Optimal. Leaf size=121 \[ -\frac {c^2 (a x+1)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}-\frac {7 c^2 (a x+1) \left (1-a^2 x^2\right )^{3/2}}{20 a}-\frac {7 c^2 \left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac {7}{8} c^2 x \sqrt {1-a^2 x^2}+\frac {7 c^2 \sin ^{-1}(a x)}{8 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6138, 671, 641, 195, 216} \[ -\frac {c^2 (a x+1)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}-\frac {7 c^2 (a x+1) \left (1-a^2 x^2\right )^{3/2}}{20 a}-\frac {7 c^2 \left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac {7}{8} c^2 x \sqrt {1-a^2 x^2}+\frac {7 c^2 \sin ^{-1}(a x)}{8 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 195
Rule 216
Rule 641
Rule 671
Rule 6138
Rubi steps
\begin {align*} \int e^{3 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx &=c^2 \int (1+a x)^3 \sqrt {1-a^2 x^2} \, dx\\ &=-\frac {c^2 (1+a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac {1}{5} \left (7 c^2\right ) \int (1+a x)^2 \sqrt {1-a^2 x^2} \, dx\\ &=-\frac {7 c^2 (1+a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}-\frac {c^2 (1+a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac {1}{4} \left (7 c^2\right ) \int (1+a x) \sqrt {1-a^2 x^2} \, dx\\ &=-\frac {7 c^2 \left (1-a^2 x^2\right )^{3/2}}{12 a}-\frac {7 c^2 (1+a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}-\frac {c^2 (1+a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac {1}{4} \left (7 c^2\right ) \int \sqrt {1-a^2 x^2} \, dx\\ &=\frac {7}{8} c^2 x \sqrt {1-a^2 x^2}-\frac {7 c^2 \left (1-a^2 x^2\right )^{3/2}}{12 a}-\frac {7 c^2 (1+a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}-\frac {c^2 (1+a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac {1}{8} \left (7 c^2\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {7}{8} c^2 x \sqrt {1-a^2 x^2}-\frac {7 c^2 \left (1-a^2 x^2\right )^{3/2}}{12 a}-\frac {7 c^2 (1+a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}-\frac {c^2 (1+a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac {7 c^2 \sin ^{-1}(a x)}{8 a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 75, normalized size = 0.62 \[ \frac {c^2 \left (\sqrt {1-a^2 x^2} \left (24 a^4 x^4+90 a^3 x^3+112 a^2 x^2+15 a x-136\right )-210 \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{120 a} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.83, size = 93, normalized size = 0.77 \[ -\frac {210 \, c^{2} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (24 \, a^{4} c^{2} x^{4} + 90 \, a^{3} c^{2} x^{3} + 112 \, a^{2} c^{2} x^{2} + 15 \, a c^{2} x - 136 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{120 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.23, size = 78, normalized size = 0.64 \[ \frac {7 \, c^{2} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{8 \, {\left | a \right |}} + \frac {1}{120} \, \sqrt {-a^{2} x^{2} + 1} {\left ({\left (15 \, c^{2} + 2 \, {\left (56 \, a c^{2} + 3 \, {\left (4 \, a^{3} c^{2} x + 15 \, a^{2} c^{2}\right )} x\right )} x\right )} x - \frac {136 \, c^{2}}{a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 183, normalized size = 1.51 \[ -\frac {c^{2} a^{5} x^{6}}{5 \sqrt {-a^{2} x^{2}+1}}-\frac {11 c^{2} a^{3} x^{4}}{15 \sqrt {-a^{2} x^{2}+1}}+\frac {31 c^{2} a \,x^{2}}{15 \sqrt {-a^{2} x^{2}+1}}-\frac {17 c^{2}}{15 a \sqrt {-a^{2} x^{2}+1}}-\frac {3 c^{2} a^{4} x^{5}}{4 \sqrt {-a^{2} x^{2}+1}}+\frac {5 c^{2} a^{2} x^{3}}{8 \sqrt {-a^{2} x^{2}+1}}+\frac {c^{2} x}{8 \sqrt {-a^{2} x^{2}+1}}+\frac {7 c^{2} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.43, size = 164, normalized size = 1.36 \[ -\frac {a^{5} c^{2} x^{6}}{5 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {3 \, a^{4} c^{2} x^{5}}{4 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {11 \, a^{3} c^{2} x^{4}}{15 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {5 \, a^{2} c^{2} x^{3}}{8 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {31 \, a c^{2} x^{2}}{15 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {c^{2} x}{8 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {7 \, c^{2} \arcsin \left (a x\right )}{8 \, a} - \frac {17 \, c^{2}}{15 \, \sqrt {-a^{2} x^{2} + 1} a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.03, size = 128, normalized size = 1.06 \[ \frac {c^2\,x\,\sqrt {1-a^2\,x^2}}{8}+\frac {7\,c^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{8\,\sqrt {-a^2}}-\frac {17\,c^2\,\sqrt {1-a^2\,x^2}}{15\,a}+\frac {14\,a\,c^2\,x^2\,\sqrt {1-a^2\,x^2}}{15}+\frac {3\,a^2\,c^2\,x^3\,\sqrt {1-a^2\,x^2}}{4}+\frac {a^3\,c^2\,x^4\,\sqrt {1-a^2\,x^2}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 20.17, size = 340, normalized size = 2.81 \[ a^{3} c^{2} \left (\begin {cases} \frac {x^{4} \sqrt {- a^{2} x^{2} + 1}}{5} - \frac {x^{2} \sqrt {- a^{2} x^{2} + 1}}{15 a^{2}} - \frac {2 \sqrt {- a^{2} x^{2} + 1}}{15 a^{4}} & \text {for}\: a \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right ) + 3 a^{2} c^{2} \left (\begin {cases} \frac {i a^{2} x^{5}}{4 \sqrt {a^{2} x^{2} - 1}} - \frac {3 i x^{3}}{8 \sqrt {a^{2} x^{2} - 1}} + \frac {i x}{8 a^{2} \sqrt {a^{2} x^{2} - 1}} - \frac {i \operatorname {acosh}{\left (a x \right )}}{8 a^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {a^{2} x^{5}}{4 \sqrt {- a^{2} x^{2} + 1}} + \frac {3 x^{3}}{8 \sqrt {- a^{2} x^{2} + 1}} - \frac {x}{8 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {\operatorname {asin}{\left (a x \right )}}{8 a^{3}} & \text {otherwise} \end {cases}\right ) + 3 a c^{2} \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\left (- a^{2} x^{2} + 1\right )^{\frac {3}{2}}}{3 a^{2}} & \text {otherwise} \end {cases}\right ) + c^{2} \left (\begin {cases} \frac {i a^{2} x^{3}}{2 \sqrt {a^{2} x^{2} - 1}} - \frac {i x}{2 \sqrt {a^{2} x^{2} - 1}} - \frac {i \operatorname {acosh}{\left (a x \right )}}{2 a} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x \sqrt {- a^{2} x^{2} + 1}}{2} + \frac {\operatorname {asin}{\left (a x \right )}}{2 a} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________