\(\int \frac {\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx\) [486]

Optimal result

Integrand size = 31, antiderivative size = 659 \[ \int \frac {\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\frac {\left (a^4 A-6 a^2 A b^2+A b^4+4 a^3 b B-4 a b^3 B\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\tan ^2(c+d x)\right ) \tan ^{1+m}(c+d x)}{\left (a^2+b^2\right )^4 d (1+m)}-\frac {b \left (a b^6 B m \left (1-m^2\right )+3 a^2 A b^5 m \left (2-5 m+m^2\right )+A b^7 m \left (2-3 m+m^2\right )+3 a^3 b^4 B \left (2+5 m+2 m^2-m^3\right )+a^7 B \left (6-11 m+6 m^2-m^3\right )-a^6 A b \left (24-26 m+9 m^2-m^3\right )+3 a^4 A b^3 \left (8+10 m-7 m^2+m^3\right )-3 a^5 b^2 B \left (12-m-4 m^2+m^3\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {b \tan (c+d x)}{a}\right ) \tan ^{1+m}(c+d x)}{6 a^4 \left (a^2+b^2\right )^4 d (1+m)}-\frac {\left (4 a^3 A b-4 a A b^3-a^4 B+6 a^2 b^2 B-b^4 B\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},-\tan ^2(c+d x)\right ) \tan ^{2+m}(c+d x)}{\left (a^2+b^2\right )^4 d (2+m)}+\frac {b (A b-a B) \tan ^{1+m}(c+d x)}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {b \left (A b^3 (2-m)-a^3 B (5-m)+a^2 A b (8-m)+a b^2 B (1+m)\right ) \tan ^{1+m}(c+d x)}{6 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {b \left (a b^4 B \left (1-m^2\right )+2 a^3 b^2 B \left (7+3 m-m^2\right )+a^4 A b \left (26-9 m+m^2\right )+2 a^2 A b^3 \left (2-6 m+m^2\right )-a^5 B \left (11-6 m+m^2\right )+A b^5 \left (2-3 m+m^2\right )\right ) \tan ^{1+m}(c+d x)}{6 a^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))} \]

[Out]

(A*a^4-6*A*a^2*b^2+A*b^4+4*B*a^3*b-4*B*a*b^3)*hypergeom([1, 1/2+1/2*m],[3/2+1/2*m],-tan(d*x+c)^2)*tan(d*x+c)^(
1+m)/(a^2+b^2)^4/d/(1+m)-1/6*b*(a*b^6*B*m*(-m^2+1)+3*a^2*A*b^5*m*(m^2-5*m+2)+A*b^7*m*(m^2-3*m+2)+3*a^3*b^4*B*(
-m^3+2*m^2+5*m+2)+a^7*B*(-m^3+6*m^2-11*m+6)-a^6*A*b*(-m^3+9*m^2-26*m+24)+3*a^4*A*b^3*(m^3-7*m^2+10*m+8)-3*a^5*
b^2*B*(m^3-4*m^2-m+12))*hypergeom([1, 1+m],[2+m],-b*tan(d*x+c)/a)*tan(d*x+c)^(1+m)/a^4/(a^2+b^2)^4/d/(1+m)-(4*
A*a^3*b-4*A*a*b^3-B*a^4+6*B*a^2*b^2-B*b^4)*hypergeom([1, 1+1/2*m],[2+1/2*m],-tan(d*x+c)^2)*tan(d*x+c)^(2+m)/(a
^2+b^2)^4/d/(2+m)+1/3*b*(A*b-B*a)*tan(d*x+c)^(1+m)/a/(a^2+b^2)/d/(a+b*tan(d*x+c))^3+1/6*b*(A*b^3*(2-m)-a^3*B*(
5-m)+a^2*A*b*(8-m)+a*b^2*B*(1+m))*tan(d*x+c)^(1+m)/a^2/(a^2+b^2)^2/d/(a+b*tan(d*x+c))^2+1/6*b*(a*b^4*B*(-m^2+1
)+2*a^3*b^2*B*(-m^2+3*m+7)+a^4*A*b*(m^2-9*m+26)+2*a^2*A*b^3*(m^2-6*m+2)-a^5*B*(m^2-6*m+11)+A*b^5*(m^2-3*m+2))*
tan(d*x+c)^(1+m)/a^3/(a^2+b^2)^3/d/(a+b*tan(d*x+c))

Rubi [A] (verified)

Time = 2.69 (sec) , antiderivative size = 659, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {3690, 3730, 3734, 3619, 3557, 371, 3715, 66} \[ \int \frac {\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\frac {b (A b-a B) \tan ^{m+1}(c+d x)}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac {b \left (a^3 (-B) (5-m)+a^2 A b (8-m)+a b^2 B (m+1)+A b^3 (2-m)\right ) \tan ^{m+1}(c+d x)}{6 a^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac {\left (a^4 A+4 a^3 b B-6 a^2 A b^2-4 a b^3 B+A b^4\right ) \tan ^{m+1}(c+d x) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\tan ^2(c+d x)\right )}{d (m+1) \left (a^2+b^2\right )^4}-\frac {\left (a^4 (-B)+4 a^3 A b+6 a^2 b^2 B-4 a A b^3-b^4 B\right ) \tan ^{m+2}(c+d x) \operatorname {Hypergeometric2F1}\left (1,\frac {m+2}{2},\frac {m+4}{2},-\tan ^2(c+d x)\right )}{d (m+2) \left (a^2+b^2\right )^4}+\frac {b \left (a^5 (-B) \left (m^2-6 m+11\right )+a^4 A b \left (m^2-9 m+26\right )+2 a^3 b^2 B \left (-m^2+3 m+7\right )+2 a^2 A b^3 \left (m^2-6 m+2\right )+a b^4 B \left (1-m^2\right )+A b^5 \left (m^2-3 m+2\right )\right ) \tan ^{m+1}(c+d x)}{6 a^3 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}-\frac {b \left (a^7 B \left (-m^3+6 m^2-11 m+6\right )-a^6 A b \left (-m^3+9 m^2-26 m+24\right )-3 a^5 b^2 B \left (m^3-4 m^2-m+12\right )+3 a^4 A b^3 \left (m^3-7 m^2+10 m+8\right )+3 a^3 b^4 B \left (-m^3+2 m^2+5 m+2\right )+3 a^2 A b^5 m \left (m^2-5 m+2\right )+a b^6 B m \left (1-m^2\right )+A b^7 m \left (m^2-3 m+2\right )\right ) \tan ^{m+1}(c+d x) \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,-\frac {b \tan (c+d x)}{a}\right )}{6 a^4 d (m+1) \left (a^2+b^2\right )^4} \]

[In]

Int[(Tan[c + d*x]^m*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^4,x]

[Out]

((a^4*A - 6*a^2*A*b^2 + A*b^4 + 4*a^3*b*B - 4*a*b^3*B)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[c + d*x
]^2]*Tan[c + d*x]^(1 + m))/((a^2 + b^2)^4*d*(1 + m)) - (b*(a*b^6*B*m*(1 - m^2) + 3*a^2*A*b^5*m*(2 - 5*m + m^2)
+ A*b^7*m*(2 - 3*m + m^2) + 3*a^3*b^4*B*(2 + 5*m + 2*m^2 - m^3) + a^7*B*(6 - 11*m + 6*m^2 - m^3) - a^6*A*b*(2
4 - 26*m + 9*m^2 - m^3) + 3*a^4*A*b^3*(8 + 10*m - 7*m^2 + m^3) - 3*a^5*b^2*B*(12 - m - 4*m^2 + m^3))*Hypergeom
etric2F1[1, 1 + m, 2 + m, -((b*Tan[c + d*x])/a)]*Tan[c + d*x]^(1 + m))/(6*a^4*(a^2 + b^2)^4*d*(1 + m)) - ((4*a
^3*A*b - 4*a*A*b^3 - a^4*B + 6*a^2*b^2*B - b^4*B)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -Tan[c + d*x]^2]*
Tan[c + d*x]^(2 + m))/((a^2 + b^2)^4*d*(2 + m)) + (b*(A*b - a*B)*Tan[c + d*x]^(1 + m))/(3*a*(a^2 + b^2)*d*(a +
b*Tan[c + d*x])^3) + (b*(A*b^3*(2 - m) - a^3*B*(5 - m) + a^2*A*b*(8 - m) + a*b^2*B*(1 + m))*Tan[c + d*x]^(1 +
m))/(6*a^2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) + (b*(a*b^4*B*(1 - m^2) + 2*a^3*b^2*B*(7 + 3*m - m^2) + a^
4*A*b*(26 - 9*m + m^2) + 2*a^2*A*b^3*(2 - 6*m + m^2) - a^5*B*(11 - 6*m + m^2) + A*b^5*(2 - 3*m + m^2))*Tan[c +
d*x]^(1 + m))/(6*a^3*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x]))

Rule 66

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x)^(m + 1)/(b*(m + 1)))*Hypergeometr
ic2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[
c, 0] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 3557

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

Rule 3619

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

Rule 3690

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(A*b - a*B)*(a + b*Tan[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[b*B*(b*c*(m + 1) + a*d*(n + 1)) + A*(a*(b*c - a*d)*(m + 1) - b^2*d*(
m + n + 2)) - (A*b - a*B)*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b*d*(A*b - a*B)*(m + n + 2)*Tan[e + f*x]^2, x], x
], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
&& LtQ[m, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) &&  !(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 {b (A b-a B) \tan ^{1+m}(c+d x)}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {\int \frac {\tan ^m(c+d x) \left (3 a^2 A+A b^2 (2-m)+a b B (1+m)-3 a (A b-a B) \tan (c+d x)+b (A b-a B) (2-m) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx}{3 a \left (a^2+b^2\right )} \\ & = \frac {b (A b-a B) \tan ^{1+m}(c+d x)}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {b \left (A b^3 (2-m)-a^3 B (5-m)+a^2 A b (8-m)+a b^2 B (1+m)\right ) \tan ^{1+m}(c+d x)}{6 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {\int \frac {\tan ^m(c+d x) \left (-a^2 b (A b-a B) (5-m) (1+m)+\left (2 a^2+b^2 (1-m)\right ) \left (3 a^2 A+A b^2 (2-m)+a b B (1+m)\right )-6 a^2 \left (2 a A b-a^2 B+b^2 B\right ) \tan (c+d x)+b (1-m) \left (A b^3 (2-m)-a^3 B (5-m)+a^2 A b (8-m)+a b^2 B (1+m)\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{6 a^2 \left (a^2+b^2\right )^2} \\ & = \frac {b (A b-a B) \tan ^{1+m}(c+d x)}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {b \left (A b^3 (2-m)-a^3 B (5-m)+a^2 A b (8-m)+a b^2 B (1+m)\right ) \tan ^{1+m}(c+d x)}{6 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {b \left (a b^4 B \left (1-m^2\right )+2 a^3 b^2 B \left (7+3 m-m^2\right )+a^4 A b \left (26-9 m+m^2\right )+2 a^2 A b^3 \left (2-6 m+m^2\right )-a^5 B \left (11-6 m+m^2\right )+A b^5 \left (2-3 m+m^2\right )\right ) \tan ^{1+m}(c+d x)}{6 a^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac {\int \frac {\tan ^m(c+d x) \left (6 a^6 A-a b^5 B m \left (1-m^2\right )-2 a^2 A b^4 m \left (2-6 m+m^2\right )-A b^6 m \left (2-3 m+m^2\right )-2 a^3 b^3 B \left (3+7 m+3 m^2-m^3\right )-a^4 A b^2 \left (18+26 m-9 m^2+m^3\right )+a^5 b B \left (18+11 m-6 m^2+m^3\right )-6 a^3 \left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) \tan (c+d x)-b m \left (a b^4 B \left (1-m^2\right )+2 a^3 b^2 B \left (7+3 m-m^2\right )+a^4 A b \left (26-9 m+m^2\right )+2 a^2 A b^3 \left (2-6 m+m^2\right )-a^5 B \left (11-6 m+m^2\right )+A b^5 \left (2-3 m+m^2\right )\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{6 a^3 \left (a^2+b^2\right )^3} \\ & = \frac {b (A b-a B) \tan ^{1+m}(c+d x)}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {b \left (A b^3 (2-m)-a^3 B (5-m)+a^2 A b (8-m)+a b^2 B (1+m)\right ) \tan ^{1+m}(c+d x)}{6 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {b \left (a b^4 B \left (1-m^2\right )+2 a^3 b^2 B \left (7+3 m-m^2\right )+a^4 A b \left (26-9 m+m^2\right )+2 a^2 A b^3 \left (2-6 m+m^2\right )-a^5 B \left (11-6 m+m^2\right )+A b^5 \left (2-3 m+m^2\right )\right ) \tan ^{1+m}(c+d x)}{6 a^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac {\int \tan ^m(c+d x) \left (6 a^3 \left (a^4 A-6 a^2 A b^2+A b^4+4 a^3 b B-4 a b^3 B\right )-6 a^3 \left (4 a^3 A b-4 a A b^3-a^4 B+6 a^2 b^2 B-b^4 B\right ) \tan (c+d x)\right ) \, dx}{6 a^3 \left (a^2+b^2\right )^4}-\frac {\left (b \left (a b^6 B m \left (1-m^2\right )+3 a^2 A b^5 m \left (2-5 m+m^2\right )+A b^7 m \left (2-3 m+m^2\right )+3 a^3 b^4 B \left (2+5 m+2 m^2-m^3\right )+a^7 B \left (6-11 m+6 m^2-m^3\right )-a^6 A b \left (24-26 m+9 m^2-m^3\right )+3 a^4 A b^3 \left (8+10 m-7 m^2+m^3\right )-3 a^5 b^2 B \left (12-m-4 m^2+m^3\right )\right )\right ) \int \frac {\tan ^m(c+d x) \left (1+\tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{6 a^3 \left (a^2+b^2\right )^4} \\ & = \frac {b (A b-a B) \tan ^{1+m}(c+d x)}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {b \left (A b^3 (2-m)-a^3 B (5-m)+a^2 A b (8-m)+a b^2 B (1+m)\right ) \tan ^{1+m}(c+d x)}{6 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {b \left (a b^4 B \left (1-m^2\right )+2 a^3 b^2 B \left (7+3 m-m^2\right )+a^4 A b \left (26-9 m+m^2\right )+2 a^2 A b^3 \left (2-6 m+m^2\right )-a^5 B \left (11-6 m+m^2\right )+A b^5 \left (2-3 m+m^2\right )\right ) \tan ^{1+m}(c+d x)}{6 a^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac {\left (a^4 A-6 a^2 A b^2+A b^4+4 a^3 b B-4 a b^3 B\right ) \int \tan ^m(c+d x) \, dx}{\left (a^2+b^2\right )^4}-\frac {\left (4 a^3 A b-4 a A b^3-a^4 B+6 a^2 b^2 B-b^4 B\right ) \int \tan ^{1+m}(c+d x) \, dx}{\left (a^2+b^2\right )^4}-\frac {\left (b \left (a b^6 B m \left (1-m^2\right )+3 a^2 A b^5 m \left (2-5 m+m^2\right )+A b^7 m \left (2-3 m+m^2\right )+3 a^3 b^4 B \left (2+5 m+2 m^2-m^3\right )+a^7 B \left (6-11 m+6 m^2-m^3\right )-a^6 A b \left (24-26 m+9 m^2-m^3\right )+3 a^4 A b^3 \left (8+10 m-7 m^2+m^3\right )-3 a^5 b^2 B \left (12-m-4 m^2+m^3\right )\right )\right ) \text {Subst}\left (\int \frac {x^m}{a+b x} \, dx,x,\tan (c+d x)\right )}{6 a^3 \left (a^2+b^2\right )^4 d} \\ & = -\frac {b \left (a b^6 B m \left (1-m^2\right )+3 a^2 A b^5 m \left (2-5 m+m^2\right )+A b^7 m \left (2-3 m+m^2\right )+3 a^3 b^4 B \left (2+5 m+2 m^2-m^3\right )+a^7 B \left (6-11 m+6 m^2-m^3\right )-a^6 A b \left (24-26 m+9 m^2-m^3\right )+3 a^4 A b^3 \left (8+10 m-7 m^2+m^3\right )-3 a^5 b^2 B \left (12-m-4 m^2+m^3\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {b \tan (c+d x)}{a}\right ) \tan ^{1+m}(c+d x)}{6 a^4 \left (a^2+b^2\right )^4 d (1+m)}+\frac {b (A b-a B) \tan ^{1+m}(c+d x)}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {b \left (A b^3 (2-m)-a^3 B (5-m)+a^2 A b (8-m)+a b^2 B (1+m)\right ) \tan ^{1+m}(c+d x)}{6 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {b \left (a b^4 B \left (1-m^2\right )+2 a^3 b^2 B \left (7+3 m-m^2\right )+a^4 A b \left (26-9 m+m^2\right )+2 a^2 A b^3 \left (2-6 m+m^2\right )-a^5 B \left (11-6 m+m^2\right )+A b^5 \left (2-3 m+m^2\right )\right ) \tan ^{1+m}(c+d x)}{6 a^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac {\left (a^4 A-6 a^2 A b^2+A b^4+4 a^3 b B-4 a b^3 B\right ) \text {Subst}\left (\int \frac {x^m}{1+x^2} \, dx,x,\tan (c+d x)\right )}{\left (a^2+b^2\right )^4 d}-\frac {\left (4 a^3 A b-4 a A b^3-a^4 B+6 a^2 b^2 B-b^4 B\right ) \text {Subst}\left (\int \frac {x^{1+m}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{\left (a^2+b^2\right )^4 d} \\ & = \frac {\left (a^4 A-6 a^2 A b^2+A b^4+4 a^3 b B-4 a b^3 B\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\tan ^2(c+d x)\right ) \tan ^{1+m}(c+d x)}{\left (a^2+b^2\right )^4 d (1+m)}-\frac {b \left (a b^6 B m \left (1-m^2\right )+3 a^2 A b^5 m \left (2-5 m+m^2\right )+A b^7 m \left (2-3 m+m^2\right )+3 a^3 b^4 B \left (2+5 m+2 m^2-m^3\right )+a^7 B \left (6-11 m+6 m^2-m^3\right )-a^6 A b \left (24-26 m+9 m^2-m^3\right )+3 a^4 A b^3 \left (8+10 m-7 m^2+m^3\right )-3 a^5 b^2 B \left (12-m-4 m^2+m^3\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {b \tan (c+d x)}{a}\right ) \tan ^{1+m}(c+d x)}{6 a^4 \left (a^2+b^2\right )^4 d (1+m)}-\frac {\left (4 a^3 A b-4 a A b^3-a^4 B+6 a^2 b^2 B-b^4 B\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},-\tan ^2(c+d x)\right ) \tan ^{2+m}(c+d x)}{\left (a^2+b^2\right )^4 d (2+m)}+\frac {b (A b-a B) \tan ^{1+m}(c+d x)}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {b \left (A b^3 (2-m)-a^3 B (5-m)+a^2 A b (8-m)+a b^2 B (1+m)\right ) \tan ^{1+m}(c+d x)}{6 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {b \left (a b^4 B \left (1-m^2\right )+2 a^3 b^2 B \left (7+3 m-m^2\right )+a^4 A b \left (26-9 m+m^2\right )+2 a^2 A b^3 \left (2-6 m+m^2\right )-a^5 B \left (11-6 m+m^2\right )+A b^5 \left (2-3 m+m^2\right )\right ) \tan ^{1+m}(c+d x)}{6 a^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1425\) vs. \(2(659)=1318\).

Time = 6.47 (sec) , antiderivative size = 1425, normalized size of antiderivative = 2.16 \[ \int \frac {\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\frac {b (A b-a B) \tan ^{1+m}(c+d x)}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {\frac {\left (-a (-3 a b (A b-a B)-a b (A b-a B) (2-m))+b^2 \left (3 a^2 A+A b^2 (2-m)+a b B (1+m)\right )\right ) \tan ^{1+m}(c+d x)}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {\frac {\left (b^2 \left (-a^2 b (A b-a B) (5-m) (1+m)+\left (2 a^2+b^2 (1-m)\right ) \left (3 a^2 A+A b^2 (2-m)+a b B (1+m)\right )\right )-a \left (-6 a^2 b \left (2 a A b-a^2 B+b^2 B\right )-a b (1-m) \left (A b^3 (2-m)-a^3 B (5-m)+a^2 A b (8-m)+a b^2 B (1+m)\right )\right )\right ) \tan ^{1+m}(c+d x)}{a \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\frac {\left (a^2 b \left (6 a^3 \left (2 a A b-a^2 B+b^2 B\right )-b^2 (1-m) \left (A b^3 (2-m)-a^3 B (5-m)+a^2 A b (8-m)+a b^2 B (1+m)\right )+b \left (-a^2 b (A b-a B) (5-m) (1+m)+\left (2 a^2+b^2 (1-m)\right ) \left (3 a^2 A+A b^2 (2-m)+a b B (1+m)\right )\right )\right )-a^2 m \left (b^2 \left (-a^2 b (A b-a B) (5-m) (1+m)+\left (2 a^2+b^2 (1-m)\right ) \left (3 a^2 A+A b^2 (2-m)+a b B (1+m)\right )\right )-a \left (-6 a^2 b \left (2 a A b-a^2 B+b^2 B\right )-a b (1-m) \left (A b^3 (2-m)-a^3 B (5-m)+a^2 A b (8-m)+a b^2 B (1+m)\right )\right )\right )+b^2 \left (\left (a^2-b^2 m\right ) \left (-a^2 b (A b-a B) (5-m) (1+m)+\left (2 a^2+b^2 (1-m)\right ) \left (3 a^2 A+A b^2 (2-m)+a b B (1+m)\right )\right )+a (1+m) \left (-6 a^2 b \left (2 a A b-a^2 B+b^2 B\right )-a b (1-m) \left (A b^3 (2-m)-a^3 B (5-m)+a^2 A b (8-m)+a b^2 B (1+m)\right )\right )\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {b \tan (c+d x)}{a}\right ) \tan ^{1+m}(c+d x)}{a \left (a^2+b^2\right ) d (1+m)}+\frac {\frac {6 a^7 A \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\tan ^2(c+d x)\right ) \tan ^{1+m}(c+d x)}{d (1+m)}-\frac {36 a^5 A b^2 \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\tan ^2(c+d x)\right ) \tan ^{1+m}(c+d x)}{d (1+m)}+\frac {6 a^3 A b^4 \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\tan ^2(c+d x)\right ) \tan ^{1+m}(c+d x)}{d (1+m)}+\frac {24 a^6 b B \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\tan ^2(c+d x)\right ) \tan ^{1+m}(c+d x)}{d (1+m)}-\frac {24 a^4 b^3 B \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\tan ^2(c+d x)\right ) \tan ^{1+m}(c+d x)}{d (1+m)}-\frac {24 a^6 A b \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},-\tan ^2(c+d x)\right ) \tan ^{2+m}(c+d x)}{d (2+m)}+\frac {24 a^4 A b^3 \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},-\tan ^2(c+d x)\right ) \tan ^{2+m}(c+d x)}{d (2+m)}+\frac {6 a^7 B \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},-\tan ^2(c+d x)\right ) \tan ^{2+m}(c+d x)}{d (2+m)}-\frac {36 a^5 b^2 B \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},-\tan ^2(c+d x)\right ) \tan ^{2+m}(c+d x)}{d (2+m)}+\frac {6 a^3 b^4 B \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},-\tan ^2(c+d x)\right ) \tan ^{2+m}(c+d x)}{d (2+m)}}{a^2+b^2}}{a \left (a^2+b^2\right )}}{2 a \left (a^2+b^2\right )}}{3 a \left (a^2+b^2\right )} \]

[In]

Integrate[(Tan[c + d*x]^m*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^4,x]

[Out]

(b*(A*b - a*B)*Tan[c + d*x]^(1 + m))/(3*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) + (((-(a*(-3*a*b*(A*b - a*B) -
a*b*(A*b - a*B)*(2 - m))) + b^2*(3*a^2*A + A*b^2*(2 - m) + a*b*B*(1 + m)))*Tan[c + d*x]^(1 + m))/(2*a*(a^2 +
b^2)*d*(a + b*Tan[c + d*x])^2) + (((b^2*(-(a^2*b*(A*b - a*B)*(5 - m)*(1 + m)) + (2*a^2 + b^2*(1 - m))*(3*a^2*A
+ A*b^2*(2 - m) + a*b*B*(1 + m))) - a*(-6*a^2*b*(2*a*A*b - a^2*B + b^2*B) - a*b*(1 - m)*(A*b^3*(2 - m) - a^3*
B*(5 - m) + a^2*A*b*(8 - m) + a*b^2*B*(1 + m))))*Tan[c + d*x]^(1 + m))/(a*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))
+ (((a^2*b*(6*a^3*(2*a*A*b - a^2*B + b^2*B) - b^2*(1 - m)*(A*b^3*(2 - m) - a^3*B*(5 - m) + a^2*A*b*(8 - m) + a
*b^2*B*(1 + m)) + b*(-(a^2*b*(A*b - a*B)*(5 - m)*(1 + m)) + (2*a^2 + b^2*(1 - m))*(3*a^2*A + A*b^2*(2 - m) + a
*b*B*(1 + m)))) - a^2*m*(b^2*(-(a^2*b*(A*b - a*B)*(5 - m)*(1 + m)) + (2*a^2 + b^2*(1 - m))*(3*a^2*A + A*b^2*(2
- m) + a*b*B*(1 + m))) - a*(-6*a^2*b*(2*a*A*b - a^2*B + b^2*B) - a*b*(1 - m)*(A*b^3*(2 - m) - a^3*B*(5 - m) +
a^2*A*b*(8 - m) + a*b^2*B*(1 + m)))) + b^2*((a^2 - b^2*m)*(-(a^2*b*(A*b - a*B)*(5 - m)*(1 + m)) + (2*a^2 + b^
2*(1 - m))*(3*a^2*A + A*b^2*(2 - m) + a*b*B*(1 + m))) + a*(1 + m)*(-6*a^2*b*(2*a*A*b - a^2*B + b^2*B) - a*b*(1
- m)*(A*b^3*(2 - m) - a^3*B*(5 - m) + a^2*A*b*(8 - m) + a*b^2*B*(1 + m)))))*Hypergeometric2F1[1, 1 + m, 2 + m
, -((b*Tan[c + d*x])/a)]*Tan[c + d*x]^(1 + m))/(a*(a^2 + b^2)*d*(1 + m)) + ((6*a^7*A*Hypergeometric2F1[1, (1 +
m)/2, (3 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(1 + m))/(d*(1 + m)) - (36*a^5*A*b^2*Hypergeometric2F1[1, (1 +
m)/2, (3 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(1 + m))/(d*(1 + m)) + (6*a^3*A*b^4*Hypergeometric2F1[1, (1 +
m)/2, (3 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(1 + m))/(d*(1 + m)) + (24*a^6*b*B*Hypergeometric2F1[1, (1 + m)
/2, (3 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(1 + m))/(d*(1 + m)) - (24*a^4*b^3*B*Hypergeometric2F1[1, (1 + m)
/2, (3 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(1 + m))/(d*(1 + m)) - (24*a^6*A*b*Hypergeometric2F1[1, (2 + m)/2
, (4 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(2 + m))/(d*(2 + m)) + (24*a^4*A*b^3*Hypergeometric2F1[1, (2 + m)/2
, (4 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(2 + m))/(d*(2 + m)) + (6*a^7*B*Hypergeometric2F1[1, (2 + m)/2, (4
+ m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(2 + m))/(d*(2 + m)) - (36*a^5*b^2*B*Hypergeometric2F1[1, (2 + m)/2, (4
+ m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(2 + m))/(d*(2 + m)) + (6*a^3*b^4*B*Hypergeometric2F1[1, (2 + m)/2, (4 +
m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(2 + m))/(d*(2 + m)))/(a^2 + b^2))/(a*(a^2 + b^2)))/(2*a*(a^2 + b^2)))/(3
*a*(a^2 + b^2))

Maple [F]

\[\int \frac {\tan \left (d x +c \right )^{m} \left (A +B \tan \left (d x +c \right )\right )}{\left (a +b \tan \left (d x +c \right )\right )^{4}}d x\]

[In]

int(tan(d*x+c)^m*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^4,x)

[Out]

int(tan(d*x+c)^m*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^4,x)

Fricas [F]

\[ \int \frac {\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\int { \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \tan \left (d x + c\right )^{m}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{4}} \,d x } \]

[In]

integrate(tan(d*x+c)^m*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^4,x, algorithm="fricas")

[Out]

integral((B*tan(d*x + c) + A)*tan(d*x + c)^m/(b^4*tan(d*x + c)^4 + 4*a*b^3*tan(d*x + c)^3 + 6*a^2*b^2*tan(d*x
+ c)^2 + 4*a^3*b*tan(d*x + c) + a^4), x)

Sympy [F]

\[ \int \frac {\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\int \frac {\left (A + B \tan {\left (c + d x \right )}\right ) \tan ^{m}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{4}}\, dx \]

[In]

integrate(tan(d*x+c)**m*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))**4,x)

[Out]

Integral((A + B*tan(c + d*x))*tan(c + d*x)**m/(a + b*tan(c + d*x))**4, x)

Maxima [F]

\[ \int \frac {\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\int { \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \tan \left (d x + c\right )^{m}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{4}} \,d x } \]

[In]

integrate(tan(d*x+c)^m*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^4,x, algorithm="maxima")

[Out]

integrate((B*tan(d*x + c) + A)*tan(d*x + c)^m/(b*tan(d*x + c) + a)^4, x)

Giac [F]

\[ \int \frac {\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\int { \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \tan \left (d x + c\right )^{m}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{4}} \,d x } \]

[In]

integrate(tan(d*x+c)^m*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^4,x, algorithm="giac")

[Out]

integrate((B*tan(d*x + c) + A)*tan(d*x + c)^m/(b*tan(d*x + c) + a)^4, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\int \frac {{\mathrm {tan}\left (c+d\,x\right )}^m\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )}{{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^4} \,d x \]

[In]

int((tan(c + d*x)^m*(A + B*tan(c + d*x)))/(a + b*tan(c + d*x))^4,x)

[Out]

int((tan(c + d*x)^m*(A + B*tan(c + d*x)))/(a + b*tan(c + d*x))^4, x)