3.347 \(\int \frac{e^{n \tan ^{-1}(a x)}}{(c+a^2 c x^2)^4} \, dx\)

Optimal. Leaf size=181 \[ \frac{360 (2 a x+n) e^{n \tan ^{-1}(a x)}}{a c^4 \left (n^2+4\right ) \left (n^2+16\right ) \left (n^2+36\right ) \left (a^2 x^2+1\right )}+\frac{30 (4 a x+n) e^{n \tan ^{-1}(a x)}}{a c^4 \left (n^2+16\right ) \left (n^2+36\right ) \left (a^2 x^2+1\right )^2}+\frac{(6 a x+n) e^{n \tan ^{-1}(a x)}}{a c^4 \left (n^2+36\right ) \left (a^2 x^2+1\right )^3}+\frac{720 e^{n \tan ^{-1}(a x)}}{a c^4 n \left (n^2+4\right ) \left (n^2+16\right ) \left (n^2+36\right )} \]

[Out]

(720*E^(n*ArcTan[a*x]))/(a*c^4*n*(4 + n^2)*(16 + n^2)*(36 + n^2)) + (E^(n*ArcTan[a*x])*(n + 6*a*x))/(a*c^4*(36
+ n^2)*(1 + a^2*x^2)^3) + (30*E^(n*ArcTan[a*x])*(n + 4*a*x))/(a*c^4*(16 + n^2)*(36 + n^2)*(1 + a^2*x^2)^2) +
(360*E^(n*ArcTan[a*x])*(n + 2*a*x))/(a*c^4*(4 + n^2)*(16 + n^2)*(36 + n^2)*(1 + a^2*x^2))

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Rubi [A]  time = 0.175117, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {5070, 5071} \[ \frac{360 (2 a x+n) e^{n \tan ^{-1}(a x)}}{a c^4 \left (n^2+4\right ) \left (n^2+16\right ) \left (n^2+36\right ) \left (a^2 x^2+1\right )}+\frac{30 (4 a x+n) e^{n \tan ^{-1}(a x)}}{a c^4 \left (n^2+16\right ) \left (n^2+36\right ) \left (a^2 x^2+1\right )^2}+\frac{(6 a x+n) e^{n \tan ^{-1}(a x)}}{a c^4 \left (n^2+36\right ) \left (a^2 x^2+1\right )^3}+\frac{720 e^{n \tan ^{-1}(a x)}}{a c^4 n \left (n^2+4\right ) \left (n^2+16\right ) \left (n^2+36\right )} \]

Antiderivative was successfully verified.

[In]

Int[E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^4,x]

[Out]

(720*E^(n*ArcTan[a*x]))/(a*c^4*n*(4 + n^2)*(16 + n^2)*(36 + n^2)) + (E^(n*ArcTan[a*x])*(n + 6*a*x))/(a*c^4*(36
+ n^2)*(1 + a^2*x^2)^3) + (30*E^(n*ArcTan[a*x])*(n + 4*a*x))/(a*c^4*(16 + n^2)*(36 + n^2)*(1 + a^2*x^2)^2) +
(360*E^(n*ArcTan[a*x])*(n + 2*a*x))/(a*c^4*(4 + n^2)*(16 + n^2)*(36 + n^2)*(1 + a^2*x^2))

Rule 5070

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]

Rule 5071

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[E^(n*ArcTan[a*x])/(a*c*n), x] /; Fre
eQ[{a, c, d, n}, x] && EqQ[d, a^2*c]

Rubi steps

\begin{align*} \int \frac{e^{n \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^4} \, dx &=\frac{e^{n \tan ^{-1}(a x)} (n+6 a x)}{a c^4 \left (36+n^2\right ) \left (1+a^2 x^2\right )^3}+\frac{30 \int \frac{e^{n \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^3} \, dx}{c \left (36+n^2\right )}\\ &=\frac{e^{n \tan ^{-1}(a x)} (n+6 a x)}{a c^4 \left (36+n^2\right ) \left (1+a^2 x^2\right )^3}+\frac{30 e^{n \tan ^{-1}(a x)} (n+4 a x)}{a c^4 \left (16+n^2\right ) \left (36+n^2\right ) \left (1+a^2 x^2\right )^2}+\frac{360 \int \frac{e^{n \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{c^2 \left (16+n^2\right ) \left (36+n^2\right )}\\ &=\frac{e^{n \tan ^{-1}(a x)} (n+6 a x)}{a c^4 \left (36+n^2\right ) \left (1+a^2 x^2\right )^3}+\frac{30 e^{n \tan ^{-1}(a x)} (n+4 a x)}{a c^4 \left (16+n^2\right ) \left (36+n^2\right ) \left (1+a^2 x^2\right )^2}+\frac{360 e^{n \tan ^{-1}(a x)} (n+2 a x)}{a c^4 \left (4+n^2\right ) \left (16+n^2\right ) \left (36+n^2\right ) \left (1+a^2 x^2\right )}+\frac{720 \int \frac{e^{n \tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx}{c^3 \left (4+n^2\right ) \left (16+n^2\right ) \left (36+n^2\right )}\\ &=\frac{720 e^{n \tan ^{-1}(a x)}}{a c^4 n \left (4+n^2\right ) \left (16+n^2\right ) \left (36+n^2\right )}+\frac{e^{n \tan ^{-1}(a x)} (n+6 a x)}{a c^4 \left (36+n^2\right ) \left (1+a^2 x^2\right )^3}+\frac{30 e^{n \tan ^{-1}(a x)} (n+4 a x)}{a c^4 \left (16+n^2\right ) \left (36+n^2\right ) \left (1+a^2 x^2\right )^2}+\frac{360 e^{n \tan ^{-1}(a x)} (n+2 a x)}{a c^4 \left (4+n^2\right ) \left (16+n^2\right ) \left (36+n^2\right ) \left (1+a^2 x^2\right )}\\ \end{align*}

Mathematica [C]  time = 0.440163, size = 165, normalized size = 0.91 \[ \frac{(6 a x+n) e^{n \tan ^{-1}(a x)}+\frac{30 \left (a^2 c x^2+c\right ) \left (12 (a x-i) (a x+i) (1-i a x)^{\frac{i n}{2}} \left (2 a^2 x^2+2 a n x+n^2+2\right ) (1+i a x)^{-\frac{i n}{2}}+n (n-2 i) (n+2 i) (4 a x+n) e^{n \tan ^{-1}(a x)}\right )}{c n \left (n^4+20 n^2+64\right )}}{a c \left (n^2+36\right ) \left (a^2 c x^2+c\right )^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^4,x]

[Out]

(E^(n*ArcTan[a*x])*(n + 6*a*x) + (30*(c + a^2*c*x^2)*(E^(n*ArcTan[a*x])*n*(-2*I + n)*(2*I + n)*(n + 4*a*x) + (
12*(1 - I*a*x)^((I/2)*n)*(-I + a*x)*(I + a*x)*(2 + n^2 + 2*a*n*x + 2*a^2*x^2))/(1 + I*a*x)^((I/2)*n)))/(c*n*(6
4 + 20*n^2 + n^4)))/(a*c*(36 + n^2)*(c + a^2*c*x^2)^3)

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Maple [A]  time = 0.04, size = 166, normalized size = 0.9 \begin{align*}{\frac{ \left ( 720\,{a}^{6}{x}^{6}+720\,{a}^{5}{x}^{5}n+360\,{a}^{4}{n}^{2}{x}^{4}+120\,{a}^{3}{n}^{3}{x}^{3}+2160\,{a}^{4}{x}^{4}+30\,{a}^{2}{n}^{4}{x}^{2}+1920\,{a}^{3}{x}^{3}n+6\,a{n}^{5}x+840\,{a}^{2}{n}^{2}{x}^{2}+{n}^{6}+240\,a{n}^{3}x+2160\,{a}^{2}{x}^{2}+50\,{n}^{4}+1584\,xna+544\,{n}^{2}+720 \right ){{\rm e}^{n\arctan \left ( ax \right ) }}}{ \left ({a}^{2}{x}^{2}+1 \right ) ^{3}{c}^{4}an \left ({n}^{6}+56\,{n}^{4}+784\,{n}^{2}+2304 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*arctan(a*x))/(a^2*c*x^2+c)^4,x)

[Out]

(720*a^6*x^6+720*a^5*n*x^5+360*a^4*n^2*x^4+120*a^3*n^3*x^3+2160*a^4*x^4+30*a^2*n^4*x^2+1920*a^3*n*x^3+6*a*n^5*
x+840*a^2*n^2*x^2+n^6+240*a*n^3*x+2160*a^2*x^2+50*n^4+1584*a*n*x+544*n^2+720)*exp(n*arctan(a*x))/(a^2*x^2+1)^3
/c^4/a/n/(n^6+56*n^4+784*n^2+2304)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctan(a*x))/(a^2*c*x^2+c)^4,x, algorithm="maxima")

[Out]

integrate(e^(n*arctan(a*x))/(a^2*c*x^2 + c)^4, x)

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Fricas [A]  time = 2.00418, size = 666, normalized size = 3.68 \begin{align*} \frac{{\left (720 \, a^{6} x^{6} + 720 \, a^{5} n x^{5} + n^{6} + 360 \,{\left (a^{4} n^{2} + 6 \, a^{4}\right )} x^{4} + 50 \, n^{4} + 120 \,{\left (a^{3} n^{3} + 16 \, a^{3} n\right )} x^{3} + 30 \,{\left (a^{2} n^{4} + 28 \, a^{2} n^{2} + 72 \, a^{2}\right )} x^{2} + 544 \, n^{2} + 6 \,{\left (a n^{5} + 40 \, a n^{3} + 264 \, a n\right )} x + 720\right )} e^{\left (n \arctan \left (a x\right )\right )}}{a c^{4} n^{7} + 56 \, a c^{4} n^{5} + 784 \, a c^{4} n^{3} +{\left (a^{7} c^{4} n^{7} + 56 \, a^{7} c^{4} n^{5} + 784 \, a^{7} c^{4} n^{3} + 2304 \, a^{7} c^{4} n\right )} x^{6} + 2304 \, a c^{4} n + 3 \,{\left (a^{5} c^{4} n^{7} + 56 \, a^{5} c^{4} n^{5} + 784 \, a^{5} c^{4} n^{3} + 2304 \, a^{5} c^{4} n\right )} x^{4} + 3 \,{\left (a^{3} c^{4} n^{7} + 56 \, a^{3} c^{4} n^{5} + 784 \, a^{3} c^{4} n^{3} + 2304 \, a^{3} c^{4} n\right )} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctan(a*x))/(a^2*c*x^2+c)^4,x, algorithm="fricas")

[Out]

(720*a^6*x^6 + 720*a^5*n*x^5 + n^6 + 360*(a^4*n^2 + 6*a^4)*x^4 + 50*n^4 + 120*(a^3*n^3 + 16*a^3*n)*x^3 + 30*(a
^2*n^4 + 28*a^2*n^2 + 72*a^2)*x^2 + 544*n^2 + 6*(a*n^5 + 40*a*n^3 + 264*a*n)*x + 720)*e^(n*arctan(a*x))/(a*c^4
*n^7 + 56*a*c^4*n^5 + 784*a*c^4*n^3 + (a^7*c^4*n^7 + 56*a^7*c^4*n^5 + 784*a^7*c^4*n^3 + 2304*a^7*c^4*n)*x^6 +
2304*a*c^4*n + 3*(a^5*c^4*n^7 + 56*a^5*c^4*n^5 + 784*a^5*c^4*n^3 + 2304*a^5*c^4*n)*x^4 + 3*(a^3*c^4*n^7 + 56*a
^3*c^4*n^5 + 784*a^3*c^4*n^3 + 2304*a^3*c^4*n)*x^2)

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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]

integrate(exp(n*atan(a*x))/(a**2*c*x**2+c)**4,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctan(a*x))/(a^2*c*x^2+c)^4,x, algorithm="giac")

[Out]

integrate(e^(n*arctan(a*x))/(a^2*c*x^2 + c)^4, x)