Optimal. Leaf size=91 \[ \frac {c^3 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{4 a}+\frac {5 c^3 \left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac {5}{8} c^3 x \sqrt {1-a^2 x^2}+\frac {5 c^3 \sin ^{-1}(a x)}{8 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {6127, 671, 641, 195, 216} \[ \frac {c^3 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{4 a}+\frac {5 c^3 \left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac {5}{8} c^3 x \sqrt {1-a^2 x^2}+\frac {5 c^3 \sin ^{-1}(a x)}{8 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 195
Rule 216
Rule 641
Rule 671
Rule 6127
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} (c-a c x)^3 \, dx &=c \int (c-a c x)^2 \sqrt {1-a^2 x^2} \, dx\\ &=\frac {c^3 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{4 a}+\frac {1}{4} \left (5 c^2\right ) \int (c-a c x) \sqrt {1-a^2 x^2} \, dx\\ &=\frac {5 c^3 \left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac {c^3 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{4 a}+\frac {1}{4} \left (5 c^3\right ) \int \sqrt {1-a^2 x^2} \, dx\\ &=\frac {5}{8} c^3 x \sqrt {1-a^2 x^2}+\frac {5 c^3 \left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac {c^3 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{4 a}+\frac {1}{8} \left (5 c^3\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {5}{8} c^3 x \sqrt {1-a^2 x^2}+\frac {5 c^3 \left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac {c^3 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{4 a}+\frac {5 c^3 \sin ^{-1}(a x)}{8 a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.09, size = 67, normalized size = 0.74 \[ \frac {c^3 \left (\sqrt {1-a^2 x^2} \left (6 a^3 x^3-16 a^2 x^2+9 a x+16\right )-30 \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{24 a} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.68, size = 82, normalized size = 0.90 \[ -\frac {30 \, c^{3} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (6 \, a^{3} c^{3} x^{3} - 16 \, a^{2} c^{3} x^{2} + 9 \, a c^{3} x + 16 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1}}{24 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.22, size = 66, normalized size = 0.73 \[ \frac {5 \, c^{3} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{8 \, {\left | a \right |}} + \frac {1}{24} \, \sqrt {-a^{2} x^{2} + 1} {\left (\frac {16 \, c^{3}}{a} + {\left (9 \, c^{3} + 2 \, {\left (3 \, a^{2} c^{3} x - 8 \, a c^{3}\right )} x\right )} x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 114, normalized size = 1.25 \[ \frac {c^{3} a^{2} x^{3} \sqrt {-a^{2} x^{2}+1}}{4}+\frac {3 c^{3} x \sqrt {-a^{2} x^{2}+1}}{8}+\frac {5 c^{3} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 \sqrt {a^{2}}}-\frac {2 c^{3} a \,x^{2} \sqrt {-a^{2} x^{2}+1}}{3}+\frac {2 c^{3} \sqrt {-a^{2} x^{2}+1}}{3 a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.42, size = 95, normalized size = 1.04 \[ \frac {1}{4} \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{3} x^{3} - \frac {2}{3} \, \sqrt {-a^{2} x^{2} + 1} a c^{3} x^{2} + \frac {3}{8} \, \sqrt {-a^{2} x^{2} + 1} c^{3} x + \frac {5 \, c^{3} \arcsin \left (a x\right )}{8 \, a} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1} c^{3}}{3 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 105, normalized size = 1.15 \[ \frac {3\,c^3\,x\,\sqrt {1-a^2\,x^2}}{8}+\frac {5\,c^3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{8\,\sqrt {-a^2}}+\frac {2\,c^3\,\sqrt {1-a^2\,x^2}}{3\,a}-\frac {2\,a\,c^3\,x^2\,\sqrt {1-a^2\,x^2}}{3}+\frac {a^2\,c^3\,x^3\,\sqrt {1-a^2\,x^2}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 6.37, size = 134, normalized size = 1.47 \[ \begin {cases} \frac {2 c^{3} \sqrt {- a^{2} x^{2} + 1} + 2 c^{3} \left (\begin {cases} \frac {\left (- a^{2} x^{2} + 1\right )^{\frac {3}{2}}}{3} - \sqrt {- a^{2} x^{2} + 1} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right ) - c^{3} \left (\begin {cases} \frac {a x \left (- 2 a^{2} x^{2} + 1\right ) \sqrt {- a^{2} x^{2} + 1}}{8} - \frac {a x \sqrt {- a^{2} x^{2} + 1}}{2} + \frac {3 \operatorname {asin}{\left (a x \right )}}{8} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right ) + c^{3} \operatorname {asin}{\left (a x \right )}}{a} & \text {for}\: a \neq 0 \\c^{3} x & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________