3.3.44 \(\int e^{\text {ArcTan}(a x)} (c+a^2 c x^2)^p \, dx\) [244]

Optimal. Leaf size=102 \[ \frac {i 2^{\left (1-\frac {i}{2}\right )+p} (1-i a x)^{\left (1+\frac {i}{2}\right )+p} \left (1+a^2 x^2\right )^{-p} \left (c+a^2 c x^2\right )^p \, _2F_1\left (\frac {i}{2}-p,\left (1+\frac {i}{2}\right )+p;\left (2+\frac {i}{2}\right )+p;\frac {1}{2} (1-i a x)\right )}{a ((2+i)+2 p)} \]

[Out]

I*2^(1-1/2*I+p)*(1-I*a*x)^(1+1/2*I+p)*(a^2*c*x^2+c)^p*hypergeom([1/2*I-p, 1+1/2*I+p],[2+1/2*I+p],1/2-1/2*I*a*x
)/a/(2+I+2*p)/((a^2*x^2+1)^p)

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Rubi [A]
time = 0.06, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {5184, 5181, 71} \begin {gather*} \frac {i 2^{p+\left (1-\frac {i}{2}\right )} (1-i a x)^{p+\left (1+\frac {i}{2}\right )} \left (a^2 x^2+1\right )^{-p} \left (a^2 c x^2+c\right )^p \, _2F_1\left (\frac {i}{2}-p,p+\left (1+\frac {i}{2}\right );p+\left (2+\frac {i}{2}\right );\frac {1}{2} (1-i a x)\right )}{a (2 p+(2+i))} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(I*2^((1 - I/2) + p)*(1 - I*a*x)^((1 + I/2) + p)*(c + a^2*c*x^2)^p*Hypergeometric2F1[I/2 - p, (1 + I/2) + p, (
2 + I/2) + p, (1 - I*a*x)/2])/(a*((2 + I) + 2*p)*(1 + a^2*x^2)^p)

Rule 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-d/(b*c - a*d), 0]))

Rule 5181

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

Rule 5184

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Dist[c^IntPart[p]*((c + d*x^2)^FracP
art[p]/(1 + a^2*x^2)^FracPart[p]), Int[(1 + a^2*x^2)^p*E^(n*ArcTan[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x]
&& EqQ[d, a^2*c] &&  !(IntegerQ[p] || GtQ[c, 0])

Rubi steps

\begin {align*} \int e^{\tan ^{-1}(a x)} \left (c+a^2 c x^2\right )^p \, dx &=\left (\left (1+a^2 x^2\right )^{-p} \left (c+a^2 c x^2\right )^p\right ) \int e^{\tan ^{-1}(a x)} \left (1+a^2 x^2\right )^p \, dx\\ &=\left (\left (1+a^2 x^2\right )^{-p} \left (c+a^2 c x^2\right )^p\right ) \int (1-i a x)^{\frac {i}{2}+p} (1+i a x)^{-\frac {i}{2}+p} \, dx\\ &=\frac {i 2^{\left (1-\frac {i}{2}\right )+p} (1-i a x)^{\left (1+\frac {i}{2}\right )+p} \left (1+a^2 x^2\right )^{-p} \left (c+a^2 c x^2\right )^p \, _2F_1\left (\frac {i}{2}-p,\left (1+\frac {i}{2}\right )+p;\left (2+\frac {i}{2}\right )+p;\frac {1}{2} (1-i a x)\right )}{a ((2+i)+2 p)}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 102, normalized size = 1.00 \begin {gather*} \frac {i 2^{-\frac {i}{2}+p} (1-i a x)^{\left (1+\frac {i}{2}\right )+p} \left (1+a^2 x^2\right )^{-p} \left (c+a^2 c x^2\right )^p \, _2F_1\left (\frac {i}{2}-p,\left (1+\frac {i}{2}\right )+p;\left (2+\frac {i}{2}\right )+p;\frac {1}{2} (1-i a x)\right )}{a \left (\left (1+\frac {i}{2}\right )+p\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(I*2^(-1/2*I + p)*(1 - I*a*x)^((1 + I/2) + p)*(c + a^2*c*x^2)^p*Hypergeometric2F1[I/2 - p, (1 + I/2) + p, (2 +
I/2) + p, (1 - I*a*x)/2])/(a*((1 + I/2) + p)*(1 + a^2*x^2)^p)

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{\arctan \left (a x \right )} \left (a^{2} c \,x^{2}+c \right )^{p}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (c \left (a^{2} x^{2} + 1\right )\right )^{p} e^{\operatorname {atan}{\left (a x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((c*(a**2*x**2 + 1))**p*exp(atan(a*x)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {e}}^{\mathrm {atan}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^p \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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