Optimal. Leaf size=120 \[ -\frac {c \sqrt {1-a^2 x^2} (a x+1)^3}{4 a^2}-\frac {c \sqrt {1-a^2 x^2} (a x+1)^2}{4 a^2}-\frac {5 c \sqrt {1-a^2 x^2} (a x+1)}{8 a^2}-\frac {15 c \sqrt {1-a^2 x^2}}{8 a^2}+\frac {15 c \sin ^{-1}(a x)}{8 a^2} \]
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Rubi [A] time = 0.10, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {6148, 795, 671, 641, 216} \[ -\frac {c \sqrt {1-a^2 x^2} (a x+1)^3}{4 a^2}-\frac {c \sqrt {1-a^2 x^2} (a x+1)^2}{4 a^2}-\frac {5 c \sqrt {1-a^2 x^2} (a x+1)}{8 a^2}-\frac {15 c \sqrt {1-a^2 x^2}}{8 a^2}+\frac {15 c \sin ^{-1}(a x)}{8 a^2} \]
Antiderivative was successfully verified.
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Rule 216
Rule 641
Rule 671
Rule 795
Rule 6148
Rubi steps
\begin {align*} \int e^{3 \tanh ^{-1}(a x)} x \left (c-a^2 c x^2\right ) \, dx &=c \int \frac {x (1+a x)^3}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {c (1+a x)^3 \sqrt {1-a^2 x^2}}{4 a^2}+\frac {(3 c) \int \frac {(1+a x)^3}{\sqrt {1-a^2 x^2}} \, dx}{4 a}\\ &=-\frac {c (1+a x)^2 \sqrt {1-a^2 x^2}}{4 a^2}-\frac {c (1+a x)^3 \sqrt {1-a^2 x^2}}{4 a^2}+\frac {(5 c) \int \frac {(1+a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{4 a}\\ &=-\frac {5 c (1+a x) \sqrt {1-a^2 x^2}}{8 a^2}-\frac {c (1+a x)^2 \sqrt {1-a^2 x^2}}{4 a^2}-\frac {c (1+a x)^3 \sqrt {1-a^2 x^2}}{4 a^2}+\frac {(15 c) \int \frac {1+a x}{\sqrt {1-a^2 x^2}} \, dx}{8 a}\\ &=-\frac {15 c \sqrt {1-a^2 x^2}}{8 a^2}-\frac {5 c (1+a x) \sqrt {1-a^2 x^2}}{8 a^2}-\frac {c (1+a x)^2 \sqrt {1-a^2 x^2}}{4 a^2}-\frac {c (1+a x)^3 \sqrt {1-a^2 x^2}}{4 a^2}+\frac {(15 c) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{8 a}\\ &=-\frac {15 c \sqrt {1-a^2 x^2}}{8 a^2}-\frac {5 c (1+a x) \sqrt {1-a^2 x^2}}{8 a^2}-\frac {c (1+a x)^2 \sqrt {1-a^2 x^2}}{4 a^2}-\frac {c (1+a x)^3 \sqrt {1-a^2 x^2}}{4 a^2}+\frac {15 c \sin ^{-1}(a x)}{8 a^2}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 54, normalized size = 0.45 \[ \frac {15 c \sin ^{-1}(a x)-c \sqrt {1-a^2 x^2} \left (2 a^3 x^3+8 a^2 x^2+15 a x+24\right )}{8 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 71, normalized size = 0.59 \[ -\frac {30 \, c \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (2 \, a^{3} c x^{3} + 8 \, a^{2} c x^{2} + 15 \, a c x + 24 \, c\right )} \sqrt {-a^{2} x^{2} + 1}}{8 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.61, size = 58, normalized size = 0.48 \[ -\frac {1}{8} \, \sqrt {-a^{2} x^{2} + 1} {\left ({\left (2 \, {\left (a c x + 4 \, c\right )} x + \frac {15 \, c}{a}\right )} x + \frac {24 \, c}{a^{2}}\right )} + \frac {15 \, c \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{8 \, a {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 148, normalized size = 1.23 \[ \frac {c \,a^{3} x^{5}}{4 \sqrt {-a^{2} x^{2}+1}}+\frac {13 c a \,x^{3}}{8 \sqrt {-a^{2} x^{2}+1}}-\frac {15 c x}{8 a \sqrt {-a^{2} x^{2}+1}}+\frac {15 c \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a \sqrt {a^{2}}}+\frac {c \,a^{2} x^{4}}{\sqrt {-a^{2} x^{2}+1}}+\frac {2 c \,x^{2}}{\sqrt {-a^{2} x^{2}+1}}-\frac {3 c}{a^{2} \sqrt {-a^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 126, normalized size = 1.05 \[ \frac {a^{3} c x^{5}}{4 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {a^{2} c x^{4}}{\sqrt {-a^{2} x^{2} + 1}} + \frac {13 \, a c x^{3}}{8 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {2 \, c x^{2}}{\sqrt {-a^{2} x^{2} + 1}} - \frac {15 \, c x}{8 \, \sqrt {-a^{2} x^{2} + 1} a} + \frac {15 \, c \arcsin \left (a x\right )}{8 \, a^{2}} - \frac {3 \, c}{\sqrt {-a^{2} x^{2} + 1} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 98, normalized size = 0.82 \[ \frac {15\,c\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{8\,a\,\sqrt {-a^2}}-c\,x^2\,\sqrt {1-a^2\,x^2}-\frac {15\,c\,x\,\sqrt {1-a^2\,x^2}}{8\,a}-\frac {a\,c\,x^3\,\sqrt {1-a^2\,x^2}}{4}-\frac {3\,c\,\sqrt {1-a^2\,x^2}}{a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 16.33, size = 326, normalized size = 2.72 \[ a^{3} c \left (\begin {cases} - \frac {i x^{5}}{4 \sqrt {a^{2} x^{2} - 1}} - \frac {i x^{3}}{8 a^{2} \sqrt {a^{2} x^{2} - 1}} + \frac {3 i x}{8 a^{4} \sqrt {a^{2} x^{2} - 1}} - \frac {3 i \operatorname {acosh}{\left (a x \right )}}{8 a^{5}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{5}}{4 \sqrt {- a^{2} x^{2} + 1}} + \frac {x^{3}}{8 a^{2} \sqrt {- a^{2} x^{2} + 1}} - \frac {3 x}{8 a^{4} \sqrt {- a^{2} x^{2} + 1}} + \frac {3 \operatorname {asin}{\left (a x \right )}}{8 a^{5}} & \text {otherwise} \end {cases}\right ) + 3 a^{2} c \left (\begin {cases} - \frac {x^{2} \sqrt {- a^{2} x^{2} + 1}}{3 a^{2}} - \frac {2 \sqrt {- a^{2} x^{2} + 1}}{3 a^{4}} & \text {for}\: a \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right ) + 3 a c \left (\begin {cases} - \frac {i x \sqrt {a^{2} x^{2} - 1}}{2 a^{2}} - \frac {i \operatorname {acosh}{\left (a x \right )}}{2 a^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{3}}{2 \sqrt {- a^{2} x^{2} + 1}} - \frac {x}{2 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {\operatorname {asin}{\left (a x \right )}}{2 a^{3}} & \text {otherwise} \end {cases}\right ) + c \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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