Integrand size = 28, antiderivative size = 105 \[ \int \frac {(e \cos (c+d x))^m}{\sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {i 2^{-\frac {1}{2}-\frac {m}{2}} a (e \cos (c+d x))^m \operatorname {Hypergeometric2F1}\left (-\frac {m}{2},\frac {3+m}{2},1-\frac {m}{2},\frac {1}{2} (1-i \tan (c+d x))\right ) (1+i \tan (c+d x))^{\frac {3+m}{2}}}{d m (a+i a \tan (c+d x))^{3/2}} \] Output:
-I*2^(-1/2-1/2*m)*a*(e*cos(d*x+c))^m*hypergeom([-1/2*m, 3/2+1/2*m],[1-1/2* m],1/2-1/2*I*tan(d*x+c))*(1+I*tan(d*x+c))^(3/2+1/2*m)/d/m/(a+I*a*tan(d*x+c ))^(3/2)
Time = 1.67 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.36 \[ \int \frac {(e \cos (c+d x))^m}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {i 4^{-m} \left (1+e^{2 i (c+d x)}\right ) \left (e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )\right )^m \left (e e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )\right )^m \cos ^{-m}(c+d x) \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {1-m}{2},-e^{2 i (c+d x)}\right )}{d (1+m) \sqrt {a+i a \tan (c+d x)}} \] Input:
Integrate[(e*Cos[c + d*x])^m/Sqrt[a + I*a*Tan[c + d*x]],x]
Output:
(I*(1 + E^((2*I)*(c + d*x)))*((1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x)))^m *((e*(1 + E^((2*I)*(c + d*x))))/E^(I*(c + d*x)))^m*Hypergeometric2F1[1, (2 + m)/2, (1 - m)/2, -E^((2*I)*(c + d*x))])/(4^m*d*(1 + m)*Cos[c + d*x]^m*S qrt[a + I*a*Tan[c + d*x]])
Time = 0.61 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.10, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 3998, 3042, 3986, 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.
\(\displaystyle \int \frac {(e \cos (c+d x))^m}{\sqrt {a+i a \tan (c+d x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(e \cos (c+d x))^m}{\sqrt {a+i a \tan (c+d x)}}dx\) |
\(\Big \downarrow \) 3998 |
\(\displaystyle (e \cos (c+d x))^m (e \sec (c+d x))^m \int \frac {(e \sec (c+d x))^{-m}}{\sqrt {i \tan (c+d x) a+a}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle (e \cos (c+d x))^m (e \sec (c+d x))^m \int \frac {(e \sec (c+d x))^{-m}}{\sqrt {i \tan (c+d x) a+a}}dx\) |
\(\Big \downarrow \) 3986 |
\(\displaystyle (a-i a \tan (c+d x))^{m/2} (a+i a \tan (c+d x))^{m/2} (e \cos (c+d x))^m \int (a-i a \tan (c+d x))^{-m/2} (i \tan (c+d x) a+a)^{\frac {1}{2} (-m-1)}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle (a-i a \tan (c+d x))^{m/2} (a+i a \tan (c+d x))^{m/2} (e \cos (c+d x))^m \int (a-i a \tan (c+d x))^{-m/2} (i \tan (c+d x) a+a)^{\frac {1}{2} (-m-1)}dx\) |
\(\Big \downarrow \) 4006 |
\(\displaystyle \frac {a^2 (a-i a \tan (c+d x))^{m/2} (a+i a \tan (c+d x))^{m/2} (e \cos (c+d x))^m \int (a-i a \tan (c+d x))^{-\frac {m}{2}-1} (i \tan (c+d x) a+a)^{\frac {1}{2} (-m-3)}d\tan (c+d x)}{d}\) |
\(\Big \downarrow \) 80 |
\(\displaystyle \frac {a 2^{-\frac {m}{2}-\frac {3}{2}} (1+i \tan (c+d x))^{\frac {m+1}{2}} (a-i a \tan (c+d x))^{m/2} (a+i a \tan (c+d x))^{\frac {1}{2} (-m-1)+\frac {m}{2}} (e \cos (c+d x))^m \int \left (\frac {1}{2} i \tan (c+d x)+\frac {1}{2}\right )^{\frac {1}{2} (-m-3)} (a-i a \tan (c+d x))^{-\frac {m}{2}-1}d\tan (c+d x)}{d}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle -\frac {i 2^{-\frac {m}{2}-\frac {1}{2}} (1+i \tan (c+d x))^{\frac {m+1}{2}} (a+i a \tan (c+d x))^{\frac {1}{2} (-m-1)+\frac {m}{2}} (e \cos (c+d x))^m \operatorname {Hypergeometric2F1}\left (-\frac {m}{2},\frac {m+3}{2},1-\frac {m}{2},\frac {1}{2} (1-i \tan (c+d x))\right )}{d m}\) |
Input:
Int[(e*Cos[c + d*x])^m/Sqrt[a + I*a*Tan[c + d*x]],x]
Output:
((-I)*2^(-1/2 - m/2)*(e*Cos[c + d*x])^m*Hypergeometric2F1[-1/2*m, (3 + m)/ 2, 1 - m/2, (1 - I*Tan[c + d*x])/2]*(1 + I*Tan[c + d*x])^((1 + m)/2)*(a + I*a*Tan[c + d*x])^((-1 - m)/2 + m/2))/(d*m)
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[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*( x_)])^(n_.), x_Symbol] :> Simp[(d*Sec[e + f*x])^m/((a + b*Tan[e + f*x])^(m/ 2)*(a - b*Tan[e + f*x])^(m/2)) Int[(a + b*Tan[e + f*x])^(m/2 + n)*(a - b* Tan[e + f*x])^(m/2), x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x _)])^(n_.), x_Symbol] :> Simp[(d*Cos[e + f*x])^m*(d*Sec[e + f*x])^m Int[( a + b*Tan[e + f*x])^n/(d*Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, m , n}, x] && !IntegerQ[m]
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]
\[\int \frac {\left (e \cos \left (d x +c \right )\right )^{m}}{\sqrt {a +i a \tan \left (d x +c \right )}}d x\]
Input:
int((e*cos(d*x+c))^m/(a+I*a*tan(d*x+c))^(1/2),x)
Output:
int((e*cos(d*x+c))^m/(a+I*a*tan(d*x+c))^(1/2),x)
\[ \int \frac {(e \cos (c+d x))^m}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{m}}{\sqrt {i \, a \tan \left (d x + c\right ) + a}} \,d x } \] Input:
integrate((e*cos(d*x+c))^m/(a+I*a*tan(d*x+c))^(1/2),x, algorithm="fricas")
Output:
integral(1/2*sqrt(2)*(1/2*(e*e^(2*I*d*x + 2*I*c) + e)*e^(-I*d*x - I*c))^m* sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*(e^(2*I*d*x + 2*I*c) + 1)*e^(-I*d*x - I* c)/a, x)
\[ \int \frac {(e \cos (c+d x))^m}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int \frac {\left (e \cos {\left (c + d x \right )}\right )^{m}}{\sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )}}\, dx \] Input:
integrate((e*cos(d*x+c))**m/(a+I*a*tan(d*x+c))**(1/2),x)
Output:
Integral((e*cos(c + d*x))**m/sqrt(I*a*(tan(c + d*x) - I)), x)
\[ \int \frac {(e \cos (c+d x))^m}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{m}}{\sqrt {i \, a \tan \left (d x + c\right ) + a}} \,d x } \] Input:
integrate((e*cos(d*x+c))^m/(a+I*a*tan(d*x+c))^(1/2),x, algorithm="maxima")
Output:
integrate((e*cos(d*x + c))^m/sqrt(I*a*tan(d*x + c) + a), x)
Exception generated. \[ \int \frac {(e \cos (c+d x))^m}{\sqrt {a+i a \tan (c+d x)}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((e*cos(d*x+c))^m/(a+I*a*tan(d*x+c))^(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. \[ \int \frac {(e \cos (c+d x))^m}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^m}{\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}} \,d x \] Input:
int((e*cos(c + d*x))^m/(a + a*tan(c + d*x)*1i)^(1/2),x)
Output:
int((e*cos(c + d*x))^m/(a + a*tan(c + d*x)*1i)^(1/2), x)
\[ \int \frac {(e \cos (c+d x))^m}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {2 e^{m} \sqrt {a}\, i \left (-\sqrt {\tan \left (d x +c \right ) i +1}\, \cos \left (d x +c \right )^{m}-\left (\int \frac {\sqrt {\tan \left (d x +c \right ) i +1}\, \cos \left (d x +c \right )^{m} \sin \left (d x +c \right )}{\cos \left (d x +c \right ) \tan \left (d x +c \right )^{2}+\cos \left (d x +c \right )}d x \right ) \tan \left (d x +c \right )^{2} d m -\left (\int \frac {\sqrt {\tan \left (d x +c \right ) i +1}\, \cos \left (d x +c \right )^{m} \sin \left (d x +c \right )}{\cos \left (d x +c \right ) \tan \left (d x +c \right )^{2}+\cos \left (d x +c \right )}d x \right ) d m -2 \left (\int \frac {\sqrt {\tan \left (d x +c \right ) i +1}\, \cos \left (d x +c \right )^{m} \tan \left (d x +c \right )}{\tan \left (d x +c \right )^{2}+1}d x \right ) \tan \left (d x +c \right )^{2} d -2 \left (\int \frac {\sqrt {\tan \left (d x +c \right ) i +1}\, \cos \left (d x +c \right )^{m} \tan \left (d x +c \right )}{\tan \left (d x +c \right )^{2}+1}d x \right ) d \right )}{a d \left (\tan \left (d x +c \right )^{2}+1\right )} \] Input:
int((e*cos(d*x+c))^m/(a+I*a*tan(d*x+c))^(1/2),x)
Output:
(2*e**m*sqrt(a)*i*( - sqrt(tan(c + d*x)*i + 1)*cos(c + d*x)**m - int((sqrt (tan(c + d*x)*i + 1)*cos(c + d*x)**m*sin(c + d*x))/(cos(c + d*x)*tan(c + d *x)**2 + cos(c + d*x)),x)*tan(c + d*x)**2*d*m - int((sqrt(tan(c + d*x)*i + 1)*cos(c + d*x)**m*sin(c + d*x))/(cos(c + d*x)*tan(c + d*x)**2 + cos(c + d*x)),x)*d*m - 2*int((sqrt(tan(c + d*x)*i + 1)*cos(c + d*x)**m*tan(c + d*x ))/(tan(c + d*x)**2 + 1),x)*tan(c + d*x)**2*d - 2*int((sqrt(tan(c + d*x)*i + 1)*cos(c + d*x)**m*tan(c + d*x))/(tan(c + d*x)**2 + 1),x)*d))/(a*d*(tan (c + d*x)**2 + 1))