (c x+1)^{7/2} \left (a+b \cosh ^{-1}(c x)\right )+168 (c x-1)^{7/2} (c x+1)^{7/2} \left (a+b \cosh ^{-1}(c x)\right )-b c x \left (-12 c^6 x^6+90 c^4 x^4-76 c^2 x^2+48 c^8 x^8 \log (x)+21\right )\right )}{1512 x^9 \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.398, size = 5006, normalized size = 15.9 \begin{align*} \text{output too large to display} \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 = 2.923, size = 1715, normalized size = 5.46 \begin{align*} \left [\frac{24 \,{\left (2 \, b c^{10} d^{2} x^{10} - b c^{8} d^{2} x^{8} - 16 \, b c^{6} d^{2} x^{6} + 34 \, b c^{4} d^{2} x^{4} - 26 \, b c^{2} d^{2} x^{2} + 7 \, b d^{2}\right )} \sqrt{-c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) + 24 \,{\left (b c^{11} d^{2} x^{11} - b c^{9} d^{2} x^{9}\right )} \sqrt{-d} \log \left (\frac{c^{2} d x^{6} + c^{2} d x^{2} - d x^{4} + \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1}{\left (x^{4} - 1\right )} \sqrt{-d} - d}{c^{2} x^{4} - x^{2}}\right ) +{\left (12 \, b c^{7} d^{2} x^{7} - 90 \, b c^{5} d^{2} x^{5} -{\left (12 \, b c^{7} - 90 \, b c^{5} + 76 \, b c^{3} - 21 \, b c\right )} d^{2} x^{9} + 76 \, b c^{3} d^{2} x^{3} - 21 \, b c d^{2} x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} + 24 \,{\left (2 \, a c^{10} d^{2} x^{10} - a c^{8} d^{2} x^{8} - 16 \, a c^{6} d^{2} x^{6} + 34 \, a c^{4} d^{2} x^{4} - 26 \, a c^{2} d^{2} x^{2} + 7 \, a d^{2}\right )} \sqrt{-c^{2} d x^{2} + d}}{1512 \,{\left (c^{2} x^{11} - x^{9}\right )}}, -\frac{48 \,{\left (b c^{11} d^{2} x^{11} - b c^{9} d^{2} x^{9}\right )} \sqrt{d} \arctan \left (\frac{\sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1}{\left (x^{2} + 1\right )} \sqrt{d}}{c^{2} d x^{4} -{\left (c^{2} + 1\right )} d x^{2} + d}\right ) - 24 \,{\left (2 \, b c^{10} d^{2} x^{10} - b c^{8} d^{2} x^{8} - 16 \, b c^{6} d^{2} x^{6} + 34 \, b c^{4} d^{2} x^{4} - 26 \, b c^{2} d^{2} x^{2} + 7 \, b d^{2}\right )} \sqrt{-c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (12 \, b c^{7} d^{2} x^{7} - 90 \, b c^{5} d^{2} x^{5} -{\left (12 \, b c^{7} - 90 \, b c^{5} + 76 \, b c^{3} - 21 \, b c\right )} d^{2} x^{9} + 76 \, b c^{3} d^{2} x^{3} - 21 \, b c d^{2} x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} - 24 \,{\left (2 \, a c^{10} d^{2} x^{10} - a c^{8} d^{2} x^{8} - 16 \, a c^{6} d^{2} x^{6} + 34 \, a c^{4} d^{2} x^{4} - 26 \, a c^{2} d^{2} x^{2} + 7 \, a d^{2}\right )} \sqrt{-c^{2} d x^{2} + d}}{1512 \,{\left (c^{2} x^{11} - x^{9}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 )}^{\frac{5}{2}}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}}{x^{10}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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