Optimal. Leaf size=120 \[ -\frac{2^{\frac{3}{2}-\frac{i n}{2}} \sqrt{a^2 c x^2+c} (1-i a x)^{\frac{1}{2} (3+i n)} \, _2F_1\left (\frac{1}{2} (i n-1),\frac{1}{2} (i n+3);\frac{1}{2} (i n+5);\frac{1}{2} (1-i a x)\right )}{a (-n+3 i) \sqrt{a^2 x^2+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0966217, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {5076, 5073, 69} \[ -\frac{2^{\frac{3}{2}-\frac{i n}{2}} \sqrt{a^2 c x^2+c} (1-i a x)^{\frac{1}{2} (3+i n)} \, _2F_1\left (\frac{1}{2} (i n-1),\frac{1}{2} (i n+3);\frac{1}{2} (i n+5);\frac{1}{2} (1-i a x)\right )}{a (-n+3 i) \sqrt{a^2 x^2+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5076
Rule 5073
Rule 69
Rubi steps
\begin{align*} \int e^{n \tan ^{-1}(a x)} \sqrt{c+a^2 c x^2} \, dx &=\frac{\sqrt{c+a^2 c x^2} \int e^{n \tan ^{-1}(a x)} \sqrt{1+a^2 x^2} \, dx}{\sqrt{1+a^2 x^2}}\\ &=\frac{\sqrt{c+a^2 c x^2} \int (1-i a x)^{\frac{1}{2}+\frac{i n}{2}} (1+i a x)^{\frac{1}{2}-\frac{i n}{2}} \, dx}{\sqrt{1+a^2 x^2}}\\ &=-\frac{2^{\frac{3}{2}-\frac{i n}{2}} (1-i a x)^{\frac{1}{2} (3+i n)} \sqrt{c+a^2 c x^2} \, _2F_1\left (\frac{1}{2} (-1+i n),\frac{1}{2} (3+i n);\frac{1}{2} (5+i n);\frac{1}{2} (1-i a x)\right )}{a (3 i-n) \sqrt{1+a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.052151, size = 117, normalized size = 0.98 \[ \frac{2^{\frac{3}{2}-\frac{i n}{2}} \sqrt{a^2 c x^2+c} (1-i a x)^{\frac{3}{2}+\frac{i n}{2}} \, _2F_1\left (\frac{1}{2} (i n+3),\frac{1}{2} i (n+i);\frac{1}{2} (i n+5);\frac{1}{2} (1-i a x)\right )}{a (n-3 i) \sqrt{a^2 x^2+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.284, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n\arctan \left ( ax \right ) }}\sqrt{{a}^{2}c{x}^{2}+c}\, dx \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 \sqrt{a^{2} c x^{2} + c} e^{\left (n \arctan \left (a x\right )\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{a^{2} c x^{2} + c} e^{\left (n \arctan \left (a x\right )\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{c \left (a^{2} x^{2} + 1\right )} e^{n \operatorname{atan}{\left (a x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]