Optimal. Leaf size=61 \[ -\frac {\left (\frac {1}{17}-\frac {4 i}{17}\right ) 2^{2+\frac {i}{2}} c (1-i a x)^{2-\frac {i}{2}} \, _2F_1\left (-1-\frac {i}{2},2-\frac {i}{2};3-\frac {i}{2};\frac {1}{2} (1-i a x)\right )}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {5181, 71}
\begin {gather*} -\frac {\left (\frac {1}{17}-\frac {4 i}{17}\right ) 2^{2+\frac {i}{2}} c (1-i a x)^{2-\frac {i}{2}} \, _2F_1\left (-1-\frac {i}{2},2-\frac {i}{2};3-\frac {i}{2};\frac {1}{2} (1-i a x)\right )}{a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 71
Rule 5181
Rubi steps
\begin {align*} \int e^{-\tan ^{-1}(a x)} \left (c+a^2 c x^2\right ) \, dx &=c \int (1-i a x)^{1-\frac {i}{2}} (1+i a x)^{1+\frac {i}{2}} \, dx\\ &=-\frac {\left (\frac {1}{17}-\frac {4 i}{17}\right ) 2^{2+\frac {i}{2}} c (1-i a x)^{2-\frac {i}{2}} \, _2F_1\left (-1-\frac {i}{2},2-\frac {i}{2};3-\frac {i}{2};\frac {1}{2} (1-i a x)\right )}{a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 61, normalized size = 1.00 \begin {gather*} -\frac {\left (\frac {1}{17}-\frac {4 i}{17}\right ) 2^{2+\frac {i}{2}} c (1-i a x)^{2-\frac {i}{2}} \, _2F_1\left (-1-\frac {i}{2},2-\frac {i}{2};3-\frac {i}{2};\frac {1}{2} (1-i a x)\right )}{a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (a^{2} c \,x^{2}+c \right ) {\mathrm e}^{-\arctan \left (a x \right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [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]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} c \left (\int a^{2} x^{2} e^{- \operatorname {atan}{\left (a x \right )}}\, dx + \int e^{- \operatorname {atan}{\left (a x \right )}}\, dx\right ) \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 [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\mathrm {e}}^{-\mathrm {atan}\left (a\,x\right )}\,\left (c\,a^2\,x^2+c\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________