Integrand size = 33, antiderivative size = 67 \[ \int (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^{5/2} \, dx=\frac {i \operatorname {Hypergeometric2F1}\left (1,\frac {5}{2}+m,\frac {7}{2},\frac {1}{2} (1-i \tan (e+f x))\right ) (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^{5/2}}{5 f} \] Output:
1/5*I*hypergeom([1, 5/2+m],[7/2],1/2-1/2*I*tan(f*x+e))*(a+I*a*tan(f*x+e))^ m*(c-I*c*tan(f*x+e))^(5/2)/f
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(141\) vs. \(2(67)=134\).
Time = 12.93 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.10 \[ \int (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^{5/2} \, dx=-\frac {i 2^{\frac {3}{2}+m} c \left (e^{i f x}\right )^m \left (\frac {c}{1+e^{2 i (e+f x)}}\right )^{3/2} \left (\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right )^m \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,1+m,-e^{2 i (e+f x)}\right ) \sec ^{-m}(e+f x) (\cos (f x)+i \sin (f x))^{-m} (a+i a \tan (e+f x))^m}{f m} \] Input:
Integrate[(a + I*a*Tan[e + f*x])^m*(c - I*c*Tan[e + f*x])^(5/2),x]
Output:
((-I)*2^(3/2 + m)*c*(E^(I*f*x))^m*(c/(1 + E^((2*I)*(e + f*x))))^(3/2)*(E^( I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^m*Hypergeometric2F1[-3/2, 1, 1 + m , -E^((2*I)*(e + f*x))]*(a + I*a*Tan[e + f*x])^m)/(f*m*Sec[e + f*x]^m*(Cos [f*x] + I*Sin[f*x])^m)
Time = 0.30 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.31, 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.
\(\displaystyle \int (c-i c \tan (e+f x))^{5/2} (a+i a \tan (e+f x))^m \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (c-i c \tan (e+f x))^{5/2} (a+i a \tan (e+f x))^mdx\) |
\(\Big \downarrow \) 4006 |
\(\displaystyle \frac {a c \int (i \tan (e+f x) a+a)^{m-1} (c-i c \tan (e+f x))^{3/2}d\tan (e+f x)}{f}\) |
\(\Big \downarrow \) 80 |
\(\displaystyle \frac {c 2^{m-1} (1+i \tan (e+f x))^{-m} (a+i a \tan (e+f x))^m \int \left (\frac {1}{2} i \tan (e+f x)+\frac {1}{2}\right )^{m-1} (c-i c \tan (e+f x))^{3/2}d\tan (e+f x)}{f}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle \frac {i 2^m (c-i c \tan (e+f x))^{5/2} (1+i \tan (e+f x))^{-m} (a+i a \tan (e+f x))^m \operatorname {Hypergeometric2F1}\left (\frac {5}{2},1-m,\frac {7}{2},\frac {1}{2} (1-i \tan (e+f x))\right )}{5 f}\) |
Input:
Int[(a + I*a*Tan[e + f*x])^m*(c - I*c*Tan[e + f*x])^(5/2),x]
Output:
((I/5)*2^m*Hypergeometric2F1[5/2, 1 - m, 7/2, (1 - I*Tan[e + f*x])/2]*(a + I*a*Tan[e + f*x])^m*(c - I*c*Tan[e + f*x])^(5/2))/(f*(1 + I*Tan[e + f*x]) ^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[((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]
\[\int \left (a +i a \tan \left (f x +e \right )\right )^{m} \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}d x\]
Input:
int((a+I*a*tan(f*x+e))^m*(c-I*c*tan(f*x+e))^(5/2),x)
Output:
int((a+I*a*tan(f*x+e))^m*(c-I*c*tan(f*x+e))^(5/2),x)
\[ \int (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^{5/2} \, dx=\int { {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m} \,d x } \] Input:
integrate((a+I*a*tan(f*x+e))^m*(c-I*c*tan(f*x+e))^(5/2),x, algorithm="fric as")
Output:
integral(4*sqrt(2)*c^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^(4*I*f*x + 4*I*e) + 2*e^(2*I*f*x + 2*I*e) + 1), x)
Timed out. \[ \int (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^{5/2} \, dx=\text {Timed out} \] Input:
integrate((a+I*a*tan(f*x+e))**m*(c-I*c*tan(f*x+e))**(5/2),x)
Output:
Timed out
\[ \int (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^{5/2} \, dx=\int { {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m} \,d x } \] Input:
integrate((a+I*a*tan(f*x+e))^m*(c-I*c*tan(f*x+e))^(5/2),x, algorithm="maxi ma")
Output:
integrate((-I*c*tan(f*x + e) + c)^(5/2)*(I*a*tan(f*x + e) + a)^m, x)
Exception generated. \[ \int (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^{5/2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+I*a*tan(f*x+e))^m*(c-I*c*tan(f*x+e))^(5/2),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 TypeError: Bad Argument TypeError: Bad Argument TypeDone
Timed out. \[ \int (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^{5/2} \, dx=\int {\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^m\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2} \,d x \] Input:
int((a + a*tan(e + f*x)*1i)^m*(c - c*tan(e + f*x)*1i)^(5/2),x)
Output:
int((a + a*tan(e + f*x)*1i)^m*(c - c*tan(e + f*x)*1i)^(5/2), x)
\[ \int (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^{5/2} \, dx=\frac {\sqrt {c}\, c^{2} \left (-4 \left (\tan \left (f x +e \right ) a i +a \right )^{m} \sqrt {-\tan \left (f x +e \right ) i +1}\, i m -2 \left (\tan \left (f x +e \right ) a i +a \right )^{m} \sqrt {-\tan \left (f x +e \right ) i +1}\, i -4 \left (\int \left (\tan \left (f x +e \right ) a i +a \right )^{m} \sqrt {-\tan \left (f x +e \right ) i +1}\, \tan \left (f x +e \right )^{2}d x \right ) f \,m^{2}+\left (\int \left (\tan \left (f x +e \right ) a i +a \right )^{m} \sqrt {-\tan \left (f x +e \right ) i +1}\, \tan \left (f x +e \right )^{2}d x \right ) f -4 \left (\int \left (\tan \left (f x +e \right ) a i +a \right )^{m} \sqrt {-\tan \left (f x +e \right ) i +1}\, \tan \left (f x +e \right )d x \right ) f i \,m^{2}+4 \left (\int \left (\tan \left (f x +e \right ) a i +a \right )^{m} \sqrt {-\tan \left (f x +e \right ) i +1}\, \tan \left (f x +e \right )d x \right ) f i m +3 \left (\int \left (\tan \left (f x +e \right ) a i +a \right )^{m} \sqrt {-\tan \left (f x +e \right ) i +1}\, \tan \left (f x +e \right )d x \right ) f i \right )}{f \left (4 m^{2}-1\right )} \] Input:
int((a+I*a*tan(f*x+e))^m*(c-I*c*tan(f*x+e))^(5/2),x)
Output:
(sqrt(c)*c**2*( - 4*(tan(e + f*x)*a*i + a)**m*sqrt( - tan(e + f*x)*i + 1)* i*m - 2*(tan(e + f*x)*a*i + a)**m*sqrt( - tan(e + f*x)*i + 1)*i - 4*int((t an(e + f*x)*a*i + a)**m*sqrt( - tan(e + f*x)*i + 1)*tan(e + f*x)**2,x)*f*m **2 + int((tan(e + f*x)*a*i + a)**m*sqrt( - tan(e + f*x)*i + 1)*tan(e + f* x)**2,x)*f - 4*int((tan(e + f*x)*a*i + a)**m*sqrt( - tan(e + f*x)*i + 1)*t an(e + f*x),x)*f*i*m**2 + 4*int((tan(e + f*x)*a*i + a)**m*sqrt( - tan(e + f*x)*i + 1)*tan(e + f*x),x)*f*i*m + 3*int((tan(e + f*x)*a*i + a)**m*sqrt( - tan(e + f*x)*i + 1)*tan(e + f*x),x)*f*i))/(f*(4*m**2 - 1))