Optimal. Leaf size=158 \[ -\frac {c^4 (1-a x)^3 \left (1-a^2 x^2\right )^{3/2}}{6 a^2}-\frac {c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{10 a^2}-\frac {7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{40 a^2}-\frac {7 c^4 \left (1-a^2 x^2\right )^{3/2}}{24 a^2}-\frac {7 c^4 x \sqrt {1-a^2 x^2}}{16 a}-\frac {7 c^4 \sin ^{-1}(a x)}{16 a^2} \]
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Rubi [A] time = 0.14, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {6128, 795, 671, 641, 195, 216} \[ -\frac {c^4 (1-a x)^3 \left (1-a^2 x^2\right )^{3/2}}{6 a^2}-\frac {c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{10 a^2}-\frac {7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{40 a^2}-\frac {7 c^4 \left (1-a^2 x^2\right )^{3/2}}{24 a^2}-\frac {7 c^4 x \sqrt {1-a^2 x^2}}{16 a}-\frac {7 c^4 \sin ^{-1}(a x)}{16 a^2} \]
Antiderivative was successfully verified.
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Rule 195
Rule 216
Rule 641
Rule 671
Rule 795
Rule 6128
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} x (c-a c x)^4 \, dx &=c \int x (c-a c x)^3 \sqrt {1-a^2 x^2} \, dx\\ &=-\frac {c^4 (1-a x)^3 \left (1-a^2 x^2\right )^{3/2}}{6 a^2}-\frac {c \int (c-a c x)^3 \sqrt {1-a^2 x^2} \, dx}{2 a}\\ &=-\frac {c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{10 a^2}-\frac {c^4 (1-a x)^3 \left (1-a^2 x^2\right )^{3/2}}{6 a^2}-\frac {\left (7 c^2\right ) \int (c-a c x)^2 \sqrt {1-a^2 x^2} \, dx}{10 a}\\ &=-\frac {7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{40 a^2}-\frac {c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{10 a^2}-\frac {c^4 (1-a x)^3 \left (1-a^2 x^2\right )^{3/2}}{6 a^2}-\frac {\left (7 c^3\right ) \int (c-a c x) \sqrt {1-a^2 x^2} \, dx}{8 a}\\ &=-\frac {7 c^4 \left (1-a^2 x^2\right )^{3/2}}{24 a^2}-\frac {7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{40 a^2}-\frac {c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{10 a^2}-\frac {c^4 (1-a x)^3 \left (1-a^2 x^2\right )^{3/2}}{6 a^2}-\frac {\left (7 c^4\right ) \int \sqrt {1-a^2 x^2} \, dx}{8 a}\\ &=-\frac {7 c^4 x \sqrt {1-a^2 x^2}}{16 a}-\frac {7 c^4 \left (1-a^2 x^2\right )^{3/2}}{24 a^2}-\frac {7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{40 a^2}-\frac {c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{10 a^2}-\frac {c^4 (1-a x)^3 \left (1-a^2 x^2\right )^{3/2}}{6 a^2}-\frac {\left (7 c^4\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{16 a}\\ &=-\frac {7 c^4 x \sqrt {1-a^2 x^2}}{16 a}-\frac {7 c^4 \left (1-a^2 x^2\right )^{3/2}}{24 a^2}-\frac {7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{40 a^2}-\frac {c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{10 a^2}-\frac {c^4 (1-a x)^3 \left (1-a^2 x^2\right )^{3/2}}{6 a^2}-\frac {7 c^4 \sin ^{-1}(a x)}{16 a^2}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 83, normalized size = 0.53 \[ -\frac {c^4 \left (\sqrt {1-a^2 x^2} \left (40 a^5 x^5-144 a^4 x^4+170 a^3 x^3-32 a^2 x^2-105 a x+176\right )-210 \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{240 a^2} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.48, size = 104, normalized size = 0.66 \[ \frac {210 \, c^{4} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (40 \, a^{5} c^{4} x^{5} - 144 \, a^{4} c^{4} x^{4} + 170 \, a^{3} c^{4} x^{3} - 32 \, a^{2} c^{4} x^{2} - 105 \, a c^{4} x + 176 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1}}{240 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 94, normalized size = 0.59 \[ -\frac {7 \, c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{16 \, a {\left | a \right |}} - \frac {1}{240} \, \sqrt {-a^{2} x^{2} + 1} {\left (\frac {176 \, c^{4}}{a^{2}} - {\left (\frac {105 \, c^{4}}{a} + 2 \, {\left (16 \, c^{4} - {\left (85 \, a c^{4} + 4 \, {\left (5 \, a^{3} c^{4} x - 18 \, a^{2} c^{4}\right )} x\right )} x\right )} x\right )} x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 163, normalized size = 1.03 \[ -\frac {c^{4} a^{3} x^{5} \sqrt {-a^{2} x^{2}+1}}{6}-\frac {17 c^{4} a \,x^{3} \sqrt {-a^{2} x^{2}+1}}{24}+\frac {7 c^{4} x \sqrt {-a^{2} x^{2}+1}}{16 a}-\frac {7 c^{4} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{16 a \sqrt {a^{2}}}+\frac {3 c^{4} a^{2} x^{4} \sqrt {-a^{2} x^{2}+1}}{5}+\frac {2 c^{4} x^{2} \sqrt {-a^{2} x^{2}+1}}{15}-\frac {11 c^{4} \sqrt {-a^{2} x^{2}+1}}{15 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 141, normalized size = 0.89 \[ -\frac {1}{6} \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{4} x^{5} + \frac {3}{5} \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{4} x^{4} - \frac {17}{24} \, \sqrt {-a^{2} x^{2} + 1} a c^{4} x^{3} + \frac {2}{15} \, \sqrt {-a^{2} x^{2} + 1} c^{4} x^{2} + \frac {7 \, \sqrt {-a^{2} x^{2} + 1} c^{4} x}{16 \, a} - \frac {7 \, c^{4} \arcsin \left (a x\right )}{16 \, a^{2}} - \frac {11 \, \sqrt {-a^{2} x^{2} + 1} c^{4}}{15 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 154, normalized size = 0.97 \[ \frac {2\,c^4\,x^2\,\sqrt {1-a^2\,x^2}}{15}-\frac {11\,c^4\,\sqrt {1-a^2\,x^2}}{15\,a^2}+\frac {7\,c^4\,x\,\sqrt {1-a^2\,x^2}}{16\,a}-\frac {17\,a\,c^4\,x^3\,\sqrt {1-a^2\,x^2}}{24}-\frac {7\,c^4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{16\,a\,\sqrt {-a^2}}+\frac {3\,a^2\,c^4\,x^4\,\sqrt {1-a^2\,x^2}}{5}-\frac {a^3\,c^4\,x^5\,\sqrt {1-a^2\,x^2}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 12.49, size = 617, normalized size = 3.91 \[ a^{5} 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 ) - 3 a^{4} 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^{3} 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 ) + 2 a^{2} 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 ) - 3 a 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 ) + c^{4} \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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