\(\int (e \cos (c+d x))^m (a+i a \tan (c+d x))^n \, dx\) [688]

Optimal result

Integrand size = 26, antiderivative size = 105 \[ \int (e \cos (c+d x))^m (a+i a \tan (c+d x))^n \, dx=-\frac {i 2^{-\frac {m}{2}+n} (e \cos (c+d x))^m \operatorname {Hypergeometric2F1}\left (-\frac {m}{2},\frac {1}{2} (2+m-2 n),1-\frac {m}{2},\frac {1}{2} (1-i \tan (c+d x))\right ) (1+i \tan (c+d x))^{\frac {1}{2} (m-2 n)} (a+i a \tan (c+d x))^n}{d m} \]

[Out]

-I*2^(-1/2*m+n)*(e*cos(d*x+c))^m*hypergeom([-1/2*m, 1+1/2*m-n],[1-1/2*m],1/2-1/2*I*tan(d*x+c))*(1+I*tan(d*x+c)
)^(1/2*m-n)*(a+I*a*tan(d*x+c))^n/d/m

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {3596, 3586, 3604, 72, 71} \[ \int (e \cos (c+d x))^m (a+i a \tan (c+d x))^n \, dx=-\frac {i 2^{n-\frac {m}{2}} (a+i a \tan (c+d x))^n (e \cos (c+d x))^m (1+i \tan (c+d x))^{\frac {1}{2} (m-2 n)} \operatorname {Hypergeometric2F1}\left (-\frac {m}{2},\frac {1}{2} (m-2 n+2),1-\frac {m}{2},\frac {1}{2} (1-i \tan (c+d x))\right )}{d m} \]

[In]

Int[(e*Cos[c + d*x])^m*(a + I*a*Tan[c + d*x])^n,x]

[Out]

((-I)*2^(-1/2*m + n)*(e*Cos[c + d*x])^m*Hypergeometric2F1[-1/2*m, (2 + m - 2*n)/2, 1 - m/2, (1 - I*Tan[c + d*x
])/2]*(1 + I*Tan[c + d*x])^((m - 2*n)/2)*(a + I*a*Tan[c + d*x])^n)/(d*m)

Rule 71

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] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-d/(b*c - a*d), 0]))

Rule 72

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), 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*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] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] &&
(RationalQ[m] ||  !SimplerQ[n + 1, m + 1])

Rule 3586

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[(d*S
ec[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]

Rule 3596

Int[(cos[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[(d*Co
s[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]

Rule 3604

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist
[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]

Rubi steps \begin{align*} \text {integral}& = \left ((e \cos (c+d x))^m (e \sec (c+d x))^m\right ) \int (e \sec (c+d x))^{-m} (a+i a \tan (c+d x))^n \, dx \\ & = \left ((e \cos (c+d x))^m (a-i a \tan (c+d x))^{m/2} (a+i a \tan (c+d x))^{m/2}\right ) \int (a-i a \tan (c+d x))^{-m/2} (a+i a \tan (c+d x))^{-\frac {m}{2}+n} \, dx \\ & = \frac {\left (a^2 (e \cos (c+d x))^m (a-i a \tan (c+d x))^{m/2} (a+i a \tan (c+d x))^{m/2}\right ) \text {Subst}\left (\int (a-i a x)^{-1-\frac {m}{2}} (a+i a x)^{-1-\frac {m}{2}+n} \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {\left (2^{-1-\frac {m}{2}+n} a (e \cos (c+d x))^m (a-i a \tan (c+d x))^{m/2} (a+i a \tan (c+d x))^n \left (\frac {a+i a \tan (c+d x)}{a}\right )^{\frac {m}{2}-n}\right ) \text {Subst}\left (\int \left (\frac {1}{2}+\frac {i x}{2}\right )^{-1-\frac {m}{2}+n} (a-i a x)^{-1-\frac {m}{2}} \, dx,x,\tan (c+d x)\right )}{d} \\ & = -\frac {i 2^{-\frac {m}{2}+n} (e \cos (c+d x))^m \operatorname {Hypergeometric2F1}\left (-\frac {m}{2},\frac {1}{2} (2+m-2 n),1-\frac {m}{2},\frac {1}{2} (1-i \tan (c+d x))\right ) (1+i \tan (c+d x))^{\frac {1}{2} (m-2 n)} (a+i a \tan (c+d x))^n}{d m} \\ \end{align*}

Mathematica [A] (verified)

Time = 14.50 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.92 \[ \int (e \cos (c+d x))^m (a+i a \tan (c+d x))^n \, dx=\frac {i 2^{-m+n} \left (e^{i d x}\right )^n \left (\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^n \left (1+e^{2 i (c+d x)}\right )^{-m+n} \left (e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )\right )^m \cos ^{-m}(c+d x) (e \cos (c+d x))^m \operatorname {Hypergeometric2F1}\left (-m+n,-\frac {m}{2}+n,1-\frac {m}{2}+n,-e^{2 i (c+d x)}\right ) \sec ^{-n}(c+d x) (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n}{d (m-2 n)} \]

[In]

Integrate[(e*Cos[c + d*x])^m*(a + I*a*Tan[c + d*x])^n,x]

[Out]

(I*2^(-m + n)*(E^(I*d*x))^n*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^n*(1 + E^((2*I)*(c + d*x)))^(-m + n)*(
(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x)))^m*(e*Cos[c + d*x])^m*Hypergeometric2F1[-m + n, -1/2*m + n, 1 - m/2
+ n, -E^((2*I)*(c + d*x))]*(a + I*a*Tan[c + d*x])^n)/(d*(m - 2*n)*Cos[c + d*x]^m*Sec[c + d*x]^n*(Cos[d*x] + I*
Sin[d*x])^n)

Maple [F]

\[\int \left (e \cos \left (d x +c \right )\right )^{m} \left (a +i a \tan \left (d x +c \right )\right )^{n}d x\]

[In]

int((e*cos(d*x+c))^m*(a+I*a*tan(d*x+c))^n,x)

[Out]

int((e*cos(d*x+c))^m*(a+I*a*tan(d*x+c))^n,x)

Fricas [F]

\[ \int (e \cos (c+d x))^m (a+i a \tan (c+d x))^n \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{m} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n} \,d x } \]

[In]

integrate((e*cos(d*x+c))^m*(a+I*a*tan(d*x+c))^n,x, algorithm="fricas")

[Out]

integral((2*a*e^(2*I*d*x + 2*I*c)/(e^(2*I*d*x + 2*I*c) + 1))^n*e^(I*d*m*x + I*c*m + m*log(a*e) - m*log(2*a*e^(
2*I*d*x + 2*I*c)/(e^(2*I*d*x + 2*I*c) + 1))), x)

Sympy [F]

\[ \int (e \cos (c+d x))^m (a+i a \tan (c+d x))^n \, dx=\int \left (e \cos {\left (c + d x \right )}\right )^{m} \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{n}\, dx \]

[In]

integrate((e*cos(d*x+c))**m*(a+I*a*tan(d*x+c))**n,x)

[Out]

Integral((e*cos(c + d*x))**m*(I*a*(tan(c + d*x) - I))**n, x)

Maxima [F]

\[ \int (e \cos (c+d x))^m (a+i a \tan (c+d x))^n \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{m} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n} \,d x } \]

[In]

integrate((e*cos(d*x+c))^m*(a+I*a*tan(d*x+c))^n,x, algorithm="maxima")

[Out]

integrate((e*cos(d*x + c))^m*(I*a*tan(d*x + c) + a)^n, x)

Giac [F]

\[ \int (e \cos (c+d x))^m (a+i a \tan (c+d x))^n \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{m} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n} \,d x } \]

[In]

integrate((e*cos(d*x+c))^m*(a+I*a*tan(d*x+c))^n,x, algorithm="giac")

[Out]

integrate((e*cos(d*x + c))^m*(I*a*tan(d*x + c) + a)^n, x)

Mupad [F(-1)]

Timed out. \[ \int (e \cos (c+d x))^m (a+i a \tan (c+d x))^n \, dx=\int {\left (e\,\cos \left (c+d\,x\right )\right )}^m\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^n \,d x \]

[In]

int((e*cos(c + d*x))^m*(a + a*tan(c + d*x)*1i)^n,x)

[Out]

int((e*cos(c + d*x))^m*(a + a*tan(c + d*x)*1i)^n, x)