3.252 \(\int \frac {e^{\tan ^{-1}(a x)}}{(c+a^2 c x^2)^5} \, dx\)

Optimal. Leaf size=149 \[ \frac {4032 (2 a x+1) e^{\tan ^{-1}(a x)}}{40885 a c^5 \left (a^2 x^2+1\right )}+\frac {336 (4 a x+1) e^{\tan ^{-1}(a x)}}{8177 a c^5 \left (a^2 x^2+1\right )^2}+\frac {56 (6 a x+1) e^{\tan ^{-1}(a x)}}{2405 a c^5 \left (a^2 x^2+1\right )^3}+\frac {(8 a x+1) e^{\tan ^{-1}(a x)}}{65 a c^5 \left (a^2 x^2+1\right )^4}+\frac {8064 e^{\tan ^{-1}(a x)}}{40885 a c^5} \]

[Out]

8064/40885*exp(arctan(a*x))/a/c^5+1/65*exp(arctan(a*x))*(8*a*x+1)/a/c^5/(a^2*x^2+1)^4+56/2405*exp(arctan(a*x))
*(6*a*x+1)/a/c^5/(a^2*x^2+1)^3+336/8177*exp(arctan(a*x))*(4*a*x+1)/a/c^5/(a^2*x^2+1)^2+4032/40885*exp(arctan(a
*x))*(2*a*x+1)/a/c^5/(a^2*x^2+1)

________________________________________________________________________________________

Rubi [A]  time = 0.14, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {5070, 5071} \[ \frac {4032 (2 a x+1) e^{\tan ^{-1}(a x)}}{40885 a c^5 \left (a^2 x^2+1\right )}+\frac {336 (4 a x+1) e^{\tan ^{-1}(a x)}}{8177 a c^5 \left (a^2 x^2+1\right )^2}+\frac {56 (6 a x+1) e^{\tan ^{-1}(a x)}}{2405 a c^5 \left (a^2 x^2+1\right )^3}+\frac {(8 a x+1) e^{\tan ^{-1}(a x)}}{65 a c^5 \left (a^2 x^2+1\right )^4}+\frac {8064 e^{\tan ^{-1}(a x)}}{40885 a c^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8064*E^ArcTan[a*x])/(40885*a*c^5) + (E^ArcTan[a*x]*(1 + 8*a*x))/(65*a*c^5*(1 + a^2*x^2)^4) + (56*E^ArcTan[a*x
]*(1 + 6*a*x))/(2405*a*c^5*(1 + a^2*x^2)^3) + (336*E^ArcTan[a*x]*(1 + 4*a*x))/(8177*a*c^5*(1 + a^2*x^2)^2) + (
4032*E^ArcTan[a*x]*(1 + 2*a*x))/(40885*a*c^5*(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^{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^5} \, dx &=\frac {e^{\tan ^{-1}(a x)} (1+8 a x)}{65 a c^5 \left (1+a^2 x^2\right )^4}+\frac {56 \int \frac {e^{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^4} \, dx}{65 c}\\ &=\frac {e^{\tan ^{-1}(a x)} (1+8 a x)}{65 a c^5 \left (1+a^2 x^2\right )^4}+\frac {56 e^{\tan ^{-1}(a x)} (1+6 a x)}{2405 a c^5 \left (1+a^2 x^2\right )^3}+\frac {336 \int \frac {e^{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^3} \, dx}{481 c^2}\\ &=\frac {e^{\tan ^{-1}(a x)} (1+8 a x)}{65 a c^5 \left (1+a^2 x^2\right )^4}+\frac {56 e^{\tan ^{-1}(a x)} (1+6 a x)}{2405 a c^5 \left (1+a^2 x^2\right )^3}+\frac {336 e^{\tan ^{-1}(a x)} (1+4 a x)}{8177 a c^5 \left (1+a^2 x^2\right )^2}+\frac {4032 \int \frac {e^{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{8177 c^3}\\ &=\frac {e^{\tan ^{-1}(a x)} (1+8 a x)}{65 a c^5 \left (1+a^2 x^2\right )^4}+\frac {56 e^{\tan ^{-1}(a x)} (1+6 a x)}{2405 a c^5 \left (1+a^2 x^2\right )^3}+\frac {336 e^{\tan ^{-1}(a x)} (1+4 a x)}{8177 a c^5 \left (1+a^2 x^2\right )^2}+\frac {4032 e^{\tan ^{-1}(a x)} (1+2 a x)}{40885 a c^5 \left (1+a^2 x^2\right )}+\frac {8064 \int \frac {e^{\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx}{40885 c^4}\\ &=\frac {8064 e^{\tan ^{-1}(a x)}}{40885 a c^5}+\frac {e^{\tan ^{-1}(a x)} (1+8 a x)}{65 a c^5 \left (1+a^2 x^2\right )^4}+\frac {56 e^{\tan ^{-1}(a x)} (1+6 a x)}{2405 a c^5 \left (1+a^2 x^2\right )^3}+\frac {336 e^{\tan ^{-1}(a x)} (1+4 a x)}{8177 a c^5 \left (1+a^2 x^2\right )^2}+\frac {4032 e^{\tan ^{-1}(a x)} (1+2 a x)}{40885 a c^5 \left (1+a^2 x^2\right )}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.32, size = 153, normalized size = 1.03 \[ \frac {629 (8 a x+1) e^{\tan ^{-1}(a x)}+\frac {56 \left (a^2 c x^2+c\right ) \left (17 c (6 a x+1) e^{\tan ^{-1}(a x)}+6 \left (a^2 c x^2+c\right ) \left (5 (4 a x+1) e^{\tan ^{-1}(a x)}+12 (1-i a x)^{\frac {i}{2}} (1+i a x)^{-\frac {i}{2}} (a x-i) (a x+i) \left (2 a^2 x^2+2 a x+3\right )\right )\right )}{c^2}}{40885 a c \left (a^2 c x^2+c\right )^4} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(629*E^ArcTan[a*x]*(1 + 8*a*x) + (56*(c + a^2*c*x^2)*(17*c*E^ArcTan[a*x]*(1 + 6*a*x) + 6*(c + a^2*c*x^2)*(5*E^
ArcTan[a*x]*(1 + 4*a*x) + (12*(1 - I*a*x)^(I/2)*(-I + a*x)*(I + a*x)*(3 + 2*a*x + 2*a^2*x^2))/(1 + I*a*x)^(I/2
))))/c^2)/(40885*a*c*(c + a^2*c*x^2)^4)

________________________________________________________________________________________

fricas [A]  time = 0.52, size = 120, normalized size = 0.81 \[ \frac {{\left (8064 \, a^{8} x^{8} + 8064 \, a^{7} x^{7} + 36288 \, a^{6} x^{6} + 30912 \, a^{5} x^{5} + 62160 \, a^{4} x^{4} + 43344 \, a^{3} x^{3} + 48664 \, a^{2} x^{2} + 25528 \, a x + 15357\right )} e^{\left (\arctan \left (a x\right )\right )}}{40885 \, {\left (a^{9} c^{5} x^{8} + 4 \, a^{7} c^{5} x^{6} + 6 \, a^{5} c^{5} x^{4} + 4 \, a^{3} c^{5} x^{2} + a c^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/40885*(8064*a^8*x^8 + 8064*a^7*x^7 + 36288*a^6*x^6 + 30912*a^5*x^5 + 62160*a^4*x^4 + 43344*a^3*x^3 + 48664*a
^2*x^2 + 25528*a*x + 15357)*e^(arctan(a*x))/(a^9*c^5*x^8 + 4*a^7*c^5*x^6 + 6*a^5*c^5*x^4 + 4*a^3*c^5*x^2 + a*c
^5)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

maple [A]  time = 0.04, size = 87, normalized size = 0.58 \[ \frac {{\mathrm e}^{\arctan \left (a x \right )} \left (8064 a^{8} x^{8}+8064 a^{7} x^{7}+36288 a^{6} x^{6}+30912 a^{5} x^{5}+62160 a^{4} x^{4}+43344 a^{3} x^{3}+48664 a^{2} x^{2}+25528 a x +15357\right )}{40885 \left (a^{2} x^{2}+1\right )^{4} c^{5} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/40885*exp(arctan(a*x))*(8064*a^8*x^8+8064*a^7*x^7+36288*a^6*x^6+30912*a^5*x^5+62160*a^4*x^4+43344*a^3*x^3+48
664*a^2*x^2+25528*a*x+15357)/(a^2*x^2+1)^4/c^5/a

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\left (\arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.74, size = 134, normalized size = 0.90 \[ \frac {{\mathrm {e}}^{\mathrm {atan}\left (a\,x\right )}\,\left (\frac {15357}{40885\,a^9\,c^5}+\frac {25528\,x}{40885\,a^8\,c^5}+\frac {8064\,x^8}{40885\,a\,c^5}+\frac {8064\,x^7}{40885\,a^2\,c^5}+\frac {36288\,x^6}{40885\,a^3\,c^5}+\frac {30912\,x^5}{40885\,a^4\,c^5}+\frac {336\,x^4}{221\,a^5\,c^5}+\frac {43344\,x^3}{40885\,a^6\,c^5}+\frac {48664\,x^2}{40885\,a^7\,c^5}\right )}{\frac {1}{a^8}+x^8+\frac {4\,x^6}{a^2}+\frac {6\,x^4}{a^4}+\frac {4\,x^2}{a^6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(exp(atan(a*x))*(15357/(40885*a^9*c^5) + (25528*x)/(40885*a^8*c^5) + (8064*x^8)/(40885*a*c^5) + (8064*x^7)/(40
885*a^2*c^5) + (36288*x^6)/(40885*a^3*c^5) + (30912*x^5)/(40885*a^4*c^5) + (336*x^4)/(221*a^5*c^5) + (43344*x^
3)/(40885*a^6*c^5) + (48664*x^2)/(40885*a^7*c^5)))/(1/a^8 + x^8 + (4*x^6)/a^2 + (6*x^4)/a^4 + (4*x^2)/a^6)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \begin {cases} \frac {8064 a^{8} x^{8} e^{\operatorname {atan}{\left (a x \right )}}}{40885 a^{9} c^{5} x^{8} + 163540 a^{7} c^{5} x^{6} + 245310 a^{5} c^{5} x^{4} + 163540 a^{3} c^{5} x^{2} + 40885 a c^{5}} + \frac {8064 a^{7} x^{7} e^{\operatorname {atan}{\left (a x \right )}}}{40885 a^{9} c^{5} x^{8} + 163540 a^{7} c^{5} x^{6} + 245310 a^{5} c^{5} x^{4} + 163540 a^{3} c^{5} x^{2} + 40885 a c^{5}} + \frac {36288 a^{6} x^{6} e^{\operatorname {atan}{\left (a x \right )}}}{40885 a^{9} c^{5} x^{8} + 163540 a^{7} c^{5} x^{6} + 245310 a^{5} c^{5} x^{4} + 163540 a^{3} c^{5} x^{2} + 40885 a c^{5}} + \frac {30912 a^{5} x^{5} e^{\operatorname {atan}{\left (a x \right )}}}{40885 a^{9} c^{5} x^{8} + 163540 a^{7} c^{5} x^{6} + 245310 a^{5} c^{5} x^{4} + 163540 a^{3} c^{5} x^{2} + 40885 a c^{5}} + \frac {62160 a^{4} x^{4} e^{\operatorname {atan}{\left (a x \right )}}}{40885 a^{9} c^{5} x^{8} + 163540 a^{7} c^{5} x^{6} + 245310 a^{5} c^{5} x^{4} + 163540 a^{3} c^{5} x^{2} + 40885 a c^{5}} + \frac {43344 a^{3} x^{3} e^{\operatorname {atan}{\left (a x \right )}}}{40885 a^{9} c^{5} x^{8} + 163540 a^{7} c^{5} x^{6} + 245310 a^{5} c^{5} x^{4} + 163540 a^{3} c^{5} x^{2} + 40885 a c^{5}} + \frac {48664 a^{2} x^{2} e^{\operatorname {atan}{\left (a x \right )}}}{40885 a^{9} c^{5} x^{8} + 163540 a^{7} c^{5} x^{6} + 245310 a^{5} c^{5} x^{4} + 163540 a^{3} c^{5} x^{2} + 40885 a c^{5}} + \frac {25528 a x e^{\operatorname {atan}{\left (a x \right )}}}{40885 a^{9} c^{5} x^{8} + 163540 a^{7} c^{5} x^{6} + 245310 a^{5} c^{5} x^{4} + 163540 a^{3} c^{5} x^{2} + 40885 a c^{5}} + \frac {15357 e^{\operatorname {atan}{\left (a x \right )}}}{40885 a^{9} c^{5} x^{8} + 163540 a^{7} c^{5} x^{6} + 245310 a^{5} c^{5} x^{4} + 163540 a^{3} c^{5} x^{2} + 40885 a c^{5}} & \text {for}\: c \neq 0 \\\tilde {\infty } \int e^{\operatorname {atan}{\left (a x \right )}}\, dx & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((8064*a**8*x**8*exp(atan(a*x))/(40885*a**9*c**5*x**8 + 163540*a**7*c**5*x**6 + 245310*a**5*c**5*x**4
+ 163540*a**3*c**5*x**2 + 40885*a*c**5) + 8064*a**7*x**7*exp(atan(a*x))/(40885*a**9*c**5*x**8 + 163540*a**7*c
**5*x**6 + 245310*a**5*c**5*x**4 + 163540*a**3*c**5*x**2 + 40885*a*c**5) + 36288*a**6*x**6*exp(atan(a*x))/(408
85*a**9*c**5*x**8 + 163540*a**7*c**5*x**6 + 245310*a**5*c**5*x**4 + 163540*a**3*c**5*x**2 + 40885*a*c**5) + 30
912*a**5*x**5*exp(atan(a*x))/(40885*a**9*c**5*x**8 + 163540*a**7*c**5*x**6 + 245310*a**5*c**5*x**4 + 163540*a*
*3*c**5*x**2 + 40885*a*c**5) + 62160*a**4*x**4*exp(atan(a*x))/(40885*a**9*c**5*x**8 + 163540*a**7*c**5*x**6 +
245310*a**5*c**5*x**4 + 163540*a**3*c**5*x**2 + 40885*a*c**5) + 43344*a**3*x**3*exp(atan(a*x))/(40885*a**9*c**
5*x**8 + 163540*a**7*c**5*x**6 + 245310*a**5*c**5*x**4 + 163540*a**3*c**5*x**2 + 40885*a*c**5) + 48664*a**2*x*
*2*exp(atan(a*x))/(40885*a**9*c**5*x**8 + 163540*a**7*c**5*x**6 + 245310*a**5*c**5*x**4 + 163540*a**3*c**5*x**
2 + 40885*a*c**5) + 25528*a*x*exp(atan(a*x))/(40885*a**9*c**5*x**8 + 163540*a**7*c**5*x**6 + 245310*a**5*c**5*
x**4 + 163540*a**3*c**5*x**2 + 40885*a*c**5) + 15357*exp(atan(a*x))/(40885*a**9*c**5*x**8 + 163540*a**7*c**5*x
**6 + 245310*a**5*c**5*x**4 + 163540*a**3*c**5*x**2 + 40885*a*c**5), Ne(c, 0)), (zoo*Integral(exp(atan(a*x)),
x), True))

________________________________________________________________________________________