\(\int \frac {(e \cos (c+d x))^m}{a+i a \tan (c+d x)} \, dx\) [691]

Optimal result

Integrand size = 26, antiderivative size = 86 \[ \int \frac {(e \cos (c+d x))^m}{a+i a \tan (c+d x)} \, dx=-\frac {i 2^{-1-\frac {m}{2}} (e \cos (c+d x))^m \operatorname {Hypergeometric2F1}\left (-\frac {m}{2},\frac {4+m}{2},1-\frac {m}{2},\frac {1}{2} (1-i \tan (c+d x))\right ) (1+i \tan (c+d x))^{m/2}}{a d m} \]

[Out]

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

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 86, 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} \[ \int \frac {(e \cos (c+d x))^m}{a+i a \tan (c+d x)} \, dx=-\frac {i 2^{-\frac {m}{2}-1} (1+i \tan (c+d x))^{m/2} (e \cos (c+d x))^m \operatorname {Hypergeometric2F1}\left (-\frac {m}{2},\frac {m+4}{2},1-\frac {m}{2},\frac {1}{2} (1-i \tan (c+d x))\right )}{a d m} \]

[In]

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

[Out]

((-I)*2^(-1 - m/2)*(e*Cos[c + d*x])^m*Hypergeometric2F1[-1/2*m, (4 + m)/2, 1 - m/2, (1 - I*Tan[c + d*x])/2]*(1
+ I*Tan[c + d*x])^(m/2))/(a*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 \frac {(e \sec (c+d x))^{-m}}{a+i a \tan (c+d x)} \, 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))^{-1-\frac {m}{2}} \, 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)^{-2-\frac {m}{2}} \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {\left (2^{-2-\frac {m}{2}} (e \cos (c+d x))^m (a-i a \tan (c+d x))^{m/2} \left (\frac {a+i a \tan (c+d x)}{a}\right )^{m/2}\right ) \text {Subst}\left (\int \left (\frac {1}{2}+\frac {i x}{2}\right )^{-2-\frac {m}{2}} (a-i a x)^{-1-\frac {m}{2}} \, dx,x,\tan (c+d x)\right )}{d} \\ & = -\frac {i 2^{-1-\frac {m}{2}} (e \cos (c+d x))^m \operatorname {Hypergeometric2F1}\left (-\frac {m}{2},\frac {4+m}{2},1-\frac {m}{2},\frac {1}{2} (1-i \tan (c+d x))\right ) (1+i \tan (c+d x))^{m/2}}{a d m} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(433\) vs. \(2(86)=172\).

Time = 7.52 (sec) , antiderivative size = 433, normalized size of antiderivative = 5.03 \[ \int \frac {(e \cos (c+d x))^m}{a+i a \tan (c+d x)} \, dx=-\frac {2^{-m/2} \cos (c+d x) (e \cos (c+d x))^m \left (1-2 \cos ^2(c+d x)+i \sin (2 (c+d x))\right )^{m/2} \left (2^{m/2} (2+m) \operatorname {Hypergeometric2F1}\left (1+m,\frac {2+m}{2},2+m,2 \cos (c+d x) (\cos (c+d x)-i \sin (c+d x))\right ) ((\cos (c)-i \sin (c)) \sin (c) (i+\tan (d x)))^{m/2}-2 (1+m) \operatorname {Hypergeometric2F1}\left (-1-\frac {m}{2},\frac {m}{2},-\frac {m}{2},\frac {1}{2} (1+i \tan (c+d x))\right ) ((\cos (c)+i \sin (c)) \sin (c) (-i+\tan (d x)))^{m/2} (1-i \tan (c+d x))^{m/2}\right )}{a d (1+m) (2+m) \left (-i \sin (c+d x) \left (\left (1-2 \cos ^2(c+d x)+i \sin (2 (c+d x))\right )^{m/2} ((\cos (c)+i \sin (c)) \sin (c) (-i+\tan (d x)))^{m/2}-((\cos (c)-i \sin (c)) \sin (c) (i+\tan (d x)))^{m/2}\right )+\cos (c+d x) \left (\left (1-2 \cos ^2(c+d x)+i \sin (2 (c+d x))\right )^{m/2} ((\cos (c)+i \sin (c)) \sin (c) (-i+\tan (d x)))^{m/2}+((\cos (c)-i \sin (c)) \sin (c) (i+\tan (d x)))^{m/2}\right )\right ) (-i+\tan (c+d x))} \]

[In]

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

[Out]

-((Cos[c + d*x]*(e*Cos[c + d*x])^m*(1 - 2*Cos[c + d*x]^2 + I*Sin[2*(c + d*x)])^(m/2)*(2^(m/2)*(2 + m)*Hypergeo
metric2F1[1 + m, (2 + m)/2, 2 + m, 2*Cos[c + d*x]*(Cos[c + d*x] - I*Sin[c + d*x])]*((Cos[c] - I*Sin[c])*Sin[c]
*(I + Tan[d*x]))^(m/2) - 2*(1 + m)*Hypergeometric2F1[-1 - m/2, m/2, -1/2*m, (1 + I*Tan[c + d*x])/2]*((Cos[c] +
I*Sin[c])*Sin[c]*(-I + Tan[d*x]))^(m/2)*(1 - I*Tan[c + d*x])^(m/2)))/(2^(m/2)*a*d*(1 + m)*(2 + m)*((-I)*Sin[c
+ d*x]*((1 - 2*Cos[c + d*x]^2 + I*Sin[2*(c + d*x)])^(m/2)*((Cos[c] + I*Sin[c])*Sin[c]*(-I + Tan[d*x]))^(m/2)
- ((Cos[c] - I*Sin[c])*Sin[c]*(I + Tan[d*x]))^(m/2)) + Cos[c + d*x]*((1 - 2*Cos[c + d*x]^2 + I*Sin[2*(c + d*x)
])^(m/2)*((Cos[c] + I*Sin[c])*Sin[c]*(-I + Tan[d*x]))^(m/2) + ((Cos[c] - I*Sin[c])*Sin[c]*(I + Tan[d*x]))^(m/2
)))*(-I + Tan[c + d*x])))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {(e \cos (c+d x))^m}{a+i a \tan (c+d x)} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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