Integrand size = 33, antiderivative size = 65 \[ \int \frac {(a+i a \tan (e+f x))^m}{\sqrt {c-i c \tan (e+f x)}} \, dx=-\frac {i \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{2}+m,\frac {1}{2},\frac {1}{2} (1-i \tan (e+f x))\right ) (a+i a \tan (e+f x))^m}{f \sqrt {c-i c \tan (e+f x)}} \] Output:
-I*hypergeom([1, -1/2+m],[1/2],1/2-1/2*I*tan(f*x+e))*(a+I*a*tan(f*x+e))^m/ f/(c-I*c*tan(f*x+e))^(1/2)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(141\) vs. \(2(65)=130\).
Time = 5.93 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.17 \[ \int \frac {(a+i a \tan (e+f x))^m}{\sqrt {c-i c \tan (e+f x)}} \, dx=-\frac {i 2^{-\frac {3}{2}+m} c \left (e^{i f x}\right )^m \left (\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right )^m \operatorname {Hypergeometric2F1}\left (1,\frac {3}{2},1+m,-e^{2 i (e+f x)}\right ) \sec ^{-m}(e+f x) (\cos (f x)+i \sin (f x))^{-m} (a+i a \tan (e+f x))^m}{\left (\frac {c}{1+e^{2 i (e+f x)}}\right )^{3/2} f m} \] Input:
Integrate[(a + I*a*Tan[e + f*x])^m/Sqrt[c - I*c*Tan[e + f*x]],x]
Output:
((-I)*2^(-3/2 + m)*c*(E^(I*f*x))^m*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x ))))^m*Hypergeometric2F1[1, 3/2, 1 + m, -E^((2*I)*(e + f*x))]*(a + I*a*Tan [e + f*x])^m)/((c/(1 + E^((2*I)*(e + f*x))))^(3/2)*f*m*Sec[e + f*x]^m*(Cos [f*x] + I*Sin[f*x])^m)
Time = 0.28 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.32, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {3042, 4006, 80, 79}
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.
\(\displaystyle \int \frac {(a+i a \tan (e+f x))^m}{\sqrt {c-i c \tan (e+f x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a+i a \tan (e+f x))^m}{\sqrt {c-i c \tan (e+f x)}}dx\) |
\(\Big \downarrow \) 4006 |
\(\displaystyle \frac {a c \int \frac {(i \tan (e+f x) a+a)^{m-1}}{(c-i c \tan (e+f x))^{3/2}}d\tan (e+f x)}{f}\) |
\(\Big \downarrow \) 80 |
\(\displaystyle \frac {c 2^{m-1} (1+i \tan (e+f x))^{-m} (a+i a \tan (e+f x))^m \int \frac {\left (\frac {1}{2} i \tan (e+f x)+\frac {1}{2}\right )^{m-1}}{(c-i c \tan (e+f x))^{3/2}}d\tan (e+f x)}{f}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle -\frac {i 2^m (1+i \tan (e+f x))^{-m} (a+i a \tan (e+f x))^m \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1-m,\frac {1}{2},\frac {1}{2} (1-i \tan (e+f x))\right )}{f \sqrt {c-i c \tan (e+f x)}}\) |
Input:
Int[(a + I*a*Tan[e + f*x])^m/Sqrt[c - I*c*Tan[e + f*x]],x]
Output:
((-I)*2^m*Hypergeometric2F1[-1/2, 1 - m, 1/2, (1 - I*Tan[e + f*x])/2]*(a + I*a*Tan[e + f*x])^m)/(f*(1 + I*Tan[e + f*x])^m*Sqrt[c - I*c*Tan[e + f*x]] )
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 , m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] || !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) ^FracPart[n]) Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) ), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !Integ erQ[n] && (RationalQ[m] || !SimplerQ[n + 1, 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]
\[\int \frac {\left (a +i a \tan \left (f x +e \right )\right )^{m}}{\sqrt {c -i c \tan \left (f x +e \right )}}d x\]
Input:
int((a+I*a*tan(f*x+e))^m/(c-I*c*tan(f*x+e))^(1/2),x)
Output:
int((a+I*a*tan(f*x+e))^m/(c-I*c*tan(f*x+e))^(1/2),x)
\[ \int \frac {(a+i a \tan (e+f x))^m}{\sqrt {c-i c \tan (e+f x)}} \, dx=\int { \frac {{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m}}{\sqrt {-i \, c \tan \left (f x + e\right ) + c}} \,d x } \] Input:
integrate((a+I*a*tan(f*x+e))^m/(c-I*c*tan(f*x+e))^(1/2),x, algorithm="fric as")
Output:
integral(1/2*sqrt(2)*(2*a*e^(2*I*f*x + 2*I*e)/(e^(2*I*f*x + 2*I*e) + 1))^m *sqrt(c/(e^(2*I*f*x + 2*I*e) + 1))*(e^(2*I*f*x + 2*I*e) + 1)/c, x)
\[ \int \frac {(a+i a \tan (e+f x))^m}{\sqrt {c-i c \tan (e+f x)}} \, dx=\int \frac {\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{m}}{\sqrt {- i c \left (\tan {\left (e + f x \right )} + i\right )}}\, dx \] Input:
integrate((a+I*a*tan(f*x+e))**m/(c-I*c*tan(f*x+e))**(1/2),x)
Output:
Integral((I*a*(tan(e + f*x) - I))**m/sqrt(-I*c*(tan(e + f*x) + I)), x)
\[ \int \frac {(a+i a \tan (e+f x))^m}{\sqrt {c-i c \tan (e+f x)}} \, dx=\int { \frac {{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m}}{\sqrt {-i \, c \tan \left (f x + e\right ) + c}} \,d x } \] Input:
integrate((a+I*a*tan(f*x+e))^m/(c-I*c*tan(f*x+e))^(1/2),x, algorithm="maxi ma")
Output:
integrate((I*a*tan(f*x + e) + a)^m/sqrt(-I*c*tan(f*x + e) + c), x)
Exception generated. \[ \int \frac {(a+i a \tan (e+f x))^m}{\sqrt {c-i c \tan (e+f x)}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+I*a*tan(f*x+e))^m/(c-I*c*tan(f*x+e))^(1/2),x, algorithm="giac ")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument TypeError: Bad Argument TypeDone
Timed out. \[ \int \frac {(a+i a \tan (e+f x))^m}{\sqrt {c-i c \tan (e+f x)}} \, dx=\int \frac {{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^m}{\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}} \,d x \] Input:
int((a + a*tan(e + f*x)*1i)^m/(c - c*tan(e + f*x)*1i)^(1/2),x)
Output:
int((a + a*tan(e + f*x)*1i)^m/(c - c*tan(e + f*x)*1i)^(1/2), x)
\[ \int \frac {(a+i a \tan (e+f x))^m}{\sqrt {c-i c \tan (e+f x)}} \, dx =\text {Too large to display} \] Input:
int((a+I*a*tan(f*x+e))^m/(c-I*c*tan(f*x+e))^(1/2),x)
Output:
(sqrt(c)*i*( - 2*(tan(e + f*x)*a*i + a)**m*sqrt( - tan(e + f*x)*i + 1) - 8 *int(( - (tan(e + f*x)*a*i + a)**m*sqrt( - tan(e + f*x)*i + 1)*tan(e + f*x ))/(4*tan(e + f*x)**2*m**2 - tan(e + f*x)**2 + 4*m**2 - 1),x)*tan(e + f*x) **2*f*m**3 + 12*int(( - (tan(e + f*x)*a*i + a)**m*sqrt( - tan(e + f*x)*i + 1)*tan(e + f*x))/(4*tan(e + f*x)**2*m**2 - tan(e + f*x)**2 + 4*m**2 - 1), x)*tan(e + f*x)**2*f*m**2 + 2*int(( - (tan(e + f*x)*a*i + a)**m*sqrt( - ta n(e + f*x)*i + 1)*tan(e + f*x))/(4*tan(e + f*x)**2*m**2 - tan(e + f*x)**2 + 4*m**2 - 1),x)*tan(e + f*x)**2*f*m - 3*int(( - (tan(e + f*x)*a*i + a)**m *sqrt( - tan(e + f*x)*i + 1)*tan(e + f*x))/(4*tan(e + f*x)**2*m**2 - tan(e + f*x)**2 + 4*m**2 - 1),x)*tan(e + f*x)**2*f - 8*int(( - (tan(e + f*x)*a* i + a)**m*sqrt( - tan(e + f*x)*i + 1)*tan(e + f*x))/(4*tan(e + f*x)**2*m** 2 - tan(e + f*x)**2 + 4*m**2 - 1),x)*f*m**3 + 12*int(( - (tan(e + f*x)*a*i + a)**m*sqrt( - tan(e + f*x)*i + 1)*tan(e + f*x))/(4*tan(e + f*x)**2*m**2 - tan(e + f*x)**2 + 4*m**2 - 1),x)*f*m**2 + 2*int(( - (tan(e + f*x)*a*i + a)**m*sqrt( - tan(e + f*x)*i + 1)*tan(e + f*x))/(4*tan(e + f*x)**2*m**2 - tan(e + f*x)**2 + 4*m**2 - 1),x)*f*m - 3*int(( - (tan(e + f*x)*a*i + a)** m*sqrt( - tan(e + f*x)*i + 1)*tan(e + f*x))/(4*tan(e + f*x)**2*m**2 - tan( e + f*x)**2 + 4*m**2 - 1),x)*f + 2*int(((tan(e + f*x)*a*i + a)**m*sqrt( - tan(e + f*x)*i + 1)*tan(e + f*x))/(tan(e + f*x)**2 + 1),x)*tan(e + f*x)**2 *f*m - int(((tan(e + f*x)*a*i + a)**m*sqrt( - tan(e + f*x)*i + 1)*tan(e...