Optimal. Leaf size=18 \[ \frac {2 \text {ArcTan}(a x)^{7/2}}{7 a c} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {5004}
\begin {gather*} \frac {2 \text {ArcTan}(a x)^{7/2}}{7 a c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 5004
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(a x)^{5/2}}{c+a^2 c x^2} \, dx &=\frac {2 \tan ^{-1}(a x)^{7/2}}{7 a c}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} \frac {2 \text {ArcTan}(a x)^{7/2}}{7 a c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.24, size = 15, normalized size = 0.83
method | result | size |
default | \(\frac {2 \arctan \left (a x \right )^{\frac {7}{2}}}{7 a c}\) | \(15\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.45, size = 14, normalized size = 0.78 \begin {gather*} \frac {2 \, \arctan \left (a x\right )^{\frac {7}{2}}}{7 \, a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 9.35, size = 15, normalized size = 0.83 \begin {gather*} \begin {cases} \frac {2 \operatorname {atan}^{\frac {7}{2}}{\left (a x \right )}}{7 a c} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.39, size = 14, normalized size = 0.78 \begin {gather*} \frac {2 \, \arctan \left (a x\right )^{\frac {7}{2}}}{7 \, a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.40, size = 14, normalized size = 0.78 \begin {gather*} \frac {2\,{\mathrm {atan}\left (a\,x\right )}^{7/2}}{7\,a\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________