Integrand size = 23, antiderivative size = 77 \[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=-\frac {e^{-\arctan (a x)} (1-3 a x)}{10 a c \left (c+a^2 c x^2\right )^{3/2}}-\frac {3 e^{-\arctan (a x)} (1-a x)}{10 a c^2 \sqrt {c+a^2 c x^2}} \] Output:
-1/10*(-3*a*x+1)/a/c/exp(arctan(a*x))/(a^2*c*x^2+c)^(3/2)-3/10*(-a*x+1)/a/ c^2/exp(arctan(a*x))/(a^2*c*x^2+c)^(1/2)
Time = 0.05 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.81 \[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {e^{-\arctan (a x)} \left (-4+6 a x-3 a^2 x^2+3 a^3 x^3\right )}{10 c^2 \left (a+a^3 x^2\right ) \sqrt {c+a^2 c x^2}} \] Input:
Integrate[1/(E^ArcTan[a*x]*(c + a^2*c*x^2)^(5/2)),x]
Output:
(-4 + 6*a*x - 3*a^2*x^2 + 3*a^3*x^3)/(10*c^2*E^ArcTan[a*x]*(a + a^3*x^2)*S qrt[c + a^2*c*x^2])
Time = 0.56 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {5593, 5592}
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.
\(\displaystyle \int \frac {e^{-\arctan (a x)}}{\left (a^2 c x^2+c\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 5593 |
\(\displaystyle \frac {3 \int \frac {e^{-\arctan (a x)}}{\left (a^2 c x^2+c\right )^{3/2}}dx}{5 c}-\frac {(1-3 a x) e^{-\arctan (a x)}}{10 a c \left (a^2 c x^2+c\right )^{3/2}}\) |
\(\Big \downarrow \) 5592 |
\(\displaystyle -\frac {3 (1-a x) e^{-\arctan (a x)}}{10 a c^2 \sqrt {a^2 c x^2+c}}-\frac {(1-3 a x) e^{-\arctan (a x)}}{10 a c \left (a^2 c x^2+c\right )^{3/2}}\) |
Input:
Int[1/(E^ArcTan[a*x]*(c + a^2*c*x^2)^(5/2)),x]
Output:
-1/10*(1 - 3*a*x)/(a*c*E^ArcTan[a*x]*(c + a^2*c*x^2)^(3/2)) - (3*(1 - a*x) )/(10*a*c^2*E^ArcTan[a*x]*Sqrt[c + a^2*c*x^2])
Int[E^(ArcTan[(a_.)*(x_)]*(n_.))/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(n + a*x)*(E^(n*ArcTan[a*x])/(a*c*(n^2 + 1)*Sqrt[c + d*x^2])), x] /; F reeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && !IntegerQ[I*n]
Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> S imp[(n - 2*a*(p + 1)*x)*(c + d*x^2)^(p + 1)*(E^(n*ArcTan[a*x])/(a*c*(n^2 + 4*(p + 1)^2))), x] + Simp[2*(p + 1)*((2*p + 3)/(c*(n^2 + 4*(p + 1)^2))) I nt[(c + d*x^2)^(p + 1)*E^(n*ArcTan[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && LtQ[p, -1] && !IntegerQ[I*n] && NeQ[n^2 + 4*(p + 1)^2, 0] && IntegerQ[2*p]
Time = 0.06 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.73
method | result | size |
gosper | \(\frac {\left (a^{2} x^{2}+1\right ) \left (3 a^{3} x^{3}-3 a^{2} x^{2}+6 a x -4\right ) {\mathrm e}^{-\arctan \left (a x \right )}}{10 a \left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}\) | \(56\) |
orering | \(\frac {\left (a^{2} x^{2}+1\right ) \left (3 a^{3} x^{3}-3 a^{2} x^{2}+6 a x -4\right ) {\mathrm e}^{-\arctan \left (a x \right )}}{10 a \left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}\) | \(56\) |
Input:
int(1/exp(arctan(a*x))/(a^2*c*x^2+c)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/10*(a^2*x^2+1)*(3*a^3*x^3-3*a^2*x^2+6*a*x-4)/a/exp(arctan(a*x))/(a^2*c*x ^2+c)^(5/2)
Time = 0.13 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.94 \[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {{\left (3 \, a^{3} x^{3} - 3 \, a^{2} x^{2} + 6 \, a x - 4\right )} \sqrt {a^{2} c x^{2} + c} e^{\left (-\arctan \left (a x\right )\right )}}{10 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )}} \] Input:
integrate(1/exp(arctan(a*x))/(a^2*c*x^2+c)^(5/2),x, algorithm="fricas")
Output:
1/10*(3*a^3*x^3 - 3*a^2*x^2 + 6*a*x - 4)*sqrt(a^2*c*x^2 + c)*e^(-arctan(a* x))/(a^5*c^3*x^4 + 2*a^3*c^3*x^2 + a*c^3)
Timed out. \[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(1/exp(atan(a*x))/(a**2*c*x**2+c)**(5/2),x)
Output:
Timed out
\[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {e^{\left (-\arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/exp(arctan(a*x))/(a^2*c*x^2+c)^(5/2),x, algorithm="maxima")
Output:
integrate(e^(-arctan(a*x))/(a^2*c*x^2 + c)^(5/2), x)
Time = 0.17 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.36 \[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (\tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{6} + 3 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{5} + 3 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right ) - 1\right )}}{5 \, {\left (a c^{\frac {5}{2}} e^{\left (\arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{6} + 3 \, a c^{\frac {5}{2}} e^{\left (\arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{4} + 3 \, a c^{\frac {5}{2}} e^{\left (\arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{2} + a c^{\frac {5}{2}} e^{\left (\arctan \left (a x\right )\right )}\right )}} \] Input:
integrate(1/exp(arctan(a*x))/(a^2*c*x^2+c)^(5/2),x, algorithm="giac")
Output:
2/5*(tan(1/2*arctan(a*x))^6 + 3*tan(1/2*arctan(a*x))^5 + 3*tan(1/2*arctan( a*x)) - 1)/(a*c^(5/2)*e^(arctan(a*x))*tan(1/2*arctan(a*x))^6 + 3*a*c^(5/2) *e^(arctan(a*x))*tan(1/2*arctan(a*x))^4 + 3*a*c^(5/2)*e^(arctan(a*x))*tan( 1/2*arctan(a*x))^2 + a*c^(5/2)*e^(arctan(a*x)))
Time = 23.29 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.05 \[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=-\frac {{\mathrm {e}}^{-\mathrm {atan}\left (a\,x\right )}\,\left (\frac {2}{5\,a^3\,c^2}-\frac {3\,x^3}{10\,c^2}-\frac {3\,x}{5\,a^2\,c^2}+\frac {3\,x^2}{10\,a\,c^2}\right )}{\frac {\sqrt {c\,a^2\,x^2+c}}{a^2}+x^2\,\sqrt {c\,a^2\,x^2+c}} \] Input:
int(exp(-atan(a*x))/(c + a^2*c*x^2)^(5/2),x)
Output:
-(exp(-atan(a*x))*(2/(5*a^3*c^2) - (3*x^3)/(10*c^2) - (3*x)/(5*a^2*c^2) + (3*x^2)/(10*a*c^2)))/((c + a^2*c*x^2)^(1/2)/a^2 + x^2*(c + a^2*c*x^2)^(1/2 ))
\[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {\int \frac {1}{e^{\mathit {atan} \left (a x \right )} \sqrt {a^{2} x^{2}+1}\, a^{4} x^{4}+2 e^{\mathit {atan} \left (a x \right )} \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}+e^{\mathit {atan} \left (a x \right )} \sqrt {a^{2} x^{2}+1}}d x}{\sqrt {c}\, c^{2}} \] Input:
int(1/exp(atan(a*x))/(a^2*c*x^2+c)^(5/2),x)
Output:
int(1/(e**atan(a*x)*sqrt(a**2*x**2 + 1)*a**4*x**4 + 2*e**atan(a*x)*sqrt(a* *2*x**2 + 1)*a**2*x**2 + e**atan(a*x)*sqrt(a**2*x**2 + 1)),x)/(sqrt(c)*c** 2)