Integrand size = 34, antiderivative size = 200 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-8 a^4 (A-i B) x-\frac {8 a^4 (A-i B) \cot (c+d x)}{d}+\frac {a^4 (148 i A+145 B) \cot ^2(c+d x)}{60 d}+\frac {8 a^4 (i A+B) \log (\sin (c+d x))}{d}-\frac {a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}-\frac {(8 i A+5 B) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{20 d}+\frac {(28 A-25 i B) \cot ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{30 d} \]
-8*a^4*(A-I*B)*x-8*a^4*(A-I*B)*cot(d*x+c)/d+1/60*a^4*(148*I*A+145*B)*cot(d *x+c)^2/d+8*a^4*(I*A+B)*ln(sin(d*x+c))/d-1/5*a*A*cot(d*x+c)^5*(a+I*a*tan(d *x+c))^3/d-1/20*(8*I*A+5*B)*cot(d*x+c)^4*(a^2+I*a^2*tan(d*x+c))^2/d+1/30*( 28*A-25*I*B)*cot(d*x+c)^3*(a^4+I*a^4*tan(d*x+c))/d
Time = 1.65 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.60 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {a^4 \left (-3 i (4 A-5 i B) (i+\cot (c+d x))^4-12 A \cot (c+d x) (i+\cot (c+d x))^4+20 (A-i B) \left (-21 \cot (c+d x)+6 i \cot ^2(c+d x)+\cot ^3(c+d x)+24 i (\log (\tan (c+d x))-\log (i+\tan (c+d x)))\right )\right )}{60 d} \]
(a^4*((-3*I)*(4*A - (5*I)*B)*(I + Cot[c + d*x])^4 - 12*A*Cot[c + d*x]*(I + Cot[c + d*x])^4 + 20*(A - I*B)*(-21*Cot[c + d*x] + (6*I)*Cot[c + d*x]^2 + Cot[c + d*x]^3 + (24*I)*(Log[Tan[c + d*x]] - Log[I + Tan[c + d*x]]))))/(6 0*d)
Time = 1.46 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.10, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 4076, 3042, 4076, 27, 3042, 4076, 3042, 4074, 27, 3042, 4012, 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.
| \(\displaystyle \int \cot ^6(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+i a \tan (c+d x))^4 (A+B \tan (c+d x))}{\tan (c+d x)^6}dx\) |
| \(\Big \downarrow \) 4076 |
| \(\displaystyle \frac {1}{5} \int \cot ^5(c+d x) (i \tan (c+d x) a+a)^3 (a (8 i A+5 B)-a (2 A-5 i B) \tan (c+d x))dx-\frac {a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \int \frac {(i \tan (c+d x) a+a)^3 (a (8 i A+5 B)-a (2 A-5 i B) \tan (c+d x))}{\tan (c+d x)^5}dx-\frac {a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}\) |
| \(\Big \downarrow \) 4076 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \int -2 \cot ^4(c+d x) (i \tan (c+d x) a+a)^2 \left ((28 A-25 i B) a^2+3 (4 i A+5 B) \tan (c+d x) a^2\right )dx-\frac {(5 B+8 i A) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{4 d}\right )-\frac {a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (-\frac {1}{2} \int \cot ^4(c+d x) (i \tan (c+d x) a+a)^2 \left ((28 A-25 i B) a^2+3 (4 i A+5 B) \tan (c+d x) a^2\right )dx-\frac {(5 B+8 i A) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{4 d}\right )-\frac {a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (-\frac {1}{2} \int \frac {(i \tan (c+d x) a+a)^2 \left ((28 A-25 i B) a^2+3 (4 i A+5 B) \tan (c+d x) a^2\right )}{\tan (c+d x)^4}dx-\frac {(5 B+8 i A) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{4 d}\right )-\frac {a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}\) |
| \(\Big \downarrow \) 4076 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{2} \left (\frac {(28 A-25 i B) \cot ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{3 d}-\frac {1}{3} \int \cot ^3(c+d x) (i \tan (c+d x) a+a) \left (a^3 (148 i A+145 B)-a^3 (92 A-95 i B) \tan (c+d x)\right )dx\right )-\frac {(5 B+8 i A) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{4 d}\right )-\frac {a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{2} \left (\frac {(28 A-25 i B) \cot ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{3 d}-\frac {1}{3} \int \frac {(i \tan (c+d x) a+a) \left (a^3 (148 i A+145 B)-a^3 (92 A-95 i B) \tan (c+d x)\right )}{\tan (c+d x)^3}dx\right )-\frac {(5 B+8 i A) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{4 d}\right )-\frac {a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}\) |
| \(\Big \downarrow \) 4074 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{2} \left (\frac {1}{3} \left (\frac {a^4 (145 B+148 i A) \cot ^2(c+d x)}{2 d}-\int -240 \cot ^2(c+d x) \left ((A-i B) a^4+(i A+B) \tan (c+d x) a^4\right )dx\right )+\frac {(28 A-25 i B) \cot ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{3 d}\right )-\frac {(5 B+8 i A) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{4 d}\right )-\frac {a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{2} \left (\frac {1}{3} \left (240 \int \cot ^2(c+d x) \left ((A-i B) a^4+(i A+B) \tan (c+d x) a^4\right )dx+\frac {a^4 (145 B+148 i A) \cot ^2(c+d x)}{2 d}\right )+\frac {(28 A-25 i B) \cot ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{3 d}\right )-\frac {(5 B+8 i A) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{4 d}\right )-\frac {a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{2} \left (\frac {1}{3} \left (240 \int \frac {(A-i B) a^4+(i A+B) \tan (c+d x) a^4}{\tan (c+d x)^2}dx+\frac {a^4 (145 B+148 i A) \cot ^2(c+d x)}{2 d}\right )+\frac {(28 A-25 i B) \cot ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{3 d}\right )-\frac {(5 B+8 i A) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{4 d}\right )-\frac {a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{2} \left (\frac {1}{3} \left (240 \left (\int \cot (c+d x) \left (a^4 (i A+B)-a^4 (A-i B) \tan (c+d x)\right )dx-\frac {a^4 (A-i B) \cot (c+d x)}{d}\right )+\frac {a^4 (145 B+148 i A) \cot ^2(c+d x)}{2 d}\right )+\frac {(28 A-25 i B) \cot ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{3 d}\right )-\frac {(5 B+8 i A) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{4 d}\right )-\frac {a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{2} \left (\frac {1}{3} \left (240 \left (\int \frac {a^4 (i A+B)-a^4 (A-i B) \tan (c+d x)}{\tan (c+d x)}dx-\frac {a^4 (A-i B) \cot (c+d x)}{d}\right )+\frac {a^4 (145 B+148 i A) \cot ^2(c+d x)}{2 d}\right )+\frac {(28 A-25 i B) \cot ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{3 d}\right )-\frac {(5 B+8 i A) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{4 d}\right )-\frac {a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}\) |
| \(\Big \downarrow \) 4014 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{2} \left (\frac {1}{3} \left (240 \left (a^4 (B+i A) \int \cot (c+d x)dx-\frac {a^4 (A-i B) \cot (c+d x)}{d}-\left (a^4 x (A-i B)\right )\right )+\frac {a^4 (145 B+148 i A) \cot ^2(c+d x)}{2 d}\right )+\frac {(28 A-25 i B) \cot ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{3 d}\right )-\frac {(5 B+8 i A) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{4 d}\right )-\frac {a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{2} \left (\frac {1}{3} \left (240 \left (a^4 (B+i A) \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx-\frac {a^4 (A-i B) \cot (c+d x)}{d}-\left (a^4 x (A-i B)\right )\right )+\frac {a^4 (145 B+148 i A) \cot ^2(c+d x)}{2 d}\right )+\frac {(28 A-25 i B) \cot ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{3 d}\right )-\frac {(5 B+8 i A) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{4 d}\right )-\frac {a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{2} \left (\frac {1}{3} \left (240 \left (-a^4 (B+i A) \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx-\frac {a^4 (A-i B) \cot (c+d x)}{d}-\left (a^4 x (A-i B)\right )\right )+\frac {a^4 (145 B+148 i A) \cot ^2(c+d x)}{2 d}\right )+\frac {(28 A-25 i B) \cot ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{3 d}\right )-\frac {(5 B+8 i A) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{4 d}\right )-\frac {a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{2} \left (\frac {(28 A-25 i B) \cot ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{3 d}+\frac {1}{3} \left (\frac {a^4 (145 B+148 i A) \cot ^2(c+d x)}{2 d}+240 \left (-\frac {a^4 (A-i B) \cot (c+d x)}{d}+\frac {a^4 (B+i A) \log (-\sin (c+d x))}{d}-\left (a^4 x (A-i B)\right )\right )\right )\right )-\frac {(5 B+8 i A) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{4 d}\right )-\frac {a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}\) |
-1/5*(a*A*Cot[c + d*x]^5*(a + I*a*Tan[c + d*x])^3)/d + (-1/4*(((8*I)*A + 5 *B)*Cot[c + d*x]^4*(a^2 + I*a^2*Tan[c + d*x])^2)/d + (((a^4*((148*I)*A + 1 45*B)*Cot[c + d*x]^2)/(2*d) + 240*(-(a^4*(A - I*B)*x) - (a^4*(A - I*B)*Cot [c + d*x])/d + (a^4*(I*A + B)*Log[-Sin[c + d*x]])/d))/3 + ((28*A - (25*I)* B)*Cot[c + d*x]^3*(a^4 + I*a^4*Tan[c + d*x]))/(3*d))/2)/5
3.1.34.3.1 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_)]), x_Symbol] :> Simp[(b *c - a*d)*(A*b - a*B)*((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[a*A*c + b*B*c + A*b*d - a*B*d - (A*b*c - a*B*c - a*A*d - b*B*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && LtQ[m , -1] && NeQ[a^2 + b^2, 0]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(-a^2)*(B*c - A*d)*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(b*c + a*d)*(n + 1))), x] - Simp[a/(d*(b*c + a*d)*(n + 1)) Int[ (a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*b*d*(m - n - 2) - B*(b*c*(m - 1) + a*d*(n + 1)) + (a*A*d*(m + n) - B*(a*c*(m - 1) + b *d*(n + 1)))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && GtQ[m, 1] && LtQ[n, -1]
Time = 0.27 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.65
| method | result | size |
| parallelrisch | \(\frac {8 a^{4} \left (\left (-\frac {i A}{2}-\frac {B}{2}\right ) \ln \left (\sec ^{2}\left (d x +c \right )\right )+\left (i A +B \right ) \ln \left (\tan \left (d x +c \right )\right )-\frac {A \left (\cot ^{5}\left (d x +c \right )\right )}{40}+\left (\cot ^{4}\left (d x +c \right )\right ) \left (-\frac {i A}{8}-\frac {B}{32}\right )+\left (\cot ^{3}\left (d x +c \right )\right ) \left (-\frac {i B}{6}+\frac {7 A}{24}\right )+\left (\cot ^{2}\left (d x +c \right )\right ) \left (\frac {i A}{2}+\frac {7 B}{16}\right )+\left (i B -A \right ) \cot \left (d x +c \right )+x d \left (i B -A \right )\right )}{d}\) | \(130\) |
| derivativedivides | \(\frac {a^{4} \left (-i A \left (\cot ^{4}\left (d x +c \right )\right )-\frac {A \left (\cot ^{5}\left (d x +c \right )\right )}{5}-\frac {4 i B \left (\cot ^{3}\left (d x +c \right )\right )}{3}-\frac {B \left (\cot ^{4}\left (d x +c \right )\right )}{4}+4 i A \left (\cot ^{2}\left (d x +c \right )\right )+\frac {7 A \left (\cot ^{3}\left (d x +c \right )\right )}{3}+8 i B \cot \left (d x +c \right )+\frac {7 B \left (\cot ^{2}\left (d x +c \right )\right )}{2}-8 A \cot \left (d x +c \right )+\frac {\left (-8 i A -8 B \right ) \ln \left (\cot ^{2}\left (d x +c \right )+1\right )}{2}+\left (-8 i B +8 A \right ) \left (\frac {\pi }{2}-\operatorname {arccot}\left (\cot \left (d x +c \right )\right )\right )\right )}{d}\) | \(151\) |
| default | \(\frac {a^{4} \left (-i A \left (\cot ^{4}\left (d x +c \right )\right )-\frac {A \left (\cot ^{5}\left (d x +c \right )\right )}{5}-\frac {4 i B \left (\cot ^{3}\left (d x +c \right )\right )}{3}-\frac {B \left (\cot ^{4}\left (d x +c \right )\right )}{4}+4 i A \left (\cot ^{2}\left (d x +c \right )\right )+\frac {7 A \left (\cot ^{3}\left (d x +c \right )\right )}{3}+8 i B \cot \left (d x +c \right )+\frac {7 B \left (\cot ^{2}\left (d x +c \right )\right )}{2}-8 A \cot \left (d x +c \right )+\frac {\left (-8 i A -8 B \right ) \ln \left (\cot ^{2}\left (d x +c \right )+1\right )}{2}+\left (-8 i B +8 A \right ) \left (\frac {\pi }{2}-\operatorname {arccot}\left (\cot \left (d x +c \right )\right )\right )\right )}{d}\) | \(151\) |
| risch | \(-\frac {16 i a^{4} B c}{d}+\frac {16 a^{4} A c}{d}-\frac {4 a^{4} \left (210 i A \,{\mathrm e}^{8 i \left (d x +c \right )}+150 B \,{\mathrm e}^{8 i \left (d x +c \right )}-555 i A \,{\mathrm e}^{6 i \left (d x +c \right )}-465 B \,{\mathrm e}^{6 i \left (d x +c \right )}+655 i A \,{\mathrm e}^{4 i \left (d x +c \right )}+565 B \,{\mathrm e}^{4 i \left (d x +c \right )}-365 i A \,{\mathrm e}^{2 i \left (d x +c \right )}-320 B \,{\mathrm e}^{2 i \left (d x +c \right )}+79 i A +70 B \right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}+\frac {8 a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B}{d}+\frac {8 i a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) A}{d}\) | \(195\) |
| norman | \(\frac {\left (8 i B \,a^{4}-8 A \,a^{4}\right ) x \left (\tan ^{5}\left (d x +c \right )\right )-\frac {A \,a^{4}}{5 d}+\frac {\left (-4 i B \,a^{4}+7 A \,a^{4}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{3 d}-\frac {\left (4 i A \,a^{4}+B \,a^{4}\right ) \tan \left (d x +c \right )}{4 d}-\frac {8 \left (-i B \,a^{4}+A \,a^{4}\right ) \left (\tan ^{4}\left (d x +c \right )\right )}{d}+\frac {\left (8 i A \,a^{4}+7 B \,a^{4}\right ) \left (\tan ^{3}\left (d x +c \right )\right )}{2 d}}{\tan \left (d x +c \right )^{5}}+\frac {8 \left (i A \,a^{4}+B \,a^{4}\right ) \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {4 \left (i A \,a^{4}+B \,a^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}\) | \(203\) |
8*a^4*((-1/2*I*A-1/2*B)*ln(sec(d*x+c)^2)+(I*A+B)*ln(tan(d*x+c))-1/40*A*cot (d*x+c)^5+cot(d*x+c)^4*(-1/8*I*A-1/32*B)+cot(d*x+c)^3*(-1/6*I*B+7/24*A)+co t(d*x+c)^2*(1/2*I*A+7/16*B)+(-A+I*B)*cot(d*x+c)+x*d*(-A+I*B))/d
Time = 0.24 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.44 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\frac {4 \, {\left (30 \, {\left (7 i \, A + 5 \, B\right )} a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 15 \, {\left (-37 i \, A - 31 \, B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 5 \, {\left (131 i \, A + 113 \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, {\left (-73 i \, A - 64 \, B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (79 i \, A + 70 \, B\right )} a^{4} + 30 \, {\left ({\left (-i \, A - B\right )} a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, {\left (i \, A + B\right )} a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, {\left (-i \, A - B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, {\left (i \, A + B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, {\left (-i \, A - B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (i \, A + B\right )} a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )\right )}}{15 \, {\left (d e^{\left (10 i \, d x + 10 i \, c\right )} - 5 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, d e^{\left (6 i \, d x + 6 i \, c\right )} - 10 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )}} \]
-4/15*(30*(7*I*A + 5*B)*a^4*e^(8*I*d*x + 8*I*c) + 15*(-37*I*A - 31*B)*a^4* e^(6*I*d*x + 6*I*c) + 5*(131*I*A + 113*B)*a^4*e^(4*I*d*x + 4*I*c) + 5*(-73 *I*A - 64*B)*a^4*e^(2*I*d*x + 2*I*c) + (79*I*A + 70*B)*a^4 + 30*((-I*A - B )*a^4*e^(10*I*d*x + 10*I*c) + 5*(I*A + B)*a^4*e^(8*I*d*x + 8*I*c) + 10*(-I *A - B)*a^4*e^(6*I*d*x + 6*I*c) + 10*(I*A + B)*a^4*e^(4*I*d*x + 4*I*c) + 5 *(-I*A - B)*a^4*e^(2*I*d*x + 2*I*c) + (I*A + B)*a^4)*log(e^(2*I*d*x + 2*I* c) - 1))/(d*e^(10*I*d*x + 10*I*c) - 5*d*e^(8*I*d*x + 8*I*c) + 10*d*e^(6*I* d*x + 6*I*c) - 10*d*e^(4*I*d*x + 4*I*c) + 5*d*e^(2*I*d*x + 2*I*c) - d)
Time = 2.52 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.48 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {8 i a^{4} \left (A - i B\right ) \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac {- 316 i A a^{4} - 280 B a^{4} + \left (1460 i A a^{4} e^{2 i c} + 1280 B a^{4} e^{2 i c}\right ) e^{2 i d x} + \left (- 2620 i A a^{4} e^{4 i c} - 2260 B a^{4} e^{4 i c}\right ) e^{4 i d x} + \left (2220 i A a^{4} e^{6 i c} + 1860 B a^{4} e^{6 i c}\right ) e^{6 i d x} + \left (- 840 i A a^{4} e^{8 i c} - 600 B a^{4} e^{8 i c}\right ) e^{8 i d x}}{15 d e^{10 i c} e^{10 i d x} - 75 d e^{8 i c} e^{8 i d x} + 150 d e^{6 i c} e^{6 i d x} - 150 d e^{4 i c} e^{4 i d x} + 75 d e^{2 i c} e^{2 i d x} - 15 d} \]
8*I*a**4*(A - I*B)*log(exp(2*I*d*x) - exp(-2*I*c))/d + (-316*I*A*a**4 - 28 0*B*a**4 + (1460*I*A*a**4*exp(2*I*c) + 1280*B*a**4*exp(2*I*c))*exp(2*I*d*x ) + (-2620*I*A*a**4*exp(4*I*c) - 2260*B*a**4*exp(4*I*c))*exp(4*I*d*x) + (2 220*I*A*a**4*exp(6*I*c) + 1860*B*a**4*exp(6*I*c))*exp(6*I*d*x) + (-840*I*A *a**4*exp(8*I*c) - 600*B*a**4*exp(8*I*c))*exp(8*I*d*x))/(15*d*exp(10*I*c)* exp(10*I*d*x) - 75*d*exp(8*I*c)*exp(8*I*d*x) + 150*d*exp(6*I*c)*exp(6*I*d* x) - 150*d*exp(4*I*c)*exp(4*I*d*x) + 75*d*exp(2*I*c)*exp(2*I*d*x) - 15*d)
Time = 0.29 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.76 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\frac {480 \, {\left (d x + c\right )} {\left (A - i \, B\right )} a^{4} + 240 \, {\left (i \, A + B\right )} a^{4} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 480 \, {\left (-i \, A - B\right )} a^{4} \log \left (\tan \left (d x + c\right )\right ) + \frac {480 \, {\left (A - i \, B\right )} a^{4} \tan \left (d x + c\right )^{4} - 30 \, {\left (8 i \, A + 7 \, B\right )} a^{4} \tan \left (d x + c\right )^{3} - 20 \, {\left (7 \, A - 4 i \, B\right )} a^{4} \tan \left (d x + c\right )^{2} - 15 \, {\left (-4 i \, A - B\right )} a^{4} \tan \left (d x + c\right ) + 12 \, A a^{4}}{\tan \left (d x + c\right )^{5}}}{60 \, d} \]
-1/60*(480*(d*x + c)*(A - I*B)*a^4 + 240*(I*A + B)*a^4*log(tan(d*x + c)^2 + 1) + 480*(-I*A - B)*a^4*log(tan(d*x + c)) + (480*(A - I*B)*a^4*tan(d*x + c)^4 - 30*(8*I*A + 7*B)*a^4*tan(d*x + c)^3 - 20*(7*A - 4*I*B)*a^4*tan(d*x + c)^2 - 15*(-4*I*A - B)*a^4*tan(d*x + c) + 12*A*a^4)/tan(d*x + c)^5)/d
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 391 vs. \(2 (174) = 348\).
Time = 1.22 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.96 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {6 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 60 i \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 15 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 310 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 160 i \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1200 i \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 900 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 4740 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4320 i \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 15360 \, {\left (i \, A a^{4} + B a^{4}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right ) - 7680 \, {\left (-i \, A a^{4} - B a^{4}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + \frac {-17536 i \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 17536 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 4740 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 4320 i \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1200 i \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 900 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 310 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 160 i \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 60 i \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 15 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, A a^{4}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{960 \, d} \]
1/960*(6*A*a^4*tan(1/2*d*x + 1/2*c)^5 - 60*I*A*a^4*tan(1/2*d*x + 1/2*c)^4 - 15*B*a^4*tan(1/2*d*x + 1/2*c)^4 - 310*A*a^4*tan(1/2*d*x + 1/2*c)^3 + 160 *I*B*a^4*tan(1/2*d*x + 1/2*c)^3 + 1200*I*A*a^4*tan(1/2*d*x + 1/2*c)^2 + 90 0*B*a^4*tan(1/2*d*x + 1/2*c)^2 + 4740*A*a^4*tan(1/2*d*x + 1/2*c) - 4320*I* B*a^4*tan(1/2*d*x + 1/2*c) - 15360*(I*A*a^4 + B*a^4)*log(tan(1/2*d*x + 1/2 *c) + I) - 7680*(-I*A*a^4 - B*a^4)*log(tan(1/2*d*x + 1/2*c)) + (-17536*I*A *a^4*tan(1/2*d*x + 1/2*c)^5 - 17536*B*a^4*tan(1/2*d*x + 1/2*c)^5 - 4740*A* a^4*tan(1/2*d*x + 1/2*c)^4 + 4320*I*B*a^4*tan(1/2*d*x + 1/2*c)^4 + 1200*I* A*a^4*tan(1/2*d*x + 1/2*c)^3 + 900*B*a^4*tan(1/2*d*x + 1/2*c)^3 + 310*A*a^ 4*tan(1/2*d*x + 1/2*c)^2 - 160*I*B*a^4*tan(1/2*d*x + 1/2*c)^2 - 60*I*A*a^4 *tan(1/2*d*x + 1/2*c) - 15*B*a^4*tan(1/2*d*x + 1/2*c) - 6*A*a^4)/tan(1/2*d *x + 1/2*c)^5)/d
Time = 7.94 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.70 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\frac {\frac {A\,a^4}{5}-{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {7\,A\,a^4}{3}-\frac {B\,a^4\,4{}\mathrm {i}}{3}\right )+{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (8\,A\,a^4-B\,a^4\,8{}\mathrm {i}\right )-{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (\frac {7\,B\,a^4}{2}+A\,a^4\,4{}\mathrm {i}\right )+\mathrm {tan}\left (c+d\,x\right )\,\left (\frac {B\,a^4}{4}+A\,a^4\,1{}\mathrm {i}\right )}{d\,{\mathrm {tan}\left (c+d\,x\right )}^5}+\frac {a^4\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B+A\,1{}\mathrm {i}\right )\,16{}\mathrm {i}}{d} \]