Optimal. Leaf size=136 \[ \frac {23 c \sin ^{-1}(a x)}{16 a^4}-\frac {17 c x^2 \sqrt {1-a^2 x^2}}{15 a^2}-\frac {1}{6} a c x^5 \sqrt {1-a^2 x^2}-\frac {3}{5} c x^4 \sqrt {1-a^2 x^2}-\frac {23 c x^3 \sqrt {1-a^2 x^2}}{24 a}-\frac {c (345 a x+544) \sqrt {1-a^2 x^2}}{240 a^4} \]
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Rubi [A] time = 0.28, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {6148, 1809, 833, 780, 216} \[ -\frac {1}{6} a c x^5 \sqrt {1-a^2 x^2}-\frac {3}{5} c x^4 \sqrt {1-a^2 x^2}-\frac {23 c x^3 \sqrt {1-a^2 x^2}}{24 a}-\frac {17 c x^2 \sqrt {1-a^2 x^2}}{15 a^2}-\frac {c (345 a x+544) \sqrt {1-a^2 x^2}}{240 a^4}+\frac {23 c \sin ^{-1}(a x)}{16 a^4} \]
Antiderivative was successfully verified.
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Rule 216
Rule 780
Rule 833
Rule 1809
Rule 6148
Rubi steps
\begin {align*} \int e^{3 \tanh ^{-1}(a x)} x^3 \left (c-a^2 c x^2\right ) \, dx &=c \int \frac {x^3 (1+a x)^3}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {1}{6} a c x^5 \sqrt {1-a^2 x^2}-\frac {c \int \frac {x^3 \left (-6 a^2-23 a^3 x-18 a^4 x^2\right )}{\sqrt {1-a^2 x^2}} \, dx}{6 a^2}\\ &=-\frac {3}{5} c x^4 \sqrt {1-a^2 x^2}-\frac {1}{6} a c x^5 \sqrt {1-a^2 x^2}+\frac {c \int \frac {x^3 \left (102 a^4+115 a^5 x\right )}{\sqrt {1-a^2 x^2}} \, dx}{30 a^4}\\ &=-\frac {23 c x^3 \sqrt {1-a^2 x^2}}{24 a}-\frac {3}{5} c x^4 \sqrt {1-a^2 x^2}-\frac {1}{6} a c x^5 \sqrt {1-a^2 x^2}-\frac {c \int \frac {x^2 \left (-345 a^5-408 a^6 x\right )}{\sqrt {1-a^2 x^2}} \, dx}{120 a^6}\\ &=-\frac {17 c x^2 \sqrt {1-a^2 x^2}}{15 a^2}-\frac {23 c x^3 \sqrt {1-a^2 x^2}}{24 a}-\frac {3}{5} c x^4 \sqrt {1-a^2 x^2}-\frac {1}{6} a c x^5 \sqrt {1-a^2 x^2}+\frac {c \int \frac {x \left (816 a^6+1035 a^7 x\right )}{\sqrt {1-a^2 x^2}} \, dx}{360 a^8}\\ &=-\frac {17 c x^2 \sqrt {1-a^2 x^2}}{15 a^2}-\frac {23 c x^3 \sqrt {1-a^2 x^2}}{24 a}-\frac {3}{5} c x^4 \sqrt {1-a^2 x^2}-\frac {1}{6} a c x^5 \sqrt {1-a^2 x^2}-\frac {c (544+345 a x) \sqrt {1-a^2 x^2}}{240 a^4}+\frac {(23 c) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{16 a^3}\\ &=-\frac {17 c x^2 \sqrt {1-a^2 x^2}}{15 a^2}-\frac {23 c x^3 \sqrt {1-a^2 x^2}}{24 a}-\frac {3}{5} c x^4 \sqrt {1-a^2 x^2}-\frac {1}{6} a c x^5 \sqrt {1-a^2 x^2}-\frac {c (544+345 a x) \sqrt {1-a^2 x^2}}{240 a^4}+\frac {23 c \sin ^{-1}(a x)}{16 a^4}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 70, normalized size = 0.51 \[ \frac {345 c \sin ^{-1}(a x)-c \sqrt {1-a^2 x^2} \left (40 a^5 x^5+144 a^4 x^4+230 a^3 x^3+272 a^2 x^2+345 a x+544\right )}{240 a^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 89, normalized size = 0.65 \[ -\frac {690 \, c \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (40 \, a^{5} c x^{5} + 144 \, a^{4} c x^{4} + 230 \, a^{3} c x^{3} + 272 \, a^{2} c x^{2} + 345 \, a c x + 544 \, c\right )} \sqrt {-a^{2} x^{2} + 1}}{240 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 78, normalized size = 0.57 \[ -\frac {1}{240} \, \sqrt {-a^{2} x^{2} + 1} {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, a c x + 18 \, c\right )} x + \frac {115 \, c}{a}\right )} x + \frac {136 \, c}{a^{2}}\right )} x + \frac {345 \, c}{a^{3}}\right )} x + \frac {544 \, c}{a^{4}}\right )} + \frac {23 \, c \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{16 \, a^{3} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 191, normalized size = 1.40 \[ \frac {c \,a^{3} x^{7}}{6 \sqrt {-a^{2} x^{2}+1}}+\frac {19 c a \,x^{5}}{24 \sqrt {-a^{2} x^{2}+1}}+\frac {23 c \,x^{3}}{48 a \sqrt {-a^{2} x^{2}+1}}-\frac {23 c x}{16 a^{3} \sqrt {-a^{2} x^{2}+1}}+\frac {23 c \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{16 a^{3} \sqrt {a^{2}}}+\frac {3 c \,a^{2} x^{6}}{5 \sqrt {-a^{2} x^{2}+1}}+\frac {8 c \,x^{4}}{15 \sqrt {-a^{2} x^{2}+1}}+\frac {17 c \,x^{2}}{15 a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {34 c}{15 a^{4} \sqrt {-a^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 169, normalized size = 1.24 \[ \frac {a^{3} c x^{7}}{6 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {3 \, a^{2} c x^{6}}{5 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {19 \, a c x^{5}}{24 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {8 \, c x^{4}}{15 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {23 \, c x^{3}}{48 \, \sqrt {-a^{2} x^{2} + 1} a} + \frac {17 \, c x^{2}}{15 \, \sqrt {-a^{2} x^{2} + 1} a^{2}} - \frac {23 \, c x}{16 \, \sqrt {-a^{2} x^{2} + 1} a^{3}} + \frac {23 \, c \arcsin \left (a x\right )}{16 \, a^{4}} - \frac {34 \, c}{15 \, \sqrt {-a^{2} x^{2} + 1} a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.91, size = 140, normalized size = 1.03 \[ \frac {23\,c\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{16\,a^3\,\sqrt {-a^2}}-\frac {3\,c\,x^4\,\sqrt {1-a^2\,x^2}}{5}-\frac {23\,c\,x^3\,\sqrt {1-a^2\,x^2}}{24\,a}-\frac {17\,c\,x^2\,\sqrt {1-a^2\,x^2}}{15\,a^2}-\frac {23\,c\,x\,\sqrt {1-a^2\,x^2}}{16\,a^3}-\frac {a\,c\,x^5\,\sqrt {1-a^2\,x^2}}{6}-\frac {34\,c\,\sqrt {1-a^2\,x^2}}{15\,a^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 21.36, size = 483, normalized size = 3.55 \[ a^{3} c \left (\begin {cases} - \frac {i x^{7}}{6 \sqrt {a^{2} x^{2} - 1}} - \frac {i x^{5}}{24 a^{2} \sqrt {a^{2} x^{2} - 1}} - \frac {5 i x^{3}}{48 a^{4} \sqrt {a^{2} x^{2} - 1}} + \frac {5 i x}{16 a^{6} \sqrt {a^{2} x^{2} - 1}} - \frac {5 i \operatorname {acosh}{\left (a x \right )}}{16 a^{7}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{7}}{6 \sqrt {- a^{2} x^{2} + 1}} + \frac {x^{5}}{24 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {5 x^{3}}{48 a^{4} \sqrt {- a^{2} x^{2} + 1}} - \frac {5 x}{16 a^{6} \sqrt {- a^{2} x^{2} + 1}} + \frac {5 \operatorname {asin}{\left (a x \right )}}{16 a^{7}} & \text {otherwise} \end {cases}\right ) + 3 a^{2} c \left (\begin {cases} - \frac {x^{4} \sqrt {- a^{2} x^{2} + 1}}{5 a^{2}} - \frac {4 x^{2} \sqrt {- a^{2} x^{2} + 1}}{15 a^{4}} - \frac {8 \sqrt {- a^{2} x^{2} + 1}}{15 a^{6}} & \text {for}\: a \neq 0 \\\frac {x^{6}}{6} & \text {otherwise} \end {cases}\right ) + 3 a c \left (\begin {cases} - \frac {i x^{5}}{4 \sqrt {a^{2} x^{2} - 1}} - \frac {i x^{3}}{8 a^{2} \sqrt {a^{2} x^{2} - 1}} + \frac {3 i x}{8 a^{4} \sqrt {a^{2} x^{2} - 1}} - \frac {3 i \operatorname {acosh}{\left (a x \right )}}{8 a^{5}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{5}}{4 \sqrt {- a^{2} x^{2} + 1}} + \frac {x^{3}}{8 a^{2} \sqrt {- a^{2} x^{2} + 1}} - \frac {3 x}{8 a^{4} \sqrt {- a^{2} x^{2} + 1}} + \frac {3 \operatorname {asin}{\left (a x \right )}}{8 a^{5}} & \text {otherwise} \end {cases}\right ) + c \left (\begin {cases} - \frac {x^{2} \sqrt {- a^{2} x^{2} + 1}}{3 a^{2}} - \frac {2 \sqrt {- a^{2} x^{2} + 1}}{3 a^{4}} & \text {for}\: a \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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