Integrand size = 23, antiderivative size = 115 \[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=-\frac {e^{-\arctan (a x)} (1-5 a x)}{26 a c \left (c+a^2 c x^2\right )^{5/2}}-\frac {e^{-\arctan (a x)} (1-3 a x)}{13 a c^2 \left (c+a^2 c x^2\right )^{3/2}}-\frac {3 e^{-\arctan (a x)} (1-a x)}{13 a c^3 \sqrt {c+a^2 c x^2}} \] Output:
-1/26*(-5*a*x+1)/a/c/exp(arctan(a*x))/(a^2*c*x^2+c)^(5/2)-1/13*(-3*a*x+1)/ a/c^2/exp(arctan(a*x))/(a^2*c*x^2+c)^(3/2)-3/13*(-a*x+1)/a/c^3/exp(arctan( a*x))/(a^2*c*x^2+c)^(1/2)
Time = 0.06 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.70 \[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\frac {e^{-\arctan (a x)} \left (-9+17 a x-14 a^2 x^2+18 a^3 x^3-6 a^4 x^4+6 a^5 x^5\right )}{26 a c^3 \left (1+a^2 x^2\right )^2 \sqrt {c+a^2 c x^2}} \] Input:
Integrate[1/(E^ArcTan[a*x]*(c + a^2*c*x^2)^(7/2)),x]
Output:
(-9 + 17*a*x - 14*a^2*x^2 + 18*a^3*x^3 - 6*a^4*x^4 + 6*a^5*x^5)/(26*a*c^3* E^ArcTan[a*x]*(1 + a^2*x^2)^2*Sqrt[c + a^2*c*x^2])
Time = 0.85 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.07, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {5593, 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.
\(\displaystyle \int \frac {e^{-\arctan (a x)}}{\left (a^2 c x^2+c\right )^{7/2}} \, dx\) |
\(\Big \downarrow \) 5593 |
\(\displaystyle \frac {10 \int \frac {e^{-\arctan (a x)}}{\left (a^2 c x^2+c\right )^{5/2}}dx}{13 c}-\frac {(1-5 a x) e^{-\arctan (a x)}}{26 a c \left (a^2 c x^2+c\right )^{5/2}}\) |
\(\Big \downarrow \) 5593 |
\(\displaystyle \frac {10 \left (\frac {3 \int \frac {e^{-\arctan (a x)}}{\left (a^2 c x^2+c\right )^{3/2}}dx}{5 c}-\frac {(1-3 a x) e^{-\arctan (a x)}}{10 a c \left (a^2 c x^2+c\right )^{3/2}}\right )}{13 c}-\frac {(1-5 a x) e^{-\arctan (a x)}}{26 a c \left (a^2 c x^2+c\right )^{5/2}}\) |
\(\Big \downarrow \) 5592 |
\(\displaystyle \frac {10 \left (-\frac {3 (1-a x) e^{-\arctan (a x)}}{10 a c^2 \sqrt {a^2 c x^2+c}}-\frac {(1-3 a x) e^{-\arctan (a x)}}{10 a c \left (a^2 c x^2+c\right )^{3/2}}\right )}{13 c}-\frac {(1-5 a x) e^{-\arctan (a x)}}{26 a c \left (a^2 c x^2+c\right )^{5/2}}\) |
Input:
Int[1/(E^ArcTan[a*x]*(c + a^2*c*x^2)^(7/2)),x]
Output:
-1/26*(1 - 5*a*x)/(a*c*E^ArcTan[a*x]*(c + a^2*c*x^2)^(5/2)) + (10*(-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])))/(13*c)
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 = 72, normalized size of antiderivative = 0.63
method | result | size |
gosper | \(\frac {\left (a^{2} x^{2}+1\right ) \left (6 a^{5} x^{5}-6 a^{4} x^{4}+18 a^{3} x^{3}-14 a^{2} x^{2}+17 a x -9\right ) {\mathrm e}^{-\arctan \left (a x \right )}}{26 a \left (a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}\) | \(72\) |
orering | \(\frac {\left (a^{2} x^{2}+1\right ) \left (6 a^{5} x^{5}-6 a^{4} x^{4}+18 a^{3} x^{3}-14 a^{2} x^{2}+17 a x -9\right ) {\mathrm e}^{-\arctan \left (a x \right )}}{26 a \left (a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}\) | \(72\) |
Input:
int(1/exp(arctan(a*x))/(a^2*c*x^2+c)^(7/2),x,method=_RETURNVERBOSE)
Output:
1/26*(a^2*x^2+1)*(6*a^5*x^5-6*a^4*x^4+18*a^3*x^3-14*a^2*x^2+17*a*x-9)/a/ex p(arctan(a*x))/(a^2*c*x^2+c)^(7/2)
Time = 0.10 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.86 \[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\frac {{\left (6 \, a^{5} x^{5} - 6 \, a^{4} x^{4} + 18 \, a^{3} x^{3} - 14 \, a^{2} x^{2} + 17 \, a x - 9\right )} \sqrt {a^{2} c x^{2} + c} e^{\left (-\arctan \left (a x\right )\right )}}{26 \, {\left (a^{7} c^{4} x^{6} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{3} c^{4} x^{2} + a c^{4}\right )}} \] Input:
integrate(1/exp(arctan(a*x))/(a^2*c*x^2+c)^(7/2),x, algorithm="fricas")
Output:
1/26*(6*a^5*x^5 - 6*a^4*x^4 + 18*a^3*x^3 - 14*a^2*x^2 + 17*a*x - 9)*sqrt(a ^2*c*x^2 + c)*e^(-arctan(a*x))/(a^7*c^4*x^6 + 3*a^5*c^4*x^4 + 3*a^3*c^4*x^ 2 + a*c^4)
Timed out. \[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\text {Timed out} \] Input:
integrate(1/exp(atan(a*x))/(a**2*c*x**2+c)**(7/2),x)
Output:
Timed out
\[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\int { \frac {e^{\left (-\arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate(1/exp(arctan(a*x))/(a^2*c*x^2+c)^(7/2),x, algorithm="maxima")
Output:
integrate(e^(-arctan(a*x))/(a^2*c*x^2 + c)^(7/2), x)
Leaf count of result is larger than twice the leaf count of optimal. 224 vs. \(2 (99) = 198\).
Time = 0.18 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.95 \[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\frac {9 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{10} + 34 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{9} + 11 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{8} + 8 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{7} + 18 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{6} + 108 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{5} - 18 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{4} + 8 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{3} - 11 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{2} + 34 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right ) - 9}{26 \, {\left (a c^{\frac {7}{2}} e^{\left (\arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{10} + 5 \, a c^{\frac {7}{2}} e^{\left (\arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{8} + 10 \, a c^{\frac {7}{2}} e^{\left (\arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{6} + 10 \, a c^{\frac {7}{2}} e^{\left (\arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{4} + 5 \, a c^{\frac {7}{2}} e^{\left (\arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{2} + a c^{\frac {7}{2}} e^{\left (\arctan \left (a x\right )\right )}\right )}} \] Input:
integrate(1/exp(arctan(a*x))/(a^2*c*x^2+c)^(7/2),x, algorithm="giac")
Output:
1/26*(9*tan(1/2*arctan(a*x))^10 + 34*tan(1/2*arctan(a*x))^9 + 11*tan(1/2*a rctan(a*x))^8 + 8*tan(1/2*arctan(a*x))^7 + 18*tan(1/2*arctan(a*x))^6 + 108 *tan(1/2*arctan(a*x))^5 - 18*tan(1/2*arctan(a*x))^4 + 8*tan(1/2*arctan(a*x ))^3 - 11*tan(1/2*arctan(a*x))^2 + 34*tan(1/2*arctan(a*x)) - 9)/(a*c^(7/2) *e^(arctan(a*x))*tan(1/2*arctan(a*x))^10 + 5*a*c^(7/2)*e^(arctan(a*x))*tan (1/2*arctan(a*x))^8 + 10*a*c^(7/2)*e^(arctan(a*x))*tan(1/2*arctan(a*x))^6 + 10*a*c^(7/2)*e^(arctan(a*x))*tan(1/2*arctan(a*x))^4 + 5*a*c^(7/2)*e^(arc tan(a*x))*tan(1/2*arctan(a*x))^2 + a*c^(7/2)*e^(arctan(a*x)))
Time = 22.97 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.07 \[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=-\frac {{\mathrm {e}}^{-\mathrm {atan}\left (a\,x\right )}\,\left (\frac {9}{26\,a^5\,c^3}-\frac {3\,x^5}{13\,c^3}-\frac {17\,x}{26\,a^4\,c^3}+\frac {3\,x^4}{13\,a\,c^3}-\frac {9\,x^3}{13\,a^2\,c^3}+\frac {7\,x^2}{13\,a^3\,c^3}\right )}{\frac {\sqrt {c\,a^2\,x^2+c}}{a^4}+x^4\,\sqrt {c\,a^2\,x^2+c}+\frac {2\,x^2\,\sqrt {c\,a^2\,x^2+c}}{a^2}} \] Input:
int(exp(-atan(a*x))/(c + a^2*c*x^2)^(7/2),x)
Output:
-(exp(-atan(a*x))*(9/(26*a^5*c^3) - (3*x^5)/(13*c^3) - (17*x)/(26*a^4*c^3) + (3*x^4)/(13*a*c^3) - (9*x^3)/(13*a^2*c^3) + (7*x^2)/(13*a^3*c^3)))/((c + a^2*c*x^2)^(1/2)/a^4 + x^4*(c + a^2*c*x^2)^(1/2) + (2*x^2*(c + a^2*c*x^2 )^(1/2))/a^2)
\[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\frac {\int \frac {1}{e^{\mathit {atan} \left (a x \right )} \sqrt {a^{2} x^{2}+1}\, a^{6} x^{6}+3 e^{\mathit {atan} \left (a x \right )} \sqrt {a^{2} x^{2}+1}\, a^{4} x^{4}+3 e^{\mathit {atan} \left (a x \right )} \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}+e^{\mathit {atan} \left (a x \right )} \sqrt {a^{2} x^{2}+1}}d x}{\sqrt {c}\, c^{3}} \] Input:
int(1/exp(atan(a*x))/(a^2*c*x^2+c)^(7/2),x)
Output:
int(1/(e**atan(a*x)*sqrt(a**2*x**2 + 1)*a**6*x**6 + 3*e**atan(a*x)*sqrt(a* *2*x**2 + 1)*a**4*x**4 + 3*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**3)