3.9.0 ∫ (a + ia tan(e + fx))1+m(A + B tan(e + fx))(c − ic tan(e + fx))−1−m dx [852]

Integrand size = 47, antiderivative size = 147 \[ \int (a+i a \tan (e+f x))^{1+m} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{-1-m} \, dx=-\frac {(i A+B) (a+i a \tan (e+f x))^{1+m} (c-i c \tan (e+f x))^{-1-m}}{2 f (1+m)}+\frac {2^m a B \operatorname {Hypergeometric2F1}\left (-m,-m,1-m,\frac {1}{2} (1-i \tan (e+f x))\right ) (1+i \tan (e+f x))^{-m} (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^{-m}}{c f m} \]

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

/2*I*tan(f*x+e))*(a+I*a*tan(f*x+e))^m/c/f/m/((1+I*tan(f*x+e))^m)/((c-I*c*tan(f*x+e))^m)

Rubi [A] (veriﬁed)

Time = 0.27 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.085, Rules used = {3669, 80, 72, 71} \[ \int (a+i a \tan (e+f x))^{1+m} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{-1-m} \, dx=\frac {a B 2^m (1+i \tan (e+f x))^{-m} (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^{-m} \operatorname {Hypergeometric2F1}\left (-m,-m,1-m,\frac {1}{2} (1-i \tan (e+f x))\right )}{c f m}-\frac {(B+i A) (a+i a \tan (e+f x))^{m+1} (c-i c \tan (e+f x))^{-m-1}}{2 f (m+1)} \]

Int[(a + I*a*Tan[e + f*x])^(1 + m)*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(-1 - m),x]

-1/2*((I*A + B)*(a + I*a*Tan[e + f*x])^(1 + m)*(c - I*c*Tan[e + f*x])^(-1 - m))/(f*(1 + m)) + (2^m*a*B*Hyperge

ometric2F1[-m, -m, 1 - m, (1 - I*Tan[e + f*x])/2]*(a + I*a*Tan[e + f*x])^m)/(c*f*m*(1 + I*Tan[e + f*x])^m*(c -

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

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] &&

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f

))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1

) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^Simplify[p + 1], x], x] /; FreeQ[{a, b, c

, d, e, f, n, p}, x] && !RationalQ[p] && SumSimplerQ[p, 1]

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((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)*(A + B*x), x

], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 + b^2, 0]

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

Mathematica [A] (veriﬁed)

Time = 5.84 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.90 \[ \int (a+i a \tan (e+f x))^{1+m} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{-1-m} \, dx=\frac {a (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^{-m} \left (\frac {2^{1+m} B \operatorname {Hypergeometric2F1}\left (-m,-m,1-m,-\frac {1}{2} i (i+\tan (e+f x))\right ) (1+i \tan (e+f x))^{-m}}{m}+\frac {(i A+B) (-i+\tan (e+f x))}{(1+m) (i+\tan (e+f x))}\right )}{2 c f} \]

Integrate[(a + I*a*Tan[e + f*x])^(1 + m)*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(-1 - m),x]

(a*(a + I*a*Tan[e + f*x])^m*((2^(1 + m)*B*Hypergeometric2F1[-m, -m, 1 - m, (-1/2*I)*(I + Tan[e + f*x])])/(m*(1

+ I*Tan[e + f*x])^m) + ((I*A + B)*(-I + Tan[e + f*x]))/((1 + m)*(I + Tan[e + f*x]))))/(2*c*f*(c - I*c*Tan[e +

Maple [F]

\[\int \left (a +i a \tan \left (f x +e \right )\right )^{1+m} \left (A +B \tan \left (f x +e \right )\right ) \left (c -i c \tan \left (f x +e \right )\right )^{-1-m}d x\]

int((a+I*a*tan(f*x+e))^(1+m)*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(-1-m),x)

Fricas [F]

\[ \int (a+i a \tan (e+f x))^{1+m} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{-1-m} \, dx=\int { {\left (B \tan \left (f x + e\right ) + A\right )} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m + 1} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{-m - 1} \,d x } \]

integrate((a+I*a*tan(f*x+e))^(1+m)*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(-1-m),x, algorithm="fricas")

integral(((A - I*B)*e^(2*I*f*x + 2*I*e) + A + I*B)*(2*c/(e^(2*I*f*x + 2*I*e) + 1))^(-m - 1)*e^(2*I*e*m - 2*(-I

*f*m - I*f)*x + (m + 1)*log(a/c) + (m + 1)*log(2*c/(e^(2*I*f*x + 2*I*e) + 1)) + 2*I*e)/(e^(2*I*f*x + 2*I*e) +

Sympy [F]

\[ \int (a+i a \tan (e+f x))^{1+m} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{-1-m} \, dx=\int \left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{m + 1} \left (- i c \left (\tan {\left (e + f x \right )} + i\right )\right )^{- m - 1} \left (A + B \tan {\left (e + f x \right )}\right )\, dx \]

integrate((a+I*a*tan(f*x+e))**(1+m)*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))**(-1-m),x)

Integral((I*a*(tan(e + f*x) - I))**(m + 1)*(-I*c*(tan(e + f*x) + I))**(-m - 1)*(A + B*tan(e + f*x)), x)

Maxima [F]

integrate((a+I*a*tan(f*x+e))^(1+m)*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(-1-m),x, algorithm="maxima")

-(2*(-I*B*a^(m + 1)*m - I*B*a^(m + 1))*cos(2*f*m*x + 2*e*m) + ((A - I*B)*a^(m + 1)*m^2 - (A - I*B)*a^(m + 1)*m

)*cos(2*e*m + 2*(f*m + 2*f)*x + 4*e) + ((A + I*B)*a^(m + 1)*m^2 - (A - I*B)*a^(m + 1)*m - 2*I*B*a^(m + 1))*cos

(2*e*m + 2*(f*m + f)*x + 2*e) - 4*(B*a^(m + 1)*c^(m + 1)*f*m^3 - B*a^(m + 1)*c^(m + 1)*f*m + (B*a^(m + 1)*c^(m

+ 1)*f*m^3 - B*a^(m + 1)*c^(m + 1)*f*m)*cos(2*f*x + 2*e) - (-I*B*a^(m + 1)*c^(m + 1)*f*m^3 + I*B*a^(m + 1)*c^

(m + 1)*f*m)*sin(2*f*x + 2*e))*integrate(((cos(4*f*x + 4*e) + 2*cos(2*f*x + 2*e) + 1)*cos(2*f*m*x + 2*e*m) + (

sin(4*f*x + 4*e) + 2*sin(2*f*x + 2*e))*sin(2*f*m*x + 2*e*m))/((c^(m + 1)*m - c^(m + 1))*cos(4*f*x + 4*e)^2 + 4

*(c^(m + 1)*m - c^(m + 1))*cos(2*f*x + 2*e)^2 + (c^(m + 1)*m - c^(m + 1))*sin(4*f*x + 4*e)^2 + 4*(c^(m + 1)*m

- c^(m + 1))*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + 4*(c^(m + 1)*m - c^(m + 1))*sin(2*f*x + 2*e)^2 + c^(m + 1)*m

+ 2*(c^(m + 1)*m + 2*(c^(m + 1)*m - c^(m + 1))*cos(2*f*x + 2*e) - c^(m + 1))*cos(4*f*x + 4*e) + 4*(c^(m + 1)*m

- c^(m + 1))*cos(2*f*x + 2*e) - c^(m + 1)), x) + 4*(-I*B*a^(m + 1)*c^(m + 1)*f*m^3 + I*B*a^(m + 1)*c^(m + 1)*

f*m + (-I*B*a^(m + 1)*c^(m + 1)*f*m^3 + I*B*a^(m + 1)*c^(m + 1)*f*m)*cos(2*f*x + 2*e) + (B*a^(m + 1)*c^(m + 1)

*f*m^3 - B*a^(m + 1)*c^(m + 1)*f*m)*sin(2*f*x + 2*e))*integrate(-((sin(4*f*x + 4*e) + 2*sin(2*f*x + 2*e))*cos(

2*f*m*x + 2*e*m) - (cos(4*f*x + 4*e) + 2*cos(2*f*x + 2*e) + 1)*sin(2*f*m*x + 2*e*m))/((c^(m + 1)*m - c^(m + 1)

)*cos(4*f*x + 4*e)^2 + 4*(c^(m + 1)*m - c^(m + 1))*cos(2*f*x + 2*e)^2 + (c^(m + 1)*m - c^(m + 1))*sin(4*f*x +

4*e)^2 + 4*(c^(m + 1)*m - c^(m + 1))*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + 4*(c^(m + 1)*m - c^(m + 1))*sin(2*f*x

+ 2*e)^2 + c^(m + 1)*m + 2*(c^(m + 1)*m + 2*(c^(m + 1)*m - c^(m + 1))*cos(2*f*x + 2*e) - c^(m + 1))*cos(4*f*x

+ 4*e) + 4*(c^(m + 1)*m - c^(m + 1))*cos(2*f*x + 2*e) - c^(m + 1)), x) + 2*(B*a^(m + 1)*m + B*a^(m + 1))*sin(

2*f*m*x + 2*e*m) - ((-I*A - B)*a^(m + 1)*m^2 + (I*A + B)*a^(m + 1)*m)*sin(2*e*m + 2*(f*m + 2*f)*x + 4*e) - ((-

I*A + B)*a^(m + 1)*m^2 + (I*A + B)*a^(m + 1)*m - 2*B*a^(m + 1))*sin(2*e*m + 2*(f*m + f)*x + 2*e))/(-2*I*c^(m +

1)*f*m^3 + 2*I*c^(m + 1)*f*m - 2*(I*c^(m + 1)*f*m^3 - I*c^(m + 1)*f*m)*cos(2*f*x + 2*e) + 2*(c^(m + 1)*f*m^3

Giac [F]

integrate((a+I*a*tan(f*x+e))^(1+m)*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(-1-m),x, algorithm="giac")

integrate((B*tan(f*x + e) + A)*(I*a*tan(f*x + e) + a)^(m + 1)*(-I*c*tan(f*x + e) + c)^(-m - 1), x)

Mupad [F(-1)]

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

int(((A + B*tan(e + f*x))*(a + a*tan(e + f*x)*1i)^(m + 1))/(c - c*tan(e + f*x)*1i)^(m + 1),x)

int(((A + B*tan(e + f*x))*(a + a*tan(e + f*x)*1i)^(m + 1))/(c - c*tan(e + f*x)*1i)^(m + 1), x)

\(\int (a+i a \tan (e+f x))^{1+m} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{-1-m} \, dx\) [852]

Optimal result