Optimal. Leaf size=395 \[ d^4 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {4}{3} d^3 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {6}{5} d^2 e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {4}{7} d e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{9} e^4 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac {4 b e^3 \left (1-c^2 x^2\right )^4 \left (9 c^2 d+7 e\right )}{441 c^9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b e^4 \left (1-c^2 x^2\right )^5}{81 c^9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b e^2 \left (1-c^2 x^2\right )^3 \left (63 c^4 d^2+90 c^2 d e+35 e^2\right )}{525 c^9 \sqrt {c x-1} \sqrt {c x+1}}-\frac {4 b e \left (1-c^2 x^2\right )^2 \left (105 c^6 d^3+189 c^4 d^2 e+135 c^2 d e^2+35 e^3\right )}{945 c^9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b \left (1-c^2 x^2\right ) \left (315 c^8 d^4+420 c^6 d^3 e+378 c^4 d^2 e^2+180 c^2 d e^3+35 e^4\right )}{315 c^9 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rubi [A] time = 0.47, antiderivative size = 395, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {194, 5705, 12, 1610, 1799, 1850} \[ \frac {6}{5} d^2 e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {4}{3} d^3 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+d^4 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {4}{7} d e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{9} e^4 x^9 \left (a+b \cosh ^{-1}(c x)\right )+\frac {2 b e^2 \left (1-c^2 x^2\right )^3 \left (63 c^4 d^2+90 c^2 d e+35 e^2\right )}{525 c^9 \sqrt {c x-1} \sqrt {c x+1}}-\frac {4 b e \left (1-c^2 x^2\right )^2 \left (189 c^4 d^2 e+105 c^6 d^3+135 c^2 d e^2+35 e^3\right )}{945 c^9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b \left (1-c^2 x^2\right ) \left (378 c^4 d^2 e^2+420 c^6 d^3 e+315 c^8 d^4+180 c^2 d e^3+35 e^4\right )}{315 c^9 \sqrt {c x-1} \sqrt {c x+1}}-\frac {4 b e^3 \left (1-c^2 x^2\right )^4 \left (9 c^2 d+7 e\right )}{441 c^9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b e^4 \left (1-c^2 x^2\right )^5}{81 c^9 \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 194
Rule 1610
Rule 1799
Rule 1850
Rule 5705
Rubi steps
\begin {align*} \int \left (d+e x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=d^4 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {4}{3} d^3 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {6}{5} d^2 e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {4}{7} d e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{9} e^4 x^9 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac {x \left (315 d^4+420 d^3 e x^2+378 d^2 e^2 x^4+180 d e^3 x^6+35 e^4 x^8\right )}{315 \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=d^4 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {4}{3} d^3 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {6}{5} d^2 e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {4}{7} d e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{9} e^4 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{315} (b c) \int \frac {x \left (315 d^4+420 d^3 e x^2+378 d^2 e^2 x^4+180 d e^3 x^6+35 e^4 x^8\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=d^4 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {4}{3} d^3 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {6}{5} d^2 e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {4}{7} d e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{9} e^4 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \int \frac {x \left (315 d^4+420 d^3 e x^2+378 d^2 e^2 x^4+180 d e^3 x^6+35 e^4 x^8\right )}{\sqrt {-1+c^2 x^2}} \, dx}{315 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=d^4 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {4}{3} d^3 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {6}{5} d^2 e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {4}{7} d e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{9} e^4 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {315 d^4+420 d^3 e x+378 d^2 e^2 x^2+180 d e^3 x^3+35 e^4 x^4}{\sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{630 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=d^4 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {4}{3} d^3 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {6}{5} d^2 e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {4}{7} d e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{9} e^4 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {315 c^8 d^4+420 c^6 d^3 e+378 c^4 d^2 e^2+180 c^2 d e^3+35 e^4}{c^8 \sqrt {-1+c^2 x}}+\frac {4 e \left (105 c^6 d^3+189 c^4 d^2 e+135 c^2 d e^2+35 e^3\right ) \sqrt {-1+c^2 x}}{c^8}+\frac {6 e^2 \left (63 c^4 d^2+90 c^2 d e+35 e^2\right ) \left (-1+c^2 x\right )^{3/2}}{c^8}+\frac {20 e^3 \left (9 c^2 d+7 e\right ) \left (-1+c^2 x\right )^{5/2}}{c^8}+\frac {35 e^4 \left (-1+c^2 x\right )^{7/2}}{c^8}\right ) \, dx,x,x^2\right )}{630 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b \left (315 c^8 d^4+420 c^6 d^3 e+378 c^4 d^2 e^2+180 c^2 d e^3+35 e^4\right ) \left (1-c^2 x^2\right )}{315 c^9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {4 b e \left (105 c^6 d^3+189 c^4 d^2 e+135 c^2 d e^2+35 e^3\right ) \left (1-c^2 x^2\right )^2}{945 c^9 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b e^2 \left (63 c^4 d^2+90 c^2 d e+35 e^2\right ) \left (1-c^2 x^2\right )^3}{525 c^9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {4 b e^3 \left (9 c^2 d+7 e\right ) \left (1-c^2 x^2\right )^4}{441 c^9 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e^4 \left (1-c^2 x^2\right )^5}{81 c^9 \sqrt {-1+c x} \sqrt {1+c x}}+d^4 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {4}{3} d^3 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {6}{5} d^2 e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {4}{7} d e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{9} e^4 x^9 \left (a+b \cosh ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A] time = 0.40, size = 265, normalized size = 0.67 \[ \frac {315 a x \left (315 d^4+420 d^3 e x^2+378 d^2 e^2 x^4+180 d e^3 x^6+35 e^4 x^8\right )-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (c^8 \left (99225 d^4+44100 d^3 e x^2+23814 d^2 e^2 x^4+8100 d e^3 x^6+1225 e^4 x^8\right )+8 c^6 e \left (11025 d^3+3969 d^2 e x^2+1215 d e^2 x^4+175 e^3 x^6\right )+48 c^4 e^2 \left (1323 d^2+270 d e x^2+35 e^2 x^4\right )+320 c^2 e^3 \left (81 d+7 e x^2\right )+4480 e^4\right )}{c^9}+315 b x \cosh ^{-1}(c x) \left (315 d^4+420 d^3 e x^2+378 d^2 e^2 x^4+180 d e^3 x^6+35 e^4 x^8\right )}{99225} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 333, normalized size = 0.84 \[ \frac {11025 \, a c^{9} e^{4} x^{9} + 56700 \, a c^{9} d e^{3} x^{7} + 119070 \, a c^{9} d^{2} e^{2} x^{5} + 132300 \, a c^{9} d^{3} e x^{3} + 99225 \, a c^{9} d^{4} x + 315 \, {\left (35 \, b c^{9} e^{4} x^{9} + 180 \, b c^{9} d e^{3} x^{7} + 378 \, b c^{9} d^{2} e^{2} x^{5} + 420 \, b c^{9} d^{3} e x^{3} + 315 \, b c^{9} d^{4} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (1225 \, b c^{8} e^{4} x^{8} + 99225 \, b c^{8} d^{4} + 88200 \, b c^{6} d^{3} e + 63504 \, b c^{4} d^{2} e^{2} + 25920 \, b c^{2} d e^{3} + 100 \, {\left (81 \, b c^{8} d e^{3} + 14 \, b c^{6} e^{4}\right )} x^{6} + 4480 \, b e^{4} + 6 \, {\left (3969 \, b c^{8} d^{2} e^{2} + 1620 \, b c^{6} d e^{3} + 280 \, b c^{4} e^{4}\right )} x^{4} + 4 \, {\left (11025 \, b c^{8} d^{3} e + 7938 \, b c^{6} d^{2} e^{2} + 3240 \, b c^{4} d e^{3} + 560 \, b c^{2} e^{4}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{99225 \, c^{9}} \]
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 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 331, normalized size = 0.84 \[ \frac {\frac {a \left (\frac {1}{9} e^{4} c^{9} x^{9}+\frac {4}{7} c^{9} d \,e^{3} x^{7}+\frac {6}{5} c^{9} d^{2} e^{2} x^{5}+\frac {4}{3} c^{9} d^{3} e \,x^{3}+c^{9} d^{4} x \right )}{c^{8}}+\frac {b \left (\frac {\mathrm {arccosh}\left (c x \right ) e^{4} c^{9} x^{9}}{9}+\frac {4 \,\mathrm {arccosh}\left (c x \right ) c^{9} d \,e^{3} x^{7}}{7}+\frac {6 \,\mathrm {arccosh}\left (c x \right ) c^{9} d^{2} e^{2} x^{5}}{5}+\frac {4 \,\mathrm {arccosh}\left (c x \right ) c^{9} d^{3} e \,x^{3}}{3}+\mathrm {arccosh}\left (c x \right ) c^{9} d^{4} x -\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (1225 c^{8} e^{4} x^{8}+8100 c^{8} d \,e^{3} x^{6}+23814 c^{8} d^{2} e^{2} x^{4}+1400 c^{6} e^{4} x^{6}+44100 c^{8} d^{3} e \,x^{2}+9720 c^{6} d \,e^{3} x^{4}+99225 c^{8} d^{4}+31752 c^{6} d^{2} e^{2} x^{2}+1680 c^{4} e^{4} x^{4}+88200 c^{6} d^{3} e +12960 c^{4} d \,e^{3} x^{2}+63504 c^{4} d^{2} e^{2}+2240 c^{2} e^{4} x^{2}+25920 c^{2} d \,e^{3}+4480 e^{4}\right )}{99225}\right )}{c^{8}}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 415, normalized size = 1.05 \[ \frac {1}{9} \, a e^{4} x^{9} + \frac {4}{7} \, a d e^{3} x^{7} + \frac {6}{5} \, a d^{2} e^{2} x^{5} + \frac {4}{3} \, a d^{3} e x^{3} + \frac {4}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b d^{3} e + \frac {2}{25} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b d^{2} e^{2} + \frac {4}{245} \, {\left (35 \, x^{7} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {c^{2} x^{2} - 1}}{c^{8}}\right )} c\right )} b d e^{3} + \frac {1}{2835} \, {\left (315 \, x^{9} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {35 \, \sqrt {c^{2} x^{2} - 1} x^{8}}{c^{2}} + \frac {40 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{6}} + \frac {64 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {c^{2} x^{2} - 1}}{c^{10}}\right )} c\right )} b e^{4} + a d^{4} x + \frac {{\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b d^{4}}{c} \]
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 \[ \int \left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^4 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 16.75, size = 600, normalized size = 1.52 \[ \begin {cases} a d^{4} x + \frac {4 a d^{3} e x^{3}}{3} + \frac {6 a d^{2} e^{2} x^{5}}{5} + \frac {4 a d e^{3} x^{7}}{7} + \frac {a e^{4} x^{9}}{9} + b d^{4} x \operatorname {acosh}{\left (c x \right )} + \frac {4 b d^{3} e x^{3} \operatorname {acosh}{\left (c x \right )}}{3} + \frac {6 b d^{2} e^{2} x^{5} \operatorname {acosh}{\left (c x \right )}}{5} + \frac {4 b d e^{3} x^{7} \operatorname {acosh}{\left (c x \right )}}{7} + \frac {b e^{4} x^{9} \operatorname {acosh}{\left (c x \right )}}{9} - \frac {b d^{4} \sqrt {c^{2} x^{2} - 1}}{c} - \frac {4 b d^{3} e x^{2} \sqrt {c^{2} x^{2} - 1}}{9 c} - \frac {6 b d^{2} e^{2} x^{4} \sqrt {c^{2} x^{2} - 1}}{25 c} - \frac {4 b d e^{3} x^{6} \sqrt {c^{2} x^{2} - 1}}{49 c} - \frac {b e^{4} x^{8} \sqrt {c^{2} x^{2} - 1}}{81 c} - \frac {8 b d^{3} e \sqrt {c^{2} x^{2} - 1}}{9 c^{3}} - \frac {8 b d^{2} e^{2} x^{2} \sqrt {c^{2} x^{2} - 1}}{25 c^{3}} - \frac {24 b d e^{3} x^{4} \sqrt {c^{2} x^{2} - 1}}{245 c^{3}} - \frac {8 b e^{4} x^{6} \sqrt {c^{2} x^{2} - 1}}{567 c^{3}} - \frac {16 b d^{2} e^{2} \sqrt {c^{2} x^{2} - 1}}{25 c^{5}} - \frac {32 b d e^{3} x^{2} \sqrt {c^{2} x^{2} - 1}}{245 c^{5}} - \frac {16 b e^{4} x^{4} \sqrt {c^{2} x^{2} - 1}}{945 c^{5}} - \frac {64 b d e^{3} \sqrt {c^{2} x^{2} - 1}}{245 c^{7}} - \frac {64 b e^{4} x^{2} \sqrt {c^{2} x^{2} - 1}}{2835 c^{7}} - \frac {128 b e^{4} \sqrt {c^{2} x^{2} - 1}}{2835 c^{9}} & \text {for}\: c \neq 0 \\\left (a + \frac {i \pi b}{2}\right ) \left (d^{4} x + \frac {4 d^{3} e x^{3}}{3} + \frac {6 d^{2} e^{2} x^{5}}{5} + \frac {4 d e^{3} x^{7}}{7} + \frac {e^{4} x^{9}}{9}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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