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

Optimal. Leaf size=351 \[ \frac{a (A b-a B) \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac{a \left (2 A b^3-a B \left (a^2+3 b^2\right )\right ) \tan ^2(c+d x)}{2 b^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac{a^2 \left (a^2 A b^3+3 a^3 b^2 B+a^5 B+6 a b^4 B-3 A b^5\right )}{b^4 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac{a \left (4 a^2 A b^5+4 a^5 b^2 B+5 a^3 b^4 B+a^7 B+10 a b^6 B-4 A b^7\right ) \log (a+b \tan (c+d x))}{b^4 d \left (a^2+b^2\right )^4}+\frac{\left (4 a^3 A b+6 a^2 b^2 B+a^4 (-B)-4 a A b^3-b^4 B\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )^4}+\frac{x \left (-6 a^2 A b^2+a^4 A+4 a^3 b B-4 a b^3 B+A b^4\right )}{\left (a^2+b^2\right )^4} \]

[Out]

((a^4*A - 6*a^2*A*b^2 + A*b^4 + 4*a^3*b*B - 4*a*b^3*B)*x)/(a^2 + b^2)^4 + ((4*a^3*A*b - 4*a*A*b^3 - a^4*B + 6*
a^2*b^2*B - b^4*B)*Log[Cos[c + d*x]])/((a^2 + b^2)^4*d) + (a*(4*a^2*A*b^5 - 4*A*b^7 + a^7*B + 4*a^5*b^2*B + 5*
a^3*b^4*B + 10*a*b^6*B)*Log[a + b*Tan[c + d*x]])/(b^4*(a^2 + b^2)^4*d) + (a*(A*b - a*B)*Tan[c + d*x]^3)/(3*b*(
a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) + (a*(2*A*b^3 - a*(a^2 + 3*b^2)*B)*Tan[c + d*x]^2)/(2*b^2*(a^2 + b^2)^2*d
*(a + b*Tan[c + d*x])^2) + (a^2*(a^2*A*b^3 - 3*A*b^5 + a^5*B + 3*a^3*b^2*B + 6*a*b^4*B))/(b^4*(a^2 + b^2)^3*d*
(a + b*Tan[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 0.823886, antiderivative size = 351, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {3605, 3645, 3635, 3626, 3617, 31, 3475} \[ \frac{a (A b-a B) \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac{a \left (2 A b^3-a B \left (a^2+3 b^2\right )\right ) \tan ^2(c+d x)}{2 b^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac{a^2 \left (a^2 A b^3+3 a^3 b^2 B+a^5 B+6 a b^4 B-3 A b^5\right )}{b^4 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac{a \left (4 a^2 A b^5+4 a^5 b^2 B+5 a^3 b^4 B+a^7 B+10 a b^6 B-4 A b^7\right ) \log (a+b \tan (c+d x))}{b^4 d \left (a^2+b^2\right )^4}+\frac{\left (4 a^3 A b+6 a^2 b^2 B+a^4 (-B)-4 a A b^3-b^4 B\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )^4}+\frac{x \left (-6 a^2 A b^2+a^4 A+4 a^3 b B-4 a b^3 B+A b^4\right )}{\left (a^2+b^2\right )^4} \]

Antiderivative was successfully verified.

[In]

Int[(Tan[c + d*x]^4*(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)*x)/(a^2 + b^2)^4 + ((4*a^3*A*b - 4*a*A*b^3 - a^4*B + 6*
a^2*b^2*B - b^4*B)*Log[Cos[c + d*x]])/((a^2 + b^2)^4*d) + (a*(4*a^2*A*b^5 - 4*A*b^7 + a^7*B + 4*a^5*b^2*B + 5*
a^3*b^4*B + 10*a*b^6*B)*Log[a + b*Tan[c + d*x]])/(b^4*(a^2 + b^2)^4*d) + (a*(A*b - a*B)*Tan[c + d*x]^3)/(3*b*(
a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) + (a*(2*A*b^3 - a*(a^2 + 3*b^2)*B)*Tan[c + d*x]^2)/(2*b^2*(a^2 + b^2)^2*d
*(a + b*Tan[c + d*x])^2) + (a^2*(a^2*A*b^3 - 3*A*b^5 + a^5*B + 3*a^3*b^2*B + 6*a*b^4*B))/(b^4*(a^2 + b^2)^3*d*
(a + b*Tan[c + d*x]))

Rule 3605

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*c - a*d)*(B*c - A*d)*(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e
+ f*x])^(n + 1))/(d*f*(n + 1)*(c^2 + d^2)), x] - Dist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m -
2)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a*A*d*(b*d*(m - 1) - a*c*(n + 1)) + (b*B*c - (A*b + a*B)*d)*(b*c*(m - 1)
+ a*d*(n + 1)) - d*((a*A - b*B)*(b*c - a*d) + (A*b + a*B)*(a*c + b*d))*(n + 1)*Tan[e + f*x] - b*(d*(A*b*c + a
*B*c - a*A*d)*(m + n) - b*B*(c^2*(m - 1) - d^2*(n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f
, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 1] && LtQ[n, -1] && (Inte
gerQ[m] || IntegersQ[2*m, 2*n])

Rule 3645

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*d^2 + c*(c*C - B*d))*(a + b*T
an[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 + d^2)), x] - Dist[1/(d*(n + 1)*(c^2 + d^2)), I
nt[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*d*(b*d*m - a*c*(n + 1)) + (c*C - B*d)*(b*c
*m + a*d*(n + 1)) - d*(n + 1)*((A - C)*(b*c - a*d) + B*(a*c + b*d))*Tan[e + f*x] - b*(d*(B*c - A*d)*(m + n + 1
) - C*(c^2*m - d^2*(n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c -
a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && LtQ[n, -1]

Rule 3635

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

Rule 3626

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

Rule 3617

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx &=\frac{a (A b-a B) \tan ^3(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac{\int \frac{\tan ^2(c+d x) \left (-3 a (A b-a B)+3 b (A b-a B) \tan (c+d x)+3 \left (a^2+b^2\right ) B \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx}{3 b \left (a^2+b^2\right )}\\ &=\frac{a (A b-a B) \tan ^3(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac{a \left (2 A b^3-a \left (a^2+3 b^2\right ) B\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac{\int \frac{\tan (c+d x) \left (-6 a \left (2 A b^3-a \left (a^2+3 b^2\right ) B\right )-6 b^2 \left (a^2 A-A b^2+2 a b B\right ) \tan (c+d x)+6 \left (a^2+b^2\right )^2 B \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{6 b^2 \left (a^2+b^2\right )^2}\\ &=\frac{a (A b-a B) \tan ^3(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac{a \left (2 A b^3-a \left (a^2+3 b^2\right ) B\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac{a^2 \left (a^2 A b^3-3 A b^5+a^5 B+3 a^3 b^2 B+6 a b^4 B\right )}{b^4 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac{\int \frac{6 a \left (a^2 A b^3-3 A b^5+a^5 B+3 a^3 b^2 B+6 a b^4 B\right )-6 b^3 \left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) \tan (c+d x)+6 \left (a^2+b^2\right )^3 B \tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{6 b^3 \left (a^2+b^2\right )^3}\\ &=\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 ) x}{\left (a^2+b^2\right )^4}+\frac{a (A b-a B) \tan ^3(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac{a \left (2 A b^3-a \left (a^2+3 b^2\right ) B\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac{a^2 \left (a^2 A b^3-3 A b^5+a^5 B+3 a^3 b^2 B+6 a b^4 B\right )}{b^4 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}-\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 (c+d x) \, dx}{\left (a^2+b^2\right )^4}+\frac{\left (a \left (4 a^2 A b^5-4 A b^7+a^7 B+4 a^5 b^2 B+5 a^3 b^4 B+10 a b^6 B\right )\right ) \int \frac{1+\tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{b^3 \left (a^2+b^2\right )^4}\\ &=\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 ) x}{\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 ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^4 d}+\frac{a (A b-a B) \tan ^3(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac{a \left (2 A b^3-a \left (a^2+3 b^2\right ) B\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac{a^2 \left (a^2 A b^3-3 A b^5+a^5 B+3 a^3 b^2 B+6 a b^4 B\right )}{b^4 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac{\left (a \left (4 a^2 A b^5-4 A b^7+a^7 B+4 a^5 b^2 B+5 a^3 b^4 B+10 a b^6 B\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,b \tan (c+d x)\right )}{b^4 \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 ) x}{\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 ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^4 d}+\frac{a \left (4 a^2 A b^5-4 A b^7+a^7 B+4 a^5 b^2 B+5 a^3 b^4 B+10 a b^6 B\right ) \log (a+b \tan (c+d x))}{b^4 \left (a^2+b^2\right )^4 d}+\frac{a (A b-a B) \tan ^3(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac{a \left (2 A b^3-a \left (a^2+3 b^2\right ) B\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac{a^2 \left (a^2 A b^3-3 A b^5+a^5 B+3 a^3 b^2 B+6 a b^4 B\right )}{b^4 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}\\ \end{align*}

Mathematica [C]  time = 6.69205, size = 1812, normalized size = 5.16 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((4*I)*a^10*A*b^8 + 4*a^9*A*b^9 + (8*I)*a^8*A*b^10 + 8*a^7*A*b^11 - (8*I)*a^4*A*b^14 - 8*a^3*A*b^15 - (4*I)*a
^2*A*b^16 - 4*a*A*b^17 + I*a^15*b^3*B + a^14*b^4*B + (7*I)*a^13*b^5*B + 7*a^12*b^6*B + (20*I)*a^11*b^7*B + 20*
a^10*b^8*B + (38*I)*a^9*b^9*B + 38*a^8*b^10*B + (49*I)*a^7*b^11*B + 49*a^6*b^12*B + (35*I)*a^5*b^13*B + 35*a^4
*b^14*B + (10*I)*a^3*b^15*B + 10*a^2*b^16*B)*(c + d*x)*Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^4*(A +
B*Tan[c + d*x]))/((a - I*b)^8*(a + I*b)^7*b^7*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + b*Tan[c + d*x])^4) - (
I*(4*a^3*A*b^5 - 4*a*A*b^7 + a^8*B + 4*a^6*b^2*B + 5*a^4*b^4*B + 10*a^2*b^6*B)*ArcTan[Tan[c + d*x]]*Sec[c + d*
x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^4*(A + B*Tan[c + d*x]))/(b^4*(a^2 + b^2)^4*d*(A*Cos[c + d*x] + B*Sin[c
+ d*x])*(a + b*Tan[c + d*x])^4) - (B*Log[Cos[c + d*x]]*Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^4*(A +
B*Tan[c + d*x]))/(b^4*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + b*Tan[c + d*x])^4) + ((4*a^3*A*b^5 - 4*a*A*b^7
+ a^8*B + 4*a^6*b^2*B + 5*a^4*b^4*B + 10*a^2*b^6*B)*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2]*Sec[c + d*x]^3*(
a*Cos[c + d*x] + b*Sin[c + d*x])^4*(A + B*Tan[c + d*x]))/(2*b^4*(a^2 + b^2)^4*d*(A*Cos[c + d*x] + B*Sin[c + d*
x])*(a + b*Tan[c + d*x])^4) + (Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])*(12*a^6*A*b^4*Cos[c + d*x] + 4
8*a^4*A*b^6*Cos[c + d*x] + 36*a^2*A*b^8*Cos[c + d*x] - 12*a^9*b*B*Cos[c + d*x] - 60*a^7*b^3*B*Cos[c + d*x] - 1
08*a^5*b^5*B*Cos[c + d*x] - 60*a^3*b^7*B*Cos[c + d*x] + 9*a^7*A*b^3*(c + d*x)*Cos[c + d*x] - 45*a^5*A*b^5*(c +
d*x)*Cos[c + d*x] - 45*a^3*A*b^7*(c + d*x)*Cos[c + d*x] + 9*a*A*b^9*(c + d*x)*Cos[c + d*x] + 36*a^6*b^4*B*(c
+ d*x)*Cos[c + d*x] - 36*a^2*b^8*B*(c + d*x)*Cos[c + d*x] + 8*a^6*A*b^4*Cos[3*(c + d*x)] - 28*a^4*A*b^6*Cos[3*
(c + d*x)] - 36*a^2*A*b^8*Cos[3*(c + d*x)] + 6*a^9*b*B*Cos[3*(c + d*x)] + 28*a^7*b^3*B*Cos[3*(c + d*x)] + 82*a
^5*b^5*B*Cos[3*(c + d*x)] + 60*a^3*b^7*B*Cos[3*(c + d*x)] + 3*a^7*A*b^3*(c + d*x)*Cos[3*(c + d*x)] - 27*a^5*A*
b^5*(c + d*x)*Cos[3*(c + d*x)] + 57*a^3*A*b^7*(c + d*x)*Cos[3*(c + d*x)] - 9*a*A*b^9*(c + d*x)*Cos[3*(c + d*x)
] + 12*a^6*b^4*B*(c + d*x)*Cos[3*(c + d*x)] - 48*a^4*b^6*B*(c + d*x)*Cos[3*(c + d*x)] + 36*a^2*b^8*B*(c + d*x)
*Cos[3*(c + d*x)] + 30*a^5*A*b^5*Sin[c + d*x] + 84*a^3*A*b^7*Sin[c + d*x] + 54*a*A*b^9*Sin[c + d*x] - 3*a^10*B
*Sin[c + d*x] - 33*a^8*b^2*B*Sin[c + d*x] - 123*a^6*b^4*B*Sin[c + d*x] - 183*a^4*b^6*B*Sin[c + d*x] - 90*a^2*b
^8*B*Sin[c + d*x] + 9*a^6*A*b^4*(c + d*x)*Sin[c + d*x] - 45*a^4*A*b^6*(c + d*x)*Sin[c + d*x] - 45*a^2*A*b^8*(c
+ d*x)*Sin[c + d*x] + 9*A*b^10*(c + d*x)*Sin[c + d*x] + 36*a^5*b^5*B*(c + d*x)*Sin[c + d*x] - 36*a*b^9*B*(c +
d*x)*Sin[c + d*x] - 4*a^7*A*b^3*Sin[3*(c + d*x)] + 18*a^5*A*b^5*Sin[3*(c + d*x)] + 4*a^3*A*b^7*Sin[3*(c + d*x
)] - 18*a*A*b^9*Sin[3*(c + d*x)] - 3*a^10*B*Sin[3*(c + d*x)] - 11*a^8*b^2*B*Sin[3*(c + d*x)] - 27*a^6*b^4*B*Si
n[3*(c + d*x)] + 11*a^4*b^6*B*Sin[3*(c + d*x)] + 30*a^2*b^8*B*Sin[3*(c + d*x)] + 9*a^6*A*b^4*(c + d*x)*Sin[3*(
c + d*x)] - 57*a^4*A*b^6*(c + d*x)*Sin[3*(c + d*x)] + 27*a^2*A*b^8*(c + d*x)*Sin[3*(c + d*x)] - 3*A*b^10*(c +
d*x)*Sin[3*(c + d*x)] + 36*a^5*b^5*B*(c + d*x)*Sin[3*(c + d*x)] - 48*a^3*b^7*B*(c + d*x)*Sin[3*(c + d*x)] + 12
*a*b^9*B*(c + d*x)*Sin[3*(c + d*x)])*(A + B*Tan[c + d*x]))/(12*(a - I*b)^4*(a + I*b)^4*b^3*d*(A*Cos[c + d*x] +
B*Sin[c + d*x])*(a + b*Tan[c + d*x])^4)

________________________________________________________________________________________

Maple [B]  time = 0.059, size = 854, normalized size = 2.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*a^8/(a^2+b^2)^4/b^4*ln(a+b*tan(d*x+c))*B-6/d*a^2*b/(a^2+b^2)^3/(a+b*tan(d*x+c))*A+3/d*a^7/b^4/(a^2+b^2)^3/
(a+b*tan(d*x+c))*B+9/d*a^5/b^2/(a^2+b^2)^3/(a+b*tan(d*x+c))*B-4/d*a/(a^2+b^2)^4*b^3*ln(a+b*tan(d*x+c))*A+4/d/(
a^2+b^2)^4*B*arctan(tan(d*x+c))*a^3*b+4/d*a^6/(a^2+b^2)^4/b^2*ln(a+b*tan(d*x+c))*B-4/d/(a^2+b^2)^4*B*arctan(ta
n(d*x+c))*a*b^3-2/d/(a^2+b^2)^4*ln(1+tan(d*x+c)^2)*A*a^3*b+10/d*a^2/(a^2+b^2)^4*b^2*ln(a+b*tan(d*x+c))*B-3/d/(
a^2+b^2)^4*ln(1+tan(d*x+c)^2)*B*a^2*b^2-6/d/(a^2+b^2)^4*A*arctan(tan(d*x+c))*a^2*b^2-1/d*a^6/b^3/(a^2+b^2)^3/(
a+b*tan(d*x+c))*A-3/d*a^4/b/(a^2+b^2)^3/(a+b*tan(d*x+c))*A+2/d/(a^2+b^2)^4*ln(1+tan(d*x+c)^2)*A*a*b^3-3/2/d*a^
6/b^4/(a^2+b^2)^2/(a+b*tan(d*x+c))^2*B-5/2/d*a^4/b^2/(a^2+b^2)^2/(a+b*tan(d*x+c))^2*B-1/3/d*a^4/b^3/(a^2+b^2)/
(a+b*tan(d*x+c))^3*A+1/3/d*a^5/b^4/(a^2+b^2)/(a+b*tan(d*x+c))^3*B+4/d*a^3/(a^2+b^2)^4*b*ln(a+b*tan(d*x+c))*A+1
/d*a^5/b^3/(a^2+b^2)^2/(a+b*tan(d*x+c))^2*A+2/d*a^3/b/(a^2+b^2)^2/(a+b*tan(d*x+c))^2*A+1/2/d/(a^2+b^2)^4*ln(1+
tan(d*x+c)^2)*B*a^4+1/2/d/(a^2+b^2)^4*ln(1+tan(d*x+c)^2)*B*b^4+1/d/(a^2+b^2)^4*A*arctan(tan(d*x+c))*a^4+1/d/(a
^2+b^2)^4*A*arctan(tan(d*x+c))*b^4+5/d*a^4/(a^2+b^2)^4*ln(a+b*tan(d*x+c))*B+10/d*a^3/(a^2+b^2)^3/(a+b*tan(d*x+
c))*B

________________________________________________________________________________________

Maxima [A]  time = 1.58703, size = 787, normalized size = 2.24 \begin{align*} \frac{\frac{6 \,{\left (A a^{4} + 4 \, B a^{3} b - 6 \, A a^{2} b^{2} - 4 \, B a b^{3} + A b^{4}\right )}{\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac{6 \,{\left (B a^{8} + 4 \, B a^{6} b^{2} + 5 \, B a^{4} b^{4} + 4 \, A a^{3} b^{5} + 10 \, B a^{2} b^{6} - 4 \, A a b^{7}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{8} b^{4} + 4 \, a^{6} b^{6} + 6 \, a^{4} b^{8} + 4 \, a^{2} b^{10} + b^{12}} + \frac{3 \,{\left (B a^{4} - 4 \, A a^{3} b - 6 \, B a^{2} b^{2} + 4 \, A a b^{3} + B b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac{11 \, B a^{9} - 2 \, A a^{8} b + 34 \, B a^{7} b^{2} - 4 \, A a^{6} b^{3} + 47 \, B a^{5} b^{4} - 26 \, A a^{4} b^{5} + 6 \,{\left (3 \, B a^{7} b^{2} - A a^{6} b^{3} + 9 \, B a^{5} b^{4} - 3 \, A a^{4} b^{5} + 10 \, B a^{3} b^{6} - 6 \, A a^{2} b^{7}\right )} \tan \left (d x + c\right )^{2} + 3 \,{\left (9 \, B a^{8} b - 2 \, A a^{7} b^{2} + 28 \, B a^{6} b^{3} - 6 \, A a^{5} b^{4} + 35 \, B a^{4} b^{5} - 20 \, A a^{3} b^{6}\right )} \tan \left (d x + c\right )}{a^{9} b^{4} + 3 \, a^{7} b^{6} + 3 \, a^{5} b^{8} + a^{3} b^{10} +{\left (a^{6} b^{7} + 3 \, a^{4} b^{9} + 3 \, a^{2} b^{11} + b^{13}\right )} \tan \left (d x + c\right )^{3} + 3 \,{\left (a^{7} b^{6} + 3 \, a^{5} b^{8} + 3 \, a^{3} b^{10} + a b^{12}\right )} \tan \left (d x + c\right )^{2} + 3 \,{\left (a^{8} b^{5} + 3 \, a^{6} b^{7} + 3 \, a^{4} b^{9} + a^{2} b^{11}\right )} \tan \left (d x + c\right )}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(6*(A*a^4 + 4*B*a^3*b - 6*A*a^2*b^2 - 4*B*a*b^3 + A*b^4)*(d*x + c)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^
6 + b^8) + 6*(B*a^8 + 4*B*a^6*b^2 + 5*B*a^4*b^4 + 4*A*a^3*b^5 + 10*B*a^2*b^6 - 4*A*a*b^7)*log(b*tan(d*x + c) +
a)/(a^8*b^4 + 4*a^6*b^6 + 6*a^4*b^8 + 4*a^2*b^10 + b^12) + 3*(B*a^4 - 4*A*a^3*b - 6*B*a^2*b^2 + 4*A*a*b^3 + B
*b^4)*log(tan(d*x + c)^2 + 1)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) + (11*B*a^9 - 2*A*a^8*b + 34*B*a
^7*b^2 - 4*A*a^6*b^3 + 47*B*a^5*b^4 - 26*A*a^4*b^5 + 6*(3*B*a^7*b^2 - A*a^6*b^3 + 9*B*a^5*b^4 - 3*A*a^4*b^5 +
10*B*a^3*b^6 - 6*A*a^2*b^7)*tan(d*x + c)^2 + 3*(9*B*a^8*b - 2*A*a^7*b^2 + 28*B*a^6*b^3 - 6*A*a^5*b^4 + 35*B*a^
4*b^5 - 20*A*a^3*b^6)*tan(d*x + c))/(a^9*b^4 + 3*a^7*b^6 + 3*a^5*b^8 + a^3*b^10 + (a^6*b^7 + 3*a^4*b^9 + 3*a^2
*b^11 + b^13)*tan(d*x + c)^3 + 3*(a^7*b^6 + 3*a^5*b^8 + 3*a^3*b^10 + a*b^12)*tan(d*x + c)^2 + 3*(a^8*b^5 + 3*a
^6*b^7 + 3*a^4*b^9 + a^2*b^11)*tan(d*x + c)))/d

________________________________________________________________________________________

Fricas [B]  time = 3.05606, size = 2438, normalized size = 6.95 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(3*B*a^9*b^2 + 6*B*a^7*b^4 + 18*A*a^6*b^5 + 47*B*a^5*b^6 - 26*A*a^4*b^7 - (11*B*a^8*b^3 - 2*A*a^7*b^4 + 42
*B*a^6*b^5 - 6*A*a^5*b^6 + 75*B*a^4*b^7 - 48*A*a^3*b^8 - 6*(A*a^4*b^7 + 4*B*a^3*b^8 - 6*A*a^2*b^9 - 4*B*a*b^10
+ A*b^11)*d*x)*tan(d*x + c)^3 + 6*(A*a^7*b^4 + 4*B*a^6*b^5 - 6*A*a^5*b^6 - 4*B*a^4*b^7 + A*a^3*b^8)*d*x - 3*(
5*B*a^9*b^2 + 18*B*a^7*b^4 + 2*A*a^6*b^5 + 37*B*a^5*b^6 - 30*A*a^4*b^7 - 20*B*a^3*b^8 + 12*A*a^2*b^9 - 6*(A*a^
5*b^6 + 4*B*a^4*b^7 - 6*A*a^3*b^8 - 4*B*a^2*b^9 + A*a*b^10)*d*x)*tan(d*x + c)^2 + 3*(B*a^11 + 4*B*a^9*b^2 + 5*
B*a^7*b^4 + 4*A*a^6*b^5 + 10*B*a^5*b^6 - 4*A*a^4*b^7 + (B*a^8*b^3 + 4*B*a^6*b^5 + 5*B*a^4*b^7 + 4*A*a^3*b^8 +
10*B*a^2*b^9 - 4*A*a*b^10)*tan(d*x + c)^3 + 3*(B*a^9*b^2 + 4*B*a^7*b^4 + 5*B*a^5*b^6 + 4*A*a^4*b^7 + 10*B*a^3*
b^8 - 4*A*a^2*b^9)*tan(d*x + c)^2 + 3*(B*a^10*b + 4*B*a^8*b^3 + 5*B*a^6*b^5 + 4*A*a^5*b^6 + 10*B*a^4*b^7 - 4*A
*a^3*b^8)*tan(d*x + c))*log((b^2*tan(d*x + c)^2 + 2*a*b*tan(d*x + c) + a^2)/(tan(d*x + c)^2 + 1)) - 3*(B*a^11
+ 4*B*a^9*b^2 + 6*B*a^7*b^4 + 4*B*a^5*b^6 + B*a^3*b^8 + (B*a^8*b^3 + 4*B*a^6*b^5 + 6*B*a^4*b^7 + 4*B*a^2*b^9 +
B*b^11)*tan(d*x + c)^3 + 3*(B*a^9*b^2 + 4*B*a^7*b^4 + 6*B*a^5*b^6 + 4*B*a^3*b^8 + B*a*b^10)*tan(d*x + c)^2 +
3*(B*a^10*b + 4*B*a^8*b^3 + 6*B*a^6*b^5 + 4*B*a^4*b^7 + B*a^2*b^9)*tan(d*x + c))*log(1/(tan(d*x + c)^2 + 1)) -
3*(2*B*a^10*b + 5*B*a^8*b^3 + 2*A*a^7*b^4 + 12*B*a^6*b^5 - 22*A*a^5*b^6 - 35*B*a^4*b^7 + 20*A*a^3*b^8 - 6*(A*
a^6*b^5 + 4*B*a^5*b^6 - 6*A*a^4*b^7 - 4*B*a^3*b^8 + A*a^2*b^9)*d*x)*tan(d*x + c))/((a^8*b^7 + 4*a^6*b^9 + 6*a^
4*b^11 + 4*a^2*b^13 + b^15)*d*tan(d*x + c)^3 + 3*(a^9*b^6 + 4*a^7*b^8 + 6*a^5*b^10 + 4*a^3*b^12 + a*b^14)*d*ta
n(d*x + c)^2 + 3*(a^10*b^5 + 4*a^8*b^7 + 6*a^6*b^9 + 4*a^4*b^11 + a^2*b^13)*d*tan(d*x + c) + (a^11*b^4 + 4*a^9
*b^6 + 6*a^7*b^8 + 4*a^5*b^10 + a^3*b^12)*d)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 2.34722, size = 971, normalized size = 2.77 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(6*(A*a^4 + 4*B*a^3*b - 6*A*a^2*b^2 - 4*B*a*b^3 + A*b^4)*(d*x + c)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^
6 + b^8) + 3*(B*a^4 - 4*A*a^3*b - 6*B*a^2*b^2 + 4*A*a*b^3 + B*b^4)*log(tan(d*x + c)^2 + 1)/(a^8 + 4*a^6*b^2 +
6*a^4*b^4 + 4*a^2*b^6 + b^8) + 6*(B*a^8 + 4*B*a^6*b^2 + 5*B*a^4*b^4 + 4*A*a^3*b^5 + 10*B*a^2*b^6 - 4*A*a*b^7)*
log(abs(b*tan(d*x + c) + a))/(a^8*b^4 + 4*a^6*b^6 + 6*a^4*b^8 + 4*a^2*b^10 + b^12) - (11*B*a^8*b^2*tan(d*x + c
)^3 + 44*B*a^6*b^4*tan(d*x + c)^3 + 55*B*a^4*b^6*tan(d*x + c)^3 + 44*A*a^3*b^7*tan(d*x + c)^3 + 110*B*a^2*b^8*
tan(d*x + c)^3 - 44*A*a*b^9*tan(d*x + c)^3 + 15*B*a^9*b*tan(d*x + c)^2 + 6*A*a^8*b^2*tan(d*x + c)^2 + 60*B*a^7
*b^3*tan(d*x + c)^2 + 24*A*a^6*b^4*tan(d*x + c)^2 + 51*B*a^5*b^5*tan(d*x + c)^2 + 186*A*a^4*b^6*tan(d*x + c)^2
+ 270*B*a^3*b^7*tan(d*x + c)^2 - 96*A*a^2*b^8*tan(d*x + c)^2 + 6*B*a^10*tan(d*x + c) + 6*A*a^9*b*tan(d*x + c)
+ 21*B*a^8*b^2*tan(d*x + c) + 24*A*a^7*b^3*tan(d*x + c) - 24*B*a^6*b^4*tan(d*x + c) + 210*A*a^5*b^5*tan(d*x +
c) + 225*B*a^4*b^6*tan(d*x + c) - 72*A*a^3*b^7*tan(d*x + c) + 2*A*a^10 - B*a^9*b + 6*A*a^8*b^2 - 26*B*a^7*b^3
+ 74*A*a^6*b^4 + 63*B*a^5*b^5 - 18*A*a^4*b^6)/((a^8*b^3 + 4*a^6*b^5 + 6*a^4*b^7 + 4*a^2*b^9 + b^11)*(b*tan(d*
x + c) + a)^3))/d