3.486 \(\int \frac{\tan ^6(c+d x)}{(a+b \tan (c+d x))^4} \, dx\)

Optimal. Leaf size=315 \[ -\frac{a^2 \tan ^4(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}-\frac{a^2 \left (2 a^2+5 b^2\right ) \tan ^3(c+d x)}{3 b^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac{2 a^2 \left (3 a^2 b^2+a^4+4 b^4\right ) \tan ^2(c+d x)}{b^3 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac{\left (12 a^4 b^2+13 a^2 b^4+4 a^6+b^6\right ) \tan (c+d x)}{b^4 d \left (a^2+b^2\right )^3}-\frac{4 a^3 \left (4 a^4 b^2+6 a^2 b^4+a^6+5 b^6\right ) \log (a+b \tan (c+d x))}{b^5 d \left (a^2+b^2\right )^4}-\frac{4 a b \left (a^2-b^2\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )^4}-\frac{x \left (-6 a^2 b^2+a^4+b^4\right )}{\left (a^2+b^2\right )^4} \]

[Out]

-(((a^4 - 6*a^2*b^2 + b^4)*x)/(a^2 + b^2)^4) - (4*a*b*(a^2 - b^2)*Log[Cos[c + d*x]])/((a^2 + b^2)^4*d) - (4*a^
3*(a^6 + 4*a^4*b^2 + 6*a^2*b^4 + 5*b^6)*Log[a + b*Tan[c + d*x]])/(b^5*(a^2 + b^2)^4*d) + ((4*a^6 + 12*a^4*b^2
+ 13*a^2*b^4 + b^6)*Tan[c + d*x])/(b^4*(a^2 + b^2)^3*d) - (a^2*Tan[c + d*x]^4)/(3*b*(a^2 + b^2)*d*(a + b*Tan[c
+ d*x])^3) - (a^2*(2*a^2 + 5*b^2)*Tan[c + d*x]^3)/(3*b^2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) - (2*a^2*(a^
4 + 3*a^2*b^2 + 4*b^4)*Tan[c + d*x]^2)/(b^3*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x]))

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Rubi [A]  time = 0.872289, antiderivative size = 315, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3565, 3645, 3647, 3626, 3617, 31, 3475} \[ -\frac{a^2 \tan ^4(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}-\frac{a^2 \left (2 a^2+5 b^2\right ) \tan ^3(c+d x)}{3 b^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac{2 a^2 \left (3 a^2 b^2+a^4+4 b^4\right ) \tan ^2(c+d x)}{b^3 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac{\left (12 a^4 b^2+13 a^2 b^4+4 a^6+b^6\right ) \tan (c+d x)}{b^4 d \left (a^2+b^2\right )^3}-\frac{4 a^3 \left (4 a^4 b^2+6 a^2 b^4+a^6+5 b^6\right ) \log (a+b \tan (c+d x))}{b^5 d \left (a^2+b^2\right )^4}-\frac{4 a b \left (a^2-b^2\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )^4}-\frac{x \left (-6 a^2 b^2+a^4+b^4\right )}{\left (a^2+b^2\right )^4} \]

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^6/(a + b*Tan[c + d*x])^4,x]

[Out]

-(((a^4 - 6*a^2*b^2 + b^4)*x)/(a^2 + b^2)^4) - (4*a*b*(a^2 - b^2)*Log[Cos[c + d*x]])/((a^2 + b^2)^4*d) - (4*a^
3*(a^6 + 4*a^4*b^2 + 6*a^2*b^4 + 5*b^6)*Log[a + b*Tan[c + d*x]])/(b^5*(a^2 + b^2)^4*d) + ((4*a^6 + 12*a^4*b^2
+ 13*a^2*b^4 + b^6)*Tan[c + d*x])/(b^4*(a^2 + b^2)^3*d) - (a^2*Tan[c + d*x]^4)/(3*b*(a^2 + b^2)*d*(a + b*Tan[c
+ d*x])^3) - (a^2*(2*a^2 + 5*b^2)*Tan[c + d*x]^3)/(3*b^2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) - (2*a^2*(a^
4 + 3*a^2*b^2 + 4*b^4)*Tan[c + d*x]^2)/(b^3*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x]))

Rule 3565

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[((b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 + d^2)), x] - D
ist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(
m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3
*a*b^2*d)*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*(n + 1)))*Tan[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && Gt
Q[m, 2] && LtQ[n, -1] && IntegerQ[2*m]

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 3647

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

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 ^6(c+d x)}{(a+b \tan (c+d x))^4} \, dx &=-\frac{a^2 \tan ^4(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac{\int \frac{\tan ^3(c+d x) \left (4 a^2-3 a b \tan (c+d x)+\left (4 a^2+3 b^2\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx}{3 b \left (a^2+b^2\right )}\\ &=-\frac{a^2 \tan ^4(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac{a^2 \left (2 a^2+5 b^2\right ) \tan ^3(c+d x)}{3 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac{\int \frac{\tan ^2(c+d x) \left (6 a^2 \left (2 a^2+5 b^2\right )-12 a b^3 \tan (c+d x)+6 \left (2 a^4+4 a^2 b^2+b^4\right ) \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^2 \tan ^4(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac{a^2 \left (2 a^2+5 b^2\right ) \tan ^3(c+d x)}{3 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac{2 a^2 \left (a^4+3 a^2 b^2+4 b^4\right ) \tan ^2(c+d x)}{b^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac{\int \frac{\tan (c+d x) \left (24 a^2 \left (a^4+3 a^2 b^2+4 b^4\right )+6 a b^3 \left (a^2-3 b^2\right ) \tan (c+d x)+6 \left (4 a^6+12 a^4 b^2+13 a^2 b^4+b^6\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{6 b^3 \left (a^2+b^2\right )^3}\\ &=\frac{\left (4 a^6+12 a^4 b^2+13 a^2 b^4+b^6\right ) \tan (c+d x)}{b^4 \left (a^2+b^2\right )^3 d}-\frac{a^2 \tan ^4(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac{a^2 \left (2 a^2+5 b^2\right ) \tan ^3(c+d x)}{3 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac{2 a^2 \left (a^4+3 a^2 b^2+4 b^4\right ) \tan ^2(c+d x)}{b^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac{\int \frac{-6 a \left (4 a^6+12 a^4 b^2+13 a^2 b^4+b^6\right )+6 b^5 \left (3 a^2-b^2\right ) \tan (c+d x)-24 a \left (a^2+b^2\right )^3 \tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{6 b^4 \left (a^2+b^2\right )^3}\\ &=-\frac{\left (a^4-6 a^2 b^2+b^4\right ) x}{\left (a^2+b^2\right )^4}+\frac{\left (4 a^6+12 a^4 b^2+13 a^2 b^4+b^6\right ) \tan (c+d x)}{b^4 \left (a^2+b^2\right )^3 d}-\frac{a^2 \tan ^4(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac{a^2 \left (2 a^2+5 b^2\right ) \tan ^3(c+d x)}{3 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac{2 a^2 \left (a^4+3 a^2 b^2+4 b^4\right ) \tan ^2(c+d x)}{b^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac{\left (4 a b \left (a^2-b^2\right )\right ) \int \tan (c+d x) \, dx}{\left (a^2+b^2\right )^4}-\frac{\left (4 a^3 \left (a^6+4 a^4 b^2+6 a^2 b^4+5 b^6\right )\right ) \int \frac{1+\tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{b^4 \left (a^2+b^2\right )^4}\\ &=-\frac{\left (a^4-6 a^2 b^2+b^4\right ) x}{\left (a^2+b^2\right )^4}-\frac{4 a b \left (a^2-b^2\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^4 d}+\frac{\left (4 a^6+12 a^4 b^2+13 a^2 b^4+b^6\right ) \tan (c+d x)}{b^4 \left (a^2+b^2\right )^3 d}-\frac{a^2 \tan ^4(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac{a^2 \left (2 a^2+5 b^2\right ) \tan ^3(c+d x)}{3 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac{2 a^2 \left (a^4+3 a^2 b^2+4 b^4\right ) \tan ^2(c+d x)}{b^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}-\frac{\left (4 a^3 \left (a^6+4 a^4 b^2+6 a^2 b^4+5 b^6\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,b \tan (c+d x)\right )}{b^5 \left (a^2+b^2\right )^4 d}\\ &=-\frac{\left (a^4-6 a^2 b^2+b^4\right ) x}{\left (a^2+b^2\right )^4}-\frac{4 a b \left (a^2-b^2\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^4 d}-\frac{4 a^3 \left (a^6+4 a^4 b^2+6 a^2 b^4+5 b^6\right ) \log (a+b \tan (c+d x))}{b^5 \left (a^2+b^2\right )^4 d}+\frac{\left (4 a^6+12 a^4 b^2+13 a^2 b^4+b^6\right ) \tan (c+d x)}{b^4 \left (a^2+b^2\right )^3 d}-\frac{a^2 \tan ^4(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac{a^2 \left (2 a^2+5 b^2\right ) \tan ^3(c+d x)}{3 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac{2 a^2 \left (a^4+3 a^2 b^2+4 b^4\right ) \tan ^2(c+d x)}{b^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}\\ \end{align*}

Mathematica [C]  time = 6.42568, size = 1281, normalized size = 4.07 \[ \frac{(a \cos (c+d x)+b \sin (c+d x)) \left (24 \sin (2 (c+d x)) a^{11}+12 \sin (4 (c+d x)) a^{11}+39 b a^{10}+12 b \cos (2 (c+d x)) a^{10}-27 b \cos (4 (c+d x)) a^{10}+158 b^2 \sin (2 (c+d x)) a^9+35 b^2 \sin (4 (c+d x)) a^9+171 b^3 a^8+32 b^3 \cos (2 (c+d x)) a^8-115 b^3 \cos (4 (c+d x)) a^8-9 b^4 (c+d x) a^7-12 b^4 (c+d x) \cos (2 (c+d x)) a^7-3 b^4 (c+d x) \cos (4 (c+d x)) a^7+396 b^4 \sin (2 (c+d x)) a^7+18 b^4 \sin (4 (c+d x)) a^7+276 b^5 a^6-16 b^5 \cos (2 (c+d x)) a^6-196 b^5 \cos (4 (c+d x)) a^6-18 b^5 (c+d x) \sin (2 (c+d x)) a^6-9 b^5 (c+d x) \sin (4 (c+d x)) a^6+45 b^6 (c+d x) a^5+72 b^6 (c+d x) \cos (2 (c+d x)) a^5+27 b^6 (c+d x) \cos (4 (c+d x)) a^5+412 b^6 \sin (2 (c+d x)) a^5-74 b^6 \sin (4 (c+d x)) a^5+180 b^7 a^4-72 b^7 \cos (2 (c+d x)) a^4-108 b^7 \cos (4 (c+d x)) a^4+102 b^7 (c+d x) \sin (2 (c+d x)) a^4+57 b^7 (c+d x) \sin (4 (c+d x)) a^4+45 b^8 (c+d x) a^3-12 b^8 (c+d x) \cos (2 (c+d x)) a^3-57 b^8 (c+d x) \cos (4 (c+d x)) a^3+168 b^8 \sin (2 (c+d x)) a^3-78 b^8 \sin (4 (c+d x)) a^3+45 b^9 a^2-48 b^9 \cos (2 (c+d x)) a^2+3 b^9 \cos (4 (c+d x)) a^2+18 b^9 (c+d x) \sin (2 (c+d x)) a^2-27 b^9 (c+d x) \sin (4 (c+d x)) a^2-9 b^{10} (c+d x) a+9 b^{10} (c+d x) \cos (4 (c+d x)) a+18 b^{10} \sin (2 (c+d x)) a-9 b^{10} \sin (4 (c+d x)) a+9 b^{11}-12 b^{11} \cos (2 (c+d x))+3 b^{11} \cos (4 (c+d x))-6 b^{11} (c+d x) \sin (2 (c+d x))+3 b^{11} (c+d x) \sin (4 (c+d x))\right ) \sec ^5(c+d x)}{24 b^4 (b-i a)^4 (i a+b)^4 d (a+b \tan (c+d x))^4}-\frac{4 i \left (-5 i a^3 b^{17}+5 a^4 b^{16}-21 i a^5 b^{15}+21 a^6 b^{14}-37 i a^7 b^{13}+37 a^8 b^{12}-36 i a^9 b^{11}+36 a^{10} b^{10}-21 i a^{11} b^9+21 a^{12} b^8-7 i a^{13} b^7+7 a^{14} b^6-i a^{15} b^5+a^{16} b^4\right ) (c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \sec ^4(c+d x)}{(a-i b)^8 (a+i b)^7 b^9 d (a+b \tan (c+d x))^4}+\frac{4 i \left (a^9+4 b^2 a^7+6 b^4 a^5+5 b^6 a^3\right ) \tan ^{-1}(\tan (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^4 \sec ^4(c+d x)}{b^5 \left (a^2+b^2\right )^4 d (a+b \tan (c+d x))^4}+\frac{4 a \log (\cos (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^4 \sec ^4(c+d x)}{b^5 d (a+b \tan (c+d x))^4}-\frac{2 \left (a^9+4 b^2 a^7+6 b^4 a^5+5 b^6 a^3\right ) \log \left ((a \cos (c+d x)+b \sin (c+d x))^2\right ) (a \cos (c+d x)+b \sin (c+d x))^4 \sec ^4(c+d x)}{b^5 \left (a^2+b^2\right )^4 d (a+b \tan (c+d x))^4} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^6/(a + b*Tan[c + d*x])^4,x]

[Out]

((-4*I)*(a^16*b^4 - I*a^15*b^5 + 7*a^14*b^6 - (7*I)*a^13*b^7 + 21*a^12*b^8 - (21*I)*a^11*b^9 + 36*a^10*b^10 -
(36*I)*a^9*b^11 + 37*a^8*b^12 - (37*I)*a^7*b^13 + 21*a^6*b^14 - (21*I)*a^5*b^15 + 5*a^4*b^16 - (5*I)*a^3*b^17)
*(c + d*x)*Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^4)/((a - I*b)^8*(a + I*b)^7*b^9*d*(a + b*Tan[c + d
*x])^4) + ((4*I)*(a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 5*a^3*b^6)*ArcTan[Tan[c + d*x]]*Sec[c + d*x]^4*(a*Cos[c + d*x]
+ b*Sin[c + d*x])^4)/(b^5*(a^2 + b^2)^4*d*(a + b*Tan[c + d*x])^4) + (4*a*Log[Cos[c + d*x]]*Sec[c + d*x]^4*(a*
Cos[c + d*x] + b*Sin[c + d*x])^4)/(b^5*d*(a + b*Tan[c + d*x])^4) - (2*(a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 5*a^3*b^6
)*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2]*Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^4)/(b^5*(a^2 + b^2
)^4*d*(a + b*Tan[c + d*x])^4) + (Sec[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x])*(39*a^10*b + 171*a^8*b^3 + 2
76*a^6*b^5 + 180*a^4*b^7 + 45*a^2*b^9 + 9*b^11 - 9*a^7*b^4*(c + d*x) + 45*a^5*b^6*(c + d*x) + 45*a^3*b^8*(c +
d*x) - 9*a*b^10*(c + d*x) + 12*a^10*b*Cos[2*(c + d*x)] + 32*a^8*b^3*Cos[2*(c + d*x)] - 16*a^6*b^5*Cos[2*(c + d
*x)] - 72*a^4*b^7*Cos[2*(c + d*x)] - 48*a^2*b^9*Cos[2*(c + d*x)] - 12*b^11*Cos[2*(c + d*x)] - 12*a^7*b^4*(c +
d*x)*Cos[2*(c + d*x)] + 72*a^5*b^6*(c + d*x)*Cos[2*(c + d*x)] - 12*a^3*b^8*(c + d*x)*Cos[2*(c + d*x)] - 27*a^1
0*b*Cos[4*(c + d*x)] - 115*a^8*b^3*Cos[4*(c + d*x)] - 196*a^6*b^5*Cos[4*(c + d*x)] - 108*a^4*b^7*Cos[4*(c + d*
x)] + 3*a^2*b^9*Cos[4*(c + d*x)] + 3*b^11*Cos[4*(c + d*x)] - 3*a^7*b^4*(c + d*x)*Cos[4*(c + d*x)] + 27*a^5*b^6
*(c + d*x)*Cos[4*(c + d*x)] - 57*a^3*b^8*(c + d*x)*Cos[4*(c + d*x)] + 9*a*b^10*(c + d*x)*Cos[4*(c + d*x)] + 24
*a^11*Sin[2*(c + d*x)] + 158*a^9*b^2*Sin[2*(c + d*x)] + 396*a^7*b^4*Sin[2*(c + d*x)] + 412*a^5*b^6*Sin[2*(c +
d*x)] + 168*a^3*b^8*Sin[2*(c + d*x)] + 18*a*b^10*Sin[2*(c + d*x)] - 18*a^6*b^5*(c + d*x)*Sin[2*(c + d*x)] + 10
2*a^4*b^7*(c + d*x)*Sin[2*(c + d*x)] + 18*a^2*b^9*(c + d*x)*Sin[2*(c + d*x)] - 6*b^11*(c + d*x)*Sin[2*(c + d*x
)] + 12*a^11*Sin[4*(c + d*x)] + 35*a^9*b^2*Sin[4*(c + d*x)] + 18*a^7*b^4*Sin[4*(c + d*x)] - 74*a^5*b^6*Sin[4*(
c + d*x)] - 78*a^3*b^8*Sin[4*(c + d*x)] - 9*a*b^10*Sin[4*(c + d*x)] - 9*a^6*b^5*(c + d*x)*Sin[4*(c + d*x)] + 5
7*a^4*b^7*(c + d*x)*Sin[4*(c + d*x)] - 27*a^2*b^9*(c + d*x)*Sin[4*(c + d*x)] + 3*b^11*(c + d*x)*Sin[4*(c + d*x
)]))/(24*b^4*((-I)*a + b)^4*(I*a + b)^4*d*(a + b*Tan[c + d*x])^4)

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Maple [A]  time = 0.039, size = 462, normalized size = 1.5 \begin{align*}{\frac{\tan \left ( dx+c \right ) }{d{b}^{4}}}-{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{4}}{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}+6\,{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{2}{b}^{2}}{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}-{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ){b}^{4}}{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}+2\,{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ){a}^{3}b}{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}-2\,{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) a{b}^{3}}{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}-{\frac{{a}^{6}}{3\,d{b}^{5} \left ({a}^{2}+{b}^{2} \right ) \left ( a+b\tan \left ( dx+c \right ) \right ) ^{3}}}-6\,{\frac{{a}^{8}}{d{b}^{5} \left ({a}^{2}+{b}^{2} \right ) ^{3} \left ( a+b\tan \left ( dx+c \right ) \right ) }}-17\,{\frac{{a}^{6}}{d{b}^{3} \left ({a}^{2}+{b}^{2} \right ) ^{3} \left ( a+b\tan \left ( dx+c \right ) \right ) }}-15\,{\frac{{a}^{4}}{bd \left ({a}^{2}+{b}^{2} \right ) ^{3} \left ( a+b\tan \left ( dx+c \right ) \right ) }}-4\,{\frac{{a}^{9}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d{b}^{5} \left ({a}^{2}+{b}^{2} \right ) ^{4}}}-16\,{\frac{{a}^{7}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d{b}^{3} \left ({a}^{2}+{b}^{2} \right ) ^{4}}}-24\,{\frac{{a}^{5}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{bd \left ({a}^{2}+{b}^{2} \right ) ^{4}}}-20\,{\frac{b{a}^{3}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}+2\,{\frac{{a}^{7}}{d{b}^{5} \left ({a}^{2}+{b}^{2} \right ) ^{2} \left ( a+b\tan \left ( dx+c \right ) \right ) ^{2}}}+3\,{\frac{{a}^{5}}{d{b}^{3} \left ({a}^{2}+{b}^{2} \right ) ^{2} \left ( a+b\tan \left ( dx+c \right ) \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^6/(a+b*tan(d*x+c))^4,x)

[Out]

1/d/b^4*tan(d*x+c)-1/d/(a^2+b^2)^4*arctan(tan(d*x+c))*a^4+6/d/(a^2+b^2)^4*arctan(tan(d*x+c))*a^2*b^2-1/d/(a^2+
b^2)^4*arctan(tan(d*x+c))*b^4+2/d/(a^2+b^2)^4*ln(1+tan(d*x+c)^2)*a^3*b-2/d/(a^2+b^2)^4*ln(1+tan(d*x+c)^2)*a*b^
3-1/3/d/b^5*a^6/(a^2+b^2)/(a+b*tan(d*x+c))^3-6/d/b^5*a^8/(a^2+b^2)^3/(a+b*tan(d*x+c))-17/d/b^3*a^6/(a^2+b^2)^3
/(a+b*tan(d*x+c))-15/d/b*a^4/(a^2+b^2)^3/(a+b*tan(d*x+c))-4/d/b^5*a^9/(a^2+b^2)^4*ln(a+b*tan(d*x+c))-16/d/b^3*
a^7/(a^2+b^2)^4*ln(a+b*tan(d*x+c))-24/d/b*a^5/(a^2+b^2)^4*ln(a+b*tan(d*x+c))-20/d*b*a^3/(a^2+b^2)^4*ln(a+b*tan
(d*x+c))+2/d/b^5*a^7/(a^2+b^2)^2/(a+b*tan(d*x+c))^2+3/d/b^3*a^5/(a^2+b^2)^2/(a+b*tan(d*x+c))^2

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Maxima [A]  time = 1.58198, size = 599, normalized size = 1.9 \begin{align*} -\frac{\frac{3 \,{\left (a^{4} - 6 \, a^{2} b^{2} + 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{12 \,{\left (a^{9} + 4 \, a^{7} b^{2} + 6 \, a^{5} b^{4} + 5 \, a^{3} b^{6}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{8} b^{5} + 4 \, a^{6} b^{7} + 6 \, a^{4} b^{9} + 4 \, a^{2} b^{11} + b^{13}} - \frac{6 \,{\left (a^{3} b - a b^{3}\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{13 \, a^{10} + 38 \, a^{8} b^{2} + 37 \, a^{6} b^{4} + 3 \,{\left (6 \, a^{8} b^{2} + 17 \, a^{6} b^{4} + 15 \, a^{4} b^{6}\right )} \tan \left (d x + c\right )^{2} + 3 \,{\left (10 \, a^{9} b + 29 \, a^{7} b^{3} + 27 \, a^{5} b^{5}\right )} \tan \left (d x + c\right )}{a^{9} b^{5} + 3 \, a^{7} b^{7} + 3 \, a^{5} b^{9} + a^{3} b^{11} +{\left (a^{6} b^{8} + 3 \, a^{4} b^{10} + 3 \, a^{2} b^{12} + b^{14}\right )} \tan \left (d x + c\right )^{3} + 3 \,{\left (a^{7} b^{7} + 3 \, a^{5} b^{9} + 3 \, a^{3} b^{11} + a b^{13}\right )} \tan \left (d x + c\right )^{2} + 3 \,{\left (a^{8} b^{6} + 3 \, a^{6} b^{8} + 3 \, a^{4} b^{10} + a^{2} b^{12}\right )} \tan \left (d x + c\right )} - \frac{3 \, \tan \left (d x + c\right )}{b^{4}}}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^6/(a+b*tan(d*x+c))^4,x, algorithm="maxima")

[Out]

-1/3*(3*(a^4 - 6*a^2*b^2 + b^4)*(d*x + c)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) + 12*(a^9 + 4*a^7*b^
2 + 6*a^5*b^4 + 5*a^3*b^6)*log(b*tan(d*x + c) + a)/(a^8*b^5 + 4*a^6*b^7 + 6*a^4*b^9 + 4*a^2*b^11 + b^13) - 6*(
a^3*b - a*b^3)*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) + (13*a^10 + 38*a^8*b^2
+ 37*a^6*b^4 + 3*(6*a^8*b^2 + 17*a^6*b^4 + 15*a^4*b^6)*tan(d*x + c)^2 + 3*(10*a^9*b + 29*a^7*b^3 + 27*a^5*b^5
)*tan(d*x + c))/(a^9*b^5 + 3*a^7*b^7 + 3*a^5*b^9 + a^3*b^11 + (a^6*b^8 + 3*a^4*b^10 + 3*a^2*b^12 + b^14)*tan(d
*x + c)^3 + 3*(a^7*b^7 + 3*a^5*b^9 + 3*a^3*b^11 + a*b^13)*tan(d*x + c)^2 + 3*(a^8*b^6 + 3*a^6*b^8 + 3*a^4*b^10
+ a^2*b^12)*tan(d*x + c)) - 3*tan(d*x + c)/b^4)/d

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Fricas [B]  time = 2.73001, size = 1933, normalized size = 6.14 \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)^6/(a+b*tan(d*x+c))^4,x, algorithm="fricas")

[Out]

-1/3*(6*a^10*b^2 + 21*a^8*b^4 + 37*a^6*b^6 - 3*(a^8*b^4 + 4*a^6*b^6 + 6*a^4*b^8 + 4*a^2*b^10 + b^12)*tan(d*x +
c)^4 - (22*a^9*b^3 + 81*a^7*b^5 + 108*a^5*b^7 + 36*a^3*b^9 + 9*a*b^11 - 3*(a^4*b^8 - 6*a^2*b^10 + b^12)*d*x)*
tan(d*x + c)^3 + 3*(a^7*b^5 - 6*a^5*b^7 + a^3*b^9)*d*x - 3*(10*a^10*b^2 + 34*a^8*b^4 + 40*a^6*b^6 - 3*a^4*b^8
+ 3*a^2*b^10 - 3*(a^5*b^7 - 6*a^3*b^9 + a*b^11)*d*x)*tan(d*x + c)^2 + 6*(a^12 + 4*a^10*b^2 + 6*a^8*b^4 + 5*a^6
*b^6 + (a^9*b^3 + 4*a^7*b^5 + 6*a^5*b^7 + 5*a^3*b^9)*tan(d*x + c)^3 + 3*(a^10*b^2 + 4*a^8*b^4 + 6*a^6*b^6 + 5*
a^4*b^8)*tan(d*x + c)^2 + 3*(a^11*b + 4*a^9*b^3 + 6*a^7*b^5 + 5*a^5*b^7)*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)) - 6*(a^12 + 4*a^10*b^2 + 6*a^8*b^4 + 4*a^6*b^6 + a^4*b^8 +
(a^9*b^3 + 4*a^7*b^5 + 6*a^5*b^7 + 4*a^3*b^9 + a*b^11)*tan(d*x + c)^3 + 3*(a^10*b^2 + 4*a^8*b^4 + 6*a^6*b^6 +
4*a^4*b^8 + a^2*b^10)*tan(d*x + c)^2 + 3*(a^11*b + 4*a^9*b^3 + 6*a^7*b^5 + 4*a^5*b^7 + a^3*b^9)*tan(d*x + c))*
log(1/(tan(d*x + c)^2 + 1)) - 3*(4*a^11*b + 10*a^9*b^3 + 4*a^7*b^5 - 23*a^5*b^7 + a^3*b^9 - 3*(a^6*b^6 - 6*a^4
*b^8 + a^2*b^10)*d*x)*tan(d*x + c))/((a^8*b^8 + 4*a^6*b^10 + 6*a^4*b^12 + 4*a^2*b^14 + b^16)*d*tan(d*x + c)^3
+ 3*(a^9*b^7 + 4*a^7*b^9 + 6*a^5*b^11 + 4*a^3*b^13 + a*b^15)*d*tan(d*x + c)^2 + 3*(a^10*b^6 + 4*a^8*b^8 + 6*a^
6*b^10 + 4*a^4*b^12 + a^2*b^14)*d*tan(d*x + c) + (a^11*b^5 + 4*a^9*b^7 + 6*a^7*b^9 + 4*a^5*b^11 + a^3*b^13)*d)

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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)**6/(a+b*tan(d*x+c))**4,x)

[Out]

Timed out

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Giac [A]  time = 4.94119, size = 637, normalized size = 2.02 \begin{align*} -\frac{\frac{3 \,{\left (a^{4} - 6 \, a^{2} b^{2} + 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 (a^{3} b - a b^{3}\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{12 \,{\left (a^{9} + 4 \, a^{7} b^{2} + 6 \, a^{5} b^{4} + 5 \, a^{3} b^{6}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{8} b^{5} + 4 \, a^{6} b^{7} + 6 \, a^{4} b^{9} + 4 \, a^{2} b^{11} + b^{13}} - \frac{22 \, a^{9} b^{3} \tan \left (d x + c\right )^{3} + 88 \, a^{7} b^{5} \tan \left (d x + c\right )^{3} + 132 \, a^{5} b^{7} \tan \left (d x + c\right )^{3} + 110 \, a^{3} b^{9} \tan \left (d x + c\right )^{3} + 48 \, a^{10} b^{2} \tan \left (d x + c\right )^{2} + 195 \, a^{8} b^{4} \tan \left (d x + c\right )^{2} + 300 \, a^{6} b^{6} \tan \left (d x + c\right )^{2} + 285 \, a^{4} b^{8} \tan \left (d x + c\right )^{2} + 36 \, a^{11} b \tan \left (d x + c\right ) + 147 \, a^{9} b^{3} \tan \left (d x + c\right ) + 228 \, a^{7} b^{5} \tan \left (d x + c\right ) + 249 \, a^{5} b^{7} \tan \left (d x + c\right ) + 9 \, a^{12} + 37 \, a^{10} b^{2} + 57 \, a^{8} b^{4} + 73 \, a^{6} b^{6}}{{\left (a^{8} b^{5} + 4 \, a^{6} b^{7} + 6 \, a^{4} b^{9} + 4 \, a^{2} b^{11} + b^{13}\right )}{\left (b \tan \left (d x + c\right ) + a\right )}^{3}} - \frac{3 \, \tan \left (d x + c\right )}{b^{4}}}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^6/(a+b*tan(d*x+c))^4,x, algorithm="giac")

[Out]

-1/3*(3*(a^4 - 6*a^2*b^2 + 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*(a^3*b - a*b^3)*
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) + 12*(a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 5*
a^3*b^6)*log(abs(b*tan(d*x + c) + a))/(a^8*b^5 + 4*a^6*b^7 + 6*a^4*b^9 + 4*a^2*b^11 + b^13) - (22*a^9*b^3*tan(
d*x + c)^3 + 88*a^7*b^5*tan(d*x + c)^3 + 132*a^5*b^7*tan(d*x + c)^3 + 110*a^3*b^9*tan(d*x + c)^3 + 48*a^10*b^2
*tan(d*x + c)^2 + 195*a^8*b^4*tan(d*x + c)^2 + 300*a^6*b^6*tan(d*x + c)^2 + 285*a^4*b^8*tan(d*x + c)^2 + 36*a^
11*b*tan(d*x + c) + 147*a^9*b^3*tan(d*x + c) + 228*a^7*b^5*tan(d*x + c) + 249*a^5*b^7*tan(d*x + c) + 9*a^12 +
37*a^10*b^2 + 57*a^8*b^4 + 73*a^6*b^6)/((a^8*b^5 + 4*a^6*b^7 + 6*a^4*b^9 + 4*a^2*b^11 + b^13)*(b*tan(d*x + c)
+ a)^3) - 3*tan(d*x + c)/b^4)/d