Optimal. Leaf size=146 \[ \frac {5 c^4 \sin ^{-1}(a x)}{16 a^3}+\frac {5 c^4 x^2 \left (1-a^2 x^2\right )^{3/2}}{7 a}+\frac {5 c^4 x \sqrt {1-a^2 x^2}}{16 a^2}+\frac {1}{7} a c^4 x^4 \left (1-a^2 x^2\right )^{3/2}-\frac {1}{2} c^4 x^3 \left (1-a^2 x^2\right )^{3/2}+\frac {5 c^4 (16-21 a x) \left (1-a^2 x^2\right )^{3/2}}{168 a^3} \]
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Rubi [A] time = 0.30, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \[ \frac {1}{7} a c^4 x^4 \left (1-a^2 x^2\right )^{3/2}-\frac {1}{2} c^4 x^3 \left (1-a^2 x^2\right )^{3/2}+\frac {5 c^4 x^2 \left (1-a^2 x^2\right )^{3/2}}{7 a}+\frac {5 c^4 (16-21 a x) \left (1-a^2 x^2\right )^{3/2}}{168 a^3}+\frac {5 c^4 x \sqrt {1-a^2 x^2}}{16 a^2}+\frac {5 c^4 \sin ^{-1}(a x)}{16 a^3} \]
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^2 (c-a c x)^4 \, dx &=c \int x^2 (c-a c x)^3 \sqrt {1-a^2 x^2} \, dx\\ &=\frac {1}{7} a c^4 x^4 \left (1-a^2 x^2\right )^{3/2}-\frac {c \int x^2 \sqrt {1-a^2 x^2} \left (-7 a^2 c^3+25 a^3 c^3 x-21 a^4 c^3 x^2\right ) \, dx}{7 a^2}\\ &=-\frac {1}{2} c^4 x^3 \left (1-a^2 x^2\right )^{3/2}+\frac {1}{7} a c^4 x^4 \left (1-a^2 x^2\right )^{3/2}+\frac {c \int x^2 \left (105 a^4 c^3-150 a^5 c^3 x\right ) \sqrt {1-a^2 x^2} \, dx}{42 a^4}\\ &=\frac {5 c^4 x^2 \left (1-a^2 x^2\right )^{3/2}}{7 a}-\frac {1}{2} c^4 x^3 \left (1-a^2 x^2\right )^{3/2}+\frac {1}{7} a c^4 x^4 \left (1-a^2 x^2\right )^{3/2}-\frac {c \int x \left (300 a^5 c^3-525 a^6 c^3 x\right ) \sqrt {1-a^2 x^2} \, dx}{210 a^6}\\ &=\frac {5 c^4 x^2 \left (1-a^2 x^2\right )^{3/2}}{7 a}-\frac {1}{2} c^4 x^3 \left (1-a^2 x^2\right )^{3/2}+\frac {1}{7} a c^4 x^4 \left (1-a^2 x^2\right )^{3/2}+\frac {5 c^4 (16-21 a x) \left (1-a^2 x^2\right )^{3/2}}{168 a^3}+\frac {\left (5 c^4\right ) \int \sqrt {1-a^2 x^2} \, dx}{8 a^2}\\ &=\frac {5 c^4 x \sqrt {1-a^2 x^2}}{16 a^2}+\frac {5 c^4 x^2 \left (1-a^2 x^2\right )^{3/2}}{7 a}-\frac {1}{2} c^4 x^3 \left (1-a^2 x^2\right )^{3/2}+\frac {1}{7} a c^4 x^4 \left (1-a^2 x^2\right )^{3/2}+\frac {5 c^4 (16-21 a x) \left (1-a^2 x^2\right )^{3/2}}{168 a^3}+\frac {\left (5 c^4\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{16 a^2}\\ &=\frac {5 c^4 x \sqrt {1-a^2 x^2}}{16 a^2}+\frac {5 c^4 x^2 \left (1-a^2 x^2\right )^{3/2}}{7 a}-\frac {1}{2} c^4 x^3 \left (1-a^2 x^2\right )^{3/2}+\frac {1}{7} a c^4 x^4 \left (1-a^2 x^2\right )^{3/2}+\frac {5 c^4 (16-21 a x) \left (1-a^2 x^2\right )^{3/2}}{168 a^3}+\frac {5 c^4 \sin ^{-1}(a x)}{16 a^3}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 91, normalized size = 0.62 \[ -\frac {c^4 \left (\sqrt {1-a^2 x^2} \left (48 a^6 x^6-168 a^5 x^5+192 a^4 x^4-42 a^3 x^3-80 a^2 x^2+105 a x-160\right )+210 \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{336 a^3} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.41, size = 114, normalized size = 0.78 \[ -\frac {210 \, c^{4} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (48 \, a^{6} c^{4} x^{6} - 168 \, a^{5} c^{4} x^{5} + 192 \, a^{4} c^{4} x^{4} - 42 \, a^{3} c^{4} x^{3} - 80 \, a^{2} c^{4} x^{2} + 105 \, a c^{4} x - 160 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1}}{336 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 104, normalized size = 0.71 \[ \frac {5 \, c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{16 \, a^{2} {\left | a \right |}} - \frac {1}{336} \, \sqrt {-a^{2} x^{2} + 1} {\left ({\left (\frac {105 \, c^{4}}{a^{2}} - 2 \, {\left (\frac {40 \, c^{4}}{a} + 3 \, {\left (7 \, c^{4} - 4 \, {\left (8 \, a c^{4} + {\left (2 \, a^{3} c^{4} x - 7 \, a^{2} c^{4}\right )} x\right )} x\right )} x\right )} x\right )} x - \frac {160 \, c^{4}}{a^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 186, normalized size = 1.27 \[ -\frac {c^{4} a^{3} x^{6} \sqrt {-a^{2} x^{2}+1}}{7}-\frac {4 c^{4} a \,x^{4} \sqrt {-a^{2} x^{2}+1}}{7}+\frac {5 c^{4} x^{2} \sqrt {-a^{2} x^{2}+1}}{21 a}+\frac {10 c^{4} \sqrt {-a^{2} x^{2}+1}}{21 a^{3}}+\frac {c^{4} a^{2} x^{5} \sqrt {-a^{2} x^{2}+1}}{2}+\frac {c^{4} x^{3} \sqrt {-a^{2} x^{2}+1}}{8}-\frac {5 c^{4} x \sqrt {-a^{2} x^{2}+1}}{16 a^{2}}+\frac {5 c^{4} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{16 a^{2} \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 164, normalized size = 1.12 \[ -\frac {1}{7} \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{4} x^{6} + \frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{4} x^{5} - \frac {4}{7} \, \sqrt {-a^{2} x^{2} + 1} a c^{4} x^{4} + \frac {1}{8} \, \sqrt {-a^{2} x^{2} + 1} c^{4} x^{3} + \frac {5 \, \sqrt {-a^{2} x^{2} + 1} c^{4} x^{2}}{21 \, a} - \frac {5 \, \sqrt {-a^{2} x^{2} + 1} c^{4} x}{16 \, a^{2}} + \frac {5 \, c^{4} \arcsin \left (a x\right )}{16 \, a^{3}} + \frac {10 \, \sqrt {-a^{2} x^{2} + 1} c^{4}}{21 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.79, size = 177, normalized size = 1.21 \[ \frac {10\,c^4\,\sqrt {1-a^2\,x^2}}{21\,a^3}+\frac {c^4\,x^3\,\sqrt {1-a^2\,x^2}}{8}-\frac {5\,c^4\,x\,\sqrt {1-a^2\,x^2}}{16\,a^2}-\frac {4\,a\,c^4\,x^4\,\sqrt {1-a^2\,x^2}}{7}+\frac {5\,c^4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{16\,a^2\,\sqrt {-a^2}}+\frac {5\,c^4\,x^2\,\sqrt {1-a^2\,x^2}}{21\,a}+\frac {a^2\,c^4\,x^5\,\sqrt {1-a^2\,x^2}}{2}-\frac {a^3\,c^4\,x^6\,\sqrt {1-a^2\,x^2}}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 13.19, size = 683, normalized size = 4.68 \[ a^{5} 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 ) - 3 a^{4} 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^{3} 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 ) + 2 a^{2} 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 ) - 3 a 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 ) + c^{4} \left (\begin {cases} - \frac {i x \sqrt {a^{2} x^{2} - 1}}{2 a^{2}} - \frac {i \operatorname {acosh}{\left (a x \right )}}{2 a^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{3}}{2 \sqrt {- a^{2} x^{2} + 1}} - \frac {x}{2 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {\operatorname {asin}{\left (a x \right )}}{2 a^{3}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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