\(\int \frac {e^{2 \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx\) [283]

Optimal result

Integrand size = 23, antiderivative size = 87 \[ \int \frac {e^{2 \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\frac {\left (\frac {2}{5}+\frac {i}{5}\right ) 2^{\frac {1}{2}-i} (1-i a x)^{\frac {1}{2}+i} \sqrt {1+a^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}+i,\frac {1}{2}+i,\frac {3}{2}+i,\frac {1}{2} (1-i a x)\right )}{a \sqrt {c+a^2 c x^2}} \] Output:

(2/5+1/5*I)*2^(1/2-I)*(1-I*a*x)^(1/2+I)*(a^2*x^2+1)^(1/2)*hypergeom([1/2+I 
, 1/2+I],[3/2+I],1/2-1/2*I*a*x)/a/(a^2*c*x^2+c)^(1/2)
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00 \[ \int \frac {e^{2 \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\frac {\left (\frac {2}{5}+\frac {i}{5}\right ) 2^{\frac {1}{2}-i} (1-i a x)^{\frac {1}{2}+i} \sqrt {1+a^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}+i,\frac {1}{2}+i,\frac {3}{2}+i,\frac {1}{2} (1-i a x)\right )}{a \sqrt {c+a^2 c x^2}} \] Input:

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

Output:

((2/5 + I/5)*2^(1/2 - I)*(1 - I*a*x)^(1/2 + I)*Sqrt[1 + a^2*x^2]*Hypergeom 
etric2F1[1/2 + I, 1/2 + I, 3/2 + I, (1 - I*a*x)/2])/(a*Sqrt[c + a^2*c*x^2] 
)
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, 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.

\(\displaystyle \int \frac {e^{2 \arctan (a x)}}{\sqrt {a^2 c x^2+c}} \, dx\)

\(\Big \downarrow \) 5599

\(\displaystyle \frac {\sqrt {a^2 x^2+1} \int \frac {e^{2 \arctan (a x)}}{\sqrt {a^2 x^2+1}}dx}{\sqrt {a^2 c x^2+c}}\)

\(\Big \downarrow \) 5596

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

\(\Big \downarrow \) 79

\(\displaystyle \frac {\left (\frac {2}{5}+\frac {i}{5}\right ) 2^{\frac {1}{2}-i} (1-i a x)^{\frac {1}{2}+i} \sqrt {a^2 x^2+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}+i,\frac {1}{2}+i,\frac {3}{2}+i,\frac {1}{2} (1-i a x)\right )}{a \sqrt {a^2 c x^2+c}}\)

Input:

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

Output:

((2/5 + I/5)*2^(1/2 - I)*(1 - I*a*x)^(1/2 + I)*Sqrt[1 + a^2*x^2]*Hypergeom 
etric2F1[1/2 + I, 1/2 + I, 3/2 + I, (1 - I*a*x)/2])/(a*Sqrt[c + a^2*c*x^2] 
)
 

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]

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

Input:

int(exp(2*arctan(a*x))/(a^2*c*x^2+c)^(1/2),x)
 

Output:

int(exp(2*arctan(a*x))/(a^2*c*x^2+c)^(1/2),x)
 

Fricas [F]

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

integrate(exp(2*arctan(a*x))/(a^2*c*x^2+c)^(1/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate(exp(2*atan(a*x))/(a**2*c*x**2+c)**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate(exp(2*arctan(a*x))/(a^2*c*x^2+c)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate(exp(2*arctan(a*x))/(a^2*c*x^2+c)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int(exp(2*atan(a*x))/(c + a^2*c*x^2)^(1/2),x)
 

Output:

int(exp(2*atan(a*x))/(c + a^2*c*x^2)^(1/2), x)
 

Reduce [F]

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

int(exp(2*atan(a*x))/(a^2*c*x^2+c)^(1/2),x)
 

Output:

int(e**(2*atan(a*x))/sqrt(a**2*x**2 + 1),x)/sqrt(c)