Optimal. Leaf size=114 \[ \frac{24 (a x+2) e^{2 \tan ^{-1}(a x)}}{377 a c^3 \sqrt{a^2 c x^2+c}}+\frac{20 (3 a x+2) e^{2 \tan ^{-1}(a x)}}{377 a c^2 \left (a^2 c x^2+c\right )^{3/2}}+\frac{(5 a x+2) e^{2 \tan ^{-1}(a x)}}{29 a c \left (a^2 c x^2+c\right )^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12535, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {5070, 5069} \[ \frac{24 (a x+2) e^{2 \tan ^{-1}(a x)}}{377 a c^3 \sqrt{a^2 c x^2+c}}+\frac{20 (3 a x+2) e^{2 \tan ^{-1}(a x)}}{377 a c^2 \left (a^2 c x^2+c\right )^{3/2}}+\frac{(5 a x+2) e^{2 \tan ^{-1}(a x)}}{29 a c \left (a^2 c x^2+c\right )^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5070
Rule 5069
Rubi steps
\begin{align*} \int \frac{e^{2 \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{7/2}} \, dx &=\frac{e^{2 \tan ^{-1}(a x)} (2+5 a x)}{29 a c \left (c+a^2 c x^2\right )^{5/2}}+\frac{20 \int \frac{e^{2 \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{29 c}\\ &=\frac{e^{2 \tan ^{-1}(a x)} (2+5 a x)}{29 a c \left (c+a^2 c x^2\right )^{5/2}}+\frac{20 e^{2 \tan ^{-1}(a x)} (2+3 a x)}{377 a c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac{120 \int \frac{e^{2 \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{377 c^2}\\ &=\frac{e^{2 \tan ^{-1}(a x)} (2+5 a x)}{29 a c \left (c+a^2 c x^2\right )^{5/2}}+\frac{20 e^{2 \tan ^{-1}(a x)} (2+3 a x)}{377 a c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac{24 e^{2 \tan ^{-1}(a x)} (2+a x)}{377 a c^3 \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0378807, size = 81, normalized size = 0.71 \[ \frac{\left (24 a^5 x^5+48 a^4 x^4+108 a^3 x^3+136 a^2 x^2+149 a x+114\right ) e^{2 \tan ^{-1}(a x)}}{377 a c^3 \left (a^2 x^2+1\right )^2 \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.037, size = 72, normalized size = 0.6 \begin{align*}{\frac{ \left ({a}^{2}{x}^{2}+1 \right ) \left ( 24\,{a}^{5}{x}^{5}+48\,{a}^{4}{x}^{4}+108\,{a}^{3}{x}^{3}+136\,{a}^{2}{x}^{2}+149\,ax+114 \right ){{\rm e}^{2\,\arctan \left ( ax \right ) }}}{377\,a} \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (2 \, \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.86729, size = 228, normalized size = 2. \begin{align*} \frac{{\left (24 \, a^{5} x^{5} + 48 \, a^{4} x^{4} + 108 \, a^{3} x^{3} + 136 \, a^{2} x^{2} + 149 \, a x + 114\right )} \sqrt{a^{2} c x^{2} + c} e^{\left (2 \, \arctan \left (a x\right )\right )}}{377 \,{\left (a^{7} c^{4} x^{6} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{3} c^{4} x^{2} + a c^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (2 \, \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]