\(\int (e \sec (c+d x))^m (a+i a \tan (c+d x))^3 \, dx\) [451]

Optimal result

Integrand size = 26, antiderivative size = 86 \[ \int (e \sec (c+d x))^m (a+i a \tan (c+d x))^3 \, dx=\frac {i 2^{3+\frac {m}{2}} a^3 \operatorname {Hypergeometric2F1}\left (-2-\frac {m}{2},\frac {m}{2},\frac {2+m}{2},\frac {1}{2} (1-i \tan (c+d x))\right ) (e \sec (c+d x))^m (1+i \tan (c+d x))^{-m/2}}{d m} \]

[Out]

I*2^(3+1/2*m)*a^3*hypergeom([1/2*m, -2-1/2*m],[1+1/2*m],1/2-1/2*I*tan(d*x+c))*(e*sec(d*x+c))^m/d/m/((1+I*tan(d
*x+c))^(1/2*m))

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3586, 3604, 72, 71} \[ \int (e \sec (c+d x))^m (a+i a \tan (c+d x))^3 \, dx=\frac {i a^3 2^{\frac {m}{2}+3} (1+i \tan (c+d x))^{-m/2} (e \sec (c+d x))^m \operatorname {Hypergeometric2F1}\left (-\frac {m}{2}-2,\frac {m}{2},\frac {m+2}{2},\frac {1}{2} (1-i \tan (c+d x))\right )}{d m} \]

[In]

Int[(e*Sec[c + d*x])^m*(a + I*a*Tan[c + d*x])^3,x]

[Out]

(I*2^(3 + m/2)*a^3*Hypergeometric2F1[-2 - m/2, m/2, (2 + m)/2, (1 - I*Tan[c + d*x])/2]*(e*Sec[c + d*x])^m)/(d*
m*(1 + I*Tan[c + d*x])^(m/2))

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 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 \sec (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))^{3+\frac {m}{2}} \, dx \\ & = \frac {\left (a^2 (e \sec (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}} a^4 (e \sec (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^{3+\frac {m}{2}} a^3 \operatorname {Hypergeometric2F1}\left (-2-\frac {m}{2},\frac {m}{2},\frac {2+m}{2},\frac {1}{2} (1-i \tan (c+d x))\right ) (e \sec (c+d x))^m (1+i \tan (c+d x))^{-m/2}}{d m} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.55 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.71 \[ \int (e \sec (c+d x))^m (a+i a \tan (c+d x))^3 \, dx=-\frac {i a^3 (e \sec (c+d x))^m \left (-3 i (2+m) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {m}{2},\frac {2+m}{2},\sec ^2(c+d x)\right ) \tan (c+d x)-i (2+m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m}{2},\frac {2+m}{2},\sec ^2(c+d x)\right ) \tan (c+d x)+\left (-8-4 m+m \sec ^2(c+d x)\right ) \sqrt {-\tan ^2(c+d x)}\right )}{d m (2+m) \sqrt {-\tan ^2(c+d x)}} \]

[In]

Integrate[(e*Sec[c + d*x])^m*(a + I*a*Tan[c + d*x])^3,x]

[Out]

((-I)*a^3*(e*Sec[c + d*x])^m*((-3*I)*(2 + m)*Hypergeometric2F1[-1/2, m/2, (2 + m)/2, Sec[c + d*x]^2]*Tan[c + d
*x] - I*(2 + m)*Hypergeometric2F1[1/2, m/2, (2 + m)/2, Sec[c + d*x]^2]*Tan[c + d*x] + (-8 - 4*m + m*Sec[c + d*
x]^2)*Sqrt[-Tan[c + d*x]^2]))/(d*m*(2 + m)*Sqrt[-Tan[c + d*x]^2])

Maple [F]

\[\int \left (e \sec \left (d x +c \right )\right )^{m} \left (a +i a \tan \left (d x +c \right )\right )^{3}d x\]

[In]

int((e*sec(d*x+c))^m*(a+I*a*tan(d*x+c))^3,x)

[Out]

int((e*sec(d*x+c))^m*(a+I*a*tan(d*x+c))^3,x)

Fricas [F]

\[ \int (e \sec (c+d x))^m (a+i a \tan (c+d x))^3 \, dx=\int { {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} \left (e \sec \left (d x + c\right )\right )^{m} \,d x } \]

[In]

integrate((e*sec(d*x+c))^m*(a+I*a*tan(d*x+c))^3,x, algorithm="fricas")

[Out]

integral(8*a^3*(2*e*e^(I*d*x + I*c)/(e^(2*I*d*x + 2*I*c) + 1))^m*e^(6*I*d*x + 6*I*c)/(e^(6*I*d*x + 6*I*c) + 3*
e^(4*I*d*x + 4*I*c) + 3*e^(2*I*d*x + 2*I*c) + 1), x)

Sympy [F]

\[ \int (e \sec (c+d x))^m (a+i a \tan (c+d x))^3 \, dx=- i a^{3} \left (\int i \left (e \sec {\left (c + d x \right )}\right )^{m}\, dx + \int \left (- 3 \left (e \sec {\left (c + d x \right )}\right )^{m} \tan {\left (c + d x \right )}\right )\, dx + \int \left (e \sec {\left (c + d x \right )}\right )^{m} \tan ^{3}{\left (c + d x \right )}\, dx + \int \left (- 3 i \left (e \sec {\left (c + d x \right )}\right )^{m} \tan ^{2}{\left (c + d x \right )}\right )\, dx\right ) \]

[In]

integrate((e*sec(d*x+c))**m*(a+I*a*tan(d*x+c))**3,x)

[Out]

-I*a**3*(Integral(I*(e*sec(c + d*x))**m, x) + Integral(-3*(e*sec(c + d*x))**m*tan(c + d*x), x) + Integral((e*s
ec(c + d*x))**m*tan(c + d*x)**3, x) + Integral(-3*I*(e*sec(c + d*x))**m*tan(c + d*x)**2, x))

Maxima [F]

\[ \int (e \sec (c+d x))^m (a+i a \tan (c+d x))^3 \, dx=\int { {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} \left (e \sec \left (d x + c\right )\right )^{m} \,d x } \]

[In]

integrate((e*sec(d*x+c))^m*(a+I*a*tan(d*x+c))^3,x, algorithm="maxima")

[Out]

integrate((I*a*tan(d*x + c) + a)^3*(e*sec(d*x + c))^m, x)

Giac [F]

\[ \int (e \sec (c+d x))^m (a+i a \tan (c+d x))^3 \, dx=\int { {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} \left (e \sec \left (d x + c\right )\right )^{m} \,d x } \]

[In]

integrate((e*sec(d*x+c))^m*(a+I*a*tan(d*x+c))^3,x, algorithm="giac")

[Out]

integrate((I*a*tan(d*x + c) + a)^3*(e*sec(d*x + c))^m, x)

Mupad [F(-1)]

Timed out. \[ \int (e \sec (c+d x))^m (a+i a \tan (c+d x))^3 \, dx=\int {\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^m\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^3 \,d x \]

[In]

int((e/cos(c + d*x))^m*(a + a*tan(c + d*x)*1i)^3,x)

[Out]

int((e/cos(c + d*x))^m*(a + a*tan(c + d*x)*1i)^3, x)