Optimal. Leaf size=341 \[ \frac{1}{4} d^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )+\frac{b x^3 \left (1-c^2 x^2\right ) \left (288 c^4 d^2+320 c^2 d e+105 e^2\right )}{4608 c^5 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b x \left (1-c^2 x^2\right ) \left (288 c^4 d^2+320 c^2 d e+105 e^2\right )}{3072 c^7 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b \sqrt{c^2 x^2-1} \left (288 c^4 d^2+320 c^2 d e+105 e^2\right ) \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )}{3072 c^8 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b e x^5 \left (1-c^2 x^2\right ) \left (64 c^2 d+21 e\right )}{1152 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b e^2 x^7 \left (1-c^2 x^2\right )}{64 c \sqrt{c x-1} \sqrt{c x+1}} \]
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Rubi [A] time = 0.360308, antiderivative size = 341, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476, Rules used = {266, 43, 5790, 12, 520, 1267, 459, 321, 217, 206} \[ \frac{1}{4} d^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )+\frac{b x^3 \left (1-c^2 x^2\right ) \left (288 c^4 d^2+320 c^2 d e+105 e^2\right )}{4608 c^5 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b x \left (1-c^2 x^2\right ) \left (288 c^4 d^2+320 c^2 d e+105 e^2\right )}{3072 c^7 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b \sqrt{c^2 x^2-1} \left (288 c^4 d^2+320 c^2 d e+105 e^2\right ) \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )}{3072 c^8 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b e x^5 \left (1-c^2 x^2\right ) \left (64 c^2 d+21 e\right )}{1152 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b e^2 x^7 \left (1-c^2 x^2\right )}{64 c \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rule 5790
Rule 12
Rule 520
Rule 1267
Rule 459
Rule 321
Rule 217
Rule 206
Rubi steps
\begin{align*} \int x^3 \left (d+e 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} d e x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac{x^4 \left (6 d^2+8 d e x^2+3 e^2 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} d e x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{24} (b c) \int \frac{x^4 \left (6 d^2+8 d e x^2+3 e^2 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} d e x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c \sqrt{-1+c^2 x^2}\right ) \int \frac{x^4 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )}{\sqrt{-1+c^2 x^2}} \, dx}{24 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b e^2 x^7 \left (1-c^2 x^2\right )}{64 c \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} d e x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b \sqrt{-1+c^2 x^2}\right ) \int \frac{x^4 \left (48 c^2 d^2+e \left (64 c^2 d+21 e\right ) x^2\right )}{\sqrt{-1+c^2 x^2}} \, dx}{192 c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b e \left (64 c^2 d+21 e\right ) x^5 \left (1-c^2 x^2\right )}{1152 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b e^2 x^7 \left (1-c^2 x^2\right )}{64 c \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} d e x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )+\frac{\left (b \left (-288 c^4 d^2-5 e \left (64 c^2 d+21 e\right )\right ) \sqrt{-1+c^2 x^2}\right ) \int \frac{x^4}{\sqrt{-1+c^2 x^2}} \, dx}{1152 c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b \left (288 c^4 d^2+5 e \left (64 c^2 d+21 e\right )\right ) x^3 \left (1-c^2 x^2\right )}{4608 c^5 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b e \left (64 c^2 d+21 e\right ) x^5 \left (1-c^2 x^2\right )}{1152 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b e^2 x^7 \left (1-c^2 x^2\right )}{64 c \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} d e x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )+\frac{\left (b \left (-288 c^4 d^2-5 e \left (64 c^2 d+21 e\right )\right ) \sqrt{-1+c^2 x^2}\right ) \int \frac{x^2}{\sqrt{-1+c^2 x^2}} \, dx}{1536 c^5 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b \left (288 c^4 d^2+5 e \left (64 c^2 d+21 e\right )\right ) x \left (1-c^2 x^2\right )}{3072 c^7 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b \left (288 c^4 d^2+5 e \left (64 c^2 d+21 e\right )\right ) x^3 \left (1-c^2 x^2\right )}{4608 c^5 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b e \left (64 c^2 d+21 e\right ) x^5 \left (1-c^2 x^2\right )}{1152 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b e^2 x^7 \left (1-c^2 x^2\right )}{64 c \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} d e x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )+\frac{\left (b \left (-288 c^4 d^2-5 e \left (64 c^2 d+21 e\right )\right ) \sqrt{-1+c^2 x^2}\right ) \int \frac{1}{\sqrt{-1+c^2 x^2}} \, dx}{3072 c^7 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b \left (288 c^4 d^2+5 e \left (64 c^2 d+21 e\right )\right ) x \left (1-c^2 x^2\right )}{3072 c^7 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b \left (288 c^4 d^2+5 e \left (64 c^2 d+21 e\right )\right ) x^3 \left (1-c^2 x^2\right )}{4608 c^5 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b e \left (64 c^2 d+21 e\right ) x^5 \left (1-c^2 x^2\right )}{1152 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b e^2 x^7 \left (1-c^2 x^2\right )}{64 c \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} d e x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )+\frac{\left (b \left (-288 c^4 d^2-5 e \left (64 c^2 d+21 e\right )\right ) \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^7 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b \left (288 c^4 d^2+5 e \left (64 c^2 d+21 e\right )\right ) x \left (1-c^2 x^2\right )}{3072 c^7 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b \left (288 c^4 d^2+5 e \left (64 c^2 d+21 e\right )\right ) x^3 \left (1-c^2 x^2\right )}{4608 c^5 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b e \left (64 c^2 d+21 e\right ) x^5 \left (1-c^2 x^2\right )}{1152 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b e^2 x^7 \left (1-c^2 x^2\right )}{64 c \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} d e x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )-\frac{b \left (288 c^4 d^2+5 e \left (64 c^2 d+21 e\right )\right ) \sqrt{-1+c^2 x^2} \tanh ^{-1}\left (\frac{c x}{\sqrt{-1+c^2 x^2}}\right )}{3072 c^8 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 0.335491, size = 214, normalized size = 0.63 \[ \frac{384 a c^8 x^4 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )-b c x \sqrt{c x-1} \sqrt{c x+1} \left (16 c^6 \left (36 d^2 x^2+32 d e x^4+9 e^2 x^6\right )+8 c^4 \left (108 d^2+80 d e x^2+21 e^2 x^4\right )+30 c^2 e \left (32 d+7 e x^2\right )+315 e^2\right )+384 b c^8 x^4 \cosh ^{-1}(c x) \left (6 d^2+8 d e x^2+3 e^2 x^4\right )-6 b \left (288 c^4 d^2+320 c^2 d e+105 e^2\right ) \tanh ^{-1}\left (\sqrt{\frac{c x-1}{c x+1}}\right )}{9216 c^8} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.019, size = 440, normalized size = 1.3 \begin{align*}{\frac{a{e}^{2}{x}^{8}}{8}}+{\frac{ade{x}^{6}}{3}}+{\frac{{d}^{2}a{x}^{4}}{4}}+{\frac{b{\rm arccosh} \left (cx\right ){e}^{2}{x}^{8}}{8}}+{\frac{b{\rm arccosh} \left (cx\right )de{x}^{6}}{3}}+{\frac{{d}^{2}b{\rm arccosh} \left (cx\right ){x}^{4}}{4}}-{\frac{b{e}^{2}{x}^{7}}{64\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{b{x}^{5}de}{18\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{b{d}^{2}{x}^{3}}{16\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{7\,b{e}^{2}{x}^{5}}{384\,{c}^{3}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{5\,bde{x}^{3}}{72\,{c}^{3}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{3\,b{d}^{2}x}{32\,{c}^{3}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{3\,b{d}^{2}}{32\,{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}}}}-{\frac{35\,b{e}^{2}{x}^{3}}{1536\,{c}^{5}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{5\,bdex}{48\,{c}^{5}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{5\,bde}{48\,{c}^{6}}\sqrt{cx-1}\sqrt{cx+1}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}}-{\frac{35\,b{e}^{2}x}{1024\,{c}^{7}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{35\,b{e}^{2}}{1024\,{c}^{8}}\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 [A] time = 1.14553, size = 485, normalized size = 1.42 \begin{align*} \frac{1}{8} \, a e^{2} x^{8} + \frac{1}{3} \, a d e x^{6} + \frac{1}{4} \, a d^{2} x^{4} + \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} + \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 d e + \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 e^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.54593, size = 539, normalized size = 1.58 \begin{align*} \frac{1152 \, a c^{8} e^{2} x^{8} + 3072 \, a c^{8} d e x^{6} + 2304 \, a c^{8} d^{2} x^{4} + 3 \,{\left (384 \, b c^{8} e^{2} x^{8} + 1024 \, b c^{8} d e x^{6} + 768 \, b c^{8} d^{2} x^{4} - 288 \, b c^{4} d^{2} - 320 \, b c^{2} d e - 105 \, b e^{2}\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (144 \, b c^{7} e^{2} x^{7} + 8 \,{\left (64 \, b c^{7} d e + 21 \, b c^{5} e^{2}\right )} x^{5} + 2 \,{\left (288 \, b c^{7} d^{2} + 320 \, b c^{5} d e + 105 \, b c^{3} e^{2}\right )} x^{3} + 3 \,{\left (288 \, b c^{5} d^{2} + 320 \, b c^{3} d e + 105 \, b c e^{2}\right )} x\right )} \sqrt{c^{2} x^{2} - 1}}{9216 \, c^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 19.7484, size = 389, normalized size = 1.14 \begin{align*} \begin{cases} \frac{a d^{2} x^{4}}{4} + \frac{a d e x^{6}}{3} + \frac{a e^{2} x^{8}}{8} + \frac{b d^{2} x^{4} \operatorname{acosh}{\left (c x \right )}}{4} + \frac{b d e x^{6} \operatorname{acosh}{\left (c x \right )}}{3} + \frac{b e^{2} x^{8} \operatorname{acosh}{\left (c x \right )}}{8} - \frac{b d^{2} x^{3} \sqrt{c^{2} x^{2} - 1}}{16 c} - \frac{b d e x^{5} \sqrt{c^{2} x^{2} - 1}}{18 c} - \frac{b e^{2} x^{7} \sqrt{c^{2} x^{2} - 1}}{64 c} - \frac{3 b d^{2} x \sqrt{c^{2} x^{2} - 1}}{32 c^{3}} - \frac{5 b d e x^{3} \sqrt{c^{2} x^{2} - 1}}{72 c^{3}} - \frac{7 b e^{2} x^{5} \sqrt{c^{2} x^{2} - 1}}{384 c^{3}} - \frac{3 b d^{2} \operatorname{acosh}{\left (c x \right )}}{32 c^{4}} - \frac{5 b d e x \sqrt{c^{2} x^{2} - 1}}{48 c^{5}} - \frac{35 b e^{2} x^{3} \sqrt{c^{2} x^{2} - 1}}{1536 c^{5}} - \frac{5 b d e \operatorname{acosh}{\left (c x \right )}}{48 c^{6}} - \frac{35 b e^{2} x \sqrt{c^{2} x^{2} - 1}}{1024 c^{7}} - \frac{35 b e^{2} \operatorname{acosh}{\left (c x \right )}}{1024 c^{8}} & \text{for}\: c \neq 0 \\\left (a + \frac{i \pi b}{2}\right ) \left (\frac{d^{2} x^{4}}{4} + \frac{d e x^{6}}{3} + \frac{e^{2} x^{8}}{8}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.44553, size = 432, normalized size = 1.27 \begin{align*} \frac{1}{4} \, a d^{2} x^{4} + \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} + \frac{1}{3072} \,{\left (384 \, a x^{8} +{\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\right )} e^{2} + \frac{1}{144} \,{\left (48 \, a d x^{6} +{\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 d\right )} e \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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