Integrand size = 40, antiderivative size = 191 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x))^3 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) x+\frac {\left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \cot (c+d x)}{d}+\frac {a \left (2 a^2 B-5 b^2 B-6 a b C\right ) \cot ^2(c+d x)}{4 d}-\frac {a^2 (3 b B+2 a C) \cot ^3(c+d x)}{6 d}+\frac {\left (a^3 B-3 a b^2 B-3 a^2 b C+b^3 C\right ) \log (\sin (c+d x))}{d}-\frac {a B \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d} \]
(3*B*a^2*b-B*b^3+C*a^3-3*C*a*b^2)*x+(3*B*a^2*b-B*b^3+C*a^3-3*C*a*b^2)*cot( d*x+c)/d+1/4*a*(2*B*a^2-5*B*b^2-6*C*a*b)*cot(d*x+c)^2/d-1/6*a^2*(3*B*b+2*C *a)*cot(d*x+c)^3/d+(B*a^3-3*B*a*b^2-3*C*a^2*b+C*b^3)*ln(sin(d*x+c))/d-1/4* a*B*cot(d*x+c)^4*(a+b*tan(d*x+c))^2/d
Result contains complex when optimal does not.
Time = 0.81 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.04 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x))^3 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {12 \left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \cot (c+d x)+6 a \left (a^2 B-3 b^2 B-3 a b C\right ) \cot ^2(c+d x)-4 a^2 (3 b B+a C) \cot ^3(c+d x)-3 a^3 B \cot ^4(c+d x)-6 (a+i b)^3 (B+i C) \log (i-\tan (c+d x))+12 \left (a^3 B-3 a b^2 B-3 a^2 b C+b^3 C\right ) \log (\tan (c+d x))-6 (a-i b)^3 (B-i C) \log (i+\tan (c+d x))}{12 d} \]
(12*(3*a^2*b*B - b^3*B + a^3*C - 3*a*b^2*C)*Cot[c + d*x] + 6*a*(a^2*B - 3* b^2*B - 3*a*b*C)*Cot[c + d*x]^2 - 4*a^2*(3*b*B + a*C)*Cot[c + d*x]^3 - 3*a ^3*B*Cot[c + d*x]^4 - 6*(a + I*b)^3*(B + I*C)*Log[I - Tan[c + d*x]] + 12*( a^3*B - 3*a*b^2*B - 3*a^2*b*C + b^3*C)*Log[Tan[c + d*x]] - 6*(a - I*b)^3*( B - I*C)*Log[I + Tan[c + d*x]])/(12*d)
Time = 1.44 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.07, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.475, Rules used = {3042, 4115, 3042, 4088, 27, 3042, 4118, 25, 3042, 4111, 27, 3042, 4012, 25, 3042, 4014, 3042, 25, 3956}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \cot ^6(c+d x) (a+b \tan (c+d x))^3 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \tan (c+d x))^3 \left (B \tan (c+d x)+C \tan (c+d x)^2\right )}{\tan (c+d x)^6}dx\) |
| \(\Big \downarrow \) 4115 |
| \(\displaystyle \int \cot ^5(c+d x) (a+b \tan (c+d x))^3 (B+C \tan (c+d x))dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \tan (c+d x))^3 (B+C \tan (c+d x))}{\tan (c+d x)^5}dx\) |
| \(\Big \downarrow \) 4088 |
| \(\displaystyle \frac {1}{4} \int 2 \cot ^4(c+d x) (a+b \tan (c+d x)) \left (-b (a B-2 b C) \tan ^2(c+d x)-2 \left (B a^2-2 b C a-b^2 B\right ) \tan (c+d x)+a (3 b B+2 a C)\right )dx-\frac {a B \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int \cot ^4(c+d x) (a+b \tan (c+d x)) \left (-b (a B-2 b C) \tan ^2(c+d x)-2 \left (B a^2-2 b C a-b^2 B\right ) \tan (c+d x)+a (3 b B+2 a C)\right )dx-\frac {a B \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \int \frac {(a+b \tan (c+d x)) \left (-b (a B-2 b C) \tan (c+d x)^2-2 \left (B a^2-2 b C a-b^2 B\right ) \tan (c+d x)+a (3 b B+2 a C)\right )}{\tan (c+d x)^4}dx-\frac {a B \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}\) |
| \(\Big \downarrow \) 4118 |
| \(\displaystyle \frac {1}{2} \left (\int -\cot ^3(c+d x) \left (b^2 (a B-2 b C) \tan ^2(c+d x)+2 \left (C a^3+3 b B a^2-3 b^2 C a-b^3 B\right ) \tan (c+d x)+a \left (2 B a^2-6 b C a-5 b^2 B\right )\right )dx-\frac {a^2 (2 a C+3 b B) \cot ^3(c+d x)}{3 d}\right )-\frac {a B \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (-\int \cot ^3(c+d x) \left (b^2 (a B-2 b C) \tan ^2(c+d x)+2 \left (C a^3+3 b B a^2-3 b^2 C a-b^3 B\right ) \tan (c+d x)+a \left (2 B a^2-6 b C a-5 b^2 B\right )\right )dx-\frac {a^2 (2 a C+3 b B) \cot ^3(c+d x)}{3 d}\right )-\frac {a B \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (-\int \frac {b^2 (a B-2 b C) \tan (c+d x)^2+2 \left (C a^3+3 b B a^2-3 b^2 C a-b^3 B\right ) \tan (c+d x)+a \left (2 B a^2-6 b C a-5 b^2 B\right )}{\tan (c+d x)^3}dx-\frac {a^2 (2 a C+3 b B) \cot ^3(c+d x)}{3 d}\right )-\frac {a B \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}\) |
| \(\Big \downarrow \) 4111 |
| \(\displaystyle \frac {1}{2} \left (-\int 2 \cot ^2(c+d x) \left (C a^3+3 b B a^2-3 b^2 C a-b^3 B-\left (B a^3-3 b C a^2-3 b^2 B a+b^3 C\right ) \tan (c+d x)\right )dx+\frac {a \left (2 a^2 B-6 a b C-5 b^2 B\right ) \cot ^2(c+d x)}{2 d}-\frac {a^2 (2 a C+3 b B) \cot ^3(c+d x)}{3 d}\right )-\frac {a B \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (-2 \int \cot ^2(c+d x) \left (C a^3+3 b B a^2-3 b^2 C a-b^3 B-\left (B a^3-3 b C a^2-3 b^2 B a+b^3 C\right ) \tan (c+d x)\right )dx+\frac {a \left (2 a^2 B-6 a b C-5 b^2 B\right ) \cot ^2(c+d x)}{2 d}-\frac {a^2 (2 a C+3 b B) \cot ^3(c+d x)}{3 d}\right )-\frac {a B \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (-2 \int \frac {C a^3+3 b B a^2-3 b^2 C a-b^3 B-\left (B a^3-3 b C a^2-3 b^2 B a+b^3 C\right ) \tan (c+d x)}{\tan (c+d x)^2}dx+\frac {a \left (2 a^2 B-6 a b C-5 b^2 B\right ) \cot ^2(c+d x)}{2 d}-\frac {a^2 (2 a C+3 b B) \cot ^3(c+d x)}{3 d}\right )-\frac {a B \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {1}{2} \left (-2 \left (\int -\cot (c+d x) \left (B a^3-3 b C a^2-3 b^2 B a+b^3 C+\left (C a^3+3 b B a^2-3 b^2 C a-b^3 B\right ) \tan (c+d x)\right )dx-\frac {\left (a^3 C+3 a^2 b B-3 a b^2 C-b^3 B\right ) \cot (c+d x)}{d}\right )+\frac {a \left (2 a^2 B-6 a b C-5 b^2 B\right ) \cot ^2(c+d x)}{2 d}-\frac {a^2 (2 a C+3 b B) \cot ^3(c+d x)}{3 d}\right )-\frac {a B \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (-2 \left (-\int \cot (c+d x) \left (B a^3-3 b C a^2-3 b^2 B a+b^3 C+\left (C a^3+3 b B a^2-3 b^2 C a-b^3 B\right ) \tan (c+d x)\right )dx-\frac {\left (a^3 C+3 a^2 b B-3 a b^2 C-b^3 B\right ) \cot (c+d x)}{d}\right )+\frac {a \left (2 a^2 B-6 a b C-5 b^2 B\right ) \cot ^2(c+d x)}{2 d}-\frac {a^2 (2 a C+3 b B) \cot ^3(c+d x)}{3 d}\right )-\frac {a B \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (-2 \left (-\int \frac {B a^3-3 b C a^2-3 b^2 B a+b^3 C+\left (C a^3+3 b B a^2-3 b^2 C a-b^3 B\right ) \tan (c+d x)}{\tan (c+d x)}dx-\frac {\left (a^3 C+3 a^2 b B-3 a b^2 C-b^3 B\right ) \cot (c+d x)}{d}\right )+\frac {a \left (2 a^2 B-6 a b C-5 b^2 B\right ) \cot ^2(c+d x)}{2 d}-\frac {a^2 (2 a C+3 b B) \cot ^3(c+d x)}{3 d}\right )-\frac {a B \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}\) |
| \(\Big \downarrow \) 4014 |
| \(\displaystyle \frac {1}{2} \left (-2 \left (-\left (a^3 B-3 a^2 b C-3 a b^2 B+b^3 C\right ) \int \cot (c+d x)dx-\frac {\left (a^3 C+3 a^2 b B-3 a b^2 C-b^3 B\right ) \cot (c+d x)}{d}-\left (x \left (a^3 C+3 a^2 b B-3 a b^2 C-b^3 B\right )\right )\right )+\frac {a \left (2 a^2 B-6 a b C-5 b^2 B\right ) \cot ^2(c+d x)}{2 d}-\frac {a^2 (2 a C+3 b B) \cot ^3(c+d x)}{3 d}\right )-\frac {a B \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (-2 \left (-\left (a^3 B-3 a^2 b C-3 a b^2 B+b^3 C\right ) \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx-\frac {\left (a^3 C+3 a^2 b B-3 a b^2 C-b^3 B\right ) \cot (c+d x)}{d}-\left (x \left (a^3 C+3 a^2 b B-3 a b^2 C-b^3 B\right )\right )\right )+\frac {a \left (2 a^2 B-6 a b C-5 b^2 B\right ) \cot ^2(c+d x)}{2 d}-\frac {a^2 (2 a C+3 b B) \cot ^3(c+d x)}{3 d}\right )-\frac {a B \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (-2 \left (\left (a^3 B-3 a^2 b C-3 a b^2 B+b^3 C\right ) \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx-\frac {\left (a^3 C+3 a^2 b B-3 a b^2 C-b^3 B\right ) \cot (c+d x)}{d}-\left (x \left (a^3 C+3 a^2 b B-3 a b^2 C-b^3 B\right )\right )\right )+\frac {a \left (2 a^2 B-6 a b C-5 b^2 B\right ) \cot ^2(c+d x)}{2 d}-\frac {a^2 (2 a C+3 b B) \cot ^3(c+d x)}{3 d}\right )-\frac {a B \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {1}{2} \left (\frac {a \left (2 a^2 B-6 a b C-5 b^2 B\right ) \cot ^2(c+d x)}{2 d}-\frac {a^2 (2 a C+3 b B) \cot ^3(c+d x)}{3 d}-2 \left (-\frac {\left (a^3 C+3 a^2 b B-3 a b^2 C-b^3 B\right ) \cot (c+d x)}{d}-\frac {\left (a^3 B-3 a^2 b C-3 a b^2 B+b^3 C\right ) \log (-\sin (c+d x))}{d}-\left (x \left (a^3 C+3 a^2 b B-3 a b^2 C-b^3 B\right )\right )\right )\right )-\frac {a B \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}\) |
((a*(2*a^2*B - 5*b^2*B - 6*a*b*C)*Cot[c + d*x]^2)/(2*d) - (a^2*(3*b*B + 2* a*C)*Cot[c + d*x]^3)/(3*d) - 2*(-((3*a^2*b*B - b^3*B + a^3*C - 3*a*b^2*C)* x) - ((3*a^2*b*B - b^3*B + a^3*C - 3*a*b^2*C)*Cot[c + d*x])/d - ((a^3*B - 3*a*b^2*B - 3*a^2*b*C + b^3*C)*Log[-Sin[c + d*x]])/d))/2 - (a*B*Cot[c + d* x]^4*(a + b*Tan[c + d*x])^2)/(4*d)
3.1.23.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/ (f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x] )^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a , b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1 ]
Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[(a*c + b*d)*(x/(a^2 + b^2)), x] + Simp[(b*c - a *d)/(a^2 + b^2) Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && N eQ[a*c + b*d, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[(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] - Simp[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] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)*(a^2 + b^2))), x ] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[b*B + a*(A - C) - (A*b - a*B - b*C)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B , C}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0 ]
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[1/b^2 Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*(b*B - a*C + b*C*Tan[e + f*x]), x], x] /; FreeQ [{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[A*b^2 - a*b*B + a^2*C, 0]
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] + Simp[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]
Time = 0.31 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.09
| method | result | size |
| parallelrisch | \(\frac {6 \left (-B \,a^{3}+3 B a \,b^{2}+3 C \,a^{2} b -C \,b^{3}\right ) \ln \left (\sec \left (d x +c \right )^{2}\right )+12 \left (B \,a^{3}-3 B a \,b^{2}-3 C \,a^{2} b +C \,b^{3}\right ) \ln \left (\tan \left (d x +c \right )\right )-3 B \,a^{3} \cot \left (d x +c \right )^{4}+4 \left (-3 B \,a^{2} b -C \,a^{3}\right ) \cot \left (d x +c \right )^{3}+6 a \cot \left (d x +c \right )^{2} \left (B \,a^{2}-3 B \,b^{2}-3 C a b \right )+12 \cot \left (d x +c \right ) \left (3 B \,a^{2} b -B \,b^{3}+C \,a^{3}-3 C a \,b^{2}\right )+36 d \left (B \,a^{2} b -\frac {1}{3} B \,b^{3}+\frac {1}{3} C \,a^{3}-C a \,b^{2}\right ) x}{12 d}\) | \(209\) |
| derivativedivides | \(\frac {-\frac {-3 B \,a^{2} b +B \,b^{3}-C \,a^{3}+3 C a \,b^{2}}{\tan \left (d x +c \right )}+\left (B \,a^{3}-3 B a \,b^{2}-3 C \,a^{2} b +C \,b^{3}\right ) \ln \left (\tan \left (d x +c \right )\right )-\frac {B \,a^{3}}{4 \tan \left (d x +c \right )^{4}}-\frac {a^{2} \left (3 B b +C a \right )}{3 \tan \left (d x +c \right )^{3}}+\frac {a \left (B \,a^{2}-3 B \,b^{2}-3 C a b \right )}{2 \tan \left (d x +c \right )^{2}}+\frac {\left (-B \,a^{3}+3 B a \,b^{2}+3 C \,a^{2} b -C \,b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (3 B \,a^{2} b -B \,b^{3}+C \,a^{3}-3 C a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\) | \(212\) |
| default | \(\frac {-\frac {-3 B \,a^{2} b +B \,b^{3}-C \,a^{3}+3 C a \,b^{2}}{\tan \left (d x +c \right )}+\left (B \,a^{3}-3 B a \,b^{2}-3 C \,a^{2} b +C \,b^{3}\right ) \ln \left (\tan \left (d x +c \right )\right )-\frac {B \,a^{3}}{4 \tan \left (d x +c \right )^{4}}-\frac {a^{2} \left (3 B b +C a \right )}{3 \tan \left (d x +c \right )^{3}}+\frac {a \left (B \,a^{2}-3 B \,b^{2}-3 C a b \right )}{2 \tan \left (d x +c \right )^{2}}+\frac {\left (-B \,a^{3}+3 B a \,b^{2}+3 C \,a^{2} b -C \,b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (3 B \,a^{2} b -B \,b^{3}+C \,a^{3}-3 C a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\) | \(212\) |
| norman | \(\frac {\frac {\left (3 B \,a^{2} b -B \,b^{3}+C \,a^{3}-3 C a \,b^{2}\right ) \tan \left (d x +c \right )^{4}}{d}+\left (3 B \,a^{2} b -B \,b^{3}+C \,a^{3}-3 C a \,b^{2}\right ) x \tan \left (d x +c \right )^{5}-\frac {B \,a^{3} \tan \left (d x +c \right )}{4 d}+\frac {a \left (B \,a^{2}-3 B \,b^{2}-3 C a b \right ) \tan \left (d x +c \right )^{3}}{2 d}-\frac {a^{2} \left (3 B b +C a \right ) \tan \left (d x +c \right )^{2}}{3 d}}{\tan \left (d x +c \right )^{5}}+\frac {\left (B \,a^{3}-3 B a \,b^{2}-3 C \,a^{2} b +C \,b^{3}\right ) \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {\left (B \,a^{3}-3 B a \,b^{2}-3 C \,a^{2} b +C \,b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d}\) | \(233\) |
| risch | \(\frac {6 i C \,a^{2} b c}{d}+\frac {6 i B a \,b^{2} c}{d}-i B \,a^{3} x +3 i C \,a^{2} b x +3 B \,a^{2} b x -B \,b^{3} x +C \,a^{3} x -3 C a \,b^{2} x -\frac {2 i C \,b^{3} c}{d}+3 i B a \,b^{2} x -\frac {2 i B \,a^{3} c}{d}-i C \,b^{3} x -\frac {2 i \left (12 B \,a^{2} b -9 C a \,b^{2}-3 B \,b^{3}+4 C \,a^{3}+6 i B \,a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-6 i B \,a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-6 i B \,a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+9 i B a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+9 i C \,a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}-18 i B a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-18 i C \,a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}+9 i B a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+9 i C \,a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}-9 B \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+3 B \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-6 C \,a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+12 C \,a^{3} {\mathrm e}^{4 i \left (d x +c \right )}+9 B \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-10 C \,a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+9 C a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+36 B \,a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}-27 C a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-30 B \,a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}+27 C a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-18 B \,a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {B \,a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}-\frac {3 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B \,b^{2}}{d}-\frac {3 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) C b}{d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) C \,b^{3}}{d}\) | \(577\) |
1/12*(6*(-B*a^3+3*B*a*b^2+3*C*a^2*b-C*b^3)*ln(sec(d*x+c)^2)+12*(B*a^3-3*B* a*b^2-3*C*a^2*b+C*b^3)*ln(tan(d*x+c))-3*B*a^3*cot(d*x+c)^4+4*(-3*B*a^2*b-C *a^3)*cot(d*x+c)^3+6*a*cot(d*x+c)^2*(B*a^2-3*B*b^2-3*C*a*b)+12*cot(d*x+c)* (3*B*a^2*b-B*b^3+C*a^3-3*C*a*b^2)+36*d*(B*a^2*b-1/3*B*b^3+1/3*C*a^3-C*a*b^ 2)*x)/d
Time = 0.26 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.18 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x))^3 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {6 \, {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2} + C b^{3}\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{4} + 3 \, {\left (3 \, B a^{3} - 6 \, C a^{2} b - 6 \, B a b^{2} + 4 \, {\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} d x\right )} \tan \left (d x + c\right )^{4} - 3 \, B a^{3} + 12 \, {\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} \tan \left (d x + c\right )^{3} + 6 \, {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2}\right )} \tan \left (d x + c\right )^{2} - 4 \, {\left (C a^{3} + 3 \, B a^{2} b\right )} \tan \left (d x + c\right )}{12 \, d \tan \left (d x + c\right )^{4}} \]
integrate(cot(d*x+c)^6*(a+b*tan(d*x+c))^3*(B*tan(d*x+c)+C*tan(d*x+c)^2),x, algorithm="fricas")
1/12*(6*(B*a^3 - 3*C*a^2*b - 3*B*a*b^2 + C*b^3)*log(tan(d*x + c)^2/(tan(d* x + c)^2 + 1))*tan(d*x + c)^4 + 3*(3*B*a^3 - 6*C*a^2*b - 6*B*a*b^2 + 4*(C* a^3 + 3*B*a^2*b - 3*C*a*b^2 - B*b^3)*d*x)*tan(d*x + c)^4 - 3*B*a^3 + 12*(C *a^3 + 3*B*a^2*b - 3*C*a*b^2 - B*b^3)*tan(d*x + c)^3 + 6*(B*a^3 - 3*C*a^2* b - 3*B*a*b^2)*tan(d*x + c)^2 - 4*(C*a^3 + 3*B*a^2*b)*tan(d*x + c))/(d*tan (d*x + c)^4)
Leaf count of result is larger than twice the leaf count of optimal. 391 vs. \(2 (187) = 374\).
Time = 5.46 (sec) , antiderivative size = 391, normalized size of antiderivative = 2.05 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x))^3 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\begin {cases} \text {NaN} & \text {for}\: c = 0 \wedge d = 0 \\x \left (a + b \tan {\left (c \right )}\right )^{3} \left (B \tan {\left (c \right )} + C \tan ^{2}{\left (c \right )}\right ) \cot ^{6}{\left (c \right )} & \text {for}\: d = 0 \\\text {NaN} & \text {for}\: c = - d x \\- \frac {B a^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {B a^{3} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + \frac {B a^{3}}{2 d \tan ^{2}{\left (c + d x \right )}} - \frac {B a^{3}}{4 d \tan ^{4}{\left (c + d x \right )}} + 3 B a^{2} b x + \frac {3 B a^{2} b}{d \tan {\left (c + d x \right )}} - \frac {B a^{2} b}{d \tan ^{3}{\left (c + d x \right )}} + \frac {3 B a b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {3 B a b^{2} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {3 B a b^{2}}{2 d \tan ^{2}{\left (c + d x \right )}} - B b^{3} x - \frac {B b^{3}}{d \tan {\left (c + d x \right )}} + C a^{3} x + \frac {C a^{3}}{d \tan {\left (c + d x \right )}} - \frac {C a^{3}}{3 d \tan ^{3}{\left (c + d x \right )}} + \frac {3 C a^{2} b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {3 C a^{2} b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {3 C a^{2} b}{2 d \tan ^{2}{\left (c + d x \right )}} - 3 C a b^{2} x - \frac {3 C a b^{2}}{d \tan {\left (c + d x \right )}} - \frac {C b^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {C b^{3} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} & \text {otherwise} \end {cases} \]
Piecewise((nan, Eq(c, 0) & Eq(d, 0)), (x*(a + b*tan(c))**3*(B*tan(c) + C*t an(c)**2)*cot(c)**6, Eq(d, 0)), (nan, Eq(c, -d*x)), (-B*a**3*log(tan(c + d *x)**2 + 1)/(2*d) + B*a**3*log(tan(c + d*x))/d + B*a**3/(2*d*tan(c + d*x)* *2) - B*a**3/(4*d*tan(c + d*x)**4) + 3*B*a**2*b*x + 3*B*a**2*b/(d*tan(c + d*x)) - B*a**2*b/(d*tan(c + d*x)**3) + 3*B*a*b**2*log(tan(c + d*x)**2 + 1) /(2*d) - 3*B*a*b**2*log(tan(c + d*x))/d - 3*B*a*b**2/(2*d*tan(c + d*x)**2) - B*b**3*x - B*b**3/(d*tan(c + d*x)) + C*a**3*x + C*a**3/(d*tan(c + d*x)) - C*a**3/(3*d*tan(c + d*x)**3) + 3*C*a**2*b*log(tan(c + d*x)**2 + 1)/(2*d ) - 3*C*a**2*b*log(tan(c + d*x))/d - 3*C*a**2*b/(2*d*tan(c + d*x)**2) - 3* C*a*b**2*x - 3*C*a*b**2/(d*tan(c + d*x)) - C*b**3*log(tan(c + d*x)**2 + 1) /(2*d) + C*b**3*log(tan(c + d*x))/d, True))
Time = 0.32 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.13 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x))^3 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {12 \, {\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} {\left (d x + c\right )} - 6 \, {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2} + C b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 12 \, {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2} + C b^{3}\right )} \log \left (\tan \left (d x + c\right )\right ) - \frac {3 \, B a^{3} - 12 \, {\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} \tan \left (d x + c\right )^{3} - 6 \, {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2}\right )} \tan \left (d x + c\right )^{2} + 4 \, {\left (C a^{3} + 3 \, B a^{2} b\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{4}}}{12 \, d} \]
integrate(cot(d*x+c)^6*(a+b*tan(d*x+c))^3*(B*tan(d*x+c)+C*tan(d*x+c)^2),x, algorithm="maxima")
1/12*(12*(C*a^3 + 3*B*a^2*b - 3*C*a*b^2 - B*b^3)*(d*x + c) - 6*(B*a^3 - 3* C*a^2*b - 3*B*a*b^2 + C*b^3)*log(tan(d*x + c)^2 + 1) + 12*(B*a^3 - 3*C*a^2 *b - 3*B*a*b^2 + C*b^3)*log(tan(d*x + c)) - (3*B*a^3 - 12*(C*a^3 + 3*B*a^2 *b - 3*C*a*b^2 - B*b^3)*tan(d*x + c)^3 - 6*(B*a^3 - 3*C*a^2*b - 3*B*a*b^2) *tan(d*x + c)^2 + 4*(C*a^3 + 3*B*a^2*b)*tan(d*x + c))/tan(d*x + c)^4)/d
Leaf count of result is larger than twice the leaf count of optimal. 528 vs. \(2 (185) = 370\).
Time = 1.48 (sec) , antiderivative size = 528, normalized size of antiderivative = 2.76 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x))^3 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=-\frac {3 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 72 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 72 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 120 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 360 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 288 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 96 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 192 \, {\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} {\left (d x + c\right )} + 192 \, {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2} + C b^{3}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) - 192 \, {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2} + C b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {400 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1200 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1200 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 400 \, C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 120 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 360 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 288 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 96 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 72 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 72 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, B a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \]
-1/192*(3*B*a^3*tan(1/2*d*x + 1/2*c)^4 - 8*C*a^3*tan(1/2*d*x + 1/2*c)^3 - 24*B*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 36*B*a^3*tan(1/2*d*x + 1/2*c)^2 + 72*C *a^2*b*tan(1/2*d*x + 1/2*c)^2 + 72*B*a*b^2*tan(1/2*d*x + 1/2*c)^2 + 120*C* a^3*tan(1/2*d*x + 1/2*c) + 360*B*a^2*b*tan(1/2*d*x + 1/2*c) - 288*C*a*b^2* tan(1/2*d*x + 1/2*c) - 96*B*b^3*tan(1/2*d*x + 1/2*c) - 192*(C*a^3 + 3*B*a^ 2*b - 3*C*a*b^2 - B*b^3)*(d*x + c) + 192*(B*a^3 - 3*C*a^2*b - 3*B*a*b^2 + C*b^3)*log(tan(1/2*d*x + 1/2*c)^2 + 1) - 192*(B*a^3 - 3*C*a^2*b - 3*B*a*b^ 2 + C*b^3)*log(abs(tan(1/2*d*x + 1/2*c))) + (400*B*a^3*tan(1/2*d*x + 1/2*c )^4 - 1200*C*a^2*b*tan(1/2*d*x + 1/2*c)^4 - 1200*B*a*b^2*tan(1/2*d*x + 1/2 *c)^4 + 400*C*b^3*tan(1/2*d*x + 1/2*c)^4 - 120*C*a^3*tan(1/2*d*x + 1/2*c)^ 3 - 360*B*a^2*b*tan(1/2*d*x + 1/2*c)^3 + 288*C*a*b^2*tan(1/2*d*x + 1/2*c)^ 3 + 96*B*b^3*tan(1/2*d*x + 1/2*c)^3 - 36*B*a^3*tan(1/2*d*x + 1/2*c)^2 + 72 *C*a^2*b*tan(1/2*d*x + 1/2*c)^2 + 72*B*a*b^2*tan(1/2*d*x + 1/2*c)^2 + 8*C* a^3*tan(1/2*d*x + 1/2*c) + 24*B*a^2*b*tan(1/2*d*x + 1/2*c) + 3*B*a^3)/tan( 1/2*d*x + 1/2*c)^4)/d
Time = 8.47 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.07 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x))^3 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (B\,a^3-3\,C\,a^2\,b-3\,B\,a\,b^2+C\,b^3\right )}{d}-\frac {{\mathrm {cot}\left (c+d\,x\right )}^4\,\left (\mathrm {tan}\left (c+d\,x\right )\,\left (\frac {C\,a^3}{3}+B\,b\,a^2\right )+\frac {B\,a^3}{4}+{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (-\frac {B\,a^3}{2}+\frac {3\,C\,a^2\,b}{2}+\frac {3\,B\,a\,b^2}{2}\right )+{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (-C\,a^3-3\,B\,a^2\,b+3\,C\,a\,b^2+B\,b^3\right )\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B-C\,1{}\mathrm {i}\right )\,{\left (b+a\,1{}\mathrm {i}\right )}^3\,1{}\mathrm {i}}{2\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (B+C\,1{}\mathrm {i}\right )\,{\left (-b+a\,1{}\mathrm {i}\right )}^3\,1{}\mathrm {i}}{2\,d} \]
(log(tan(c + d*x))*(B*a^3 + C*b^3 - 3*B*a*b^2 - 3*C*a^2*b))/d - (cot(c + d *x)^4*(tan(c + d*x)*((C*a^3)/3 + B*a^2*b) + (B*a^3)/4 + tan(c + d*x)^2*((3 *B*a*b^2)/2 - (B*a^3)/2 + (3*C*a^2*b)/2) + tan(c + d*x)^3*(B*b^3 - C*a^3 - 3*B*a^2*b + 3*C*a*b^2)))/d - (log(tan(c + d*x) + 1i)*(B - C*1i)*(a*1i + b )^3*1i)/(2*d) - (log(tan(c + d*x) - 1i)*(B + C*1i)*(a*1i - b)^3*1i)/(2*d)