\(\int (a+i a \tan (e+f x))^m \sqrt {c+d \tan (e+f x)} \, dx\) [1187]
Optimal result
Integrand size = 30, antiderivative size = 116 \[
\int (a+i a \tan (e+f x))^m \sqrt {c+d \tan (e+f x)} \, dx=-\frac {i \operatorname {AppellF1}\left (m,-\frac {1}{2},1,1+m,-\frac {d (1+i \tan (e+f x))}{i c-d},\frac {1}{2} (1+i \tan (e+f x))\right ) (a+i a \tan (e+f x))^m \sqrt {c+d \tan (e+f x)}}{2 f m \sqrt {\frac {c+d \tan (e+f x)}{c+i d}}}
\]
[Out]
-1/2*I*AppellF1(m,-1/2,1,1+m,-d*(1+I*tan(f*x+e))/(I*c-d),1/2+1/2*I*tan(f*x+e))*(c+d*tan(f*x+e))^(1/2)*(a+I*a*t
an(f*x+e))^m/f/m/((c+d*tan(f*x+e))/(c+I*d))^(1/2)
Rubi [A] (verified)
Time = 0.21 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of
steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3645, 142, 141}
\[
\int (a+i a \tan (e+f x))^m \sqrt {c+d \tan (e+f x)} \, dx=-\frac {i (a+i a \tan (e+f x))^m \sqrt {c+d \tan (e+f x)} \operatorname {AppellF1}\left (m,-\frac {1}{2},1,m+1,-\frac {d (i \tan (e+f x)+1)}{i c-d},\frac {1}{2} (i \tan (e+f x)+1)\right )}{2 f m \sqrt {\frac {c+d \tan (e+f x)}{c+i d}}}
\]
[In]
Int[(a + I*a*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]],x]
[Out]
((-1/2*I)*AppellF1[m, -1/2, 1, 1 + m, -((d*(1 + I*Tan[e + f*x]))/(I*c - d)), (1 + I*Tan[e + f*x])/2]*(a + I*a*
Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]])/(f*m*Sqrt[(c + d*Tan[e + f*x])/(c + I*d)])
Rule 141
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[(b*e - a*f
)^p*((a + b*x)^(m + 1)/(b^(p + 1)*(m + 1)*(b/(b*c - a*d))^n))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/(
b*c - a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && !IntegerQ[m] && !Int
egerQ[n] && IntegerQ[p] && GtQ[b/(b*c - a*d), 0] && !(GtQ[d/(d*a - c*b), 0] && SimplerQ[c + d*x, a + b*x])
Rule 142
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[(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*(b*(c/(b*c -
a*d)) + b*d*(x/(b*c - a*d)))^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && !IntegerQ[m] &&
!IntegerQ[n] && IntegerQ[p] && !GtQ[b/(b*c - a*d), 0] && !SimplerQ[c + d*x, a + b*x]
Rule 3645
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dis
t[a*(b/f), Subst[Int[(a + x)^(m - 1)*((c + (d/b)*x)^n/(b^2 + a*x)), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b,
c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
Rubi steps \begin{align*}
\text {integral}& = \frac {\left (i a^2\right ) \text {Subst}\left (\int \frac {(a+x)^{-1+m} \sqrt {c-\frac {i d x}{a}}}{-a^2+a x} \, dx,x,i a \tan (e+f x)\right )}{f} \\ & = \frac {\left (i a^2 \sqrt {c+d \tan (e+f x)}\right ) \text {Subst}\left (\int \frac {(a+x)^{-1+m} \sqrt {\frac {c}{c+i d}-\frac {i d x}{a (c+i d)}}}{-a^2+a x} \, dx,x,i a \tan (e+f x)\right )}{f \sqrt {\frac {c+d \tan (e+f x)}{c+i d}}} \\ & = -\frac {i \operatorname {AppellF1}\left (m,-\frac {1}{2},1,1+m,-\frac {d (1+i \tan (e+f x))}{i c-d},\frac {1}{2} (1+i \tan (e+f x))\right ) (a+i a \tan (e+f x))^m \sqrt {c+d \tan (e+f x)}}{2 f m \sqrt {\frac {c+d \tan (e+f x)}{c+i d}}} \\
\end{align*}
Mathematica [F]
\[
\int (a+i a \tan (e+f x))^m \sqrt {c+d \tan (e+f x)} \, dx=\int (a+i a \tan (e+f x))^m \sqrt {c+d \tan (e+f x)} \, dx
\]
[In]
Integrate[(a + I*a*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]],x]
[Out]
Integrate[(a + I*a*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]], x]
Maple [F]
\[\int \left (a +i a \tan \left (f x +e \right )\right )^{m} \sqrt {c +d \tan \left (f x +e \right )}d x\]
[In]
int((a+I*a*tan(f*x+e))^m*(c+d*tan(f*x+e))^(1/2),x)
[Out]
int((a+I*a*tan(f*x+e))^m*(c+d*tan(f*x+e))^(1/2),x)
Fricas [F]
\[
\int (a+i a \tan (e+f x))^m \sqrt {c+d \tan (e+f x)} \, dx=\int { \sqrt {d \tan \left (f x + e\right ) + c} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m} \,d x }
\]
[In]
integrate((a+I*a*tan(f*x+e))^m*(c+d*tan(f*x+e))^(1/2),x, algorithm="fricas")
[Out]
integral((2*a*e^(2*I*f*x + 2*I*e)/(e^(2*I*f*x + 2*I*e) + 1))^m*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/
(e^(2*I*f*x + 2*I*e) + 1)), x)
Sympy [F]
\[
\int (a+i a \tan (e+f x))^m \sqrt {c+d \tan (e+f x)} \, dx=\int \left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{m} \sqrt {c + d \tan {\left (e + f x \right )}}\, dx
\]
[In]
integrate((a+I*a*tan(f*x+e))**m*(c+d*tan(f*x+e))**(1/2),x)
[Out]
Integral((I*a*(tan(e + f*x) - I))**m*sqrt(c + d*tan(e + f*x)), x)
Maxima [F]
\[
\int (a+i a \tan (e+f x))^m \sqrt {c+d \tan (e+f x)} \, dx=\int { \sqrt {d \tan \left (f x + e\right ) + c} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m} \,d x }
\]
[In]
integrate((a+I*a*tan(f*x+e))^m*(c+d*tan(f*x+e))^(1/2),x, algorithm="maxima")
[Out]
integrate(sqrt(d*tan(f*x + e) + c)*(I*a*tan(f*x + e) + a)^m, x)
Giac [F(-2)]
Exception generated. \[
\int (a+i a \tan (e+f x))^m \sqrt {c+d \tan (e+f x)} \, dx=\text {Exception raised: TypeError}
\]
[In]
integrate((a+I*a*tan(f*x+e))^m*(c+d*tan(f*x+e))^(1/2),x, algorithm="giac")
[Out]
Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(c
onst gen &
Mupad [F(-1)]
Timed out. \[
\int (a+i a \tan (e+f x))^m \sqrt {c+d \tan (e+f x)} \, dx=\int {\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^m\,\sqrt {c+d\,\mathrm {tan}\left (e+f\,x\right )} \,d x
\]
[In]
int((a + a*tan(e + f*x)*1i)^m*(c + d*tan(e + f*x))^(1/2),x)
[Out]
int((a + a*tan(e + f*x)*1i)^m*(c + d*tan(e + f*x))^(1/2), x)