Integrand size = 23, antiderivative size = 120 \[ \int \frac {e^{n \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=-\frac {2^{\frac {1}{2}-\frac {i n}{2}} (1-i a x)^{\frac {1}{2} (1+i n)} \sqrt {1+a^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (1+i n),\frac {1}{2} (1+i n),\frac {1}{2} (3+i n),\frac {1}{2} (1-i a x)\right )}{a (i-n) \sqrt {c+a^2 c x^2}} \] Output:
-2^(1/2-1/2*I*n)*(1-I*a*x)^(1/2+1/2*I*n)*(a^2*x^2+1)^(1/2)*hypergeom([1/2+ 1/2*I*n, 1/2+1/2*I*n],[3/2+1/2*I*n],1/2-1/2*I*a*x)/a/(I-n)/(a^2*c*x^2+c)^( 1/2)
Time = 0.05 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.98 \[ \int \frac {e^{n \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\frac {2^{\frac {1}{2}-\frac {i n}{2}} (1-i a x)^{\frac {1}{2}+\frac {i n}{2}} \sqrt {1+a^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}+\frac {i n}{2},\frac {1}{2}+\frac {i n}{2},\frac {3}{2}+\frac {i n}{2},\frac {1}{2}-\frac {i a x}{2}\right )}{a (-i+n) \sqrt {c+a^2 c x^2}} \] Input:
Integrate[E^(n*ArcTan[a*x])/Sqrt[c + a^2*c*x^2],x]
Output:
(2^(1/2 - (I/2)*n)*(1 - I*a*x)^(1/2 + (I/2)*n)*Sqrt[1 + a^2*x^2]*Hypergeom etric2F1[1/2 + (I/2)*n, 1/2 + (I/2)*n, 3/2 + (I/2)*n, 1/2 - (I/2)*a*x])/(a *(-I + n)*Sqrt[c + a^2*c*x^2])
Time = 0.61 (sec) , antiderivative size = 120, 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.
\(\displaystyle \int \frac {e^{n \arctan (a x)}}{\sqrt {a^2 c x^2+c}} \, dx\) |
\(\Big \downarrow \) 5599 |
\(\displaystyle \frac {\sqrt {a^2 x^2+1} \int \frac {e^{n \arctan (a x)}}{\sqrt {a^2 x^2+1}}dx}{\sqrt {a^2 c x^2+c}}\) |
\(\Big \downarrow \) 5596 |
\(\displaystyle \frac {\sqrt {a^2 x^2+1} \int (1-i a x)^{\frac {1}{2} (i n-1)} (i a x+1)^{\frac {1}{2} (-i n-1)}dx}{\sqrt {a^2 c x^2+c}}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle -\frac {2^{\frac {1}{2}-\frac {i n}{2}} \sqrt {a^2 x^2+1} (1-i a x)^{\frac {1}{2} (1+i n)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (i n+1),\frac {1}{2} (i n+1),\frac {1}{2} (i n+3),\frac {1}{2} (1-i a x)\right )}{a (-n+i) \sqrt {a^2 c x^2+c}}\) |
Input:
Int[E^(n*ArcTan[a*x])/Sqrt[c + a^2*c*x^2],x]
Output:
-((2^(1/2 - (I/2)*n)*(1 - I*a*x)^((1 + I*n)/2)*Sqrt[1 + a^2*x^2]*Hypergeom etric2F1[(1 + I*n)/2, (1 + I*n)/2, (3 + I*n)/2, (1 - I*a*x)/2])/(a*(I - n) *Sqrt[c + a^2*c*x^2]))
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]))
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])
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])
\[\int \frac {{\mathrm e}^{n \arctan \left (a x \right )}}{\sqrt {a^{2} c \,x^{2}+c}}d x\]
Input:
int(exp(n*arctan(a*x))/(a^2*c*x^2+c)^(1/2),x)
Output:
int(exp(n*arctan(a*x))/(a^2*c*x^2+c)^(1/2),x)
\[ \int \frac {e^{n \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {e^{\left (n \arctan \left (a x\right )\right )}}{\sqrt {a^{2} c x^{2} + c}} \,d x } \] Input:
integrate(exp(n*arctan(a*x))/(a^2*c*x^2+c)^(1/2),x, algorithm="fricas")
Output:
integral(e^(n*arctan(a*x))/sqrt(a^2*c*x^2 + c), x)
\[ \int \frac {e^{n \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {e^{n \operatorname {atan}{\left (a x \right )}}}{\sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \] Input:
integrate(exp(n*atan(a*x))/(a**2*c*x**2+c)**(1/2),x)
Output:
Integral(exp(n*atan(a*x))/sqrt(c*(a**2*x**2 + 1)), x)
\[ \int \frac {e^{n \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {e^{\left (n \arctan \left (a x\right )\right )}}{\sqrt {a^{2} c x^{2} + c}} \,d x } \] Input:
integrate(exp(n*arctan(a*x))/(a^2*c*x^2+c)^(1/2),x, algorithm="maxima")
Output:
integrate(e^(n*arctan(a*x))/sqrt(a^2*c*x^2 + c), x)
\[ \int \frac {e^{n \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {e^{\left (n \arctan \left (a x\right )\right )}}{\sqrt {a^{2} c x^{2} + c}} \,d x } \] Input:
integrate(exp(n*arctan(a*x))/(a^2*c*x^2+c)^(1/2),x, algorithm="giac")
Output:
integrate(e^(n*arctan(a*x))/sqrt(a^2*c*x^2 + c), x)
Timed out. \[ \int \frac {e^{n \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{\sqrt {c\,a^2\,x^2+c}} \,d x \] Input:
int(exp(n*atan(a*x))/(c + a^2*c*x^2)^(1/2),x)
Output:
int(exp(n*atan(a*x))/(c + a^2*c*x^2)^(1/2), x)
\[ \int \frac {e^{n \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\frac {\int \frac {e^{\mathit {atan} \left (a x \right ) n}}{\sqrt {a^{2} x^{2}+1}}d x}{\sqrt {c}} \] Input:
int(exp(n*atan(a*x))/(a^2*c*x^2+c)^(1/2),x)
Output:
int(e**(atan(a*x)*n)/sqrt(a**2*x**2 + 1),x)/sqrt(c)