Optimal. Leaf size=74 \[ \frac {2 (1+a x)}{5 a \left (c-a^2 c x^2\right )^{5/2}}+\frac {x}{5 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x}{5 c^2 \sqrt {c-a^2 c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6276, 667, 198,
197} \begin {gather*} \frac {2 x}{5 c^2 \sqrt {c-a^2 c x^2}}+\frac {x}{5 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 (a x+1)}{5 a \left (c-a^2 c x^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 197
Rule 198
Rule 667
Rule 6276
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=c \int \frac {(1+a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx\\ &=\frac {2 (1+a x)}{5 a \left (c-a^2 c x^2\right )^{5/2}}+\frac {3}{5} \int \frac {1}{\left (c-a^2 c x^2\right )^{5/2}} \, dx\\ &=\frac {2 (1+a x)}{5 a \left (c-a^2 c x^2\right )^{5/2}}+\frac {x}{5 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 \int \frac {1}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{5 c}\\ &=\frac {2 (1+a x)}{5 a \left (c-a^2 c x^2\right )^{5/2}}+\frac {x}{5 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x}{5 c^2 \sqrt {c-a^2 c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 53, normalized size = 0.72 \begin {gather*} \frac {2+a x-4 a^2 x^2+2 a^3 x^3}{5 a c^2 (-1+a x)^2 \sqrt {c-a^2 c x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(205\) vs.
\(2(62)=124\).
time = 0.06, size = 206, normalized size = 2.78
method | result | size |
gosper | \(\frac {\left (2 a^{3} x^{3}-4 a^{2} x^{2}+a x +2\right ) \left (a x +1\right )^{2}}{5 \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} a}\) | \(47\) |
trager | \(-\frac {\left (2 a^{3} x^{3}-4 a^{2} x^{2}+a x +2\right ) \sqrt {-a^{2} c \,x^{2}+c}}{5 c^{3} \left (a x -1\right )^{3} a \left (a x +1\right )}\) | \(57\) |
default | \(-\frac {x}{3 c \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}-\frac {2 x}{3 c^{2} \sqrt {-a^{2} c \,x^{2}+c}}-\frac {2 \left (\frac {1}{5 a c \left (x -\frac {1}{a}\right ) \left (-c \,a^{2} \left (x -\frac {1}{a}\right )^{2}-2 c a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}-\frac {4 a \left (-\frac {-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c}{6 a^{2} c^{2} \left (-c \,a^{2} \left (x -\frac {1}{a}\right )^{2}-2 c a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}-\frac {-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c}{3 a^{2} c^{3} \sqrt {-c \,a^{2} \left (x -\frac {1}{a}\right )^{2}-2 c a \left (x -\frac {1}{a}\right )}}\right )}{5}\right )}{a}\) | \(206\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 218 vs.
\(2 (62) = 124\).
time = 0.30, size = 218, normalized size = 2.95 \begin {gather*} \frac {1}{5} \, a {\left (\frac {a}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a^{4} c x + {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a^{3} c} - \frac {a}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a^{4} c x - {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a^{3} c} - \frac {1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a^{3} c x + {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a^{2} c} - \frac {1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a^{3} c x - {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a^{2} c} + \frac {2 \, x}{\sqrt {-a^{2} c x^{2} + c} a c^{2}} + \frac {x}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a c}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.37, size = 75, normalized size = 1.01 \begin {gather*} -\frac {{\left (2 \, a^{3} x^{3} - 4 \, a^{2} x^{2} + a x + 2\right )} \sqrt {-a^{2} c x^{2} + c}}{5 \, {\left (a^{5} c^{3} x^{4} - 2 \, a^{4} c^{3} x^{3} + 2 \, a^{2} c^{3} x - a c^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {a x}{a^{5} c^{2} x^{5} \sqrt {- a^{2} c x^{2} + c} - a^{4} c^{2} x^{4} \sqrt {- a^{2} c x^{2} + c} - 2 a^{3} c^{2} x^{3} \sqrt {- a^{2} c x^{2} + c} + 2 a^{2} c^{2} x^{2} \sqrt {- a^{2} c x^{2} + c} + a c^{2} x \sqrt {- a^{2} c x^{2} + c} - c^{2} \sqrt {- a^{2} c x^{2} + c}}\, dx - \int \frac {1}{a^{5} c^{2} x^{5} \sqrt {- a^{2} c x^{2} + c} - a^{4} c^{2} x^{4} \sqrt {- a^{2} c x^{2} + c} - 2 a^{3} c^{2} x^{3} \sqrt {- a^{2} c x^{2} + c} + 2 a^{2} c^{2} x^{2} \sqrt {- a^{2} c x^{2} + c} + a c^{2} x \sqrt {- a^{2} c x^{2} + c} - c^{2} \sqrt {- a^{2} c x^{2} + c}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.06, size = 56, normalized size = 0.76 \begin {gather*} -\frac {\sqrt {c-a^2\,c\,x^2}\,\left (2\,a^3\,x^3-4\,a^2\,x^2+a\,x+2\right )}{5\,a\,c^3\,{\left (a\,x-1\right )}^3\,\left (a\,x+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________