Optimal. Leaf size=230 \[ -\frac {d^3 (c x-1)^5 (c x+1)^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}-\frac {d^3 (c x-1)^4 (c x+1)^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}+\frac {49 b d^3 \cosh ^{-1}(c x)}{5120 c^4}+\frac {b d^3 x (c x-1)^{9/2} (c x+1)^{9/2}}{100 c^3}+\frac {7 b d^3 x (c x-1)^{7/2} (c x+1)^{7/2}}{1600 c^3}-\frac {49 b d^3 x (c x-1)^{5/2} (c x+1)^{5/2}}{9600 c^3}+\frac {49 b d^3 x (c x-1)^{3/2} (c x+1)^{3/2}}{7680 c^3}-\frac {49 b d^3 x \sqrt {c x-1} \sqrt {c x+1}}{5120 c^3} \]
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Rubi [A] time = 0.28, 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 = {266, 43, 5731, 12, 566, 21, 388, 195, 217, 206} \[ \frac {d^3 \left (1-c^2 x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}-\frac {d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}-\frac {b d^3 x \left (1-c^2 x^2\right )^5}{100 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {7 b d^3 x \left (1-c^2 x^2\right )^4}{1600 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {49 b d^3 x \left (1-c^2 x^2\right )^3}{9600 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {49 b d^3 x \left (1-c^2 x^2\right )^2}{7680 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {49 b d^3 x \left (1-c^2 x^2\right )}{5120 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {49 b d^3 \sqrt {c^2 x^2-1} \tanh ^{-1}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{5120 c^4 \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 21
Rule 43
Rule 195
Rule 206
Rule 217
Rule 266
Rule 388
Rule 566
Rule 5731
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 ) \operatorname {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 = 0.37, size = 162, normalized size = 0.70 \[ -\frac {d^3 \left (1920 a c^4 x^4 \left (4 c^6 x^6-15 c^4 x^4+20 c^2 x^2-10\right )+1920 b c^4 x^4 \left (4 c^6 x^6-15 c^4 x^4+20 c^2 x^2-10\right ) \cosh ^{-1}(c x)+b c x \sqrt {c x-1} \sqrt {c x+1} \left (-768 c^8 x^8+2736 c^6 x^6-3208 c^4 x^4+790 c^2 x^2+1185\right )+2370 b \tanh ^{-1}\left (\sqrt {\frac {c x-1}{c x+1}}\right )\right )}{76800 c^4} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.59, size = 197, normalized size = 0.86 \[ -\frac {7680 \, a c^{10} d^{3} x^{10} - 28800 \, a c^{8} d^{3} x^{8} + 38400 \, a c^{6} d^{3} x^{6} - 19200 \, a c^{4} d^{3} x^{4} + 15 \, {\left (512 \, b c^{10} d^{3} x^{10} - 1920 \, b c^{8} d^{3} x^{8} + 2560 \, b c^{6} d^{3} x^{6} - 1280 \, b c^{4} d^{3} x^{4} + 79 \, b d^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (768 \, b c^{9} d^{3} x^{9} - 2736 \, b c^{7} d^{3} x^{7} + 3208 \, b c^{5} d^{3} x^{5} - 790 \, b c^{3} d^{3} x^{3} - 1185 \, b c d^{3} x\right )} \sqrt {c^{2} x^{2} - 1}}{76800 \, c^{4}} \]
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 = 284, normalized size = 1.23 \[ -\frac {c^{6} d^{3} a \,x^{10}}{10}+\frac {3 c^{4} d^{3} a \,x^{8}}{8}-\frac {c^{2} d^{3} a \,x^{6}}{2}+\frac {d^{3} a \,x^{4}}{4}-\frac {c^{6} d^{3} b \,\mathrm {arccosh}\left (c x \right ) x^{10}}{10}+\frac {3 c^{4} d^{3} b \,\mathrm {arccosh}\left (c x \right ) x^{8}}{8}-\frac {c^{2} d^{3} b \,\mathrm {arccosh}\left (c x \right ) x^{6}}{2}+\frac {d^{3} b \,\mathrm {arccosh}\left (c x \right ) x^{4}}{4}+\frac {c^{5} d^{3} b \sqrt {c x -1}\, \sqrt {c x +1}\, x^{9}}{100}-\frac {57 c^{3} d^{3} b \sqrt {c x -1}\, \sqrt {c x +1}\, x^{7}}{1600}+\frac {401 c \,d^{3} b \sqrt {c x -1}\, \sqrt {c x +1}\, x^{5}}{9600}-\frac {79 d^{3} b \sqrt {c x -1}\, \sqrt {c x +1}\, x^{3}}{7680 c}-\frac {79 b \,d^{3} x \sqrt {c x -1}\, \sqrt {c x +1}}{5120 c^{3}}-\frac {79 d^{3} b \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{5120 c^{4} \sqrt {c^{2} x^{2}-1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.37, size = 501, normalized size = 2.18 \[ -\frac {1}{10} \, a c^{6} d^{3} x^{10} + \frac {3}{8} \, a c^{4} d^{3} x^{8} - \frac {1}{2} \, a c^{2} d^{3} x^{6} - \frac {1}{12800} \, {\left (1280 \, x^{10} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {128 \, \sqrt {c^{2} x^{2} - 1} x^{9}}{c^{2}} + \frac {144 \, \sqrt {c^{2} x^{2} - 1} x^{7}}{c^{4}} + \frac {168 \, \sqrt {c^{2} x^{2} - 1} x^{5}}{c^{6}} + \frac {210 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{8}} + \frac {315 \, \sqrt {c^{2} x^{2} - 1} x}{c^{10}} + \frac {315 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{11}}\right )} c\right )} b c^{6} d^{3} + \frac {1}{1024} \, {\left (384 \, x^{8} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {48 \, \sqrt {c^{2} x^{2} - 1} x^{7}}{c^{2}} + \frac {56 \, \sqrt {c^{2} x^{2} - 1} x^{5}}{c^{4}} + \frac {70 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{6}} + \frac {105 \, \sqrt {c^{2} x^{2} - 1} x}{c^{8}} + \frac {105 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{9}}\right )} c\right )} b c^{4} d^{3} + \frac {1}{4} \, a d^{3} x^{4} - \frac {1}{96} \, {\left (48 \, x^{6} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {c^{2} x^{2} - 1} x}{c^{6}} + \frac {15 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{7}}\right )} c\right )} b c^{2} d^{3} + \frac {1}{32} \, {\left (8 \, x^{4} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {c^{2} x^{2} - 1} x}{c^{4}} + \frac {3 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{5}}\right )} c\right )} b d^{3} \]
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 \[ \int x^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 28.39, size = 287, normalized size = 1.25 \[ \begin {cases} - \frac {a c^{6} d^{3} x^{10}}{10} + \frac {3 a c^{4} d^{3} x^{8}}{8} - \frac {a c^{2} d^{3} x^{6}}{2} + \frac {a d^{3} x^{4}}{4} - \frac {b c^{6} d^{3} x^{10} \operatorname {acosh}{\left (c x \right )}}{10} + \frac {b c^{5} d^{3} x^{9} \sqrt {c^{2} x^{2} - 1}}{100} + \frac {3 b c^{4} d^{3} x^{8} \operatorname {acosh}{\left (c x \right )}}{8} - \frac {57 b c^{3} d^{3} x^{7} \sqrt {c^{2} x^{2} - 1}}{1600} - \frac {b c^{2} d^{3} x^{6} \operatorname {acosh}{\left (c x \right )}}{2} + \frac {401 b c d^{3} x^{5} \sqrt {c^{2} x^{2} - 1}}{9600} + \frac {b d^{3} x^{4} \operatorname {acosh}{\left (c x \right )}}{4} - \frac {79 b d^{3} x^{3} \sqrt {c^{2} x^{2} - 1}}{7680 c} - \frac {79 b d^{3} x \sqrt {c^{2} x^{2} - 1}}{5120 c^{3}} - \frac {79 b d^{3} \operatorname {acosh}{\left (c x \right )}}{5120 c^{4}} & \text {for}\: c \neq 0 \\\frac {d^{3} x^{4} \left (a + \frac {i \pi b}{2}\right )}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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