\(\int \frac {(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^3} \, dx\) [1311]

Optimal result

Integrand size = 25, antiderivative size = 455 \[ \int \frac {(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^3} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a-i b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (i a+b) (c-i d)^3 f (1+m)}+\frac {\operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a+i b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (a+i b) (i c-d)^3 f (1+m)}+\frac {d^2 \left (2 a^2 d^2 \left (3 c^2-d^2\right )-4 a b c d \left (c^2 (3-m)-d^2 (1+m)\right )-b^2 \left (d^4 (1-m) m+2 c^2 d^2 \left (1+3 m-m^2\right )-c^4 \left (6-5 m+m^2\right )\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {d (a+b \tan (e+f x))}{b c-a d}\right ) (a+b \tan (e+f x))^{1+m}}{2 (b c-a d)^3 \left (c^2+d^2\right )^3 f (1+m)}+\frac {d^2 (a+b \tan (e+f x))^{1+m}}{2 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {d^2 \left (4 a c d-b \left (d^2 (1-m)+c^2 (5-m)\right )\right ) (a+b \tan (e+f x))^{1+m}}{2 (b c-a d)^2 \left (c^2+d^2\right )^2 f (c+d \tan (e+f x))} \]

[Out]

1/2*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)/(c-I*d)^3/f/(1+m)+1/2*hy
pergeom([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)/(I*c-d)^3/f/(1+m)+1/2*d^2*(2*a
^2*d^2*(3*c^2-d^2)-4*a*b*c*d*(c^2*(3-m)-d^2*(1+m))-b^2*(d^4*(1-m)*m+2*c^2*d^2*(-m^2+3*m+1)-c^4*(m^2-5*m+6)))*h
ypergeom([1, 1+m],[2+m],-d*(a+b*tan(f*x+e))/(-a*d+b*c))*(a+b*tan(f*x+e))^(1+m)/(-a*d+b*c)^3/(c^2+d^2)^3/f/(1+m
)+1/2*d^2*(a+b*tan(f*x+e))^(1+m)/(-a*d+b*c)/(c^2+d^2)/f/(c+d*tan(f*x+e))^2-1/2*d^2*(4*a*c*d-b*(d^2*(1-m)+c^2*(
5-m)))*(a+b*tan(f*x+e))^(1+m)/(-a*d+b*c)^2/(c^2+d^2)^2/f/(c+d*tan(f*x+e))

Rubi [A] (verified)

Time = 1.89 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3650, 3730, 3734, 3620, 3618, 70, 3715} \[ \int \frac {(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^3} \, dx=\frac {d^2 \left (2 a^2 d^2 \left (3 c^2-d^2\right )-4 a b c d \left (c^2 (3-m)-d^2 (m+1)\right )-\left (b^2 \left (-\left (c^4 \left (m^2-5 m+6\right )\right )+2 c^2 d^2 \left (-m^2+3 m+1\right )+d^4 (1-m) m\right )\right )\right ) (a+b \tan (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,-\frac {d (a+b \tan (e+f x))}{b c-a d}\right )}{2 f (m+1) \left (c^2+d^2\right )^3 (b c-a d)^3}-\frac {d^2 \left (4 a c d-b c^2 (5-m)-b d^2 (1-m)\right ) (a+b \tan (e+f x))^{m+1}}{2 f \left (c^2+d^2\right )^2 (b c-a d)^2 (c+d \tan (e+f x))}+\frac {d^2 (a+b \tan (e+f x))^{m+1}}{2 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^2}+\frac {(a+b \tan (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {a+b \tan (e+f x)}{a-i b}\right )}{2 f (m+1) (b+i a) (c-i d)^3}+\frac {(a+b \tan (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {a+b \tan (e+f x)}{a+i b}\right )}{2 f (m+1) (a+i b) (-d+i c)^3} \]

[In]

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

[Out]

(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)
*(c - I*d)^3*f*(1 + m)) + (Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a + I*b)]*(a + b*Tan[e + f
*x])^(1 + m))/(2*(a + I*b)*(I*c - d)^3*f*(1 + m)) + (d^2*(2*a^2*d^2*(3*c^2 - d^2) - 4*a*b*c*d*(c^2*(3 - m) - d
^2*(1 + m)) - b^2*(d^4*(1 - m)*m + 2*c^2*d^2*(1 + 3*m - m^2) - c^4*(6 - 5*m + m^2)))*Hypergeometric2F1[1, 1 +
m, 2 + m, -((d*(a + b*Tan[e + f*x]))/(b*c - a*d))]*(a + b*Tan[e + f*x])^(1 + m))/(2*(b*c - a*d)^3*(c^2 + d^2)^
3*f*(1 + m)) + (d^2*(a + b*Tan[e + f*x])^(1 + m))/(2*(b*c - a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^2) - (d^2*
(4*a*c*d - b*d^2*(1 - m) - b*c^2*(5 - m))*(a + b*Tan[e + f*x])^(1 + m))/(2*(b*c - a*d)^2*(c^2 + d^2)^2*f*(c +
d*Tan[e + f*x]))

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]

Rule 3650

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[b^2*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d))), x] + D
ist[1/((m + 1)*(a^2 + b^2)*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c -
a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[e + f*x]^2, x],
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && I
ntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || IntegerQ[m]) &&  !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] &&
NeQ[a, 0])))

Rule 3715

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

Rule 3730

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t
an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Ta
n[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Dist[1/((m + 1)*(
b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1)
- b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e
+ f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C,
n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] &&  !(ILtQ[n, -1] && ( !I
ntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3734

Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (
f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/(a^2 + b^2), Int[(c + d*Tan[e + f*
x])^n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Dist[(A*b^2 - a*b*B + a^2*C)/(a^2 +
b^2), Int[(c + d*Tan[e + f*x])^n*((1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e,
f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] &&  !GtQ[n, 0] &&  !LeQ[n, -
1]

Rubi steps \begin{align*} \text {integral}& = \frac {d^2 (a+b \tan (e+f x))^{1+m}}{2 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}+\frac {\int \frac {(a+b \tan (e+f x))^m \left (2 c (b c-a d)+b d^2 (1-m)-2 d (b c-a d) \tan (e+f x)+b d^2 (1-m) \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^2} \, dx}{2 (b c-a d) \left (c^2+d^2\right )} \\ & = \frac {d^2 (a+b \tan (e+f x))^{1+m}}{2 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {d^2 \left (4 a c d-b d^2 (1-m)-b c^2 (5-m)\right ) (a+b \tan (e+f x))^{1+m}}{2 (b c-a d)^2 \left (c^2+d^2\right )^2 f (c+d \tan (e+f x))}+\frac {\int \frac {(a+b \tan (e+f x))^m \left (-d^2 (2 a d-b c (3-m)) (a d-b c (1+m))-\left (2 b c^2-2 a c d+b d^2 (1-m)\right ) \left (a c d-b \left (c^2-d^2 m\right )\right )-4 c d (b c-a d)^2 \tan (e+f x)+b d^2 \left (4 a c d-b d^2 (1-m)-b c^2 (5-m)\right ) m \tan ^2(e+f x)\right )}{c+d \tan (e+f x)} \, dx}{2 (b c-a d)^2 \left (c^2+d^2\right )^2} \\ & = \frac {d^2 (a+b \tan (e+f x))^{1+m}}{2 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {d^2 \left (4 a c d-b d^2 (1-m)-b c^2 (5-m)\right ) (a+b \tan (e+f x))^{1+m}}{2 (b c-a d)^2 \left (c^2+d^2\right )^2 f (c+d \tan (e+f x))}+\frac {\int (a+b \tan (e+f x))^m \left (2 c (b c-a d)^2 \left (c^2-3 d^2\right )-2 d (b c-a d)^2 \left (3 c^2-d^2\right ) \tan (e+f x)\right ) \, dx}{2 (b c-a d)^2 \left (c^2+d^2\right )^3}+\frac {\left (d^2 \left (2 a^2 d^2 \left (3 c^2-d^2\right )-4 a b c d \left (c^2 (3-m)-d^2 (1+m)\right )-b^2 \left (d^4 (1-m) m+2 c^2 d^2 \left (1+3 m-m^2\right )-c^4 \left (6-5 m+m^2\right )\right )\right )\right ) \int \frac {(a+b \tan (e+f x))^m \left (1+\tan ^2(e+f x)\right )}{c+d \tan (e+f x)} \, dx}{2 (b c-a d)^2 \left (c^2+d^2\right )^3} \\ & = \frac {d^2 (a+b \tan (e+f x))^{1+m}}{2 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {d^2 \left (4 a c d-b d^2 (1-m)-b c^2 (5-m)\right ) (a+b \tan (e+f x))^{1+m}}{2 (b c-a d)^2 \left (c^2+d^2\right )^2 f (c+d \tan (e+f x))}+\frac {\int (1+i \tan (e+f x)) (a+b \tan (e+f x))^m \, dx}{2 (c-i d)^3}+\frac {\int (1-i \tan (e+f x)) (a+b \tan (e+f x))^m \, dx}{2 (c+i d)^3}+\frac {\left (d^2 \left (2 a^2 d^2 \left (3 c^2-d^2\right )-4 a b c d \left (c^2 (3-m)-d^2 (1+m)\right )-b^2 \left (d^4 (1-m) m+2 c^2 d^2 \left (1+3 m-m^2\right )-c^4 \left (6-5 m+m^2\right )\right )\right )\right ) \text {Subst}\left (\int \frac {(a+b x)^m}{c+d x} \, dx,x,\tan (e+f x)\right )}{2 (b c-a d)^2 \left (c^2+d^2\right )^3 f} \\ & = \frac {d^2 \left (2 a^2 d^2 \left (3 c^2-d^2\right )-4 a b c d \left (c^2 (3-m)-d^2 (1+m)\right )-b^2 \left (d^4 (1-m) m+2 c^2 d^2 \left (1+3 m-m^2\right )-c^4 \left (6-5 m+m^2\right )\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {d (a+b \tan (e+f x))}{b c-a d}\right ) (a+b \tan (e+f x))^{1+m}}{2 (b c-a d)^3 \left (c^2+d^2\right )^3 f (1+m)}+\frac {d^2 (a+b \tan (e+f x))^{1+m}}{2 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {d^2 \left (4 a c d-b d^2 (1-m)-b c^2 (5-m)\right ) (a+b \tan (e+f x))^{1+m}}{2 (b c-a d)^2 \left (c^2+d^2\right )^2 f (c+d \tan (e+f x))}-\frac {\text {Subst}\left (\int \frac {(a+i b x)^m}{-1+x} \, dx,x,-i \tan (e+f x)\right )}{2 (i c-d)^3 f}+\frac {\text {Subst}\left (\int \frac {(a-i b x)^m}{-1+x} \, dx,x,i \tan (e+f x)\right )}{2 (i c+d)^3 f} \\ & = -\frac {\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) (i c+d)^3 f (1+m)}-\frac {\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) (c+i d)^3 f (1+m)}+\frac {d^2 \left (2 a^2 d^2 \left (3 c^2-d^2\right )-4 a b c d \left (c^2 (3-m)-d^2 (1+m)\right )-b^2 \left (d^4 (1-m) m+2 c^2 d^2 \left (1+3 m-m^2\right )-c^4 \left (6-5 m+m^2\right )\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {d (a+b \tan (e+f x))}{b c-a d}\right ) (a+b \tan (e+f x))^{1+m}}{2 (b c-a d)^3 \left (c^2+d^2\right )^3 f (1+m)}+\frac {d^2 (a+b \tan (e+f x))^{1+m}}{2 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {d^2 \left (4 a c d-b d^2 (1-m)-b c^2 (5-m)\right ) (a+b \tan (e+f x))^{1+m}}{2 (b c-a d)^2 \left (c^2+d^2\right )^2 f (c+d \tan (e+f x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.31 (sec) , antiderivative size = 670, normalized size of antiderivative = 1.47 \[ \int \frac {(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^3} \, dx=-\frac {d^2 (a+b \tan (e+f x))^{1+m}}{2 (-b c+a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {-\frac {\left (d^2 \left (2 c (b c-a d)+b d^2 (1-m)\right )-c \left (-2 d^2 (b c-a d)-b c d^2 (1-m)\right )\right ) (a+b \tan (e+f x))^{1+m}}{(-b c+a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))}-\frac {-\frac {\left (4 c^2 d^2 (b c-a d)^2+b c^2 d^2 \left (4 a c d-b d^2 (1-m)-b c^2 (5-m)\right ) m+d^2 \left (-d^2 (2 a d-b c (3-m)) (a d-b c (1+m))-\left (2 b c^2-2 a c d+b d^2 (1-m)\right ) \left (a c d-b \left (c^2-d^2 m\right )\right )\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {d (a+b \tan (e+f x))}{-b c+a d}\right ) (a+b \tan (e+f x))^{1+m}}{(-b c+a d) \left (c^2+d^2\right ) f (1+m)}+\frac {\frac {i \left (2 c (b c-a d)^2 \left (c^2-3 d^2\right )-2 i d (b c-a d)^2 \left (3 c^2-d^2\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {-i a-i b \tan (e+f x)}{-i a+b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (a+i b) f (1+m)}-\frac {i \left (2 c (b c-a d)^2 \left (c^2-3 d^2\right )+2 i d (b c-a d)^2 \left (3 c^2-d^2\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {i a+i b \tan (e+f x)}{-i a-b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (a-i b) f (1+m)}}{c^2+d^2}}{(-b c+a d) \left (c^2+d^2\right )}}{2 (-b c+a d) \left (c^2+d^2\right )} \]

[In]

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

[Out]

-1/2*(d^2*(a + b*Tan[e + f*x])^(1 + m))/((-(b*c) + a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^2) - (-(((d^2*(2*c*
(b*c - a*d) + b*d^2*(1 - m)) - c*(-2*d^2*(b*c - a*d) - b*c*d^2*(1 - m)))*(a + b*Tan[e + f*x])^(1 + m))/((-(b*c
) + a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x]))) - (-(((4*c^2*d^2*(b*c - a*d)^2 + b*c^2*d^2*(4*a*c*d - b*d^2*(1 -
m) - b*c^2*(5 - m))*m + d^2*(-(d^2*(2*a*d - b*c*(3 - m))*(a*d - b*c*(1 + m))) - (2*b*c^2 - 2*a*c*d + b*d^2*(1
- m))*(a*c*d - b*(c^2 - d^2*m))))*Hypergeometric2F1[1, 1 + m, 2 + m, (d*(a + b*Tan[e + f*x]))/(-(b*c) + a*d)]
*(a + b*Tan[e + f*x])^(1 + m))/((-(b*c) + a*d)*(c^2 + d^2)*f*(1 + m))) + (((I/2)*(2*c*(b*c - a*d)^2*(c^2 - 3*d
^2) - (2*I)*d*(b*c - a*d)^2*(3*c^2 - d^2))*Hypergeometric2F1[1, 1 + m, 2 + m, ((-I)*a - I*b*Tan[e + f*x])/((-I
)*a + b)]*(a + b*Tan[e + f*x])^(1 + m))/((a + I*b)*f*(1 + m)) - ((I/2)*(2*c*(b*c - a*d)^2*(c^2 - 3*d^2) + (2*I
)*d*(b*c - a*d)^2*(3*c^2 - d^2))*Hypergeometric2F1[1, 1 + m, 2 + m, -((I*a + I*b*Tan[e + f*x])/((-I)*a - b))]*
(a + b*Tan[e + f*x])^(1 + m))/((a - I*b)*f*(1 + m)))/(c^2 + d^2))/((-(b*c) + a*d)*(c^2 + d^2)))/(2*(-(b*c) + a
*d)*(c^2 + d^2))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral((b*tan(f*x + e) + a)^m/(d^3*tan(f*x + e)^3 + 3*c*d^2*tan(f*x + e)^2 + 3*c^2*d*tan(f*x + e) + c^3), x)

Sympy [F]

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

[In]

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

[Out]

Integral((a + b*tan(e + f*x))**m/(c + d*tan(e + f*x))**3, x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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