Integrand size = 23, antiderivative size = 114 \[ \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\frac {e^{2 \arctan (a x)} (2+5 a x)}{29 a c \left (c+a^2 c x^2\right )^{5/2}}+\frac {20 e^{2 \arctan (a x)} (2+3 a x)}{377 a c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {24 e^{2 \arctan (a x)} (2+a x)}{377 a c^3 \sqrt {c+a^2 c x^2}} \] Output:
1/29*exp(2*arctan(a*x))*(5*a*x+2)/a/c/(a^2*c*x^2+c)^(5/2)+20/377*exp(2*arc tan(a*x))*(3*a*x+2)/a/c^2/(a^2*c*x^2+c)^(3/2)+24/377*exp(2*arctan(a*x))*(a *x+2)/a/c^3/(a^2*c*x^2+c)^(1/2)
Time = 0.05 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.71 \[ \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\frac {e^{2 \arctan (a x)} \left (114+149 a x+136 a^2 x^2+108 a^3 x^3+48 a^4 x^4+24 a^5 x^5\right )}{377 a c^3 \left (1+a^2 x^2\right )^2 \sqrt {c+a^2 c x^2}} \] Input:
Integrate[E^(2*ArcTan[a*x])/(c + a^2*c*x^2)^(7/2),x]
Output:
(E^(2*ArcTan[a*x])*(114 + 149*a*x + 136*a^2*x^2 + 108*a^3*x^3 + 48*a^4*x^4 + 24*a^5*x^5))/(377*a*c^3*(1 + a^2*x^2)^2*Sqrt[c + a^2*c*x^2])
Time = 0.76 (sec) , antiderivative size = 122, 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.
\(\displaystyle \int \frac {e^{2 \arctan (a x)}}{\left (a^2 c x^2+c\right )^{7/2}} \, dx\) |
\(\Big \downarrow \) 5593 |
\(\displaystyle \frac {20 \int \frac {e^{2 \arctan (a x)}}{\left (a^2 c x^2+c\right )^{5/2}}dx}{29 c}+\frac {(5 a x+2) e^{2 \arctan (a x)}}{29 a c \left (a^2 c x^2+c\right )^{5/2}}\) |
\(\Big \downarrow \) 5593 |
\(\displaystyle \frac {20 \left (\frac {6 \int \frac {e^{2 \arctan (a x)}}{\left (a^2 c x^2+c\right )^{3/2}}dx}{13 c}+\frac {(3 a x+2) e^{2 \arctan (a x)}}{13 a c \left (a^2 c x^2+c\right )^{3/2}}\right )}{29 c}+\frac {(5 a x+2) e^{2 \arctan (a x)}}{29 a c \left (a^2 c x^2+c\right )^{5/2}}\) |
\(\Big \downarrow \) 5592 |
\(\displaystyle \frac {20 \left (\frac {6 (a x+2) e^{2 \arctan (a x)}}{65 a c^2 \sqrt {a^2 c x^2+c}}+\frac {(3 a x+2) e^{2 \arctan (a x)}}{13 a c \left (a^2 c x^2+c\right )^{3/2}}\right )}{29 c}+\frac {(5 a x+2) e^{2 \arctan (a x)}}{29 a c \left (a^2 c x^2+c\right )^{5/2}}\) |
Input:
Int[E^(2*ArcTan[a*x])/(c + a^2*c*x^2)^(7/2),x]
Output:
(E^(2*ArcTan[a*x])*(2 + 5*a*x))/(29*a*c*(c + a^2*c*x^2)^(5/2)) + (20*((E^( 2*ArcTan[a*x])*(2 + 3*a*x))/(13*a*c*(c + a^2*c*x^2)^(3/2)) + (6*E^(2*ArcTa n[a*x])*(2 + a*x))/(65*a*c^2*Sqrt[c + a^2*c*x^2])))/(29*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 | \(\frac {\left (a^{2} x^{2}+1\right ) \left (24 a^{5} x^{5}+48 a^{4} x^{4}+108 a^{3} x^{3}+136 a^{2} x^{2}+149 a x +114\right ) {\mathrm e}^{2 \arctan \left (a x \right )}}{377 a \left (a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}\) | \(72\) |
orering | \(\frac {\left (a^{2} x^{2}+1\right ) \left (24 a^{5} x^{5}+48 a^{4} x^{4}+108 a^{3} x^{3}+136 a^{2} x^{2}+149 a x +114\right ) {\mathrm e}^{2 \arctan \left (a x \right )}}{377 a \left (a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}\) | \(72\) |
Input:
int(exp(2*arctan(a*x))/(a^2*c*x^2+c)^(7/2),x,method=_RETURNVERBOSE)
Output:
1/377*(a^2*x^2+1)*(24*a^5*x^5+48*a^4*x^4+108*a^3*x^3+136*a^2*x^2+149*a*x+1 14)*exp(2*arctan(a*x))/a/(a^2*c*x^2+c)^(7/2)
Time = 0.09 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.87 \[ \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\frac {{\left (24 \, a^{5} x^{5} + 48 \, a^{4} x^{4} + 108 \, a^{3} x^{3} + 136 \, a^{2} x^{2} + 149 \, a x + 114\right )} \sqrt {a^{2} c x^{2} + c} e^{\left (2 \, \arctan \left (a x\right )\right )}}{377 \, {\left (a^{7} c^{4} x^{6} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{3} c^{4} x^{2} + a c^{4}\right )}} \] Input:
integrate(exp(2*arctan(a*x))/(a^2*c*x^2+c)^(7/2),x, algorithm="fricas")
Output:
1/377*(24*a^5*x^5 + 48*a^4*x^4 + 108*a^3*x^3 + 136*a^2*x^2 + 149*a*x + 114 )*sqrt(a^2*c*x^2 + c)*e^(2*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. \[ \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\text {Timed out} \] Input:
integrate(exp(2*atan(a*x))/(a**2*c*x**2+c)**(7/2),x)
Output:
Timed out
\[ \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\int { \frac {e^{\left (2 \, \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate(exp(2*arctan(a*x))/(a^2*c*x^2+c)^(7/2),x, algorithm="maxima")
Output:
integrate(e^(2*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. 272 vs. \(2 (99) = 198\).
Time = 0.16 (sec) , antiderivative size = 272, normalized size of antiderivative = 2.39 \[ \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=-\frac {2 \, {\left (57 \, e^{\left (2 \, \arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{10} - 149 \, e^{\left (2 \, \arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{9} - 13 \, e^{\left (2 \, \arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{8} + 164 \, e^{\left (2 \, \arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{7} + 138 \, e^{\left (2 \, \arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{6} - 414 \, e^{\left (2 \, \arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{5} - 138 \, e^{\left (2 \, \arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{4} + 164 \, e^{\left (2 \, \arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{3} + 13 \, e^{\left (2 \, \arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{2} - 149 \, e^{\left (2 \, \arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right ) - 57 \, e^{\left (2 \, \arctan \left (a x\right )\right )}\right )}}{377 \, {\left (a c^{\frac {7}{2}} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{10} + 5 \, a c^{\frac {7}{2}} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{8} + 10 \, a c^{\frac {7}{2}} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{6} + 10 \, a c^{\frac {7}{2}} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{4} + 5 \, a c^{\frac {7}{2}} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{2} + a c^{\frac {7}{2}}\right )}} \] Input:
integrate(exp(2*arctan(a*x))/(a^2*c*x^2+c)^(7/2),x, algorithm="giac")
Output:
-2/377*(57*e^(2*arctan(a*x))*tan(1/2*arctan(a*x))^10 - 149*e^(2*arctan(a*x ))*tan(1/2*arctan(a*x))^9 - 13*e^(2*arctan(a*x))*tan(1/2*arctan(a*x))^8 + 164*e^(2*arctan(a*x))*tan(1/2*arctan(a*x))^7 + 138*e^(2*arctan(a*x))*tan(1 /2*arctan(a*x))^6 - 414*e^(2*arctan(a*x))*tan(1/2*arctan(a*x))^5 - 138*e^( 2*arctan(a*x))*tan(1/2*arctan(a*x))^4 + 164*e^(2*arctan(a*x))*tan(1/2*arct an(a*x))^3 + 13*e^(2*arctan(a*x))*tan(1/2*arctan(a*x))^2 - 149*e^(2*arctan (a*x))*tan(1/2*arctan(a*x)) - 57*e^(2*arctan(a*x)))/(a*c^(7/2)*tan(1/2*arc tan(a*x))^10 + 5*a*c^(7/2)*tan(1/2*arctan(a*x))^8 + 10*a*c^(7/2)*tan(1/2*a rctan(a*x))^6 + 10*a*c^(7/2)*tan(1/2*arctan(a*x))^4 + 5*a*c^(7/2)*tan(1/2* arctan(a*x))^2 + a*c^(7/2))
Time = 22.95 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.07 \[ \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\frac {{\mathrm {e}}^{2\,\mathrm {atan}\left (a\,x\right )}\,\left (\frac {114}{377\,a^5\,c^3}+\frac {24\,x^5}{377\,c^3}+\frac {149\,x}{377\,a^4\,c^3}+\frac {48\,x^4}{377\,a\,c^3}+\frac {108\,x^3}{377\,a^2\,c^3}+\frac {136\,x^2}{377\,a^3\,c^3}\right )}{\frac {\sqrt {c\,a^2\,x^2+c}}{a^4}+x^4\,\sqrt {c\,a^2\,x^2+c}+\frac {2\,x^2\,\sqrt {c\,a^2\,x^2+c}}{a^2}} \] Input:
int(exp(2*atan(a*x))/(c + a^2*c*x^2)^(7/2),x)
Output:
(exp(2*atan(a*x))*(114/(377*a^5*c^3) + (24*x^5)/(377*c^3) + (149*x)/(377*a ^4*c^3) + (48*x^4)/(377*a*c^3) + (108*x^3)/(377*a^2*c^3) + (136*x^2)/(377* 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)
Time = 0.19 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.82 \[ \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\frac {e^{2 \mathit {atan} \left (a x \right )} \sqrt {c}\, \sqrt {a^{2} x^{2}+1}\, \left (24 a^{5} x^{5}+48 a^{4} x^{4}+108 a^{3} x^{3}+136 a^{2} x^{2}+149 a x +114\right )}{377 a \,c^{4} \left (a^{6} x^{6}+3 a^{4} x^{4}+3 a^{2} x^{2}+1\right )} \] Input:
int(exp(2*atan(a*x))/(a^2*c*x^2+c)^(7/2),x)
Output:
(e**(2*atan(a*x))*sqrt(c)*sqrt(a**2*x**2 + 1)*(24*a**5*x**5 + 48*a**4*x**4 + 108*a**3*x**3 + 136*a**2*x**2 + 149*a*x + 114))/(377*a*c**4*(a**6*x**6 + 3*a**4*x**4 + 3*a**2*x**2 + 1))