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*exp(n*arctan(a*x))/a/c^4/n/(n^2+36)/(n^4+20*n^2+64)+exp(n*arctan(a*x))*(6*a*x+n)/a/c^4/(n^2+36)/(a^2*x^2+1
)^3+30*exp(n*arctan(a*x))*(4*a*x+n)/a/c^4/(n^2+16)/(n^2+36)/(a^2*x^2+1)^2+360*exp(n*arctan(a*x))*(2*a*x+n)/a/c
^4/(n^2+36)/(n^4+20*n^2+64)/(a^2*x^2+1)

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Rubi [A]  time = 0.18, antiderivative size = 181, normalized size of antiderivative = 1.00, 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*}

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Mathematica [C]  time = 0.52, 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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fricas [A]  time = 0.45, size = 298, normalized size = 1.65 \[ \frac {{\left (720 \, a^{6} x^{6} + 720 \, a^{5} n x^{5} + n^{6} + 360 \, {\left (a^{4} n^{2} + 6 \, a^{4}\right )} x^{4} + 50 \, n^{4} + 120 \, {\left (a^{3} n^{3} + 16 \, a^{3} n\right )} x^{3} + 30 \, {\left (a^{2} n^{4} + 28 \, a^{2} n^{2} + 72 \, a^{2}\right )} x^{2} + 544 \, n^{2} + 6 \, {\left (a n^{5} + 40 \, a n^{3} + 264 \, a n\right )} x + 720\right )} e^{\left (n \arctan \left (a x\right )\right )}}{a c^{4} n^{7} + 56 \, a c^{4} n^{5} + 784 \, a c^{4} n^{3} + {\left (a^{7} c^{4} n^{7} + 56 \, a^{7} c^{4} n^{5} + 784 \, a^{7} c^{4} n^{3} + 2304 \, a^{7} c^{4} n\right )} x^{6} + 2304 \, a c^{4} n + 3 \, {\left (a^{5} c^{4} n^{7} + 56 \, a^{5} c^{4} n^{5} + 784 \, a^{5} c^{4} n^{3} + 2304 \, a^{5} c^{4} n\right )} x^{4} + 3 \, {\left (a^{3} c^{4} n^{7} + 56 \, a^{3} c^{4} n^{5} + 784 \, a^{3} c^{4} n^{3} + 2304 \, a^{3} c^{4} n\right )} x^{2}} \]

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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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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]

Timed out

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maple [A]  time = 0.04, size = 166, normalized size = 0.92 \[ \frac {\left (720 a^{6} x^{6}+720 a^{5} x^{5} n +360 a^{4} n^{2} x^{4}+120 a^{3} n^{3} x^{3}+2160 a^{4} x^{4}+30 a^{2} n^{4} x^{2}+1920 x^{3} a^{3} n +6 a \,n^{5} x +840 a^{2} n^{2} x^{2}+n^{6}+240 a \,n^{3} x +2160 a^{2} x^{2}+50 n^{4}+1584 n a x +544 n^{2}+720\right ) {\mathrm e}^{n \arctan \left (a x \right )}}{\left (a^{2} x^{2}+1\right )^{3} c^{4} a n \left (n^{6}+56 n^{4}+784 n^{2}+2304\right )} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{4}}\,{d x} \]

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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mupad [B]  time = 0.87, size = 281, normalized size = 1.55 \[ \frac {{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}\,\left (\frac {720\,x^5}{a^2\,c^4\,\left (n^6+56\,n^4+784\,n^2+2304\right )}+\frac {n^6+50\,n^4+544\,n^2+720}{a^7\,c^4\,n\,\left (n^6+56\,n^4+784\,n^2+2304\right )}+\frac {720\,x^6}{a\,c^4\,n\,\left (n^6+56\,n^4+784\,n^2+2304\right )}+\frac {6\,x\,\left (n^4+40\,n^2+264\right )}{a^6\,c^4\,\left (n^6+56\,n^4+784\,n^2+2304\right )}+\frac {120\,x^3\,\left (n^2+16\right )}{a^4\,c^4\,\left (n^6+56\,n^4+784\,n^2+2304\right )}+\frac {360\,x^4\,\left (n^2+6\right )}{a^3\,c^4\,n\,\left (n^6+56\,n^4+784\,n^2+2304\right )}+\frac {30\,x^2\,\left (n^4+28\,n^2+72\right )}{a^5\,c^4\,n\,\left (n^6+56\,n^4+784\,n^2+2304\right )}\right )}{\frac {1}{a^6}+x^6+\frac {3\,x^4}{a^2}+\frac {3\,x^2}{a^4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(exp(n*atan(a*x))*((720*x^5)/(a^2*c^4*(784*n^2 + 56*n^4 + n^6 + 2304)) + (544*n^2 + 50*n^4 + n^6 + 720)/(a^7*c
^4*n*(784*n^2 + 56*n^4 + n^6 + 2304)) + (720*x^6)/(a*c^4*n*(784*n^2 + 56*n^4 + n^6 + 2304)) + (6*x*(40*n^2 + n
^4 + 264))/(a^6*c^4*(784*n^2 + 56*n^4 + n^6 + 2304)) + (120*x^3*(n^2 + 16))/(a^4*c^4*(784*n^2 + 56*n^4 + n^6 +
2304)) + (360*x^4*(n^2 + 6))/(a^3*c^4*n*(784*n^2 + 56*n^4 + n^6 + 2304)) + (30*x^2*(28*n^2 + n^4 + 72))/(a^5*
c^4*n*(784*n^2 + 56*n^4 + n^6 + 2304))))/(1/a^6 + x^6 + (3*x^4)/a^2 + (3*x^2)/a^4)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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