Optimal. Leaf size=454 \[ \frac{1}{32} d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{128 c^2}-\frac{3 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{256 c^4}-\frac{3 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{512 b c^5 \sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{16} d x^5 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{b c^5 d^2 x^{10} \sqrt{d-c^2 d x^2}}{100 \sqrt{c x-1} \sqrt{c x+1}}+\frac{21 b c^3 d^2 x^8 \sqrt{d-c^2 d x^2}}{640 \sqrt{c x-1} \sqrt{c x+1}}-\frac{31 b c d^2 x^6 \sqrt{d-c^2 d x^2}}{960 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b d^2 x^4 \sqrt{d-c^2 d x^2}}{512 c \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 b d^2 x^2 \sqrt{d-c^2 d x^2}}{512 c^3 \sqrt{c x-1} \sqrt{c x+1}} \]
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Rubi [A] time = 1.38481, antiderivative size = 485, normalized size of antiderivative = 1.07, number of steps used = 15, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5798, 5745, 5743, 5759, 5676, 30, 14, 266, 43} \[ \frac{1}{32} d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{10} d^2 x^5 (1-c x)^2 (c x+1)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{16} d^2 x^5 (1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{128 c^2}-\frac{3 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{256 c^4}-\frac{3 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{512 b c^5 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^5 d^2 x^{10} \sqrt{d-c^2 d x^2}}{100 \sqrt{c x-1} \sqrt{c x+1}}+\frac{21 b c^3 d^2 x^8 \sqrt{d-c^2 d x^2}}{640 \sqrt{c x-1} \sqrt{c x+1}}-\frac{31 b c d^2 x^6 \sqrt{d-c^2 d x^2}}{960 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b d^2 x^4 \sqrt{d-c^2 d x^2}}{512 c \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 b d^2 x^2 \sqrt{d-c^2 d x^2}}{512 c^3 \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
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Rule 5798
Rule 5745
Rule 5743
Rule 5759
Rule 5676
Rule 30
Rule 14
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^4 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \int x^4 (-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{10} d^2 x^5 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \int x^4 (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b c d^2 \sqrt{d-c^2 d x^2}\right ) \int x^5 \left (-1+c^2 x^2\right )^2 \, dx}{10 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{16} d^2 x^5 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{10} d^2 x^5 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{\left (3 d^2 \sqrt{d-c^2 d x^2}\right ) \int x^4 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{16 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b c d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int x^2 \left (-1+c^2 x\right )^2 \, dx,x,x^2\right )}{20 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b c d^2 \sqrt{d-c^2 d x^2}\right ) \int x^5 \left (-1+c^2 x^2\right ) \, dx}{16 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{32} d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{16} d^2 x^5 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{10} d^2 x^5 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{32 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b c d^2 \sqrt{d-c^2 d x^2}\right ) \int x^5 \, dx}{32 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b c d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (x^2-2 c^2 x^3+c^4 x^4\right ) \, dx,x,x^2\right )}{20 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b c d^2 \sqrt{d-c^2 d x^2}\right ) \int \left (-x^5+c^2 x^7\right ) \, dx}{16 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{31 b c d^2 x^6 \sqrt{d-c^2 d x^2}}{960 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{21 b c^3 d^2 x^8 \sqrt{d-c^2 d x^2}}{640 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^5 d^2 x^{10} \sqrt{d-c^2 d x^2}}{100 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{128 c^2}+\frac{1}{32} d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{16} d^2 x^5 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{10} d^2 x^5 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (3 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{128 c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b d^2 \sqrt{d-c^2 d x^2}\right ) \int x^3 \, dx}{128 c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b d^2 x^4 \sqrt{d-c^2 d x^2}}{512 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{31 b c d^2 x^6 \sqrt{d-c^2 d x^2}}{960 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{21 b c^3 d^2 x^8 \sqrt{d-c^2 d x^2}}{640 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^5 d^2 x^{10} \sqrt{d-c^2 d x^2}}{100 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{3 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{256 c^4}-\frac{d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{128 c^2}+\frac{1}{32} d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{16} d^2 x^5 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{10} d^2 x^5 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (3 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{256 c^4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (3 b d^2 \sqrt{d-c^2 d x^2}\right ) \int x \, dx}{256 c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{3 b d^2 x^2 \sqrt{d-c^2 d x^2}}{512 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b d^2 x^4 \sqrt{d-c^2 d x^2}}{512 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{31 b c d^2 x^6 \sqrt{d-c^2 d x^2}}{960 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{21 b c^3 d^2 x^8 \sqrt{d-c^2 d x^2}}{640 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^5 d^2 x^{10} \sqrt{d-c^2 d x^2}}{100 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{3 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{256 c^4}-\frac{d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{128 c^2}+\frac{1}{32} d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{16} d^2 x^5 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{10} d^2 x^5 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{3 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{512 b c^5 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 6.54835, size = 581, normalized size = 1.28 \[ \sqrt{-d \left (c^2 x^2-1\right )} \left (\frac{1}{10} a c^4 d^2 x^9-\frac{21}{80} a c^2 d^2 x^7-\frac{a d^2 x^3}{128 c^2}-\frac{3 a d^2 x}{256 c^4}+\frac{31}{160} a d^2 x^5\right )-\frac{3 a d^{5/2} \tan ^{-1}\left (\frac{c x \sqrt{-d \left (c^2 x^2-1\right )}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )}{256 c^5}+\frac{b d^2 \sqrt{-d (c x-1) (c x+1)} \left (18 \cosh \left (2 \cosh ^{-1}(c x)\right )-9 \cosh \left (4 \cosh ^{-1}(c x)\right )-2 \left (36 \cosh ^{-1}(c x)^2+\cosh \left (6 \cosh ^{-1}(c x)\right )+18 \cosh ^{-1}(c x) \sinh \left (2 \cosh ^{-1}(c x)\right )-18 \cosh ^{-1}(c x) \sinh \left (4 \cosh ^{-1}(c x)\right )-6 \cosh ^{-1}(c x) \sinh \left (6 \cosh ^{-1}(c x)\right )\right )\right )}{2304 c^5 \sqrt{\frac{c x-1}{c x+1}} (c x+1)}+\frac{b d^2 \sqrt{-d (c x-1) (c x+1)} \left (1440 \cosh ^{-1}(c x)^2-576 \cosh \left (2 \cosh ^{-1}(c x)\right )+144 \cosh \left (4 \cosh ^{-1}(c x)\right )+64 \cosh \left (6 \cosh ^{-1}(c x)\right )+9 \cosh \left (8 \cosh ^{-1}(c x)\right )+1152 \cosh ^{-1}(c x) \sinh \left (2 \cosh ^{-1}(c x)\right )-576 \cosh ^{-1}(c x) \sinh \left (4 \cosh ^{-1}(c x)\right )-384 \cosh ^{-1}(c x) \sinh \left (6 \cosh ^{-1}(c x)\right )-72 \cosh ^{-1}(c x) \sinh \left (8 \cosh ^{-1}(c x)\right )\right )}{36864 c^5 \sqrt{\frac{c x-1}{c x+1}} (c x+1)}-\frac{b d^2 \sqrt{-d (c x-1) (c x+1)} \left (50400 \cosh ^{-1}(c x)^2-25200 \cosh \left (2 \cosh ^{-1}(c x)\right )+3600 \cosh \left (4 \cosh ^{-1}(c x)\right )+2600 \cosh \left (6 \cosh ^{-1}(c x)\right )+675 \cosh \left (8 \cosh ^{-1}(c x)\right )+72 \cosh \left (10 \cosh ^{-1}(c x)\right )+50400 \cosh ^{-1}(c x) \sinh \left (2 \cosh ^{-1}(c x)\right )-14400 \cosh ^{-1}(c x) \sinh \left (4 \cosh ^{-1}(c x)\right )-15600 \cosh ^{-1}(c x) \sinh \left (6 \cosh ^{-1}(c x)\right )-5400 \cosh ^{-1}(c x) \sinh \left (8 \cosh ^{-1}(c x)\right )-720 \cosh ^{-1}(c x) \sinh \left (10 \cosh ^{-1}(c x)\right )\right )}{3686400 c^5 \sqrt{\frac{c x-1}{c x+1}} (c x+1)} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.49, size = 690, normalized size = 1.5 \begin{align*} \text{result 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a c^{4} d^{2} x^{8} - 2 \, a c^{2} d^{2} x^{6} + a d^{2} x^{4} +{\left (b c^{4} d^{2} x^{8} - 2 \, b c^{2} d^{2} x^{6} + b d^{2} x^{4}\right )} \operatorname{arcosh}\left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}, x\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{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )} x^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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