3.364 \(\int e^{n \tan ^{-1}(a x)} x^m (c+a^2 c x^2) \, dx\)

Optimal. Leaf size=49 \[ \frac {c x^{m+1} F_1\left (m+1;-\frac {i n}{2}-1,\frac {i n}{2}-1;m+2;i a x,-i a x\right )}{m+1} \]

[Out]

c*x^(1+m)*AppellF1(1+m,-1+1/2*I*n,-1-1/2*I*n,2+m,-I*a*x,I*a*x)/(1+m)

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Rubi [A]  time = 0.07, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {5082, 133} \[ \frac {c x^{m+1} F_1\left (m+1;-\frac {i n}{2}-1,\frac {i n}{2}-1;m+2;i a x,-i a x\right )}{m+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x^(1 + m)*AppellF1[1 + m, -1 - (I/2)*n, -1 + (I/2)*n, 2 + m, I*a*x, (-I)*a*x])/(1 + m)

Rule 133

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[(c^n*e^p*(b*x)^(m +
1)*AppellF1[m + 1, -n, -p, m + 2, -((d*x)/c), -((f*x)/e)])/(b*(m + 1)), x] /; FreeQ[{b, c, d, e, f, m, n, p},
x] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])

Rule 5082

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[x^m*(1 - I
*a*x)^(p + (I*n)/2)*(1 + I*a*x)^(p - (I*n)/2), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[d, a^2*c] && (Int
egerQ[p] || GtQ[c, 0])

Rubi steps

\begin {align*} \int e^{n \tan ^{-1}(a x)} x^m \left (c+a^2 c x^2\right ) \, dx &=c \int x^m (1-i a x)^{1+\frac {i n}{2}} (1+i a x)^{1-\frac {i n}{2}} \, dx\\ &=\frac {c x^{1+m} F_1\left (1+m;-1-\frac {i n}{2},-1+\frac {i n}{2};2+m;i a x,-i a x\right )}{1+m}\\ \end {align*}

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Mathematica [F]  time = 0.52, size = 0, normalized size = 0.00 \[ \int e^{n \tan ^{-1}(a x)} x^m \left (c+a^2 c x^2\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [F]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{2} c x^{2} + c\right )} x^{m} e^{\left (n \arctan \left (a x\right )\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a^2*c*x^2 + c)*x^m*e^(n*arctan(a*x)), x)

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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(n*arctan(a*x))*x^m*(a^2*c*x^2+c),x, algorithm="giac")

[Out]

sage0*x

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maple [F]  time = 0.19, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \arctan \left (a x \right )} x^{m} \left (a^{2} c \,x^{2}+c \right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int x^m\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}\,\left (c\,a^2\,x^2+c\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ c \left (\int x^{m} e^{n \operatorname {atan}{\left (a x \right )}}\, dx + \int a^{2} x^{2} x^{m} e^{n \operatorname {atan}{\left (a x \right )}}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*(Integral(x**m*exp(n*atan(a*x)), x) + Integral(a**2*x**2*x**m*exp(n*atan(a*x)), x))

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