Integrand size = 31, antiderivative size = 52 \[ \int \frac {(a+i a \tan (e+f x))^m}{(c-i c \tan (e+f x))^2} \, dx=-\frac {i \operatorname {Hypergeometric2F1}\left (3,m,1+m,\frac {1}{2} (1+i \tan (e+f x))\right ) (a+i a \tan (e+f x))^m}{8 c^2 f m} \] Output:
-1/8*I*hypergeom([3, m],[1+m],1/2+1/2*I*tan(f*x+e))*(a+I*a*tan(f*x+e))^m/c ^2/f/m
Time = 0.76 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00 \[ \int \frac {(a+i a \tan (e+f x))^m}{(c-i c \tan (e+f x))^2} \, dx=-\frac {i \operatorname {Hypergeometric2F1}\left (3,m,1+m,\frac {1}{2} (1+i \tan (e+f x))\right ) (a+i a \tan (e+f x))^m}{8 c^2 f m} \] Input:
Integrate[(a + I*a*Tan[e + f*x])^m/(c - I*c*Tan[e + f*x])^2,x]
Output:
((-1/8*I)*Hypergeometric2F1[3, m, 1 + m, (1 + I*Tan[e + f*x])/2]*(a + I*a* Tan[e + f*x])^m)/(c^2*f*m)
Time = 0.35 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {3042, 4005, 3042, 3968, 78}
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.
\(\displaystyle \int \frac {(a+i a \tan (e+f x))^m}{(c-i c \tan (e+f x))^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a+i a \tan (e+f x))^m}{(c-i c \tan (e+f x))^2}dx\) |
\(\Big \downarrow \) 4005 |
\(\displaystyle \frac {\int \cos ^4(e+f x) (i \tan (e+f x) a+a)^{m+2}dx}{a^2 c^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {(i \tan (e+f x) a+a)^{m+2}}{\sec (e+f x)^4}dx}{a^2 c^2}\) |
\(\Big \downarrow \) 3968 |
\(\displaystyle -\frac {i a^3 \int \frac {(i \tan (e+f x) a+a)^{m-1}}{(a-i a \tan (e+f x))^3}d(i a \tan (e+f x))}{c^2 f}\) |
\(\Big \downarrow \) 78 |
\(\displaystyle -\frac {i (a+i a \tan (e+f x))^m \operatorname {Hypergeometric2F1}\left (3,m,m+1,\frac {i \tan (e+f x) a+a}{2 a}\right )}{8 c^2 f m}\) |
Input:
Int[(a + I*a*Tan[e + f*x])^m/(c - I*c*Tan[e + f*x])^2,x]
Output:
((-1/8*I)*Hypergeometric2F1[3, m, 1 + m, (a + I*a*Tan[e + f*x])/(2*a)]*(a + I*a*Tan[e + f*x])^m)/(c^2*f*m)
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b *c - a*d)^n*((a + b*x)^(m + 1)/(b^(n + 1)*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m}, x] && !IntegerQ[m] && IntegerQ[n]
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_ ), x_Symbol] :> Simp[1/(a^(m - 2)*b*f) Subst[Int[(a - x)^(m/2 - 1)*(a + x )^(n + m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && EqQ[a^2 + b^2, 0] && IntegerQ[m/2]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[a^m*c^m Int[Sec[e + f*x]^(2*m)*(c + d*Tan[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[ b*c + a*d, 0] && EqQ[a^2 + b^2, 0] && IntegerQ[m] && !(IGtQ[n, 0] && (LtQ[ m, 0] || GtQ[m, n]))
\[\int \frac {\left (a +i a \tan \left (f x +e \right )\right )^{m}}{\left (c -i c \tan \left (f x +e \right )\right )^{2}}d x\]
Input:
int((a+I*a*tan(f*x+e))^m/(c-I*c*tan(f*x+e))^2,x)
Output:
int((a+I*a*tan(f*x+e))^m/(c-I*c*tan(f*x+e))^2,x)
\[ \int \frac {(a+i a \tan (e+f x))^m}{(c-i c \tan (e+f x))^2} \, dx=\int { \frac {{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{2}} \,d x } \] Input:
integrate((a+I*a*tan(f*x+e))^m/(c-I*c*tan(f*x+e))^2,x, algorithm="fricas")
Output:
integral(1/4*(2*a*e^(2*I*f*x + 2*I*e)/(e^(2*I*f*x + 2*I*e) + 1))^m*(e^(4*I *f*x + 4*I*e) + 2*e^(2*I*f*x + 2*I*e) + 1)/c^2, x)
\[ \int \frac {(a+i a \tan (e+f x))^m}{(c-i c \tan (e+f x))^2} \, dx=- \frac {\int \frac {\left (i a \tan {\left (e + f x \right )} + a\right )^{m}}{\tan ^{2}{\left (e + f x \right )} + 2 i \tan {\left (e + f x \right )} - 1}\, dx}{c^{2}} \] Input:
integrate((a+I*a*tan(f*x+e))**m/(c-I*c*tan(f*x+e))**2,x)
Output:
-Integral((I*a*tan(e + f*x) + a)**m/(tan(e + f*x)**2 + 2*I*tan(e + f*x) - 1), x)/c**2
Exception generated. \[ \int \frac {(a+i a \tan (e+f x))^m}{(c-i c \tan (e+f x))^2} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((a+I*a*tan(f*x+e))^m/(c-I*c*tan(f*x+e))^2,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
\[ \int \frac {(a+i a \tan (e+f x))^m}{(c-i c \tan (e+f x))^2} \, dx=\int { \frac {{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{2}} \,d x } \] Input:
integrate((a+I*a*tan(f*x+e))^m/(c-I*c*tan(f*x+e))^2,x, algorithm="giac")
Output:
integrate((I*a*tan(f*x + e) + a)^m/(-I*c*tan(f*x + e) + c)^2, x)
Timed out. \[ \int \frac {(a+i a \tan (e+f x))^m}{(c-i c \tan (e+f x))^2} \, dx=\int \frac {{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^m}{{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \] Input:
int((a + a*tan(e + f*x)*1i)^m/(c - c*tan(e + f*x)*1i)^2,x)
Output:
int((a + a*tan(e + f*x)*1i)^m/(c - c*tan(e + f*x)*1i)^2, x)
\[ \int \frac {(a+i a \tan (e+f x))^m}{(c-i c \tan (e+f x))^2} \, dx=-\frac {\int \frac {\left (\tan \left (f x +e \right ) a i +a \right )^{m}}{\tan \left (f x +e \right )^{2}+2 \tan \left (f x +e \right ) i -1}d x}{c^{2}} \] Input:
int((a+I*a*tan(f*x+e))^m/(c-I*c*tan(f*x+e))^2,x)
Output:
( - int((tan(e + f*x)*a*i + a)**m/(tan(e + f*x)**2 + 2*tan(e + f*x)*i - 1) ,x))/c**2