∫ e2ArcTan(ax) ____________________(c+a2 cx2)5∕2 dx [271]

Optimal. Leaf size=76 \[ \frac {e^{2 \text {ArcTan}(a x)} (2+3 a x)}{13 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {6 e^{2 \text {ArcTan}(a x)} (2+a x)}{65 a c^2 \sqrt {c+a^2 c x^2}} \]

1/13*exp(2*arctan(a*x))*(3*a*x+2)/a/c/(a^2*c*x^2+c)^(3/2)+6/65*exp(2*arctan(a*x))*(a*x+2)/a/c^2/(a^2*c*x^2+c)^

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Rubi [A]
time = 0.06, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {5178, 5177} \begin {gather*} \frac {6 (a x+2) e^{2 \text {ArcTan}(a x)}}{65 a c^2 \sqrt {a^2 c x^2+c}}+\frac {(3 a x+2) e^{2 \text {ArcTan}(a x)}}{13 a c \left (a^2 c x^2+c\right )^{3/2}} \end {gather*}

Int[E^(2*ArcTan[a*x])/(c + a^2*c*x^2)^(5/2),x]

(E^(2*ArcTan[a*x])*(2 + 3*a*x))/(13*a*c*(c + a^2*c*x^2)^(3/2)) + (6*E^(2*ArcTan[a*x])*(2 + a*x))/(65*a*c^2*Sqr

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(n + a*x)*(E^(n*ArcTan[a*x])/(

a*c*(n^2 + 1)*Sqrt[c + d*x^2])), x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && !IntegerQ[I*n]

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(n - 2*a*(p + 1)*x)*(c + d*x^2)

^(p + 1)*(E^(n*ArcTan[a*x])/(a*c*(n^2 + 4*(p + 1)^2))), x] + Dist[2*(p + 1)*((2*p + 3)/(c*(n^2 + 4*(p + 1)^2))

), Int[(c + d*x^2)^(p + 1)*E^(n*ArcTan[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && LtQ[p, -1]

&& !IntegerQ[I*n] && NeQ[n^2 + 4*(p + 1)^2, 0] && IntegerQ[2*p]

\begin {align*} \int \frac {e^{2 \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx &=\frac {e^{2 \tan ^{-1}(a x)} (2+3 a x)}{13 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {6 \int \frac {e^{2 \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{13 c}\\ &=\frac {e^{2 \tan ^{-1}(a x)} (2+3 a x)}{13 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {6 e^{2 \tan ^{-1}(a x)} (2+a x)}{65 a c^2 \sqrt {c+a^2 c x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 62, normalized size = 0.82 \begin {gather*} \frac {e^{2 \text {ArcTan}(a x)} \left (22+21 a x+12 a^2 x^2+6 a^3 x^3\right )}{65 c^2 \left (a+a^3 x^2\right ) \sqrt {c+a^2 c x^2}} \end {gather*}

Integrate[E^(2*ArcTan[a*x])/(c + a^2*c*x^2)^(5/2),x]

(E^(2*ArcTan[a*x])*(22 + 21*a*x + 12*a^2*x^2 + 6*a^3*x^3))/(65*c^2*(a + a^3*x^2)*Sqrt[c + a^2*c*x^2])

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method	result	size

gosper	\(\frac {\left (a^{2} x^{2}+1\right ) \left (6 a^{3} x^{3}+12 a^{2} x^{2}+21 a x +22\right ) {\mathrm e}^{2 \arctan \left (a x \right )}}{65 a \left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}\)	\(56\)

int(exp(2*arctan(a*x))/(a^2*c*x^2+c)^(5/2),x,method=_RETURNVERBOSE)

1/65*(a^2*x^2+1)*(6*a^3*x^3+12*a^2*x^2+21*a*x+22)*exp(2*arctan(a*x))/a/(a^2*c*x^2+c)^(5/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

integrate(exp(2*arctan(a*x))/(a^2*c*x^2+c)^(5/2),x, algorithm="maxima")

integrate(e^(2*arctan(a*x))/(a^2*c*x^2 + c)^(5/2), x)

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Fricas [A]
time = 3.27, size = 72, normalized size = 0.95 \begin {gather*} \frac {{\left (6 \, a^{3} x^{3} + 12 \, a^{2} x^{2} + 21 \, a x + 22\right )} \sqrt {a^{2} c x^{2} + c} e^{\left (2 \, \arctan \left (a x\right )\right )}}{65 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )}} \end {gather*}

integrate(exp(2*arctan(a*x))/(a^2*c*x^2+c)^(5/2),x, algorithm="fricas")

1/65*(6*a^3*x^3 + 12*a^2*x^2 + 21*a*x + 22)*sqrt(a^2*c*x^2 + c)*e^(2*arctan(a*x))/(a^5*c^3*x^4 + 2*a^3*c^3*x^2

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{2 \operatorname {atan}{\left (a x \right )}}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}

integrate(exp(2*atan(a*x))/(a**2*c*x**2+c)**(5/2),x)

Integral(exp(2*atan(a*x))/(c*(a**2*x**2 + 1))**(5/2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

integrate(exp(2*arctan(a*x))/(a^2*c*x^2+c)^(5/2),x, algorithm="giac")

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Mupad [B]
time = 0.22, size = 80, normalized size = 1.05 \begin {gather*} \frac {{\mathrm {e}}^{2\,\mathrm {atan}\left (a\,x\right )}\,\left (\frac {22}{65\,a^3\,c^2}+\frac {6\,x^3}{65\,c^2}+\frac {21\,x}{65\,a^2\,c^2}+\frac {12\,x^2}{65\,a\,c^2}\right )}{\frac {\sqrt {c\,a^2\,x^2+c}}{a^2}+x^2\,\sqrt {c\,a^2\,x^2+c}} \end {gather*}

int(exp(2*atan(a*x))/(c + a^2*c*x^2)^(5/2),x)

(exp(2*atan(a*x))*(22/(65*a^3*c^2) + (6*x^3)/(65*c^2) + (21*x)/(65*a^2*c^2) + (12*x^2)/(65*a*c^2)))/((c + a^2*

c*x^2)^(1/2)/a^2 + x^2*(c + a^2*c*x^2)^(1/2))

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3.3.71 \(\int \frac {e^{2 \text {ArcTan}(a x)}}{(c+a^2 c x^2)^{5/2}} \, dx\) [271]