Optimal. Leaf size=230 \[ -\frac {49 b d^3 x \sqrt {-1+c x} \sqrt {1+c x}}{5120 c^3}+\frac {49 b d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}}{7680 c^3}-\frac {49 b d^3 x (-1+c x)^{5/2} (1+c x)^{5/2}}{9600 c^3}+\frac {7 b d^3 x (-1+c x)^{7/2} (1+c x)^{7/2}}{1600 c^3}+\frac {b d^3 x (-1+c x)^{9/2} (1+c x)^{9/2}}{100 c^3}+\frac {49 b d^3 \cosh ^{-1}(c x)}{5120 c^4}-\frac {d^3 (-1+c x)^4 (1+c x)^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}-\frac {d^3 (-1+c x)^5 (1+c x)^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4} \]
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Rubi [A]
time = 0.18, antiderivative size = 328, normalized size of antiderivative = 1.43, number of steps
used = 11, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {272, 45,
5921, 12, 580, 21, 396, 201, 223, 212} \begin {gather*} \frac {d^3 \left (1-c^2 x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}-\frac {d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}+\frac {49 b d^3 \sqrt {c^2 x^2-1} \tanh ^{-1}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{5120 c^4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b d^3 x \left (1-c^2 x^2\right )^5}{100 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {7 b d^3 x \left (1-c^2 x^2\right )^4}{1600 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {49 b d^3 x \left (1-c^2 x^2\right )^3}{9600 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {49 b d^3 x \left (1-c^2 x^2\right )^2}{7680 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {49 b d^3 x \left (1-c^2 x^2\right )}{5120 c^3 \sqrt {c x-1} \sqrt {c x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 21
Rule 45
Rule 201
Rule 212
Rule 223
Rule 272
Rule 396
Rule 580
Rule 5921
Rubi steps
\begin {align*} \int x^3 \left (d-c^2 d x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=-\frac {d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}+\frac {d^3 \left (1-c^2 x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}-(b c) \int \frac {d^3 \left (-1-4 c^2 x^2\right ) \left (1-c^2 x^2\right )^4}{40 c^4 \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}+\frac {d^3 \left (1-c^2 x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}-\frac {\left (b d^3\right ) \int \frac {\left (-1-4 c^2 x^2\right ) \left (1-c^2 x^2\right )^4}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{40 c^3}\\ &=-\frac {d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}+\frac {d^3 \left (1-c^2 x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}-\frac {\left (b d^3 \sqrt {-1+c^2 x^2}\right ) \int \frac {\left (-1-4 c^2 x^2\right ) \left (1-c^2 x^2\right )^4}{\sqrt {-1+c^2 x^2}} \, dx}{40 c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}+\frac {d^3 \left (1-c^2 x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}-\frac {\left (b d^3 \sqrt {-1+c^2 x^2}\right ) \int \left (-1-4 c^2 x^2\right ) \left (-1+c^2 x^2\right )^{7/2} \, dx}{40 c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b d^3 x \left (1-c^2 x^2\right )^5}{100 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}+\frac {d^3 \left (1-c^2 x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}+\frac {\left (7 b d^3 \sqrt {-1+c^2 x^2}\right ) \int \left (-1+c^2 x^2\right )^{7/2} \, dx}{200 c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {7 b d^3 x \left (1-c^2 x^2\right )^4}{1600 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b d^3 x \left (1-c^2 x^2\right )^5}{100 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}+\frac {d^3 \left (1-c^2 x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}-\frac {\left (49 b d^3 \sqrt {-1+c^2 x^2}\right ) \int \left (-1+c^2 x^2\right )^{5/2} \, dx}{1600 c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {49 b d^3 x \left (1-c^2 x^2\right )^3}{9600 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {7 b d^3 x \left (1-c^2 x^2\right )^4}{1600 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b d^3 x \left (1-c^2 x^2\right )^5}{100 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}+\frac {d^3 \left (1-c^2 x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}+\frac {\left (49 b d^3 \sqrt {-1+c^2 x^2}\right ) \int \left (-1+c^2 x^2\right )^{3/2} \, dx}{1920 c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {49 b d^3 x \left (1-c^2 x^2\right )^2}{7680 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {49 b d^3 x \left (1-c^2 x^2\right )^3}{9600 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {7 b d^3 x \left (1-c^2 x^2\right )^4}{1600 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b d^3 x \left (1-c^2 x^2\right )^5}{100 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}+\frac {d^3 \left (1-c^2 x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}-\frac {\left (49 b d^3 \sqrt {-1+c^2 x^2}\right ) \int \sqrt {-1+c^2 x^2} \, dx}{2560 c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {49 b d^3 x \left (1-c^2 x^2\right )}{5120 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {49 b d^3 x \left (1-c^2 x^2\right )^2}{7680 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {49 b d^3 x \left (1-c^2 x^2\right )^3}{9600 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {7 b d^3 x \left (1-c^2 x^2\right )^4}{1600 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b d^3 x \left (1-c^2 x^2\right )^5}{100 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}+\frac {d^3 \left (1-c^2 x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}+\frac {\left (49 b d^3 \sqrt {-1+c^2 x^2}\right ) \int \frac {1}{\sqrt {-1+c^2 x^2}} \, dx}{5120 c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {49 b d^3 x \left (1-c^2 x^2\right )}{5120 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {49 b d^3 x \left (1-c^2 x^2\right )^2}{7680 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {49 b d^3 x \left (1-c^2 x^2\right )^3}{9600 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {7 b d^3 x \left (1-c^2 x^2\right )^4}{1600 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b d^3 x \left (1-c^2 x^2\right )^5}{100 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}+\frac {d^3 \left (1-c^2 x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}+\frac {\left (49 b d^3 \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\frac {x}{\sqrt {-1+c^2 x^2}}\right )}{5120 c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {49 b d^3 x \left (1-c^2 x^2\right )}{5120 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {49 b d^3 x \left (1-c^2 x^2\right )^2}{7680 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {49 b d^3 x \left (1-c^2 x^2\right )^3}{9600 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {7 b d^3 x \left (1-c^2 x^2\right )^4}{1600 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b d^3 x \left (1-c^2 x^2\right )^5}{100 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}+\frac {d^3 \left (1-c^2 x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}+\frac {49 b d^3 \sqrt {-1+c^2 x^2} \tanh ^{-1}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{5120 c^4 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 168, normalized size = 0.73 \begin {gather*} -\frac {d^3 \left (1920 a c^4 x^4 \left (-10+20 c^2 x^2-15 c^4 x^4+4 c^6 x^6\right )+b c x \sqrt {-1+c x} \sqrt {1+c x} \left (1185+790 c^2 x^2-3208 c^4 x^4+2736 c^6 x^6-768 c^8 x^8\right )+1920 b c^4 x^4 \left (-10+20 c^2 x^2-15 c^4 x^4+4 c^6 x^6\right ) \cosh ^{-1}(c x)+1185 b \log \left (c x+\sqrt {-1+c x} \sqrt {1+c x}\right )\right )}{76800 c^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.70, size = 286, normalized size = 1.24
method | result | size |
derivativedivides | \(\frac {-d^{3} a \left (\frac {\left (c^{2} x^{2}-1\right )^{5}}{10}+\frac {\left (c^{2} x^{2}-1\right )^{4}}{8}\right )-\frac {d^{3} b \,\mathrm {arccosh}\left (c x \right ) c^{10} x^{10}}{10}+\frac {3 d^{3} b \,\mathrm {arccosh}\left (c x \right ) c^{8} x^{8}}{8}-\frac {d^{3} b \,\mathrm {arccosh}\left (c x \right ) c^{6} x^{6}}{2}+\frac {d^{3} b \,\mathrm {arccosh}\left (c x \right ) c^{4} x^{4}}{4}-\frac {b \,d^{3} \mathrm {arccosh}\left (c x \right )}{40}+\frac {d^{3} b \sqrt {c x -1}\, \sqrt {c x +1}\, c^{9} x^{9}}{100}-\frac {57 d^{3} b \sqrt {c x -1}\, \sqrt {c x +1}\, c^{7} x^{7}}{1600}+\frac {401 d^{3} b \sqrt {c x -1}\, \sqrt {c x +1}\, c^{5} x^{5}}{9600}-\frac {79 d^{3} b \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}}{7680}-\frac {79 b c \,d^{3} x \sqrt {c x -1}\, \sqrt {c x +1}}{5120}+\frac {49 d^{3} b \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{5120 \sqrt {c^{2} x^{2}-1}}}{c^{4}}\) | \(286\) |
default | \(\frac {-d^{3} a \left (\frac {\left (c^{2} x^{2}-1\right )^{5}}{10}+\frac {\left (c^{2} x^{2}-1\right )^{4}}{8}\right )-\frac {d^{3} b \,\mathrm {arccosh}\left (c x \right ) c^{10} x^{10}}{10}+\frac {3 d^{3} b \,\mathrm {arccosh}\left (c x \right ) c^{8} x^{8}}{8}-\frac {d^{3} b \,\mathrm {arccosh}\left (c x \right ) c^{6} x^{6}}{2}+\frac {d^{3} b \,\mathrm {arccosh}\left (c x \right ) c^{4} x^{4}}{4}-\frac {b \,d^{3} \mathrm {arccosh}\left (c x \right )}{40}+\frac {d^{3} b \sqrt {c x -1}\, \sqrt {c x +1}\, c^{9} x^{9}}{100}-\frac {57 d^{3} b \sqrt {c x -1}\, \sqrt {c x +1}\, c^{7} x^{7}}{1600}+\frac {401 d^{3} b \sqrt {c x -1}\, \sqrt {c x +1}\, c^{5} x^{5}}{9600}-\frac {79 d^{3} b \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}}{7680}-\frac {79 b c \,d^{3} x \sqrt {c x -1}\, \sqrt {c x +1}}{5120}+\frac {49 d^{3} b \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{5120 \sqrt {c^{2} x^{2}-1}}}{c^{4}}\) | \(286\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 501 vs.
\(2 (194) = 388\).
time = 0.27, size = 501, normalized size = 2.18 \begin {gather*} -\frac {1}{10} \, a c^{6} d^{3} x^{10} + \frac {3}{8} \, a c^{4} d^{3} x^{8} - \frac {1}{2} \, a c^{2} d^{3} x^{6} - \frac {1}{12800} \, {\left (1280 \, x^{10} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {128 \, \sqrt {c^{2} x^{2} - 1} x^{9}}{c^{2}} + \frac {144 \, \sqrt {c^{2} x^{2} - 1} x^{7}}{c^{4}} + \frac {168 \, \sqrt {c^{2} x^{2} - 1} x^{5}}{c^{6}} + \frac {210 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{8}} + \frac {315 \, \sqrt {c^{2} x^{2} - 1} x}{c^{10}} + \frac {315 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{11}}\right )} c\right )} b c^{6} d^{3} + \frac {1}{1024} \, {\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 c^{4} d^{3} + \frac {1}{4} \, a d^{3} x^{4} - \frac {1}{96} \, {\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 c^{2} d^{3} + \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^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 197, normalized size = 0.86 \begin {gather*} -\frac {7680 \, a c^{10} d^{3} x^{10} - 28800 \, a c^{8} d^{3} x^{8} + 38400 \, a c^{6} d^{3} x^{6} - 19200 \, a c^{4} d^{3} x^{4} + 15 \, {\left (512 \, b c^{10} d^{3} x^{10} - 1920 \, b c^{8} d^{3} x^{8} + 2560 \, b c^{6} d^{3} x^{6} - 1280 \, b c^{4} d^{3} x^{4} + 79 \, b d^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (768 \, b c^{9} d^{3} x^{9} - 2736 \, b c^{7} d^{3} x^{7} + 3208 \, b c^{5} d^{3} x^{5} - 790 \, b c^{3} d^{3} x^{3} - 1185 \, b c d^{3} x\right )} \sqrt {c^{2} x^{2} - 1}}{76800 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 2.07, size = 287, normalized size = 1.25 \begin {gather*} \begin {cases} - \frac {a c^{6} d^{3} x^{10}}{10} + \frac {3 a c^{4} d^{3} x^{8}}{8} - \frac {a c^{2} d^{3} x^{6}}{2} + \frac {a d^{3} x^{4}}{4} - \frac {b c^{6} d^{3} x^{10} \operatorname {acosh}{\left (c x \right )}}{10} + \frac {b c^{5} d^{3} x^{9} \sqrt {c^{2} x^{2} - 1}}{100} + \frac {3 b c^{4} d^{3} x^{8} \operatorname {acosh}{\left (c x \right )}}{8} - \frac {57 b c^{3} d^{3} x^{7} \sqrt {c^{2} x^{2} - 1}}{1600} - \frac {b c^{2} d^{3} x^{6} \operatorname {acosh}{\left (c x \right )}}{2} + \frac {401 b c d^{3} x^{5} \sqrt {c^{2} x^{2} - 1}}{9600} + \frac {b d^{3} x^{4} \operatorname {acosh}{\left (c x \right )}}{4} - \frac {79 b d^{3} x^{3} \sqrt {c^{2} x^{2} - 1}}{7680 c} - \frac {79 b d^{3} x \sqrt {c^{2} x^{2} - 1}}{5120 c^{3}} - \frac {79 b d^{3} \operatorname {acosh}{\left (c x \right )}}{5120 c^{4}} & \text {for}\: c \neq 0 \\\frac {d^{3} x^{4} \left (a + \frac {i \pi b}{2}\right )}{4} & \text {otherwise} \end {cases} \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int x^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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