Optimal. Leaf size=123 \[ \frac {c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac {7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}+\frac {7 c^4 \left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac {7}{8} c^4 x \sqrt {1-a^2 x^2}+\frac {7 c^4 \sin ^{-1}(a x)}{8 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ \frac {c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac {7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}+\frac {7 c^4 \left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac {7}{8} c^4 x \sqrt {1-a^2 x^2}+\frac {7 c^4 \sin ^{-1}(a x)}{8 a} \]
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)^4 \, dx &=c \int (c-a c x)^3 \sqrt {1-a^2 x^2} \, dx\\ &=\frac {c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac {1}{5} \left (7 c^2\right ) \int (c-a c x)^2 \sqrt {1-a^2 x^2} \, dx\\ &=\frac {7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}+\frac {c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac {1}{4} \left (7 c^3\right ) \int (c-a c x) \sqrt {1-a^2 x^2} \, dx\\ &=\frac {7 c^4 \left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac {7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}+\frac {c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac {1}{4} \left (7 c^4\right ) \int \sqrt {1-a^2 x^2} \, dx\\ &=\frac {7}{8} c^4 x \sqrt {1-a^2 x^2}+\frac {7 c^4 \left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac {7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}+\frac {c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac {1}{8} \left (7 c^4\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {7}{8} c^4 x \sqrt {1-a^2 x^2}+\frac {7 c^4 \left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac {7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}+\frac {c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac {7 c^4 \sin ^{-1}(a x)}{8 a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 75, normalized size = 0.61 \[ -\frac {c^4 \left (\sqrt {1-a^2 x^2} \left (24 a^4 x^4-90 a^3 x^3+112 a^2 x^2-15 a x-136\right )+210 \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{120 a} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.47, size = 92, normalized size = 0.75 \[ -\frac {210 \, c^{4} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (24 \, a^{4} c^{4} x^{4} - 90 \, a^{3} c^{4} x^{3} + 112 \, a^{2} c^{4} x^{2} - 15 \, a c^{4} x - 136 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1}}{120 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.19, size = 78, normalized size = 0.63 \[ \frac {7 \, c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{8 \, {\left | a \right |}} + \frac {1}{120} \, \sqrt {-a^{2} x^{2} + 1} {\left (\frac {136 \, c^{4}}{a} + {\left (15 \, c^{4} - 2 \, {\left (56 \, a c^{4} + 3 \, {\left (4 \, a^{3} c^{4} x - 15 \, a^{2} c^{4}\right )} x\right )} x\right )} x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 137, normalized size = 1.11 \[ -\frac {c^{4} a^{3} x^{4} \sqrt {-a^{2} x^{2}+1}}{5}-\frac {14 c^{4} a \,x^{2} \sqrt {-a^{2} x^{2}+1}}{15}+\frac {17 c^{4} \sqrt {-a^{2} x^{2}+1}}{15 a}+\frac {3 c^{4} a^{2} x^{3} \sqrt {-a^{2} x^{2}+1}}{4}+\frac {c^{4} x \sqrt {-a^{2} x^{2}+1}}{8}+\frac {7 c^{4} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.46, size = 118, normalized size = 0.96 \[ -\frac {1}{5} \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{4} x^{4} + \frac {3}{4} \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{4} x^{3} - \frac {14}{15} \, \sqrt {-a^{2} x^{2} + 1} a c^{4} x^{2} + \frac {1}{8} \, \sqrt {-a^{2} x^{2} + 1} c^{4} x + \frac {7 \, c^{4} \arcsin \left (a x\right )}{8 \, a} + \frac {17 \, \sqrt {-a^{2} x^{2} + 1} c^{4}}{15 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.00, size = 128, normalized size = 1.04 \[ \frac {c^4\,x\,\sqrt {1-a^2\,x^2}}{8}+\frac {7\,c^4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{8\,\sqrt {-a^2}}+\frac {17\,c^4\,\sqrt {1-a^2\,x^2}}{15\,a}-\frac {14\,a\,c^4\,x^2\,\sqrt {1-a^2\,x^2}}{15}+\frac {3\,a^2\,c^4\,x^3\,\sqrt {1-a^2\,x^2}}{4}-\frac {a^3\,c^4\,x^4\,\sqrt {1-a^2\,x^2}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 8.98, size = 226, normalized size = 1.84 \[ \begin {cases} \frac {3 c^{4} \sqrt {- a^{2} x^{2} + 1} + 2 c^{4} \left (\begin {cases} - \frac {a x \sqrt {- a^{2} x^{2} + 1}}{2} + \frac {\operatorname {asin}{\left (a x \right )}}{2} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right ) + 2 c^{4} \left (\begin {cases} \frac {\left (- a^{2} x^{2} + 1\right )^{\frac {3}{2}}}{3} - \sqrt {- a^{2} x^{2} + 1} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right ) - 3 c^{4} \left (\begin {cases} \frac {a x \left (- 2 a^{2} x^{2} + 1\right ) \sqrt {- a^{2} x^{2} + 1}}{8} - \frac {a x \sqrt {- a^{2} x^{2} + 1}}{2} + \frac {3 \operatorname {asin}{\left (a x \right )}}{8} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right ) + c^{4} \left (\begin {cases} - \frac {\left (- a^{2} x^{2} + 1\right )^{\frac {5}{2}}}{5} + \frac {2 \left (- a^{2} x^{2} + 1\right )^{\frac {3}{2}}}{3} - \sqrt {- a^{2} x^{2} + 1} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right ) + c^{4} \operatorname {asin}{\left (a x \right )}}{a} & \text {for}\: a \neq 0 \\c^{4} x & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________