\(\int (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^3 \, dx\) [1180]

Optimal result

Integrand size = 28, antiderivative size = 192 \[ \int (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^3 \, dx=-\frac {2 d \left (d^2+i c d m-c^2 (3+m)\right ) (a+i a \tan (e+f x))^m}{f m (2+m)}+\frac {(i c+d)^3 \operatorname {Hypergeometric2F1}\left (1,m,1+m,\frac {1}{2} (1+i \tan (e+f x))\right ) (a+i a \tan (e+f x))^m}{2 f m}-\frac {d^2 (d m+i c (4+m)) (a+i a \tan (e+f x))^{1+m}}{a f (1+m) (2+m)}+\frac {d (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^2}{f (2+m)} \]

[Out]

-2*d*(d^2+I*c*d*m-c^2*(3+m))*(a+I*a*tan(f*x+e))^m/f/m/(2+m)+1/2*(I*c+d)^3*hypergeom([1, m],[1+m],1/2+1/2*I*tan
(f*x+e))*(a+I*a*tan(f*x+e))^m/f/m-d^2*(d*m+I*c*(4+m))*(a+I*a*tan(f*x+e))^(1+m)/a/f/(1+m)/(2+m)+d*(a+I*a*tan(f*
x+e))^m*(c+d*tan(f*x+e))^2/f/(2+m)

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3641, 3673, 3608, 3562, 70} \[ \int (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^3 \, dx=-\frac {2 d \left (-\left (c^2 (m+3)\right )+i c d m+d^2\right ) (a+i a \tan (e+f x))^m}{f m (m+2)}-\frac {d^2 (d m+i c (m+4)) (a+i a \tan (e+f x))^{m+1}}{a f (m+1) (m+2)}+\frac {(d+i c)^3 (a+i a \tan (e+f x))^m \operatorname {Hypergeometric2F1}\left (1,m,m+1,\frac {1}{2} (i \tan (e+f x)+1)\right )}{2 f m}+\frac {d (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^2}{f (m+2)} \]

[In]

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

[Out]

(-2*d*(d^2 + I*c*d*m - c^2*(3 + m))*(a + I*a*Tan[e + f*x])^m)/(f*m*(2 + m)) + ((I*c + d)^3*Hypergeometric2F1[1
, m, 1 + m, (1 + I*Tan[e + f*x])/2]*(a + I*a*Tan[e + f*x])^m)/(2*f*m) - (d^2*(d*m + I*c*(4 + m))*(a + I*a*Tan[
e + f*x])^(1 + m))/(a*f*(1 + m)*(2 + m)) + (d*(a + I*a*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^2)/(f*(2 + 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 3562

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[-b/d, Subst[Int[(a + x)^(n - 1)/(a - x), x]
, x, b*Tan[c + d*x]], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 + b^2, 0]

Rule 3608

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

Rule 3641

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

Rule 3673

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

Rubi steps \begin{align*} \text {integral}& = \frac {d (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^2}{f (2+m)}-\frac {\int (a+i a \tan (e+f x))^m (c+d \tan (e+f x)) \left (-a \left (c^2 (2+m)-d (2 d+i c m)\right )+a d (i d m-c (4+m)) \tan (e+f x)\right ) \, dx}{a (2+m)} \\ & = -\frac {d^2 (d m+i c (4+m)) (a+i a \tan (e+f x))^{1+m}}{a f (1+m) (2+m)}+\frac {d (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^2}{f (2+m)}-\frac {\int (a+i a \tan (e+f x))^m \left (a \left (i c^2 d m-i d^3 m-c^3 (2+m)+c d^2 (6+m)\right )+2 a d \left (d^2+i c d m-c^2 (3+m)\right ) \tan (e+f x)\right ) \, dx}{a (2+m)} \\ & = -\frac {2 d \left (d^2+i c d m-c^2 (3+m)\right ) (a+i a \tan (e+f x))^m}{f m (2+m)}-\frac {d^2 (d m+i c (4+m)) (a+i a \tan (e+f x))^{1+m}}{a f (1+m) (2+m)}+\frac {d (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^2}{f (2+m)}+(c-i d)^3 \int (a+i a \tan (e+f x))^m \, dx \\ & = -\frac {2 d \left (d^2+i c d m-c^2 (3+m)\right ) (a+i a \tan (e+f x))^m}{f m (2+m)}-\frac {d^2 (d m+i c (4+m)) (a+i a \tan (e+f x))^{1+m}}{a f (1+m) (2+m)}+\frac {d (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^2}{f (2+m)}+\frac {\left (a (i c+d)^3\right ) \text {Subst}\left (\int \frac {(a+x)^{-1+m}}{a-x} \, dx,x,i a \tan (e+f x)\right )}{f} \\ & = -\frac {2 d \left (d^2+i c d m-c^2 (3+m)\right ) (a+i a \tan (e+f x))^m}{f m (2+m)}+\frac {(i c+d)^3 \operatorname {Hypergeometric2F1}\left (1,m,1+m,\frac {1}{2} (1+i \tan (e+f x))\right ) (a+i a \tan (e+f x))^m}{2 f m}-\frac {d^2 (d m+i c (4+m)) (a+i a \tan (e+f x))^{1+m}}{a f (1+m) (2+m)}+\frac {d (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^2}{f (2+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.27 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.76 \[ \int (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^3 \, dx=\frac {(a+i a \tan (e+f x))^m \left (-4 d (1+m) \left (d^2+i c d m-c^2 (3+m)\right )+(i c+d)^3 (1+m) (2+m) \operatorname {Hypergeometric2F1}\left (1,m,1+m,\frac {1}{2} (1+i \tan (e+f x))\right )+2 d^2 m (-i d m+c (4+m)) (-i+\tan (e+f x))+2 d m (1+m) (c+d \tan (e+f x))^2\right )}{2 f m (1+m) (2+m)} \]

[In]

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

[Out]

((a + I*a*Tan[e + f*x])^m*(-4*d*(1 + m)*(d^2 + I*c*d*m - c^2*(3 + m)) + (I*c + d)^3*(1 + m)*(2 + m)*Hypergeome
tric2F1[1, m, 1 + m, (1 + I*Tan[e + f*x])/2] + 2*d^2*m*((-I)*d*m + c*(4 + m))*(-I + Tan[e + f*x]) + 2*d*m*(1 +
m)*(c + d*Tan[e + f*x])^2))/(2*f*m*(1 + m)*(2 + m))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral((I*a*(tan(e + f*x) - I))**m*(c + d*tan(e + f*x))**3, x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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