Integrand size = 35, antiderivative size = 43 \[ \int \frac {(a+i a \tan (e+f x))^{5/2}}{(c-i c \tan (e+f x))^{5/2}} \, dx=-\frac {i (a+i a \tan (e+f x))^{5/2}}{5 f (c-i c \tan (e+f x))^{5/2}} \] Output:
-1/5*I*(a+I*a*tan(f*x+e))^(5/2)/f/(c-I*c*tan(f*x+e))^(5/2)
Time = 1.78 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.70 \[ \int \frac {(a+i a \tan (e+f x))^{5/2}}{(c-i c \tan (e+f x))^{5/2}} \, dx=-\frac {i a^2 (-i+\tan (e+f x))^2 \sqrt {a+i a \tan (e+f x)}}{5 c^2 f (i+\tan (e+f x))^2 \sqrt {c-i c \tan (e+f x)}} \] Input:
Integrate[(a + I*a*Tan[e + f*x])^(5/2)/(c - I*c*Tan[e + f*x])^(5/2),x]
Output:
((-1/5*I)*a^2*(-I + Tan[e + f*x])^2*Sqrt[a + I*a*Tan[e + f*x]])/(c^2*f*(I + Tan[e + f*x])^2*Sqrt[c - I*c*Tan[e + f*x]])
Time = 0.28 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {3042, 4006, 48}
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.
\(\displaystyle \int \frac {(a+i a \tan (e+f x))^{5/2}}{(c-i c \tan (e+f x))^{5/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a+i a \tan (e+f x))^{5/2}}{(c-i c \tan (e+f x))^{5/2}}dx\) |
\(\Big \downarrow \) 4006 |
\(\displaystyle \frac {a c \int \frac {(i \tan (e+f x) a+a)^{3/2}}{(c-i c \tan (e+f x))^{7/2}}d\tan (e+f x)}{f}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle -\frac {i (a+i a \tan (e+f x))^{5/2}}{5 f (c-i c \tan (e+f x))^{5/2}}\) |
Input:
Int[(a + I*a*Tan[e + f*x])^(5/2)/(c - I*c*Tan[e + f*x])^(5/2),x]
Output:
((-1/5*I)*(a + I*a*Tan[e + f*x])^(5/2))/(f*(c - I*c*Tan[e + f*x])^(5/2))
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[a*(c/f) Subst[Int[(a + b*x)^(m - 1)*( c + d*x)^(n - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n }, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 + b^2, 0]
Time = 0.89 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.81
method | result | size |
orering | \(-\frac {i \left (a +i a \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5 f \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}\) | \(35\) |
risch | \(-\frac {i a^{2} \sqrt {\frac {a \,{\mathrm e}^{2 i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, {\mathrm e}^{4 i \left (f x +e \right )}}{5 c^{2} \sqrt {\frac {c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\) | \(65\) |
derivativedivides | \(-\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{2} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (-\tan \left (f x +e \right )+i\right )}{5 f \,c^{3} \left (i+\tan \left (f x +e \right )\right )^{4}}\) | \(75\) |
default | \(-\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{2} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (-\tan \left (f x +e \right )+i\right )}{5 f \,c^{3} \left (i+\tan \left (f x +e \right )\right )^{4}}\) | \(75\) |
Input:
int((a+I*a*tan(f*x+e))^(5/2)/(c-I*c*tan(f*x+e))^(5/2),x,method=_RETURNVERB OSE)
Output:
-1/5*I*(a+I*a*tan(f*x+e))^(5/2)/f/(c-I*c*tan(f*x+e))^(5/2)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (31) = 62\).
Time = 0.10 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.65 \[ \int \frac {(a+i a \tan (e+f x))^{5/2}}{(c-i c \tan (e+f x))^{5/2}} \, dx=\frac {{\left (-i \, a^{2} e^{\left (7 i \, f x + 7 i \, e\right )} - i \, a^{2} e^{\left (5 i \, f x + 5 i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{5 \, c^{3} f} \] Input:
integrate((a+I*a*tan(f*x+e))^(5/2)/(c-I*c*tan(f*x+e))^(5/2),x, algorithm=" fricas")
Output:
1/5*(-I*a^2*e^(7*I*f*x + 7*I*e) - I*a^2*e^(5*I*f*x + 5*I*e))*sqrt(a/(e^(2* I*f*x + 2*I*e) + 1))*sqrt(c/(e^(2*I*f*x + 2*I*e) + 1))/(c^3*f)
\[ \int \frac {(a+i a \tan (e+f x))^{5/2}}{(c-i c \tan (e+f x))^{5/2}} \, dx=\int \frac {\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {5}{2}}}{\left (- i c \left (\tan {\left (e + f x \right )} + i\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((a+I*a*tan(f*x+e))**(5/2)/(c-I*c*tan(f*x+e))**(5/2),x)
Output:
Integral((I*a*(tan(e + f*x) - I))**(5/2)/(-I*c*(tan(e + f*x) + I))**(5/2), x)
Time = 0.21 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.91 \[ \int \frac {(a+i a \tan (e+f x))^{5/2}}{(c-i c \tan (e+f x))^{5/2}} \, dx=\frac {{\left (-i \, a^{2} \cos \left (5 \, f x + 5 \, e\right ) + a^{2} \sin \left (5 \, f x + 5 \, e\right )\right )} \sqrt {a}}{5 \, c^{\frac {5}{2}} f} \] Input:
integrate((a+I*a*tan(f*x+e))^(5/2)/(c-I*c*tan(f*x+e))^(5/2),x, algorithm=" maxima")
Output:
1/5*(-I*a^2*cos(5*f*x + 5*e) + a^2*sin(5*f*x + 5*e))*sqrt(a)/(c^(5/2)*f)
\[ \int \frac {(a+i a \tan (e+f x))^{5/2}}{(c-i c \tan (e+f x))^{5/2}} \, dx=\int { \frac {{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((a+I*a*tan(f*x+e))^(5/2)/(c-I*c*tan(f*x+e))^(5/2),x, algorithm=" giac")
Output:
integrate((I*a*tan(f*x + e) + a)^(5/2)/(-I*c*tan(f*x + e) + c)^(5/2), x)
Time = 2.44 (sec) , antiderivative size = 114, normalized size of antiderivative = 2.65 \[ \int \frac {(a+i a \tan (e+f x))^{5/2}}{(c-i c \tan (e+f x))^{5/2}} \, dx=-\frac {a^2\,\left (\cos \left (4\,e+4\,f\,x\right )+\sin \left (4\,e+4\,f\,x\right )\,1{}\mathrm {i}\right )\,\sqrt {\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,1{}\mathrm {i}}{5\,c^2\,f\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}} \] Input:
int((a + a*tan(e + f*x)*1i)^(5/2)/(c - c*tan(e + f*x)*1i)^(5/2),x)
Output:
-(a^2*(cos(4*e + 4*f*x) + sin(4*e + 4*f*x)*1i)*((a*(cos(2*e + 2*f*x) + sin (2*e + 2*f*x)*1i + 1))/(cos(2*e + 2*f*x) + 1))^(1/2)*1i)/(5*c^2*f*((c*(cos (2*e + 2*f*x) - sin(2*e + 2*f*x)*1i + 1))/(cos(2*e + 2*f*x) + 1))^(1/2))
\[ \int \frac {(a+i a \tan (e+f x))^{5/2}}{(c-i c \tan (e+f x))^{5/2}} \, dx=\frac {\sqrt {a}\, a^{2} \left (-\left (\int \frac {\sqrt {\tan \left (f x +e \right ) i +1}}{\sqrt {-\tan \left (f x +e \right ) i +1}\, \tan \left (f x +e \right )^{2}+2 \sqrt {-\tan \left (f x +e \right ) i +1}\, \tan \left (f x +e \right ) i -\sqrt {-\tan \left (f x +e \right ) i +1}}d x \right )+\int \frac {\sqrt {\tan \left (f x +e \right ) i +1}\, \tan \left (f x +e \right )^{2}}{\sqrt {-\tan \left (f x +e \right ) i +1}\, \tan \left (f x +e \right )^{2}+2 \sqrt {-\tan \left (f x +e \right ) i +1}\, \tan \left (f x +e \right ) i -\sqrt {-\tan \left (f x +e \right ) i +1}}d x -2 \left (\int \frac {\sqrt {\tan \left (f x +e \right ) i +1}\, \tan \left (f x +e \right )}{\sqrt {-\tan \left (f x +e \right ) i +1}\, \tan \left (f x +e \right )^{2}+2 \sqrt {-\tan \left (f x +e \right ) i +1}\, \tan \left (f x +e \right ) i -\sqrt {-\tan \left (f x +e \right ) i +1}}d x \right ) i \right )}{\sqrt {c}\, c^{2}} \] Input:
int((a+I*a*tan(f*x+e))^(5/2)/(c-I*c*tan(f*x+e))^(5/2),x)
Output:
(sqrt(a)*a**2*( - int(sqrt(tan(e + f*x)*i + 1)/(sqrt( - tan(e + f*x)*i + 1 )*tan(e + f*x)**2 + 2*sqrt( - tan(e + f*x)*i + 1)*tan(e + f*x)*i - sqrt( - tan(e + f*x)*i + 1)),x) + int((sqrt(tan(e + f*x)*i + 1)*tan(e + f*x)**2)/ (sqrt( - tan(e + f*x)*i + 1)*tan(e + f*x)**2 + 2*sqrt( - tan(e + f*x)*i + 1)*tan(e + f*x)*i - sqrt( - tan(e + f*x)*i + 1)),x) - 2*int((sqrt(tan(e + f*x)*i + 1)*tan(e + f*x))/(sqrt( - tan(e + f*x)*i + 1)*tan(e + f*x)**2 + 2 *sqrt( - tan(e + f*x)*i + 1)*tan(e + f*x)*i - sqrt( - tan(e + f*x)*i + 1)) ,x)*i))/(sqrt(c)*c**2)