\(\int e^{\arctan (a x)} (c+a^2 c x^2)^p \, dx\) [258]

Optimal result

Integrand size = 19, antiderivative size = 102 \[ \int e^{\arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\frac {i 2^{\left (1-\frac {i}{2}\right )+p} (1-i a x)^{\left (1+\frac {i}{2}\right )+p} \left (1+a^2 x^2\right )^{-p} \left (c+a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (\frac {i}{2}-p,\left (1+\frac {i}{2}\right )+p,\left (2+\frac {i}{2}\right )+p,\frac {1}{2} (1-i a x)\right )}{a ((2+i)+2 p)} \] Output:

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)
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00 \[ \int e^{\arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\frac {i 2^{-\frac {i}{2}+p} (1-i a x)^{\left (1+\frac {i}{2}\right )+p} \left (1+a^2 x^2\right )^{-p} \left (c+a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (\frac {i}{2}-p,\left (1+\frac {i}{2}\right )+p,\left (2+\frac {i}{2}\right )+p,\frac {1}{2} (1-i a x)\right )}{a \left (\left (1+\frac {i}{2}\right )+p\right )} \] Input:

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

Output:

(I*2^(-1/2*I + p)*(1 - I*a*x)^((1 + I/2) + p)*(c + a^2*c*x^2)^p*Hypergeome 
tric2F1[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)
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {5599, 5596, 79}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int e^{\arctan (a x)} \left (a^2 c x^2+c\right )^p \, dx\)

\(\Big \downarrow \) 5599

\(\displaystyle \left (a^2 x^2+1\right )^{-p} \left (a^2 c x^2+c\right )^p \int e^{\arctan (a x)} \left (a^2 x^2+1\right )^pdx\)

\(\Big \downarrow \) 5596

\(\displaystyle \left (a^2 x^2+1\right )^{-p} \left (a^2 c x^2+c\right )^p \int (1-i a x)^{p+\frac {i}{2}} (i a x+1)^{p-\frac {i}{2}}dx\)

\(\Big \downarrow \) 79

\(\displaystyle \frac {i 2^{p+\left (1-\frac {i}{2}\right )} (1-i a x)^{p+\left (1+\frac {i}{2}\right )} \left (a^2 x^2+1\right )^{-p} \left (a^2 c x^2+c\right )^p \operatorname {Hypergeometric2F1}\left (\frac {i}{2}-p,p+\left (1+\frac {i}{2}\right ),p+\left (2+\frac {i}{2}\right ),\frac {1}{2} (1-i a x)\right )}{a (2 p+(2+i))}\)

Input:

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

Output:

(I*2^((1 - I/2) + p)*(1 - I*a*x)^((1 + I/2) + p)*(c + a^2*c*x^2)^p*Hyperge 
ometric2F1[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)
 

Defintions of rubi rules used

rule 79

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] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
 

rule 5596

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> 
Simp[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 5599

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> S 
imp[c^IntPart[p]*((c + d*x^2)^FracPart[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] && E 
qQ[d, a^2*c] &&  !(IntegerQ[p] || GtQ[c, 0])
 

Maple [F]

\[\int {\mathrm e}^{\arctan \left (a x \right )} \left (a^{2} c \,x^{2}+c \right )^{p}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int e^{\arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{p} e^{\left (\arctan \left (a x\right )\right )} \,d x } \] Input:

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

Output:

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

Sympy [F]

\[ \int e^{\arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\int \left (c \left (a^{2} x^{2} + 1\right )\right )^{p} e^{\operatorname {atan}{\left (a x \right )}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int e^{\arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{p} e^{\left (\arctan \left (a x\right )\right )} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int e^{\arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{p} e^{\left (\arctan \left (a x\right )\right )} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int e^{\arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\int {\mathrm {e}}^{\mathrm {atan}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^p \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int e^{\arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\int e^{\mathit {atan} \left (a x \right )} \left (a^{2} c \,x^{2}+c \right )^{p}d x \] Input:

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

Output:

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