Optimal. Leaf size=200 \[ \frac{1}{8} c^4 d^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{3} c^2 d^2 x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} d^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{64} b c^3 d^2 x^7 \sqrt{c x-1} \sqrt{c x+1}-\frac{73 b d^2 x \sqrt{c x-1} \sqrt{c x+1}}{3072 c^3}-\frac{73 b d^2 \cosh ^{-1}(c x)}{3072 c^4}+\frac{43 b c d^2 x^5 \sqrt{c x-1} \sqrt{c x+1}}{1152}-\frac{73 b d^2 x^3 \sqrt{c x-1} \sqrt{c x+1}}{4608 c} \]
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Rubi [A] time = 0.276914, antiderivative size = 284, normalized size of antiderivative = 1.42, number of steps used = 9, number of rules used = 10, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {266, 43, 5731, 12, 520, 1267, 459, 321, 217, 206} \[ \frac{1}{8} c^4 d^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{3} c^2 d^2 x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} d^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac{b c^3 d^2 x^7 \left (1-c^2 x^2\right )}{64 \sqrt{c x-1} \sqrt{c x+1}}-\frac{43 b c d^2 x^5 \left (1-c^2 x^2\right )}{1152 \sqrt{c x-1} \sqrt{c x+1}}+\frac{73 b d^2 x^3 \left (1-c^2 x^2\right )}{4608 c \sqrt{c x-1} \sqrt{c x+1}}+\frac{73 b d^2 x \left (1-c^2 x^2\right )}{3072 c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{73 b d^2 \sqrt{c^2 x^2-1} \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )}{3072 c^4 \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rule 5731
Rule 12
Rule 520
Rule 1267
Rule 459
Rule 321
Rule 217
Rule 206
Rubi steps
\begin{align*} \int x^3 \left (d-c^2 d x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac{1}{4} d^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{3} c^2 d^2 x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{8} c^4 d^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac{d^2 x^4 \left (6-8 c^2 x^2+3 c^4 x^4\right )}{24 \sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=\frac{1}{4} d^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{3} c^2 d^2 x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{8} c^4 d^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{24} \left (b c d^2\right ) \int \frac{x^4 \left (6-8 c^2 x^2+3 c^4 x^4\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=\frac{1}{4} d^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{3} c^2 d^2 x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{8} c^4 d^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c d^2 \sqrt{-1+c^2 x^2}\right ) \int \frac{x^4 \left (6-8 c^2 x^2+3 c^4 x^4\right )}{\sqrt{-1+c^2 x^2}} \, dx}{24 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b c^3 d^2 x^7 \left (1-c^2 x^2\right )}{64 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{1}{4} d^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{3} c^2 d^2 x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{8} c^4 d^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b d^2 \sqrt{-1+c^2 x^2}\right ) \int \frac{x^4 \left (48 c^2-43 c^4 x^2\right )}{\sqrt{-1+c^2 x^2}} \, dx}{192 c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{43 b c d^2 x^5 \left (1-c^2 x^2\right )}{1152 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d^2 x^7 \left (1-c^2 x^2\right )}{64 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{1}{4} d^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{3} c^2 d^2 x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{8} c^4 d^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (73 b c d^2 \sqrt{-1+c^2 x^2}\right ) \int \frac{x^4}{\sqrt{-1+c^2 x^2}} \, dx}{1152 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{73 b d^2 x^3 \left (1-c^2 x^2\right )}{4608 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{43 b c d^2 x^5 \left (1-c^2 x^2\right )}{1152 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d^2 x^7 \left (1-c^2 x^2\right )}{64 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{1}{4} d^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{3} c^2 d^2 x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{8} c^4 d^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (73 b d^2 \sqrt{-1+c^2 x^2}\right ) \int \frac{x^2}{\sqrt{-1+c^2 x^2}} \, dx}{1536 c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{73 b d^2 x \left (1-c^2 x^2\right )}{3072 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{73 b d^2 x^3 \left (1-c^2 x^2\right )}{4608 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{43 b c d^2 x^5 \left (1-c^2 x^2\right )}{1152 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d^2 x^7 \left (1-c^2 x^2\right )}{64 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{1}{4} d^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{3} c^2 d^2 x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{8} c^4 d^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (73 b d^2 \sqrt{-1+c^2 x^2}\right ) \int \frac{1}{\sqrt{-1+c^2 x^2}} \, dx}{3072 c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{73 b d^2 x \left (1-c^2 x^2\right )}{3072 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{73 b d^2 x^3 \left (1-c^2 x^2\right )}{4608 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{43 b c d^2 x^5 \left (1-c^2 x^2\right )}{1152 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d^2 x^7 \left (1-c^2 x^2\right )}{64 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{1}{4} d^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{3} c^2 d^2 x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{8} c^4 d^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (73 b d^2 \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{1-c^2 x^2} \, dx,x,\frac{x}{\sqrt{-1+c^2 x^2}}\right )}{3072 c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{73 b d^2 x \left (1-c^2 x^2\right )}{3072 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{73 b d^2 x^3 \left (1-c^2 x^2\right )}{4608 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{43 b c d^2 x^5 \left (1-c^2 x^2\right )}{1152 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d^2 x^7 \left (1-c^2 x^2\right )}{64 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{1}{4} d^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{3} c^2 d^2 x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{8} c^4 d^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )-\frac{73 b d^2 \sqrt{-1+c^2 x^2} \tanh ^{-1}\left (\frac{c x}{\sqrt{-1+c^2 x^2}}\right )}{3072 c^4 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 0.226475, size = 194, normalized size = 0.97 \[ \frac{d^2 \left (1152 a c^8 x^8-3072 a c^6 x^6+2304 a c^4 x^4-144 b c^7 x^7 \sqrt{c x-1} \sqrt{c x+1}+344 b c^5 x^5 \sqrt{c x-1} \sqrt{c x+1}-146 b c^3 x^3 \sqrt{c x-1} \sqrt{c x+1}+384 b c^4 x^4 \left (3 c^4 x^4-8 c^2 x^2+6\right ) \cosh ^{-1}(c x)-219 b c x \sqrt{c x-1} \sqrt{c x+1}-438 b \tanh ^{-1}\left (\sqrt{\frac{c x-1}{c x+1}}\right )\right )}{9216 c^4} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.017, size = 230, normalized size = 1.2 \begin{align*}{\frac{{c}^{4}{d}^{2}a{x}^{8}}{8}}-{\frac{{c}^{2}{d}^{2}a{x}^{6}}{3}}+{\frac{{d}^{2}a{x}^{4}}{4}}+{\frac{{c}^{4}{d}^{2}b{\rm arccosh} \left (cx\right ){x}^{8}}{8}}-{\frac{{c}^{2}{d}^{2}b{\rm arccosh} \left (cx\right ){x}^{6}}{3}}+{\frac{{d}^{2}b{\rm arccosh} \left (cx\right ){x}^{4}}{4}}-{\frac{{d}^{2}b{c}^{3}{x}^{7}}{64}\sqrt{cx-1}\sqrt{cx+1}}+{\frac{43\,{d}^{2}bc{x}^{5}}{1152}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{73\,{d}^{2}b{x}^{3}}{4608\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{73\,{d}^{2}bx}{3072\,{c}^{3}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{73\,{d}^{2}b}{3072\,{c}^{4}}\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 = 1.41441, size = 504, normalized size = 2.52 \begin{align*} \frac{1}{8} \, a c^{4} d^{2} x^{8} - \frac{1}{3} \, a c^{2} d^{2} 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^{4} d^{2} + \frac{1}{4} \, a d^{2} x^{4} - \frac{1}{144} \,{\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^{2} d^{2} + \frac{1}{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 d^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82347, size = 373, normalized size = 1.86 \begin{align*} \frac{1152 \, a c^{8} d^{2} x^{8} - 3072 \, a c^{6} d^{2} x^{6} + 2304 \, a c^{4} d^{2} x^{4} + 3 \,{\left (384 \, b c^{8} d^{2} x^{8} - 1024 \, b c^{6} d^{2} x^{6} + 768 \, b c^{4} d^{2} x^{4} - 73 \, b d^{2}\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (144 \, b c^{7} d^{2} x^{7} - 344 \, b c^{5} d^{2} x^{5} + 146 \, b c^{3} d^{2} x^{3} + 219 \, b c d^{2} x\right )} \sqrt{c^{2} x^{2} - 1}}{9216 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 17.0226, size = 224, normalized size = 1.12 \begin{align*} \begin{cases} \frac{a c^{4} d^{2} x^{8}}{8} - \frac{a c^{2} d^{2} x^{6}}{3} + \frac{a d^{2} x^{4}}{4} + \frac{b c^{4} d^{2} x^{8} \operatorname{acosh}{\left (c x \right )}}{8} - \frac{b c^{3} d^{2} x^{7} \sqrt{c^{2} x^{2} - 1}}{64} - \frac{b c^{2} d^{2} x^{6} \operatorname{acosh}{\left (c x \right )}}{3} + \frac{43 b c d^{2} x^{5} \sqrt{c^{2} x^{2} - 1}}{1152} + \frac{b d^{2} x^{4} \operatorname{acosh}{\left (c x \right )}}{4} - \frac{73 b d^{2} x^{3} \sqrt{c^{2} x^{2} - 1}}{4608 c} - \frac{73 b d^{2} x \sqrt{c^{2} x^{2} - 1}}{3072 c^{3}} - \frac{73 b d^{2} \operatorname{acosh}{\left (c x \right )}}{3072 c^{4}} & \text{for}\: c \neq 0 \\\frac{d^{2} x^{4} \left (a + \frac{i \pi b}{2}\right )}{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.57489, size = 451, normalized size = 2.25 \begin{align*} \frac{1}{8} \, a c^{4} d^{2} x^{8} - \frac{1}{3} \, a c^{2} d^{2} 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^{4} d^{2} + \frac{1}{4} \, a d^{2} x^{4} - \frac{1}{144} \,{\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^{2} d^{2} + \frac{1}{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 d^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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