\(\int \frac {(a+i a \tan (e+f x))^m}{(c+d \tan (e+f x))^{5/2}} \, dx\) [1190]
Optimal result
Integrand size = 30, antiderivative size = 125 \[
\int \frac {(a+i a \tan (e+f x))^m}{(c+d \tan (e+f x))^{5/2}} \, dx=-\frac {i \operatorname {AppellF1}\left (m,\frac {5}{2},1,1+m,-\frac {d (1+i \tan (e+f x))}{i c-d},\frac {1}{2} (1+i \tan (e+f x))\right ) (a+i a \tan (e+f x))^m \sqrt {\frac {c+d \tan (e+f x)}{c+i d}}}{2 (c+i d)^2 f m \sqrt {c+d \tan (e+f x)}}
\]
[Out]
-1/2*I*AppellF1(m,5/2,1,1+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))/(c+I*d))^(1/2)
*(a+I*a*tan(f*x+e))^m/(c+I*d)^2/f/m/(c+d*tan(f*x+e))^(1/2)
Rubi [A] (verified)
Time = 0.20 (sec) , antiderivative size = 125, 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}
\[
\int \frac {(a+i a \tan (e+f x))^m}{(c+d \tan (e+f x))^{5/2}} \, dx=-\frac {i (a+i a \tan (e+f x))^m \sqrt {\frac {c+d \tan (e+f x)}{c+i d}} \operatorname {AppellF1}\left (m,\frac {5}{2},1,m+1,-\frac {d (i \tan (e+f x)+1)}{i c-d},\frac {1}{2} (i \tan (e+f x)+1)\right )}{2 f m (c+i d)^2 \sqrt {c+d \tan (e+f x)}}
\]
[In]
Int[(a + I*a*Tan[e + f*x])^m/(c + d*Tan[e + f*x])^(5/2),x]
[Out]
((-1/2*I)*AppellF1[m, 5/2, 1, 1 + m, -((d*(1 + I*Tan[e + f*x]))/(I*c - d)), (1 + I*Tan[e + f*x])/2]*(a + I*a*T
an[e + f*x])^m*Sqrt[(c + d*Tan[e + f*x])/(c + I*d)])/((c + I*d)^2*f*m*Sqrt[c + d*Tan[e + f*x]])
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}}{\left (-a^2+a x\right ) \left (c-\frac {i d x}{a}\right )^{5/2}} \, dx,x,i a \tan (e+f x)\right )}{f} \\ & = \frac {\left (i a^2 \sqrt {\frac {c+d \tan (e+f x)}{c+i d}}\right ) \text {Subst}\left (\int \frac {(a+x)^{-1+m}}{\left (-a^2+a x\right ) \left (\frac {c}{c+i d}-\frac {i d x}{a (c+i d)}\right )^{5/2}} \, dx,x,i a \tan (e+f x)\right )}{(c+i d)^2 f \sqrt {c+d \tan (e+f x)}} \\ & = -\frac {i \operatorname {AppellF1}\left (m,\frac {5}{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 {\frac {c+d \tan (e+f x)}{c+i d}}}{2 (c+i d)^2 f m \sqrt {c+d \tan (e+f x)}} \\
\end{align*}
Mathematica [F]
\[
\int \frac {(a+i a \tan (e+f x))^m}{(c+d \tan (e+f x))^{5/2}} \, dx=\int \frac {(a+i a \tan (e+f x))^m}{(c+d \tan (e+f x))^{5/2}} \, dx
\]
[In]
Integrate[(a + I*a*Tan[e + f*x])^m/(c + d*Tan[e + f*x])^(5/2),x]
[Out]
Integrate[(a + I*a*Tan[e + f*x])^m/(c + d*Tan[e + f*x])^(5/2), x]
Maple [F]
\[\int \frac {\left (a +i a \tan \left (f x +e \right )\right )^{m}}{\left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}d x\]
[In]
int((a+I*a*tan(f*x+e))^m/(c+d*tan(f*x+e))^(5/2),x)
[Out]
int((a+I*a*tan(f*x+e))^m/(c+d*tan(f*x+e))^(5/2),x)
Fricas [F]
\[
\int \frac {(a+i a \tan (e+f x))^m}{(c+d \tan (e+f x))^{5/2}} \, dx=\int { \frac {{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m}}{{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x }
\]
[In]
integrate((a+I*a*tan(f*x+e))^m/(c+d*tan(f*x+e))^(5/2),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))*(-I*e^(6*I*f*x + 6*I*e) - 3*I*e^(4*I*f*x + 4*I*e) - 3*I*e^(2*I*f*x + 2*I*e) - I)/(-
I*c^3 + 3*c^2*d + 3*I*c*d^2 - d^3 + (-I*c^3 - 3*c^2*d + 3*I*c*d^2 + d^3)*e^(6*I*f*x + 6*I*e) - 3*(I*c^3 + c^2*
d + I*c*d^2 + d^3)*e^(4*I*f*x + 4*I*e) - 3*(I*c^3 - c^2*d + I*c*d^2 - d^3)*e^(2*I*f*x + 2*I*e)), x)
Sympy [F]
\[
\int \frac {(a+i a \tan (e+f x))^m}{(c+d \tan (e+f x))^{5/2}} \, dx=\int \frac {\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{m}}{\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx
\]
[In]
integrate((a+I*a*tan(f*x+e))**m/(c+d*tan(f*x+e))**(5/2),x)
[Out]
Integral((I*a*(tan(e + f*x) - I))**m/(c + d*tan(e + f*x))**(5/2), x)
Maxima [F]
\[
\int \frac {(a+i a \tan (e+f x))^m}{(c+d \tan (e+f x))^{5/2}} \, dx=\int { \frac {{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m}}{{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x }
\]
[In]
integrate((a+I*a*tan(f*x+e))^m/(c+d*tan(f*x+e))^(5/2),x, algorithm="maxima")
[Out]
integrate((I*a*tan(f*x + e) + a)^m/(d*tan(f*x + e) + c)^(5/2), x)
Giac [F(-2)]
Exception generated. \[
\int \frac {(a+i a \tan (e+f x))^m}{(c+d \tan (e+f x))^{5/2}} \, dx=\text {Exception raised: TypeError}
\]
[In]
integrate((a+I*a*tan(f*x+e))^m/(c+d*tan(f*x+e))^(5/2),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. \[
\int \frac {(a+i a \tan (e+f x))^m}{(c+d \tan (e+f x))^{5/2}} \, dx=\int \frac {{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^m}{{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}} \,d x
\]
[In]
int((a + a*tan(e + f*x)*1i)^m/(c + d*tan(e + f*x))^(5/2),x)
[Out]
int((a + a*tan(e + f*x)*1i)^m/(c + d*tan(e + f*x))^(5/2), x)