Integrand size = 16, antiderivative size = 82 \[ \int \frac {e^{\cot ^{-1}(x)}}{\left (a+a x^2\right )^{7/2}} \, dx=-\frac {e^{\cot ^{-1}(x)} (1-5 x)}{26 a \left (a+a x^2\right )^{5/2}}-\frac {e^{\cot ^{-1}(x)} (1-3 x)}{13 a^2 \left (a+a x^2\right )^{3/2}}-\frac {3 e^{\cot ^{-1}(x)} (1-x)}{13 a^3 \sqrt {a+a x^2}} \] Output:
-1/26*exp(arccot(x))*(1-5*x)/a/(a*x^2+a)^(5/2)-1/13*exp(arccot(x))*(1-3*x) /a^2/(a*x^2+a)^(3/2)-3/13*exp(arccot(x))*(1-x)/a^3/(a*x^2+a)^(1/2)
Time = 0.34 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.16 \[ \int \frac {e^{\cot ^{-1}(x)}}{\left (a+a x^2\right )^{7/2}} \, dx=\frac {e^{\cot ^{-1}(x)} \left (-130+130 x-39 \sqrt {1+\frac {1}{x^2}} x \cos \left (3 \cot ^{-1}(x)\right )+5 \sqrt {1+\frac {1}{x^2}} x \cos \left (5 \cot ^{-1}(x)\right )+13 \sqrt {1+\frac {1}{x^2}} x \sin \left (3 \cot ^{-1}(x)\right )-\sqrt {1+\frac {1}{x^2}} x \sin \left (5 \cot ^{-1}(x)\right )\right )}{416 a^3 \sqrt {a \left (1+x^2\right )}} \] Input:
Integrate[E^ArcCot[x]/(a + a*x^2)^(7/2),x]
Output:
(E^ArcCot[x]*(-130 + 130*x - 39*Sqrt[1 + x^(-2)]*x*Cos[3*ArcCot[x]] + 5*Sq rt[1 + x^(-2)]*x*Cos[5*ArcCot[x]] + 13*Sqrt[1 + x^(-2)]*x*Sin[3*ArcCot[x]] - Sqrt[1 + x^(-2)]*x*Sin[5*ArcCot[x]]))/(416*a^3*Sqrt[a*(1 + x^2)])
Time = 0.66 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.10, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {5638, 5638, 5637}
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.
\(\displaystyle \int \frac {e^{\cot ^{-1}(x)}}{\left (a x^2+a\right )^{7/2}} \, dx\) |
\(\Big \downarrow \) 5638 |
\(\displaystyle \frac {10 \int \frac {e^{\cot ^{-1}(x)}}{\left (a x^2+a\right )^{5/2}}dx}{13 a}-\frac {(1-5 x) e^{\cot ^{-1}(x)}}{26 a \left (a x^2+a\right )^{5/2}}\) |
\(\Big \downarrow \) 5638 |
\(\displaystyle \frac {10 \left (\frac {3 \int \frac {e^{\cot ^{-1}(x)}}{\left (a x^2+a\right )^{3/2}}dx}{5 a}-\frac {(1-3 x) e^{\cot ^{-1}(x)}}{10 a \left (a x^2+a\right )^{3/2}}\right )}{13 a}-\frac {(1-5 x) e^{\cot ^{-1}(x)}}{26 a \left (a x^2+a\right )^{5/2}}\) |
\(\Big \downarrow \) 5637 |
\(\displaystyle \frac {10 \left (-\frac {3 (1-x) e^{\cot ^{-1}(x)}}{10 a^2 \sqrt {a x^2+a}}-\frac {(1-3 x) e^{\cot ^{-1}(x)}}{10 a \left (a x^2+a\right )^{3/2}}\right )}{13 a}-\frac {(1-5 x) e^{\cot ^{-1}(x)}}{26 a \left (a x^2+a\right )^{5/2}}\) |
Input:
Int[E^ArcCot[x]/(a + a*x^2)^(7/2),x]
Output:
-1/26*(E^ArcCot[x]*(1 - 5*x))/(a*(a + a*x^2)^(5/2)) + (10*(-1/10*(E^ArcCot [x]*(1 - 3*x))/(a*(a + a*x^2)^(3/2)) - (3*E^ArcCot[x]*(1 - x))/(10*a^2*Sqr t[a + a*x^2])))/(13*a)
Int[E^(ArcCot[(a_.)*(x_)]*(n_.))/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(-(n - a*x))*(E^(n*ArcCot[a*x])/(a*c*(n^2 + 1)*Sqrt[c + d*x^2])), x] / ; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && !IntegerQ[(I*n - 1)/2]
Int[E^(ArcCot[(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*ArcCot[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))) Int[(c + d*x^2)^(p + 1)*E^(n*ArcCot[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && LtQ[p, -1] && NeQ[p, -3/2] && NeQ[n^2 + 4*(p + 1)^2, 0] && !(IntegerQ[p] && IntegerQ[I*(n/2)]) && !( !IntegerQ[p] && IntegerQ [(I*n - 1)/2])
Time = 0.07 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.55
method | result | size |
gosper | \(\frac {\left (x^{2}+1\right ) \left (6 x^{5}-6 x^{4}+18 x^{3}-14 x^{2}+17 x -9\right ) {\mathrm e}^{\operatorname {arccot}\left (x \right )}}{26 \left (a \,x^{2}+a \right )^{\frac {7}{2}}}\) | \(45\) |
orering | \(\frac {\left (x^{2}+1\right ) \left (6 x^{5}-6 x^{4}+18 x^{3}-14 x^{2}+17 x -9\right ) {\mathrm e}^{\operatorname {arccot}\left (x \right )}}{26 \left (a \,x^{2}+a \right )^{\frac {7}{2}}}\) | \(45\) |
Input:
int(exp(arccot(x))/(a*x^2+a)^(7/2),x,method=_RETURNVERBOSE)
Output:
1/26*(x^2+1)*(6*x^5-6*x^4+18*x^3-14*x^2+17*x-9)*exp(arccot(x))/(a*x^2+a)^( 7/2)
Time = 0.08 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.83 \[ \int \frac {e^{\cot ^{-1}(x)}}{\left (a+a x^2\right )^{7/2}} \, dx=\frac {{\left (6 \, x^{5} - 6 \, x^{4} + 18 \, x^{3} - 14 \, x^{2} + 17 \, x - 9\right )} \sqrt {a x^{2} + a} e^{\operatorname {arccot}\left (x\right )}}{26 \, {\left (a^{4} x^{6} + 3 \, a^{4} x^{4} + 3 \, a^{4} x^{2} + a^{4}\right )}} \] Input:
integrate(exp(arccot(x))/(a*x^2+a)^(7/2),x, algorithm="fricas")
Output:
1/26*(6*x^5 - 6*x^4 + 18*x^3 - 14*x^2 + 17*x - 9)*sqrt(a*x^2 + a)*e^arccot (x)/(a^4*x^6 + 3*a^4*x^4 + 3*a^4*x^2 + a^4)
Timed out. \[ \int \frac {e^{\cot ^{-1}(x)}}{\left (a+a x^2\right )^{7/2}} \, dx=\text {Timed out} \] Input:
integrate(exp(acot(x))/(a*x**2+a)**(7/2),x)
Output:
Timed out
\[ \int \frac {e^{\cot ^{-1}(x)}}{\left (a+a x^2\right )^{7/2}} \, dx=\int { \frac {e^{\operatorname {arccot}\left (x\right )}}{{\left (a x^{2} + a\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate(exp(arccot(x))/(a*x^2+a)^(7/2),x, algorithm="maxima")
Output:
integrate(e^arccot(x)/(a*x^2 + a)^(7/2), x)
\[ \int \frac {e^{\cot ^{-1}(x)}}{\left (a+a x^2\right )^{7/2}} \, dx=\int { \frac {e^{\operatorname {arccot}\left (x\right )}}{{\left (a x^{2} + a\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate(exp(arccot(x))/(a*x^2+a)^(7/2),x, algorithm="giac")
Output:
integrate(e^arccot(x)/(a*x^2 + a)^(7/2), x)
Time = 0.16 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.71 \[ \int \frac {e^{\cot ^{-1}(x)}}{\left (a+a x^2\right )^{7/2}} \, dx=-\frac {9\,{\mathrm {e}}^{\mathrm {acot}\left (x\right )}-17\,x\,{\mathrm {e}}^{\mathrm {acot}\left (x\right )}+14\,x^2\,{\mathrm {e}}^{\mathrm {acot}\left (x\right )}-18\,x^3\,{\mathrm {e}}^{\mathrm {acot}\left (x\right )}+6\,x^4\,{\mathrm {e}}^{\mathrm {acot}\left (x\right )}-6\,x^5\,{\mathrm {e}}^{\mathrm {acot}\left (x\right )}}{26\,a\,{\left (a\,x^2+a\right )}^{5/2}} \] Input:
int(exp(acot(x))/(a + a*x^2)^(7/2),x)
Output:
-(9*exp(acot(x)) - 17*x*exp(acot(x)) + 14*x^2*exp(acot(x)) - 18*x^3*exp(ac ot(x)) + 6*x^4*exp(acot(x)) - 6*x^5*exp(acot(x)))/(26*a*(a + a*x^2)^(5/2))
Time = 0.18 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.74 \[ \int \frac {e^{\cot ^{-1}(x)}}{\left (a+a x^2\right )^{7/2}} \, dx=\frac {e^{\mathit {atan} \left (\frac {1}{x}\right )} \sqrt {a}\, \sqrt {x^{2}+1}\, \left (6 x^{5}-6 x^{4}+18 x^{3}-14 x^{2}+17 x -9\right )}{26 a^{4} \left (x^{6}+3 x^{4}+3 x^{2}+1\right )} \] Input:
int(exp(acot(x))/(a*x^2+a)^(7/2),x)
Output:
(e**atan(1/x)*sqrt(a)*sqrt(x**2 + 1)*(6*x**5 - 6*x**4 + 18*x**3 - 14*x**2 + 17*x - 9))/(26*a**4*(x**6 + 3*x**4 + 3*x**2 + 1))