Optimal. Leaf size=173 \[ -\frac {29 c^4 \sin ^{-1}(a x)}{128 a^4}-\frac {19 c^4 x^2 \left (1-a^2 x^2\right )^{3/2}}{35 a^2}+\frac {1}{8} a c^4 x^5 \left (1-a^2 x^2\right )^{3/2}-\frac {3}{7} c^4 x^4 \left (1-a^2 x^2\right )^{3/2}+\frac {29 c^4 x^3 \left (1-a^2 x^2\right )^{3/2}}{48 a}-\frac {c^4 (2432-3045 a x) \left (1-a^2 x^2\right )^{3/2}}{6720 a^4}-\frac {29 c^4 x \sqrt {1-a^2 x^2}}{128 a^3} \]
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Rubi [A] time = 0.33, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6128, 1809, 833, 780, 195, 216} \[ \frac {1}{8} a c^4 x^5 \left (1-a^2 x^2\right )^{3/2}-\frac {3}{7} c^4 x^4 \left (1-a^2 x^2\right )^{3/2}+\frac {29 c^4 x^3 \left (1-a^2 x^2\right )^{3/2}}{48 a}-\frac {19 c^4 x^2 \left (1-a^2 x^2\right )^{3/2}}{35 a^2}-\frac {29 c^4 x \sqrt {1-a^2 x^2}}{128 a^3}-\frac {c^4 (2432-3045 a x) \left (1-a^2 x^2\right )^{3/2}}{6720 a^4}-\frac {29 c^4 \sin ^{-1}(a x)}{128 a^4} \]
Antiderivative was successfully verified.
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Rule 195
Rule 216
Rule 780
Rule 833
Rule 1809
Rule 6128
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} x^3 (c-a c x)^4 \, dx &=c \int x^3 (c-a c x)^3 \sqrt {1-a^2 x^2} \, dx\\ &=\frac {1}{8} a c^4 x^5 \left (1-a^2 x^2\right )^{3/2}-\frac {c \int x^3 \sqrt {1-a^2 x^2} \left (-8 a^2 c^3+29 a^3 c^3 x-24 a^4 c^3 x^2\right ) \, dx}{8 a^2}\\ &=-\frac {3}{7} c^4 x^4 \left (1-a^2 x^2\right )^{3/2}+\frac {1}{8} a c^4 x^5 \left (1-a^2 x^2\right )^{3/2}+\frac {c \int x^3 \left (152 a^4 c^3-203 a^5 c^3 x\right ) \sqrt {1-a^2 x^2} \, dx}{56 a^4}\\ &=\frac {29 c^4 x^3 \left (1-a^2 x^2\right )^{3/2}}{48 a}-\frac {3}{7} c^4 x^4 \left (1-a^2 x^2\right )^{3/2}+\frac {1}{8} a c^4 x^5 \left (1-a^2 x^2\right )^{3/2}-\frac {c \int x^2 \left (609 a^5 c^3-912 a^6 c^3 x\right ) \sqrt {1-a^2 x^2} \, dx}{336 a^6}\\ &=-\frac {19 c^4 x^2 \left (1-a^2 x^2\right )^{3/2}}{35 a^2}+\frac {29 c^4 x^3 \left (1-a^2 x^2\right )^{3/2}}{48 a}-\frac {3}{7} c^4 x^4 \left (1-a^2 x^2\right )^{3/2}+\frac {1}{8} a c^4 x^5 \left (1-a^2 x^2\right )^{3/2}+\frac {c \int x \left (1824 a^6 c^3-3045 a^7 c^3 x\right ) \sqrt {1-a^2 x^2} \, dx}{1680 a^8}\\ &=-\frac {19 c^4 x^2 \left (1-a^2 x^2\right )^{3/2}}{35 a^2}+\frac {29 c^4 x^3 \left (1-a^2 x^2\right )^{3/2}}{48 a}-\frac {3}{7} c^4 x^4 \left (1-a^2 x^2\right )^{3/2}+\frac {1}{8} a c^4 x^5 \left (1-a^2 x^2\right )^{3/2}-\frac {c^4 (2432-3045 a x) \left (1-a^2 x^2\right )^{3/2}}{6720 a^4}-\frac {\left (29 c^4\right ) \int \sqrt {1-a^2 x^2} \, dx}{64 a^3}\\ &=-\frac {29 c^4 x \sqrt {1-a^2 x^2}}{128 a^3}-\frac {19 c^4 x^2 \left (1-a^2 x^2\right )^{3/2}}{35 a^2}+\frac {29 c^4 x^3 \left (1-a^2 x^2\right )^{3/2}}{48 a}-\frac {3}{7} c^4 x^4 \left (1-a^2 x^2\right )^{3/2}+\frac {1}{8} a c^4 x^5 \left (1-a^2 x^2\right )^{3/2}-\frac {c^4 (2432-3045 a x) \left (1-a^2 x^2\right )^{3/2}}{6720 a^4}-\frac {\left (29 c^4\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{128 a^3}\\ &=-\frac {29 c^4 x \sqrt {1-a^2 x^2}}{128 a^3}-\frac {19 c^4 x^2 \left (1-a^2 x^2\right )^{3/2}}{35 a^2}+\frac {29 c^4 x^3 \left (1-a^2 x^2\right )^{3/2}}{48 a}-\frac {3}{7} c^4 x^4 \left (1-a^2 x^2\right )^{3/2}+\frac {1}{8} a c^4 x^5 \left (1-a^2 x^2\right )^{3/2}-\frac {c^4 (2432-3045 a x) \left (1-a^2 x^2\right )^{3/2}}{6720 a^4}-\frac {29 c^4 \sin ^{-1}(a x)}{128 a^4}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 99, normalized size = 0.57 \[ -\frac {c^4 \left (\sqrt {1-a^2 x^2} \left (1680 a^7 x^7-5760 a^6 x^6+6440 a^5 x^5-1536 a^4 x^4-2030 a^3 x^3+2432 a^2 x^2-3045 a x+4864\right )-6090 \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{13440 a^4} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.44, size = 126, normalized size = 0.73 \[ \frac {6090 \, c^{4} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (1680 \, a^{7} c^{4} x^{7} - 5760 \, a^{6} c^{4} x^{6} + 6440 \, a^{5} c^{4} x^{5} - 1536 \, a^{4} c^{4} x^{4} - 2030 \, a^{3} c^{4} x^{3} + 2432 \, a^{2} c^{4} x^{2} - 3045 \, a c^{4} x + 4864 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1}}{13440 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 117, normalized size = 0.68 \[ -\frac {29 \, c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{128 \, a^{3} {\left | a \right |}} - \frac {1}{13440} \, \sqrt {-a^{2} x^{2} + 1} {\left ({\left (2 \, {\left (\frac {1216 \, c^{4}}{a^{2}} - {\left (\frac {1015 \, c^{4}}{a} + 4 \, {\left (192 \, c^{4} - 5 \, {\left (161 \, a c^{4} + 6 \, {\left (7 \, a^{3} c^{4} x - 24 \, a^{2} c^{4}\right )} x\right )} x\right )} x\right )} x\right )} x - \frac {3045 \, c^{4}}{a^{3}}\right )} x + \frac {4864 \, c^{4}}{a^{4}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 209, normalized size = 1.21 \[ -\frac {c^{4} a^{3} x^{7} \sqrt {-a^{2} x^{2}+1}}{8}-\frac {23 c^{4} a \,x^{5} \sqrt {-a^{2} x^{2}+1}}{48}+\frac {29 c^{4} x^{3} \sqrt {-a^{2} x^{2}+1}}{192 a}+\frac {29 c^{4} x \sqrt {-a^{2} x^{2}+1}}{128 a^{3}}-\frac {29 c^{4} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{128 a^{3} \sqrt {a^{2}}}+\frac {3 c^{4} a^{2} x^{6} \sqrt {-a^{2} x^{2}+1}}{7}+\frac {4 c^{4} x^{4} \sqrt {-a^{2} x^{2}+1}}{35}-\frac {19 c^{4} x^{2} \sqrt {-a^{2} x^{2}+1}}{105 a^{2}}-\frac {38 c^{4} \sqrt {-a^{2} x^{2}+1}}{105 a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 187, normalized size = 1.08 \[ -\frac {1}{8} \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{4} x^{7} + \frac {3}{7} \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{4} x^{6} - \frac {23}{48} \, \sqrt {-a^{2} x^{2} + 1} a c^{4} x^{5} + \frac {4}{35} \, \sqrt {-a^{2} x^{2} + 1} c^{4} x^{4} + \frac {29 \, \sqrt {-a^{2} x^{2} + 1} c^{4} x^{3}}{192 \, a} - \frac {19 \, \sqrt {-a^{2} x^{2} + 1} c^{4} x^{2}}{105 \, a^{2}} + \frac {29 \, \sqrt {-a^{2} x^{2} + 1} c^{4} x}{128 \, a^{3}} - \frac {29 \, c^{4} \arcsin \left (a x\right )}{128 \, a^{4}} - \frac {38 \, \sqrt {-a^{2} x^{2} + 1} c^{4}}{105 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 200, normalized size = 1.16 \[ \frac {4\,c^4\,x^4\,\sqrt {1-a^2\,x^2}}{35}-\frac {38\,c^4\,\sqrt {1-a^2\,x^2}}{105\,a^4}+\frac {29\,c^4\,x\,\sqrt {1-a^2\,x^2}}{128\,a^3}-\frac {23\,a\,c^4\,x^5\,\sqrt {1-a^2\,x^2}}{48}-\frac {29\,c^4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{128\,a^3\,\sqrt {-a^2}}+\frac {29\,c^4\,x^3\,\sqrt {1-a^2\,x^2}}{192\,a}-\frac {19\,c^4\,x^2\,\sqrt {1-a^2\,x^2}}{105\,a^2}+\frac {3\,a^2\,c^4\,x^6\,\sqrt {1-a^2\,x^2}}{7}-\frac {a^3\,c^4\,x^7\,\sqrt {1-a^2\,x^2}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 18.31, size = 842, normalized size = 4.87 \[ a^{5} c^{4} \left (\begin {cases} - \frac {i x^{9}}{8 \sqrt {a^{2} x^{2} - 1}} - \frac {i x^{7}}{48 a^{2} \sqrt {a^{2} x^{2} - 1}} - \frac {7 i x^{5}}{192 a^{4} \sqrt {a^{2} x^{2} - 1}} - \frac {35 i x^{3}}{384 a^{6} \sqrt {a^{2} x^{2} - 1}} + \frac {35 i x}{128 a^{8} \sqrt {a^{2} x^{2} - 1}} - \frac {35 i \operatorname {acosh}{\left (a x \right )}}{128 a^{9}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{9}}{8 \sqrt {- a^{2} x^{2} + 1}} + \frac {x^{7}}{48 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {7 x^{5}}{192 a^{4} \sqrt {- a^{2} x^{2} + 1}} + \frac {35 x^{3}}{384 a^{6} \sqrt {- a^{2} x^{2} + 1}} - \frac {35 x}{128 a^{8} \sqrt {- a^{2} x^{2} + 1}} + \frac {35 \operatorname {asin}{\left (a x \right )}}{128 a^{9}} & \text {otherwise} \end {cases}\right ) - 3 a^{4} c^{4} \left (\begin {cases} - \frac {x^{6} \sqrt {- a^{2} x^{2} + 1}}{7 a^{2}} - \frac {6 x^{4} \sqrt {- a^{2} x^{2} + 1}}{35 a^{4}} - \frac {8 x^{2} \sqrt {- a^{2} x^{2} + 1}}{35 a^{6}} - \frac {16 \sqrt {- a^{2} x^{2} + 1}}{35 a^{8}} & \text {for}\: a \neq 0 \\\frac {x^{8}}{8} & \text {otherwise} \end {cases}\right ) + 2 a^{3} c^{4} \left (\begin {cases} - \frac {i x^{7}}{6 \sqrt {a^{2} x^{2} - 1}} - \frac {i x^{5}}{24 a^{2} \sqrt {a^{2} x^{2} - 1}} - \frac {5 i x^{3}}{48 a^{4} \sqrt {a^{2} x^{2} - 1}} + \frac {5 i x}{16 a^{6} \sqrt {a^{2} x^{2} - 1}} - \frac {5 i \operatorname {acosh}{\left (a x \right )}}{16 a^{7}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{7}}{6 \sqrt {- a^{2} x^{2} + 1}} + \frac {x^{5}}{24 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {5 x^{3}}{48 a^{4} \sqrt {- a^{2} x^{2} + 1}} - \frac {5 x}{16 a^{6} \sqrt {- a^{2} x^{2} + 1}} + \frac {5 \operatorname {asin}{\left (a x \right )}}{16 a^{7}} & \text {otherwise} \end {cases}\right ) + 2 a^{2} c^{4} \left (\begin {cases} - \frac {x^{4} \sqrt {- a^{2} x^{2} + 1}}{5 a^{2}} - \frac {4 x^{2} \sqrt {- a^{2} x^{2} + 1}}{15 a^{4}} - \frac {8 \sqrt {- a^{2} x^{2} + 1}}{15 a^{6}} & \text {for}\: a \neq 0 \\\frac {x^{6}}{6} & \text {otherwise} \end {cases}\right ) - 3 a c^{4} \left (\begin {cases} - \frac {i x^{5}}{4 \sqrt {a^{2} x^{2} - 1}} - \frac {i x^{3}}{8 a^{2} \sqrt {a^{2} x^{2} - 1}} + \frac {3 i x}{8 a^{4} \sqrt {a^{2} x^{2} - 1}} - \frac {3 i \operatorname {acosh}{\left (a x \right )}}{8 a^{5}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{5}}{4 \sqrt {- a^{2} x^{2} + 1}} + \frac {x^{3}}{8 a^{2} \sqrt {- a^{2} x^{2} + 1}} - \frac {3 x}{8 a^{4} \sqrt {- a^{2} x^{2} + 1}} + \frac {3 \operatorname {asin}{\left (a x \right )}}{8 a^{5}} & \text {otherwise} \end {cases}\right ) + c^{4} \left (\begin {cases} - \frac {x^{2} \sqrt {- a^{2} x^{2} + 1}}{3 a^{2}} - \frac {2 \sqrt {- a^{2} x^{2} + 1}}{3 a^{4}} & \text {for}\: a \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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