Optimal. Leaf size=341 \[ \frac {1}{4} d^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )+\frac {b e^2 x^7 \left (1-c^2 x^2\right )}{64 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {b e x^5 \left (1-c^2 x^2\right ) \left (64 c^2 d+21 e\right )}{1152 c^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b \sqrt {c^2 x^2-1} \left (288 c^4 d^2+320 c^2 d e+105 e^2\right ) \tanh ^{-1}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{3072 c^8 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b x \left (1-c^2 x^2\right ) \left (288 c^4 d^2+320 c^2 d e+105 e^2\right )}{3072 c^7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b x^3 \left (1-c^2 x^2\right ) \left (288 c^4 d^2+320 c^2 d e+105 e^2\right )}{4608 c^5 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rubi [A] time = 0.36, antiderivative size = 341, normalized size of antiderivative = 1.00, 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 12
Rule 43
Rule 206
Rule 217
Rule 266
Rule 321
Rule 459
Rule 520
Rule 1267
Rule 5790
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*}
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Mathematica [A] time = 0.35, size = 214, normalized size = 0.63 \[ \frac {384 a c^8 x^4 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )+384 b c^8 x^4 \cosh ^{-1}(c x) \left (6 d^2+8 d e x^2+3 e^2 x^4\right )-6 b \left (288 c^4 d^2+320 c^2 d e+105 e^2\right ) \tanh ^{-1}\left (\sqrt {\frac {c x-1}{c x+1}}\right )-b c x \sqrt {c x-1} \sqrt {c x+1} \left (16 c^6 \left (36 d^2 x^2+32 d e x^4+9 e^2 x^6\right )+8 c^4 \left (108 d^2+80 d e x^2+21 e^2 x^4\right )+30 c^2 e \left (32 d+7 e x^2\right )+315 e^2\right )}{9216 c^8} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.70, size = 227, normalized size = 0.67 \[ \frac {1152 \, a c^{8} e^{2} x^{8} + 3072 \, a c^{8} d e x^{6} + 2304 \, a c^{8} d^{2} x^{4} + 3 \, {\left (384 \, b c^{8} e^{2} x^{8} + 1024 \, b c^{8} d e x^{6} + 768 \, b c^{8} d^{2} x^{4} - 288 \, b c^{4} d^{2} - 320 \, b c^{2} d e - 105 \, b e^{2}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (144 \, b c^{7} e^{2} x^{7} + 8 \, {\left (64 \, b c^{7} d e + 21 \, b c^{5} e^{2}\right )} x^{5} + 2 \, {\left (288 \, b c^{7} d^{2} + 320 \, b c^{5} d e + 105 \, b c^{3} e^{2}\right )} x^{3} + 3 \, {\left (288 \, b c^{5} d^{2} + 320 \, b c^{3} d e + 105 \, b c e^{2}\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{9216 \, c^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 440, normalized size = 1.29 \[ \frac {a \,e^{2} x^{8}}{8}+\frac {a d e \,x^{6}}{3}+\frac {a \,x^{4} d^{2}}{4}+\frac {b \,\mathrm {arccosh}\left (c x \right ) e^{2} x^{8}}{8}+\frac {b \,\mathrm {arccosh}\left (c x \right ) d e \,x^{6}}{3}+\frac {b \,\mathrm {arccosh}\left (c x \right ) x^{4} d^{2}}{4}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, e^{2} x^{7}}{64 c}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, x^{5} d e}{18 c}-\frac {b \,d^{2} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}}{16 c}-\frac {7 b \sqrt {c x -1}\, \sqrt {c x +1}\, e^{2} x^{5}}{384 c^{3}}-\frac {5 b \sqrt {c x -1}\, \sqrt {c x +1}\, d e \,x^{3}}{72 c^{3}}-\frac {3 b \,d^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{32 c^{3}}-\frac {3 b \sqrt {c x -1}\, \sqrt {c x +1}\, d^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{32 c^{4} \sqrt {c^{2} x^{2}-1}}-\frac {35 b \sqrt {c x -1}\, \sqrt {c x +1}\, e^{2} x^{3}}{1536 c^{5}}-\frac {5 b \sqrt {c x -1}\, \sqrt {c x +1}\, d e x}{48 c^{5}}-\frac {5 b \sqrt {c x -1}\, \sqrt {c x +1}\, d e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{48 c^{6} \sqrt {c^{2} x^{2}-1}}-\frac {35 b \sqrt {c x -1}\, \sqrt {c x +1}\, e^{2} x}{1024 c^{7}}-\frac {35 b \sqrt {c x -1}\, \sqrt {c x +1}\, e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{1024 c^{8} \sqrt {c^{2} x^{2}-1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 332, normalized size = 0.97 \[ \frac {1}{8} \, a e^{2} x^{8} + \frac {1}{3} \, a d e x^{6} + \frac {1}{4} \, a d^{2} x^{4} + \frac {1}{32} \, {\left (8 \, x^{4} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {c^{2} x^{2} - 1} x}{c^{4}} + \frac {3 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{5}}\right )} c\right )} b d^{2} + \frac {1}{144} \, {\left (48 \, x^{6} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {c^{2} x^{2} - 1} x}{c^{6}} + \frac {15 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{7}}\right )} c\right )} b d e + \frac {1}{3072} \, {\left (384 \, x^{8} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {48 \, \sqrt {c^{2} x^{2} - 1} x^{7}}{c^{2}} + \frac {56 \, \sqrt {c^{2} x^{2} - 1} x^{5}}{c^{4}} + \frac {70 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{6}} + \frac {105 \, \sqrt {c^{2} x^{2} - 1} x}{c^{8}} + \frac {105 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{9}}\right )} c\right )} b e^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.23, size = 389, normalized size = 1.14 \[ \begin {cases} \frac {a d^{2} x^{4}}{4} + \frac {a d e x^{6}}{3} + \frac {a e^{2} x^{8}}{8} + \frac {b d^{2} x^{4} \operatorname {acosh}{\left (c x \right )}}{4} + \frac {b d e x^{6} \operatorname {acosh}{\left (c x \right )}}{3} + \frac {b e^{2} x^{8} \operatorname {acosh}{\left (c x \right )}}{8} - \frac {b d^{2} x^{3} \sqrt {c^{2} x^{2} - 1}}{16 c} - \frac {b d e x^{5} \sqrt {c^{2} x^{2} - 1}}{18 c} - \frac {b e^{2} x^{7} \sqrt {c^{2} x^{2} - 1}}{64 c} - \frac {3 b d^{2} x \sqrt {c^{2} x^{2} - 1}}{32 c^{3}} - \frac {5 b d e x^{3} \sqrt {c^{2} x^{2} - 1}}{72 c^{3}} - \frac {7 b e^{2} x^{5} \sqrt {c^{2} x^{2} - 1}}{384 c^{3}} - \frac {3 b d^{2} \operatorname {acosh}{\left (c x \right )}}{32 c^{4}} - \frac {5 b d e x \sqrt {c^{2} x^{2} - 1}}{48 c^{5}} - \frac {35 b e^{2} x^{3} \sqrt {c^{2} x^{2} - 1}}{1536 c^{5}} - \frac {5 b d e \operatorname {acosh}{\left (c x \right )}}{48 c^{6}} - \frac {35 b e^{2} x \sqrt {c^{2} x^{2} - 1}}{1024 c^{7}} - \frac {35 b e^{2} \operatorname {acosh}{\left (c x \right )}}{1024 c^{8}} & \text {for}\: c \neq 0 \\\left (a + \frac {i \pi b}{2}\right ) \left (\frac {d^{2} x^{4}}{4} + \frac {d e x^{6}}{3} + \frac {e^{2} x^{8}}{8}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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