Integrand size = 24, antiderivative size = 162 \[ \int \cot ^7(c+d x) (a+i a \tan (c+d x))^4 \, dx=-8 i a^4 x-\frac {8 i a^4 \cot (c+d x)}{d}-\frac {4 a^4 \cot ^2(c+d x)}{d}+\frac {8 i a^4 \cot ^3(c+d x)}{3 d}+\frac {67 a^4 \cot ^4(c+d x)}{60 d}-\frac {8 a^4 \log (\sin (c+d x))}{d}-\frac {\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}-\frac {7 i \cot ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{15 d} \]
-8*I*a^4*x-8*I*a^4*cot(d*x+c)/d-4*a^4*cot(d*x+c)^2/d+8/3*I*a^4*cot(d*x+c)^ 3/d+67/60*a^4*cot(d*x+c)^4/d-8*a^4*ln(sin(d*x+c))/d-1/6*cot(d*x+c)^6*(a^2+ I*a^2*tan(d*x+c))^2/d-7/15*I*cot(d*x+c)^5*(a^4+I*a^4*tan(d*x+c))/d
Time = 0.34 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.76 \[ \int \cot ^7(c+d x) (a+i a \tan (c+d x))^4 \, dx=a^4 \left (-\frac {8 i \cot (c+d x)}{d}-\frac {4 \cot ^2(c+d x)}{d}+\frac {8 i \cot ^3(c+d x)}{3 d}+\frac {7 \cot ^4(c+d x)}{4 d}-\frac {4 i \cot ^5(c+d x)}{5 d}-\frac {\cot ^6(c+d x)}{6 d}-\frac {8 \log (\tan (c+d x))}{d}+\frac {8 \log (i+\tan (c+d x))}{d}\right ) \]
a^4*(((-8*I)*Cot[c + d*x])/d - (4*Cot[c + d*x]^2)/d + (((8*I)/3)*Cot[c + d *x]^3)/d + (7*Cot[c + d*x]^4)/(4*d) - (((4*I)/5)*Cot[c + d*x]^5)/d - Cot[c + d*x]^6/(6*d) - (8*Log[Tan[c + d*x]])/d + (8*Log[I + Tan[c + d*x]])/d)
Time = 1.25 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.10, number of steps used = 22, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {3042, 4036, 27, 3042, 4076, 25, 3042, 4074, 27, 3042, 4012, 25, 3042, 4012, 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.
| \(\displaystyle \int \cot ^7(c+d x) (a+i a \tan (c+d x))^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+i a \tan (c+d x))^4}{\tan (c+d x)^7}dx\) |
| \(\Big \downarrow \) 4036 |
| \(\displaystyle -\frac {1}{6} \int -2 \cot ^6(c+d x) (i \tan (c+d x) a+a)^2 \left (7 i a^2-5 a^2 \tan (c+d x)\right )dx-\frac {\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \int \cot ^6(c+d x) (i \tan (c+d x) a+a)^2 \left (7 i a^2-5 a^2 \tan (c+d x)\right )dx-\frac {\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \int \frac {(i \tan (c+d x) a+a)^2 \left (7 i a^2-5 a^2 \tan (c+d x)\right )}{\tan (c+d x)^6}dx-\frac {\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}\) |
| \(\Big \downarrow \) 4076 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{5} \int -\cot ^5(c+d x) (i \tan (c+d x) a+a) \left (53 i \tan (c+d x) a^3+67 a^3\right )dx-\frac {7 i \cot ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{5 d}\right )-\frac {\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{3} \left (-\frac {1}{5} \int \cot ^5(c+d x) (i \tan (c+d x) a+a) \left (53 i \tan (c+d x) a^3+67 a^3\right )dx-\frac {7 i \cot ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{5 d}\right )-\frac {\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (-\frac {1}{5} \int \frac {(i \tan (c+d x) a+a) \left (53 i \tan (c+d x) a^3+67 a^3\right )}{\tan (c+d x)^5}dx-\frac {7 i \cot ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{5 d}\right )-\frac {\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}\) |
| \(\Big \downarrow \) 4074 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{5} \left (\frac {67 a^4 \cot ^4(c+d x)}{4 d}-\int 120 \cot ^4(c+d x) \left (i a^4-a^4 \tan (c+d x)\right )dx\right )-\frac {7 i \cot ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{5 d}\right )-\frac {\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{5} \left (\frac {67 a^4 \cot ^4(c+d x)}{4 d}-120 \int \cot ^4(c+d x) \left (i a^4-a^4 \tan (c+d x)\right )dx\right )-\frac {7 i \cot ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{5 d}\right )-\frac {\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{5} \left (\frac {67 a^4 \cot ^4(c+d x)}{4 d}-120 \int \frac {i a^4-a^4 \tan (c+d x)}{\tan (c+d x)^4}dx\right )-\frac {7 i \cot ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{5 d}\right )-\frac {\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{5} \left (\frac {67 a^4 \cot ^4(c+d x)}{4 d}-120 \left (\int -\cot ^3(c+d x) \left (i \tan (c+d x) a^4+a^4\right )dx-\frac {i a^4 \cot ^3(c+d x)}{3 d}\right )\right )-\frac {7 i \cot ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{5 d}\right )-\frac {\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{5} \left (\frac {67 a^4 \cot ^4(c+d x)}{4 d}-120 \left (-\int \cot ^3(c+d x) \left (i \tan (c+d x) a^4+a^4\right )dx-\frac {i a^4 \cot ^3(c+d x)}{3 d}\right )\right )-\frac {7 i \cot ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{5 d}\right )-\frac {\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{5} \left (\frac {67 a^4 \cot ^4(c+d x)}{4 d}-120 \left (-\int \frac {i \tan (c+d x) a^4+a^4}{\tan (c+d x)^3}dx-\frac {i a^4 \cot ^3(c+d x)}{3 d}\right )\right )-\frac {7 i \cot ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{5 d}\right )-\frac {\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{5} \left (\frac {67 a^4 \cot ^4(c+d x)}{4 d}-120 \left (-\int \cot ^2(c+d x) \left (i a^4-a^4 \tan (c+d x)\right )dx-\frac {i a^4 \cot ^3(c+d x)}{3 d}+\frac {a^4 \cot ^2(c+d x)}{2 d}\right )\right )-\frac {7 i \cot ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{5 d}\right )-\frac {\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{5} \left (\frac {67 a^4 \cot ^4(c+d x)}{4 d}-120 \left (-\int \frac {i a^4-a^4 \tan (c+d x)}{\tan (c+d x)^2}dx-\frac {i a^4 \cot ^3(c+d x)}{3 d}+\frac {a^4 \cot ^2(c+d x)}{2 d}\right )\right )-\frac {7 i \cot ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{5 d}\right )-\frac {\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{5} \left (\frac {67 a^4 \cot ^4(c+d x)}{4 d}-120 \left (-\int -\cot (c+d x) \left (i \tan (c+d x) a^4+a^4\right )dx-\frac {i a^4 \cot ^3(c+d x)}{3 d}+\frac {a^4 \cot ^2(c+d x)}{2 d}+\frac {i a^4 \cot (c+d x)}{d}\right )\right )-\frac {7 i \cot ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{5 d}\right )-\frac {\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{5} \left (\frac {67 a^4 \cot ^4(c+d x)}{4 d}-120 \left (\int \cot (c+d x) \left (i \tan (c+d x) a^4+a^4\right )dx-\frac {i a^4 \cot ^3(c+d x)}{3 d}+\frac {a^4 \cot ^2(c+d x)}{2 d}+\frac {i a^4 \cot (c+d x)}{d}\right )\right )-\frac {7 i \cot ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{5 d}\right )-\frac {\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{5} \left (\frac {67 a^4 \cot ^4(c+d x)}{4 d}-120 \left (\int \frac {i \tan (c+d x) a^4+a^4}{\tan (c+d x)}dx-\frac {i a^4 \cot ^3(c+d x)}{3 d}+\frac {a^4 \cot ^2(c+d x)}{2 d}+\frac {i a^4 \cot (c+d x)}{d}\right )\right )-\frac {7 i \cot ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{5 d}\right )-\frac {\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}\) |
| \(\Big \downarrow \) 4014 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{5} \left (\frac {67 a^4 \cot ^4(c+d x)}{4 d}-120 \left (a^4 \int \cot (c+d x)dx-\frac {i a^4 \cot ^3(c+d x)}{3 d}+\frac {a^4 \cot ^2(c+d x)}{2 d}+\frac {i a^4 \cot (c+d x)}{d}+i a^4 x\right )\right )-\frac {7 i \cot ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{5 d}\right )-\frac {\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{5} \left (\frac {67 a^4 \cot ^4(c+d x)}{4 d}-120 \left (a^4 \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx-\frac {i a^4 \cot ^3(c+d x)}{3 d}+\frac {a^4 \cot ^2(c+d x)}{2 d}+\frac {i a^4 \cot (c+d x)}{d}+i a^4 x\right )\right )-\frac {7 i \cot ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{5 d}\right )-\frac {\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{5} \left (\frac {67 a^4 \cot ^4(c+d x)}{4 d}-120 \left (-a^4 \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx-\frac {i a^4 \cot ^3(c+d x)}{3 d}+\frac {a^4 \cot ^2(c+d x)}{2 d}+\frac {i a^4 \cot (c+d x)}{d}+i a^4 x\right )\right )-\frac {7 i \cot ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{5 d}\right )-\frac {\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{5} \left (\frac {67 a^4 \cot ^4(c+d x)}{4 d}-120 \left (-\frac {i a^4 \cot ^3(c+d x)}{3 d}+\frac {a^4 \cot ^2(c+d x)}{2 d}+\frac {i a^4 \cot (c+d x)}{d}+\frac {a^4 \log (-\sin (c+d x))}{d}+i a^4 x\right )\right )-\frac {7 i \cot ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{5 d}\right )-\frac {\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}\) |
-1/6*(Cot[c + d*x]^6*(a^2 + I*a^2*Tan[c + d*x])^2)/d + (((67*a^4*Cot[c + d *x]^4)/(4*d) - 120*(I*a^4*x + (I*a^4*Cot[c + d*x])/d + (a^4*Cot[c + d*x]^2 )/(2*d) - ((I/3)*a^4*Cot[c + d*x]^3)/d + (a^4*Log[-Sin[c + d*x]])/d))/5 - (((7*I)/5)*Cot[c + d*x]^5*(a^4 + I*a^4*Tan[c + d*x]))/d)/3
3.1.44.3.1 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_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-a^2)*(b*c - a*d)*(a + b*Tan[e + f*x] )^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(b*c + a*d)*(n + 1))), x] + Si mp[a/(d*(b*c + a*d)*(n + 1)) Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[ e + f*x])^(n + 1)*Simp[b*(b*c*(m - 2) - a*d*(m - 2*n - 4)) + (a*b*c*(m - 2) + b^2*d*(n + 1) - a^2*d*(m + n - 1))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[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_.) + (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 = 0.41 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.58
| method | result | size |
| parallelrisch | \(-\frac {8 a^{4} \left (\frac {i \left (\cot ^{5}\left (d x +c \right )\right )}{10}+\frac {\left (\cot ^{6}\left (d x +c \right )\right )}{48}-\frac {i \left (\cot ^{3}\left (d x +c \right )\right )}{3}-\frac {7 \left (\cot ^{4}\left (d x +c \right )\right )}{32}+i d x +i \cot \left (d x +c \right )+\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\tan \left (d x +c \right )\right )-\frac {\ln \left (\sec ^{2}\left (d x +c \right )\right )}{2}\right )}{d}\) | \(94\) |
| derivativedivides | \(\frac {a^{4} \left (-8 i \cot \left (d x +c \right )-\frac {\left (\cot ^{6}\left (d x +c \right )\right )}{6}-\frac {4 i \left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {7 \left (\cot ^{4}\left (d x +c \right )\right )}{4}+\frac {8 i \left (\cot ^{3}\left (d x +c \right )\right )}{3}-4 \left (\cot ^{2}\left (d x +c \right )\right )+4 \ln \left (\cot ^{2}\left (d x +c \right )+1\right )+8 i \left (\frac {\pi }{2}-\operatorname {arccot}\left (\cot \left (d x +c \right )\right )\right )\right )}{d}\) | \(99\) |
| default | \(\frac {a^{4} \left (-8 i \cot \left (d x +c \right )-\frac {\left (\cot ^{6}\left (d x +c \right )\right )}{6}-\frac {4 i \left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {7 \left (\cot ^{4}\left (d x +c \right )\right )}{4}+\frac {8 i \left (\cot ^{3}\left (d x +c \right )\right )}{3}-4 \left (\cot ^{2}\left (d x +c \right )\right )+4 \ln \left (\cot ^{2}\left (d x +c \right )+1\right )+8 i \left (\frac {\pi }{2}-\operatorname {arccot}\left (\cot \left (d x +c \right )\right )\right )\right )}{d}\) | \(99\) |
| risch | \(\frac {16 i a^{4} c}{d}+\frac {4 a^{4} \left (270 \,{\mathrm e}^{10 i \left (d x +c \right )}-855 \,{\mathrm e}^{8 i \left (d x +c \right )}+1350 \,{\mathrm e}^{6 i \left (d x +c \right )}-1125 \,{\mathrm e}^{4 i \left (d x +c \right )}+486 \,{\mathrm e}^{2 i \left (d x +c \right )}-86\right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}-\frac {8 a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(110\) |
| norman | \(\frac {-\frac {a^{4}}{6 d}+\frac {7 a^{4} \left (\tan ^{2}\left (d x +c \right )\right )}{4 d}-\frac {4 a^{4} \left (\tan ^{4}\left (d x +c \right )\right )}{d}-\frac {4 i a^{4} \tan \left (d x +c \right )}{5 d}+\frac {8 i a^{4} \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}-\frac {8 i a^{4} \left (\tan ^{5}\left (d x +c \right )\right )}{d}-8 i a^{4} x \left (\tan ^{6}\left (d x +c \right )\right )}{\tan \left (d x +c \right )^{6}}-\frac {8 a^{4} \ln \left (\tan \left (d x +c \right )\right )}{d}+\frac {4 a^{4} \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}\) | \(150\) |
-8*a^4*(1/10*I*cot(d*x+c)^5+1/48*cot(d*x+c)^6-1/3*I*cot(d*x+c)^3-7/32*cot( d*x+c)^4+I*d*x+I*cot(d*x+c)+1/2*cot(d*x+c)^2+ln(tan(d*x+c))-1/2*ln(sec(d*x +c)^2))/d
Time = 0.25 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.57 \[ \int \cot ^7(c+d x) (a+i a \tan (c+d x))^4 \, dx=\frac {4 \, {\left (270 \, a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} - 855 \, a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 1350 \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} - 1125 \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 486 \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - 86 \, a^{4} - 30 \, {\left (a^{4} e^{\left (12 i \, d x + 12 i \, c\right )} - 6 \, a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} - 20 \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 6 \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )\right )}}{15 \, {\left (d e^{\left (12 i \, d x + 12 i \, c\right )} - 6 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, d e^{\left (8 i \, d x + 8 i \, c\right )} - 20 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, d e^{\left (4 i \, d x + 4 i \, c\right )} - 6 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
4/15*(270*a^4*e^(10*I*d*x + 10*I*c) - 855*a^4*e^(8*I*d*x + 8*I*c) + 1350*a ^4*e^(6*I*d*x + 6*I*c) - 1125*a^4*e^(4*I*d*x + 4*I*c) + 486*a^4*e^(2*I*d*x + 2*I*c) - 86*a^4 - 30*(a^4*e^(12*I*d*x + 12*I*c) - 6*a^4*e^(10*I*d*x + 1 0*I*c) + 15*a^4*e^(8*I*d*x + 8*I*c) - 20*a^4*e^(6*I*d*x + 6*I*c) + 15*a^4* e^(4*I*d*x + 4*I*c) - 6*a^4*e^(2*I*d*x + 2*I*c) + a^4)*log(e^(2*I*d*x + 2* I*c) - 1))/(d*e^(12*I*d*x + 12*I*c) - 6*d*e^(10*I*d*x + 10*I*c) + 15*d*e^( 8*I*d*x + 8*I*c) - 20*d*e^(6*I*d*x + 6*I*c) + 15*d*e^(4*I*d*x + 4*I*c) - 6 *d*e^(2*I*d*x + 2*I*c) + d)
Time = 4.27 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.52 \[ \int \cot ^7(c+d x) (a+i a \tan (c+d x))^4 \, dx=- \frac {8 a^{4} \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac {1080 a^{4} e^{10 i c} e^{10 i d x} - 3420 a^{4} e^{8 i c} e^{8 i d x} + 5400 a^{4} e^{6 i c} e^{6 i d x} - 4500 a^{4} e^{4 i c} e^{4 i d x} + 1944 a^{4} e^{2 i c} e^{2 i d x} - 344 a^{4}}{15 d e^{12 i c} e^{12 i d x} - 90 d e^{10 i c} e^{10 i d x} + 225 d e^{8 i c} e^{8 i d x} - 300 d e^{6 i c} e^{6 i d x} + 225 d e^{4 i c} e^{4 i d x} - 90 d e^{2 i c} e^{2 i d x} + 15 d} \]
-8*a**4*log(exp(2*I*d*x) - exp(-2*I*c))/d + (1080*a**4*exp(10*I*c)*exp(10* I*d*x) - 3420*a**4*exp(8*I*c)*exp(8*I*d*x) + 5400*a**4*exp(6*I*c)*exp(6*I* d*x) - 4500*a**4*exp(4*I*c)*exp(4*I*d*x) + 1944*a**4*exp(2*I*c)*exp(2*I*d* x) - 344*a**4)/(15*d*exp(12*I*c)*exp(12*I*d*x) - 90*d*exp(10*I*c)*exp(10*I *d*x) + 225*d*exp(8*I*c)*exp(8*I*d*x) - 300*d*exp(6*I*c)*exp(6*I*d*x) + 22 5*d*exp(4*I*c)*exp(4*I*d*x) - 90*d*exp(2*I*c)*exp(2*I*d*x) + 15*d)
Time = 0.59 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.76 \[ \int \cot ^7(c+d x) (a+i a \tan (c+d x))^4 \, dx=-\frac {480 i \, {\left (d x + c\right )} a^{4} - 240 \, a^{4} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 480 \, a^{4} \log \left (\tan \left (d x + c\right )\right ) - \frac {-480 i \, a^{4} \tan \left (d x + c\right )^{5} - 240 \, a^{4} \tan \left (d x + c\right )^{4} + 160 i \, a^{4} \tan \left (d x + c\right )^{3} + 105 \, a^{4} \tan \left (d x + c\right )^{2} - 48 i \, a^{4} \tan \left (d x + c\right ) - 10 \, a^{4}}{\tan \left (d x + c\right )^{6}}}{60 \, d} \]
-1/60*(480*I*(d*x + c)*a^4 - 240*a^4*log(tan(d*x + c)^2 + 1) + 480*a^4*log (tan(d*x + c)) - (-480*I*a^4*tan(d*x + c)^5 - 240*a^4*tan(d*x + c)^4 + 160 *I*a^4*tan(d*x + c)^3 + 105*a^4*tan(d*x + c)^2 - 48*I*a^4*tan(d*x + c) - 1 0*a^4)/tan(d*x + c)^6)/d
Time = 0.86 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.51 \[ \int \cot ^7(c+d x) (a+i a \tan (c+d x))^4 \, dx=-\frac {5 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 48 i \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 240 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 880 i \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2835 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 30720 \, a^{4} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right ) + 15360 \, a^{4} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - 10080 i \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {37632 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 10080 i \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2835 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 880 i \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 240 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 48 i \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, a^{4}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}}}{1920 \, d} \]
-1/1920*(5*a^4*tan(1/2*d*x + 1/2*c)^6 - 48*I*a^4*tan(1/2*d*x + 1/2*c)^5 - 240*a^4*tan(1/2*d*x + 1/2*c)^4 + 880*I*a^4*tan(1/2*d*x + 1/2*c)^3 + 2835*a ^4*tan(1/2*d*x + 1/2*c)^2 - 30720*a^4*log(tan(1/2*d*x + 1/2*c) + I) + 1536 0*a^4*log(tan(1/2*d*x + 1/2*c)) - 10080*I*a^4*tan(1/2*d*x + 1/2*c) - (3763 2*a^4*tan(1/2*d*x + 1/2*c)^6 - 10080*I*a^4*tan(1/2*d*x + 1/2*c)^5 - 2835*a ^4*tan(1/2*d*x + 1/2*c)^4 + 880*I*a^4*tan(1/2*d*x + 1/2*c)^3 + 240*a^4*tan (1/2*d*x + 1/2*c)^2 - 48*I*a^4*tan(1/2*d*x + 1/2*c) - 5*a^4)/tan(1/2*d*x + 1/2*c)^6)/d
Time = 5.33 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.66 \[ \int \cot ^7(c+d x) (a+i a \tan (c+d x))^4 \, dx=-\frac {a^4\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,16{}\mathrm {i}}{d}-\frac {a^4\,{\mathrm {tan}\left (c+d\,x\right )}^5\,8{}\mathrm {i}+4\,a^4\,{\mathrm {tan}\left (c+d\,x\right )}^4-\frac {a^4\,{\mathrm {tan}\left (c+d\,x\right )}^3\,8{}\mathrm {i}}{3}-\frac {7\,a^4\,{\mathrm {tan}\left (c+d\,x\right )}^2}{4}+\frac {a^4\,\mathrm {tan}\left (c+d\,x\right )\,4{}\mathrm {i}}{5}+\frac {a^4}{6}}{d\,{\mathrm {tan}\left (c+d\,x\right )}^6} \]