\(\int (a+b \tan (e+f x))^m (c+d \tan (e+f x)) \, dx\) [1307]

Optimal result

Integrand size = 23, antiderivative size = 143 \[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x)) \, dx=\frac {(c-i d) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a-i b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (i a+b) f (1+m)}+\frac {(i c-d) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a+i b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (a+i b) f (1+m)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3620, 3618, 70} \[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x)) \, dx=\frac {(c-i d) (a+b \tan (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {a+b \tan (e+f x)}{a-i b}\right )}{2 f (m+1) (b+i a)}+\frac {(-d+i c) (a+b \tan (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {a+b \tan (e+f x)}{a+i b}\right )}{2 f (m+1) (a+i b)} \]

[In]

Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x]),x]

[Out]

((c - I*d)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a - I*b)]*(a + b*Tan[e + f*x])^(1 + m))/(2
*(I*a + b)*f*(1 + m)) + ((I*c - d)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a + I*b)]*(a + b*T
an[e + f*x])^(1 + m))/(2*(a + I*b)*f*(1 + m))

Rule 70

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b*c - a*d)^n*((a + b*x)^(m + 1)/(b^(
n + 1)*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rule 3618

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c*(
d/f), Subst[Int[(a + (b/d)*x)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] &&
NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]

Rule 3620

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c
+ I*d)/2, Int[(a + b*Tan[e + f*x])^m*(1 - I*Tan[e + f*x]), x], x] + Dist[(c - I*d)/2, Int[(a + b*Tan[e + f*x]
)^m*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0]
&& NeQ[c^2 + d^2, 0] &&  !IntegerQ[m]

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

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.84 \[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x)) \, dx=\frac {i \left (-\frac {(c-i d) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a-i b}\right )}{a-i b}+\frac {(c+i d) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a+i b}\right )}{a+i b}\right ) (a+b \tan (e+f x))^{1+m}}{2 f (1+m)} \]

[In]

Integrate[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x]),x]

[Out]

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

Maple [F]

\[\int \left (a +b \tan \left (f x +e \right )\right )^{m} \left (c +d \tan \left (f x +e \right )\right )d x\]

[In]

int((a+b*tan(f*x+e))^m*(c+d*tan(f*x+e)),x)

[Out]

int((a+b*tan(f*x+e))^m*(c+d*tan(f*x+e)),x)

Fricas [F]

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

[In]

integrate((a+b*tan(f*x+e))^m*(c+d*tan(f*x+e)),x, algorithm="fricas")

[Out]

integral((d*tan(f*x + e) + c)*(b*tan(f*x + e) + a)^m, x)

Sympy [F]

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

[In]

integrate((a+b*tan(f*x+e))**m*(c+d*tan(f*x+e)),x)

[Out]

Integral((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x)), x)

Maxima [F]

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

[In]

integrate((a+b*tan(f*x+e))^m*(c+d*tan(f*x+e)),x, algorithm="maxima")

[Out]

integrate((d*tan(f*x + e) + c)*(b*tan(f*x + e) + a)^m, x)

Giac [F]

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

[In]

integrate((a+b*tan(f*x+e))^m*(c+d*tan(f*x+e)),x, algorithm="giac")

[Out]

integrate((d*tan(f*x + e) + c)*(b*tan(f*x + e) + a)^m, x)

Mupad [F(-1)]

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

[In]

int((a + b*tan(e + f*x))^m*(c + d*tan(e + f*x)),x)

[Out]

int((a + b*tan(e + f*x))^m*(c + d*tan(e + f*x)), x)