Optimal. Leaf size=148 \[ -\frac {c^3 \sin ^{-1}(a x)}{8 a^4}-\frac {11 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}{35 a^2}-\frac {1}{7} c^3 x^4 \left (1-a^2 x^2\right )^{3/2}+\frac {c^3 x^3 \left (1-a^2 x^2\right )^{3/2}}{3 a}-\frac {c^3 (88-105 a x) \left (1-a^2 x^2\right )^{3/2}}{420 a^4}-\frac {c^3 x \sqrt {1-a^2 x^2}}{8 a^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.24, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6128, 1809, 833, 780, 195, 216} \[ -\frac {1}{7} c^3 x^4 \left (1-a^2 x^2\right )^{3/2}+\frac {c^3 x^3 \left (1-a^2 x^2\right )^{3/2}}{3 a}-\frac {11 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}{35 a^2}-\frac {c^3 x \sqrt {1-a^2 x^2}}{8 a^3}-\frac {c^3 (88-105 a x) \left (1-a^2 x^2\right )^{3/2}}{420 a^4}-\frac {c^3 \sin ^{-1}(a x)}{8 a^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 195
Rule 216
Rule 780
Rule 833
Rule 1809
Rule 6128
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} x^3 (c-a c x)^3 \, dx &=c \int x^3 (c-a c x)^2 \sqrt {1-a^2 x^2} \, dx\\ &=-\frac {1}{7} c^3 x^4 \left (1-a^2 x^2\right )^{3/2}-\frac {c \int x^3 \left (-11 a^2 c^2+14 a^3 c^2 x\right ) \sqrt {1-a^2 x^2} \, dx}{7 a^2}\\ &=\frac {c^3 x^3 \left (1-a^2 x^2\right )^{3/2}}{3 a}-\frac {1}{7} c^3 x^4 \left (1-a^2 x^2\right )^{3/2}+\frac {c \int x^2 \left (-42 a^3 c^2+66 a^4 c^2 x\right ) \sqrt {1-a^2 x^2} \, dx}{42 a^4}\\ &=-\frac {11 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}{35 a^2}+\frac {c^3 x^3 \left (1-a^2 x^2\right )^{3/2}}{3 a}-\frac {1}{7} c^3 x^4 \left (1-a^2 x^2\right )^{3/2}-\frac {c \int x \left (-132 a^4 c^2+210 a^5 c^2 x\right ) \sqrt {1-a^2 x^2} \, dx}{210 a^6}\\ &=-\frac {11 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}{35 a^2}+\frac {c^3 x^3 \left (1-a^2 x^2\right )^{3/2}}{3 a}-\frac {1}{7} c^3 x^4 \left (1-a^2 x^2\right )^{3/2}-\frac {c^3 (88-105 a x) \left (1-a^2 x^2\right )^{3/2}}{420 a^4}-\frac {c^3 \int \sqrt {1-a^2 x^2} \, dx}{4 a^3}\\ &=-\frac {c^3 x \sqrt {1-a^2 x^2}}{8 a^3}-\frac {11 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}{35 a^2}+\frac {c^3 x^3 \left (1-a^2 x^2\right )^{3/2}}{3 a}-\frac {1}{7} c^3 x^4 \left (1-a^2 x^2\right )^{3/2}-\frac {c^3 (88-105 a x) \left (1-a^2 x^2\right )^{3/2}}{420 a^4}-\frac {c^3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{8 a^3}\\ &=-\frac {c^3 x \sqrt {1-a^2 x^2}}{8 a^3}-\frac {11 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}{35 a^2}+\frac {c^3 x^3 \left (1-a^2 x^2\right )^{3/2}}{3 a}-\frac {1}{7} c^3 x^4 \left (1-a^2 x^2\right )^{3/2}-\frac {c^3 (88-105 a x) \left (1-a^2 x^2\right )^{3/2}}{420 a^4}-\frac {c^3 \sin ^{-1}(a x)}{8 a^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.13, size = 91, normalized size = 0.61 \[ \frac {c^3 \left (\sqrt {1-a^2 x^2} \left (120 a^6 x^6-280 a^5 x^5+144 a^4 x^4+70 a^3 x^3-88 a^2 x^2+105 a x-176\right )+210 \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{840 a^4} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 114, normalized size = 0.77 \[ \frac {210 \, c^{3} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (120 \, a^{6} c^{3} x^{6} - 280 \, a^{5} c^{3} x^{5} + 144 \, a^{4} c^{3} x^{4} + 70 \, a^{3} c^{3} x^{3} - 88 \, a^{2} c^{3} x^{2} + 105 \, a c^{3} x - 176 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1}}{840 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.21, size = 104, normalized size = 0.70 \[ \frac {1}{840} \, \sqrt {-a^{2} x^{2} + 1} {\left ({\left (2 \, {\left ({\left (\frac {35 \, c^{3}}{a} + 4 \, {\left (18 \, c^{3} + 5 \, {\left (3 \, a^{2} c^{3} x - 7 \, a c^{3}\right )} x\right )} x\right )} x - \frac {44 \, c^{3}}{a^{2}}\right )} x + \frac {105 \, c^{3}}{a^{3}}\right )} x - \frac {176 \, c^{3}}{a^{4}}\right )} - \frac {c^{3} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{8 \, a^{3} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 186, normalized size = 1.26 \[ \frac {c^{3} a^{2} x^{6} \sqrt {-a^{2} x^{2}+1}}{7}+\frac {6 c^{3} x^{4} \sqrt {-a^{2} x^{2}+1}}{35}-\frac {11 c^{3} x^{2} \sqrt {-a^{2} x^{2}+1}}{105 a^{2}}-\frac {22 c^{3} \sqrt {-a^{2} x^{2}+1}}{105 a^{4}}-\frac {c^{3} a \,x^{5} \sqrt {-a^{2} x^{2}+1}}{3}+\frac {c^{3} x^{3} \sqrt {-a^{2} x^{2}+1}}{12 a}+\frac {c^{3} x \sqrt {-a^{2} x^{2}+1}}{8 a^{3}}-\frac {c^{3} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{3} \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.40, size = 164, normalized size = 1.11 \[ \frac {1}{7} \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{3} x^{6} - \frac {1}{3} \, \sqrt {-a^{2} x^{2} + 1} a c^{3} x^{5} + \frac {6}{35} \, \sqrt {-a^{2} x^{2} + 1} c^{3} x^{4} + \frac {\sqrt {-a^{2} x^{2} + 1} c^{3} x^{3}}{12 \, a} - \frac {11 \, \sqrt {-a^{2} x^{2} + 1} c^{3} x^{2}}{105 \, a^{2}} + \frac {\sqrt {-a^{2} x^{2} + 1} c^{3} x}{8 \, a^{3}} - \frac {c^{3} \arcsin \left (a x\right )}{8 \, a^{4}} - \frac {22 \, \sqrt {-a^{2} x^{2} + 1} c^{3}}{105 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 177, normalized size = 1.20 \[ \frac {6\,c^3\,x^4\,\sqrt {1-a^2\,x^2}}{35}-\frac {22\,c^3\,\sqrt {1-a^2\,x^2}}{105\,a^4}+\frac {c^3\,x\,\sqrt {1-a^2\,x^2}}{8\,a^3}-\frac {a\,c^3\,x^5\,\sqrt {1-a^2\,x^2}}{3}-\frac {c^3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{8\,a^3\,\sqrt {-a^2}}+\frac {c^3\,x^3\,\sqrt {1-a^2\,x^2}}{12\,a}-\frac {11\,c^3\,x^2\,\sqrt {1-a^2\,x^2}}{105\,a^2}+\frac {a^2\,c^3\,x^6\,\sqrt {1-a^2\,x^2}}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 11.22, size = 512, normalized size = 3.46 \[ - a^{4} c^{3} \left (\begin {cases} - \frac {x^{6} \sqrt {- a^{2} x^{2} + 1}}{7 a^{2}} - \frac {6 x^{4} \sqrt {- a^{2} x^{2} + 1}}{35 a^{4}} - \frac {8 x^{2} \sqrt {- a^{2} x^{2} + 1}}{35 a^{6}} - \frac {16 \sqrt {- a^{2} x^{2} + 1}}{35 a^{8}} & \text {for}\: a \neq 0 \\\frac {x^{8}}{8} & \text {otherwise} \end {cases}\right ) + 2 a^{3} c^{3} \left (\begin {cases} - \frac {i x^{7}}{6 \sqrt {a^{2} x^{2} - 1}} - \frac {i x^{5}}{24 a^{2} \sqrt {a^{2} x^{2} - 1}} - \frac {5 i x^{3}}{48 a^{4} \sqrt {a^{2} x^{2} - 1}} + \frac {5 i x}{16 a^{6} \sqrt {a^{2} x^{2} - 1}} - \frac {5 i \operatorname {acosh}{\left (a x \right )}}{16 a^{7}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{7}}{6 \sqrt {- a^{2} x^{2} + 1}} + \frac {x^{5}}{24 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {5 x^{3}}{48 a^{4} \sqrt {- a^{2} x^{2} + 1}} - \frac {5 x}{16 a^{6} \sqrt {- a^{2} x^{2} + 1}} + \frac {5 \operatorname {asin}{\left (a x \right )}}{16 a^{7}} & \text {otherwise} \end {cases}\right ) - 2 a c^{3} \left (\begin {cases} - \frac {i x^{5}}{4 \sqrt {a^{2} x^{2} - 1}} - \frac {i x^{3}}{8 a^{2} \sqrt {a^{2} x^{2} - 1}} + \frac {3 i x}{8 a^{4} \sqrt {a^{2} x^{2} - 1}} - \frac {3 i \operatorname {acosh}{\left (a x \right )}}{8 a^{5}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{5}}{4 \sqrt {- a^{2} x^{2} + 1}} + \frac {x^{3}}{8 a^{2} \sqrt {- a^{2} x^{2} + 1}} - \frac {3 x}{8 a^{4} \sqrt {- a^{2} x^{2} + 1}} + \frac {3 \operatorname {asin}{\left (a x \right )}}{8 a^{5}} & \text {otherwise} \end {cases}\right ) + c^{3} \left (\begin {cases} - \frac {x^{2} \sqrt {- a^{2} x^{2} + 1}}{3 a^{2}} - \frac {2 \sqrt {- a^{2} x^{2} + 1}}{3 a^{4}} & \text {for}\: a \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________