3.250 \(\int \frac {e^{\tan ^{-1}(a x)}}{(c+a^2 c x^2)^3} \, dx\)

Optimal. Leaf size=83 \[ \frac {12 (2 a x+1) e^{\tan ^{-1}(a x)}}{85 a c^3 \left (a^2 x^2+1\right )}+\frac {(4 a x+1) e^{\tan ^{-1}(a x)}}{17 a c^3 \left (a^2 x^2+1\right )^2}+\frac {24 e^{\tan ^{-1}(a x)}}{85 a c^3} \]

[Out]

24/85*exp(arctan(a*x))/a/c^3+1/17*exp(arctan(a*x))*(4*a*x+1)/a/c^3/(a^2*x^2+1)^2+12/85*exp(arctan(a*x))*(2*a*x
+1)/a/c^3/(a^2*x^2+1)

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Rubi [A]  time = 0.08, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {5070, 5071} \[ \frac {12 (2 a x+1) e^{\tan ^{-1}(a x)}}{85 a c^3 \left (a^2 x^2+1\right )}+\frac {(4 a x+1) e^{\tan ^{-1}(a x)}}{17 a c^3 \left (a^2 x^2+1\right )^2}+\frac {24 e^{\tan ^{-1}(a x)}}{85 a c^3} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTan[a*x]/(c + a^2*c*x^2)^3,x]

[Out]

(24*E^ArcTan[a*x])/(85*a*c^3) + (E^ArcTan[a*x]*(1 + 4*a*x))/(17*a*c^3*(1 + a^2*x^2)^2) + (12*E^ArcTan[a*x]*(1
+ 2*a*x))/(85*a*c^3*(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 )^3} \, dx &=\frac {e^{\tan ^{-1}(a x)} (1+4 a x)}{17 a c^3 \left (1+a^2 x^2\right )^2}+\frac {12 \int \frac {e^{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{17 c}\\ &=\frac {e^{\tan ^{-1}(a x)} (1+4 a x)}{17 a c^3 \left (1+a^2 x^2\right )^2}+\frac {12 e^{\tan ^{-1}(a x)} (1+2 a x)}{85 a c^3 \left (1+a^2 x^2\right )}+\frac {24 \int \frac {e^{\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx}{85 c^2}\\ &=\frac {24 e^{\tan ^{-1}(a x)}}{85 a c^3}+\frac {e^{\tan ^{-1}(a x)} (1+4 a x)}{17 a c^3 \left (1+a^2 x^2\right )^2}+\frac {12 e^{\tan ^{-1}(a x)} (1+2 a x)}{85 a c^3 \left (1+a^2 x^2\right )}\\ \end {align*}

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Mathematica [C]  time = 0.29, size = 114, normalized size = 1.37 \[ \frac {\frac {5 (4 a x+1) e^{\tan ^{-1}(a x)}}{\left (a^2 x^2+1\right )^2}+\frac {24 (1-i a x)^{\frac {i}{2}} (1+i a x)^{-\frac {i}{2}} (a x+(1-i))}{a x-i}+(12-24 i) (1-i a x)^{-1+\frac {i}{2}} (1+i a x)^{-1-\frac {i}{2}}}{85 a c^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcTan[a*x]/(c + a^2*c*x^2)^3,x]

[Out]

((12 - 24*I)/((1 - I*a*x)^(1 - I/2)*(1 + I*a*x)^(1 + I/2)) + (24*(1 - I*a*x)^(I/2)*((1 - I) + a*x))/((1 + I*a*
x)^(I/2)*(-I + a*x)) + (5*E^ArcTan[a*x]*(1 + 4*a*x))/(1 + a^2*x^2)^2)/(85*a*c^3)

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fricas [A]  time = 0.54, size = 66, normalized size = 0.80 \[ \frac {{\left (24 \, a^{4} x^{4} + 24 \, a^{3} x^{3} + 60 \, a^{2} x^{2} + 44 \, a x + 41\right )} e^{\left (\arctan \left (a x\right )\right )}}{85 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arctan(a*x))/(a^2*c*x^2+c)^3,x, algorithm="fricas")

[Out]

1/85*(24*a^4*x^4 + 24*a^3*x^3 + 60*a^2*x^2 + 44*a*x + 41)*e^(arctan(a*x))/(a^5*c^3*x^4 + 2*a^3*c^3*x^2 + a*c^3
)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arctan(a*x))/(a^2*c*x^2+c)^3,x, algorithm="giac")

[Out]

sage0*x

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maple [A]  time = 0.04, size = 55, normalized size = 0.66 \[ \frac {{\mathrm e}^{\arctan \left (a x \right )} \left (24 a^{4} x^{4}+24 a^{3} x^{3}+60 a^{2} x^{2}+44 a x +41\right )}{85 \left (a^{2} x^{2}+1\right )^{2} a \,c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(arctan(a*x))/(a^2*c*x^2+c)^3,x)

[Out]

1/85*exp(arctan(a*x))*(24*a^4*x^4+24*a^3*x^3+60*a^2*x^2+44*a*x+41)/(a^2*x^2+1)^2/a/c^3

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\left (\arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arctan(a*x))/(a^2*c*x^2+c)^3,x, algorithm="maxima")

[Out]

integrate(e^(arctan(a*x))/(a^2*c*x^2 + c)^3, x)

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mupad [B]  time = 0.62, size = 74, normalized size = 0.89 \[ \frac {24\,{\mathrm {e}}^{\mathrm {atan}\left (a\,x\right )}}{85\,a\,c^3}+\frac {12\,{\mathrm {e}}^{\mathrm {atan}\left (a\,x\right )}\,\left (2\,a\,x+1\right )}{85\,a\,c^3\,\left (a^2\,x^2+1\right )}+\frac {{\mathrm {e}}^{\mathrm {atan}\left (a\,x\right )}\,\left (4\,a\,x+1\right )}{17\,a\,c^3\,{\left (a^2\,x^2+1\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(atan(a*x))/(c + a^2*c*x^2)^3,x)

[Out]

(24*exp(atan(a*x)))/(85*a*c^3) + (12*exp(atan(a*x))*(2*a*x + 1))/(85*a*c^3*(a^2*x^2 + 1)) + (exp(atan(a*x))*(4
*a*x + 1))/(17*a*c^3*(a^2*x^2 + 1)^2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \begin {cases} \frac {24 a^{4} x^{4} e^{\operatorname {atan}{\left (a x \right )}}}{85 a^{5} c^{3} x^{4} + 170 a^{3} c^{3} x^{2} + 85 a c^{3}} + \frac {24 a^{3} x^{3} e^{\operatorname {atan}{\left (a x \right )}}}{85 a^{5} c^{3} x^{4} + 170 a^{3} c^{3} x^{2} + 85 a c^{3}} + \frac {60 a^{2} x^{2} e^{\operatorname {atan}{\left (a x \right )}}}{85 a^{5} c^{3} x^{4} + 170 a^{3} c^{3} x^{2} + 85 a c^{3}} + \frac {44 a x e^{\operatorname {atan}{\left (a x \right )}}}{85 a^{5} c^{3} x^{4} + 170 a^{3} c^{3} x^{2} + 85 a c^{3}} + \frac {41 e^{\operatorname {atan}{\left (a x \right )}}}{85 a^{5} c^{3} x^{4} + 170 a^{3} c^{3} x^{2} + 85 a c^{3}} & \text {for}\: c \neq 0 \\\tilde {\infty } \int e^{\operatorname {atan}{\left (a x \right )}}\, dx & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(atan(a*x))/(a**2*c*x**2+c)**3,x)

[Out]

Piecewise((24*a**4*x**4*exp(atan(a*x))/(85*a**5*c**3*x**4 + 170*a**3*c**3*x**2 + 85*a*c**3) + 24*a**3*x**3*exp
(atan(a*x))/(85*a**5*c**3*x**4 + 170*a**3*c**3*x**2 + 85*a*c**3) + 60*a**2*x**2*exp(atan(a*x))/(85*a**5*c**3*x
**4 + 170*a**3*c**3*x**2 + 85*a*c**3) + 44*a*x*exp(atan(a*x))/(85*a**5*c**3*x**4 + 170*a**3*c**3*x**2 + 85*a*c
**3) + 41*exp(atan(a*x))/(85*a**5*c**3*x**4 + 170*a**3*c**3*x**2 + 85*a*c**3), Ne(c, 0)), (zoo*Integral(exp(at
an(a*x)), x), True))

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