Integrand size = 28, antiderivative size = 234 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))} \, dx=\frac {\left (c^4+4 i c^3 d-6 c^2 d^2-4 i c d^3-7 d^4\right ) x}{8 a^3 (c-i d) (c+i d)^4}+\frac {d^4 \log (c \cos (e+f x)+d \sin (e+f x))}{a^3 (c+i d)^4 (i c+d) f}-\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3}+\frac {i c-3 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2}+\frac {c^2+4 i c d-7 d^2}{8 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right )} \]
1/8*(c^4+4*I*c^3*d-6*c^2*d^2-4*I*c*d^3-7*d^4)*x/a^3/(c-I*d)/(c+I*d)^4+d^4* ln(c*cos(f*x+e)+d*sin(f*x+e))/a^3/(c+I*d)^4/(I*c+d)/f-1/6/(I*c-d)/f/(a+I*a *tan(f*x+e))^3+1/8*(I*c-3*d)/a/(c+I*d)^2/f/(a+I*a*tan(f*x+e))^2+1/8*(c^2+4 *I*c*d-7*d^2)/(I*c-d)^3/f/(a^3+I*a^3*tan(f*x+e))
Time = 1.67 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))} \, dx=-\frac {\frac {3 i \left (\left (c^4+4 i c^3 d-6 c^2 d^2-4 i c d^3-15 d^4\right ) \log (i-\tan (e+f x))-(c+i d)^4 \log (i+\tan (e+f x))+16 d^4 \log (c+d \tan (e+f x))\right )}{(c-i d) (c+i d)^2}+\frac {8 (c+i d)}{(-i+\tan (e+f x))^3}+\frac {6 i (c+3 i d)}{(-i+\tan (e+f x))^2}-\frac {6 \left (c^2+4 i c d-7 d^2\right )}{(c+i d) (-i+\tan (e+f x))}}{48 a^3 (c+i d)^2 f} \]
-1/48*(((3*I)*((c^4 + (4*I)*c^3*d - 6*c^2*d^2 - (4*I)*c*d^3 - 15*d^4)*Log[ I - Tan[e + f*x]] - (c + I*d)^4*Log[I + Tan[e + f*x]] + 16*d^4*Log[c + d*T an[e + f*x]]))/((c - I*d)*(c + I*d)^2) + (8*(c + I*d))/(-I + Tan[e + f*x]) ^3 + ((6*I)*(c + (3*I)*d))/(-I + Tan[e + f*x])^2 - (6*(c^2 + (4*I)*c*d - 7 *d^2))/((c + I*d)*(-I + Tan[e + f*x])))/(a^3*(c + I*d)^2*f)
Time = 1.32 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.15, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {3042, 4042, 27, 3042, 4079, 27, 3042, 4079, 25, 3042, 4014, 3042, 4013}
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.
| \(\displaystyle \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))}dx\) |
| \(\Big \downarrow \) 4042 |
| \(\displaystyle -\frac {\int -\frac {3 (a (i c-2 d)+i a d \tan (e+f x))}{(i \tan (e+f x) a+a)^2 (c+d \tan (e+f x))}dx}{6 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a (i c-2 d)+i a d \tan (e+f x)}{(i \tan (e+f x) a+a)^2 (c+d \tan (e+f x))}dx}{2 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a (i c-2 d)+i a d \tan (e+f x)}{(i \tan (e+f x) a+a)^2 (c+d \tan (e+f x))}dx}{2 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3}\) |
| \(\Big \downarrow \) 4079 |
| \(\displaystyle \frac {-\frac {\int \frac {2 \left (\left (c^2+3 i d c-4 d^2\right ) a^2+(c+3 i d) d \tan (e+f x) a^2\right )}{(i \tan (e+f x) a+a) (c+d \tan (e+f x))}dx}{4 a^2 (-d+i c)}-\frac {a (c+3 i d)}{4 f (c+i d) (a+i a \tan (e+f x))^2}}{2 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {\left (c^2+3 i d c-4 d^2\right ) a^2+(c+3 i d) d \tan (e+f x) a^2}{(i \tan (e+f x) a+a) (c+d \tan (e+f x))}dx}{2 a^2 (-d+i c)}-\frac {a (c+3 i d)}{4 f (c+i d) (a+i a \tan (e+f x))^2}}{2 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {\left (c^2+3 i d c-4 d^2\right ) a^2+(c+3 i d) d \tan (e+f x) a^2}{(i \tan (e+f x) a+a) (c+d \tan (e+f x))}dx}{2 a^2 (-d+i c)}-\frac {a (c+3 i d)}{4 f (c+i d) (a+i a \tan (e+f x))^2}}{2 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3}\) |
| \(\Big \downarrow \) 4079 |
| \(\displaystyle \frac {-\frac {-\frac {\int -\frac {\left (i c^3-4 d c^2-7 i d^2 c+8 d^3\right ) a^3+d \left (i c^2-4 d c-7 i d^2\right ) \tan (e+f x) a^3}{c+d \tan (e+f x)}dx}{2 a^2 (-d+i c)}-\frac {a^2 \left (c^2+4 i c d-7 d^2\right )}{2 f (-d+i c) (a+i a \tan (e+f x))}}{2 a^2 (-d+i c)}-\frac {a (c+3 i d)}{4 f (c+i d) (a+i a \tan (e+f x))^2}}{2 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {\int \frac {a^3 \left (i c^3-4 d c^2-7 i d^2 c+8 d^3\right )-a^3 d \left (4 c d-i \left (c^2-7 d^2\right )\right ) \tan (e+f x)}{c+d \tan (e+f x)}dx}{2 a^2 (-d+i c)}-\frac {a^2 \left (c^2+4 i c d-7 d^2\right )}{2 f (-d+i c) (a+i a \tan (e+f x))}}{2 a^2 (-d+i c)}-\frac {a (c+3 i d)}{4 f (c+i d) (a+i a \tan (e+f x))^2}}{2 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {\int \frac {a^3 \left (i c^3-4 d c^2-7 i d^2 c+8 d^3\right )-a^3 d \left (4 c d-i \left (c^2-7 d^2\right )\right ) \tan (e+f x)}{c+d \tan (e+f x)}dx}{2 a^2 (-d+i c)}-\frac {a^2 \left (c^2+4 i c d-7 d^2\right )}{2 f (-d+i c) (a+i a \tan (e+f x))}}{2 a^2 (-d+i c)}-\frac {a (c+3 i d)}{4 f (c+i d) (a+i a \tan (e+f x))^2}}{2 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3}\) |
| \(\Big \downarrow \) 4014 |
| \(\displaystyle \frac {-\frac {\frac {\frac {8 a^3 d^4 \int \frac {d-c \tan (e+f x)}{c+d \tan (e+f x)}dx}{c^2+d^2}+\frac {a^3 x \left (i c^4-4 c^3 d-6 i c^2 d^2+4 c d^3-7 i d^4\right )}{c^2+d^2}}{2 a^2 (-d+i c)}-\frac {a^2 \left (c^2+4 i c d-7 d^2\right )}{2 f (-d+i c) (a+i a \tan (e+f x))}}{2 a^2 (-d+i c)}-\frac {a (c+3 i d)}{4 f (c+i d) (a+i a \tan (e+f x))^2}}{2 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {\frac {8 a^3 d^4 \int \frac {d-c \tan (e+f x)}{c+d \tan (e+f x)}dx}{c^2+d^2}+\frac {a^3 x \left (i c^4-4 c^3 d-6 i c^2 d^2+4 c d^3-7 i d^4\right )}{c^2+d^2}}{2 a^2 (-d+i c)}-\frac {a^2 \left (c^2+4 i c d-7 d^2\right )}{2 f (-d+i c) (a+i a \tan (e+f x))}}{2 a^2 (-d+i c)}-\frac {a (c+3 i d)}{4 f (c+i d) (a+i a \tan (e+f x))^2}}{2 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3}\) |
| \(\Big \downarrow \) 4013 |
| \(\displaystyle \frac {-\frac {\frac {\frac {8 a^3 d^4 \log (c \cos (e+f x)+d \sin (e+f x))}{f \left (c^2+d^2\right )}+\frac {a^3 x \left (i c^4-4 c^3 d-6 i c^2 d^2+4 c d^3-7 i d^4\right )}{c^2+d^2}}{2 a^2 (-d+i c)}-\frac {a^2 \left (c^2+4 i c d-7 d^2\right )}{2 f (-d+i c) (a+i a \tan (e+f x))}}{2 a^2 (-d+i c)}-\frac {a (c+3 i d)}{4 f (c+i d) (a+i a \tan (e+f x))^2}}{2 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3}\) |
-1/6*1/((I*c - d)*f*(a + I*a*Tan[e + f*x])^3) + (-1/4*(a*(c + (3*I)*d))/(( c + I*d)*f*(a + I*a*Tan[e + f*x])^2) - (((a^3*(I*c^4 - 4*c^3*d - (6*I)*c^2 *d^2 + 4*c*d^3 - (7*I)*d^4)*x)/(c^2 + d^2) + (8*a^3*d^4*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((c^2 + d^2)*f))/(2*a^2*(I*c - d)) - (a^2*(c^2 + (4*I) *c*d - 7*d^2))/(2*(I*c - d)*f*(a + I*a*Tan[e + f*x])))/(2*a^2*(I*c - d)))/ (2*a^2*(I*c - d))
3.11.88.3.1 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[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)* (x_)]), x_Symbol] :> Simp[(c/(b*f))*Log[RemoveContent[a*Cos[e + f*x] + b*Si n[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] && EqQ[a*c + b*d, 0]
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*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(2*f*m*(b*c - a*d))), x] + Simp[1/(2*a*m*(b*c - a*d)) In t[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[b*c*m - a*d*(2*m + n + 1) + b*d*(m + n + 1)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, 0] && (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_)])^(n_), x_Symbol] :> Sim p[(a*A + b*B)*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(2*f*m*( b*c - a*d))), x] + Simp[1/(2*a*m*(b*c - a*d)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(b*c*m - a*d*(2*m + n + 1)) + B*(a*c*m - b*d*(n + 1)) + d*(A*b - a*B)*(m + n + 1)*Tan[e + f*x], x], x], x] /; Free Q[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0] && !GtQ[n, 0]
Time = 0.69 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.58
| method | result | size |
| risch | \(-\frac {x}{8 a^{3} \left (i d -c \right )}-\frac {5 \,{\mathrm e}^{-2 i \left (f x +e \right )} c d}{8 a^{3} \left (i d +c \right )^{3} f}+\frac {3 i {\mathrm e}^{-2 i \left (f x +e \right )} c^{2}}{16 a^{3} \left (i d +c \right )^{3} f}-\frac {11 i {\mathrm e}^{-2 i \left (f x +e \right )} d^{2}}{16 a^{3} \left (i d +c \right )^{3} f}-\frac {5 \,{\mathrm e}^{-4 i \left (f x +e \right )} d}{32 a^{3} \left (i d +c \right )^{2} f}+\frac {3 i {\mathrm e}^{-4 i \left (f x +e \right )} c}{32 a^{3} \left (i d +c \right )^{2} f}+\frac {i {\mathrm e}^{-6 i \left (f x +e \right )}}{48 a^{3} \left (i d +c \right ) f}-\frac {2 d^{4} x}{a^{3} \left (3 i c^{4} d +2 i c^{2} d^{3}-i d^{5}+c^{5}-2 c^{3} d^{2}-3 c \,d^{4}\right )}-\frac {2 d^{4} e}{a^{3} f \left (3 i c^{4} d +2 i c^{2} d^{3}-i d^{5}+c^{5}-2 c^{3} d^{2}-3 c \,d^{4}\right )}-\frac {i d^{4} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {i d +c}{i d -c}\right )}{a^{3} f \left (3 i c^{4} d +2 i c^{2} d^{3}-i d^{5}+c^{5}-2 c^{3} d^{2}-3 c \,d^{4}\right )}\) | \(370\) |
| derivativedivides | \(\frac {i d^{4} \ln \left (c +d \tan \left (f x +e \right )\right )}{f \,a^{3} \left (i d -c \right ) \left (i d +c \right )^{4}}+\frac {11 i \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) c \,d^{2}}{32 f \,a^{3} \left (i d +c \right )^{4}}+\frac {5 \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) c^{2} d}{32 f \,a^{3} \left (i d +c \right )^{4}}-\frac {15 \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) d^{3}}{32 f \,a^{3} \left (i d +c \right )^{4}}+\frac {\arctan \left (\tan \left (f x +e \right )\right ) c^{3}}{16 f \,a^{3} \left (i d +c \right )^{4}}-\frac {11 \arctan \left (\tan \left (f x +e \right )\right ) c \,d^{2}}{16 f \,a^{3} \left (i d +c \right )^{4}}+\frac {7 i c \,d^{2}}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {i c^{3}}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{2}}+\frac {5 i \arctan \left (\tan \left (f x +e \right )\right ) c^{2} d}{16 f \,a^{3} \left (i d +c \right )^{4}}-\frac {7 i d^{3}}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )}-\frac {c^{3}}{6 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{3}}+\frac {c \,d^{2}}{2 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{3}}-\frac {15 i \arctan \left (\tan \left (f x +e \right )\right ) d^{3}}{16 f \,a^{3} \left (i d +c \right )^{4}}+\frac {i d^{3}}{6 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{3}}+\frac {c^{3}}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )}-\frac {11 c \,d^{2}}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )}-\frac {i c^{2} d}{2 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{3}}+\frac {5 i c^{2} d}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )}+\frac {5 c^{2} d}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {3 d^{3}}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {i \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) c^{3}}{32 f \,a^{3} \left (i d +c \right )^{4}}-\frac {i \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f \,a^{3} \left (16 i d -16 c \right )}-\frac {\arctan \left (\tan \left (f x +e \right )\right )}{f \,a^{3} \left (16 i d -16 c \right )}\) | \(702\) |
| default | \(\frac {i d^{4} \ln \left (c +d \tan \left (f x +e \right )\right )}{f \,a^{3} \left (i d -c \right ) \left (i d +c \right )^{4}}+\frac {11 i \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) c \,d^{2}}{32 f \,a^{3} \left (i d +c \right )^{4}}+\frac {5 \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) c^{2} d}{32 f \,a^{3} \left (i d +c \right )^{4}}-\frac {15 \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) d^{3}}{32 f \,a^{3} \left (i d +c \right )^{4}}+\frac {\arctan \left (\tan \left (f x +e \right )\right ) c^{3}}{16 f \,a^{3} \left (i d +c \right )^{4}}-\frac {11 \arctan \left (\tan \left (f x +e \right )\right ) c \,d^{2}}{16 f \,a^{3} \left (i d +c \right )^{4}}+\frac {7 i c \,d^{2}}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {i c^{3}}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{2}}+\frac {5 i \arctan \left (\tan \left (f x +e \right )\right ) c^{2} d}{16 f \,a^{3} \left (i d +c \right )^{4}}-\frac {7 i d^{3}}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )}-\frac {c^{3}}{6 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{3}}+\frac {c \,d^{2}}{2 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{3}}-\frac {15 i \arctan \left (\tan \left (f x +e \right )\right ) d^{3}}{16 f \,a^{3} \left (i d +c \right )^{4}}+\frac {i d^{3}}{6 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{3}}+\frac {c^{3}}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )}-\frac {11 c \,d^{2}}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )}-\frac {i c^{2} d}{2 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{3}}+\frac {5 i c^{2} d}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )}+\frac {5 c^{2} d}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {3 d^{3}}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {i \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) c^{3}}{32 f \,a^{3} \left (i d +c \right )^{4}}-\frac {i \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f \,a^{3} \left (16 i d -16 c \right )}-\frac {\arctan \left (\tan \left (f x +e \right )\right )}{f \,a^{3} \left (16 i d -16 c \right )}\) | \(702\) |
| norman | \(\frac {\frac {-16 i c d -5 c^{2}+17 d^{2}}{12 a f \left (i c^{3}-3 i c \,d^{2}-3 c^{2} d +d^{3}\right )}+\frac {\left (20 i c d +7 c^{2}-17 d^{2}\right ) \tan \left (f x +e \right )}{8 a f \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}+\frac {\left (4 i c d +c^{2}-7 d^{2}\right ) \left (\tan ^{5}\left (f x +e \right )\right )}{8 a f \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}+\frac {\left (5 i c d +c^{2}-7 d^{2}\right ) \left (\tan ^{3}\left (f x +e \right )\right )}{3 a f \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}+\frac {\left (4 i c^{3} d -4 i c \,d^{3}+c^{4}-6 c^{2} d^{2}-7 d^{4}\right ) x}{8 \left (c^{2}+d^{2}\right ) a \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}-\frac {i d^{2} \left (\tan ^{4}\left (f x +e \right )\right )}{2 a f \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}+\frac {\left (c^{2}+5 d^{2}\right ) \left (\tan ^{2}\left (f x +e \right )\right )}{4 a f \left (i c^{3}-3 i c \,d^{2}-3 c^{2} d +d^{3}\right )}+\frac {3 \left (4 i c^{3} d -4 i c \,d^{3}+c^{4}-6 c^{2} d^{2}-7 d^{4}\right ) x \left (\tan ^{2}\left (f x +e \right )\right )}{8 \left (c^{2}+d^{2}\right ) a \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}+\frac {3 \left (4 i c^{3} d -4 i c \,d^{3}+c^{4}-6 c^{2} d^{2}-7 d^{4}\right ) x \left (\tan ^{4}\left (f x +e \right )\right )}{8 \left (c^{2}+d^{2}\right ) a \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}+\frac {\left (4 i c^{3} d -4 i c \,d^{3}+c^{4}-6 c^{2} d^{2}-7 d^{4}\right ) x \left (\tan ^{6}\left (f x +e \right )\right )}{8 \left (c^{2}+d^{2}\right ) a \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}}{a^{2} \left (1+\tan ^{2}\left (f x +e \right )\right )^{3}}+\frac {d^{4} \ln \left (c +d \tan \left (f x +e \right )\right )}{a^{3} f \left (i c^{5}-2 i c^{3} d^{2}-3 i c \,d^{4}-3 c^{4} d -2 c^{2} d^{3}+d^{5}\right )}-\frac {d^{4} \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 a^{3} f \left (i c^{5}-2 i c^{3} d^{2}-3 i c \,d^{4}-3 c^{4} d -2 c^{2} d^{3}+d^{5}\right )}\) | \(763\) |
-1/8*x/a^3/(I*d-c)-5/8/a^3/(c+I*d)^3/f*exp(-2*I*(f*x+e))*c*d+3/16*I/a^3/(c +I*d)^3/f*exp(-2*I*(f*x+e))*c^2-11/16*I/a^3/(c+I*d)^3/f*exp(-2*I*(f*x+e))* d^2-5/32/a^3/(c+I*d)^2/f*exp(-4*I*(f*x+e))*d+3/32*I/a^3/(c+I*d)^2/f*exp(-4 *I*(f*x+e))*c+1/48*I/a^3/(c+I*d)/f*exp(-6*I*(f*x+e))-2*d^4/a^3/(3*I*c^4*d+ 2*I*c^2*d^3-I*d^5+c^5-2*c^3*d^2-3*c*d^4)*x-2*d^4/a^3/f/(3*I*c^4*d+2*I*c^2* d^3-I*d^5+c^5-2*c^3*d^2-3*c*d^4)*e-I*d^4/a^3/f/(3*I*c^4*d+2*I*c^2*d^3-I*d^ 5+c^5-2*c^3*d^2-3*c*d^4)*ln(exp(2*I*(f*x+e))-(c+I*d)/(I*d-c))
Time = 0.25 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.15 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))} \, dx=-\frac {{\left (96 \, d^{4} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (\frac {{\left (i \, c + d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + i \, c - d}{i \, c + d}\right ) - 2 \, c^{4} - 4 i \, c^{3} d - 4 i \, c d^{3} + 2 \, d^{4} - 12 \, {\left (-i \, c^{4} + 4 \, c^{3} d + 6 i \, c^{2} d^{2} - 4 \, c d^{3} + 15 i \, d^{4}\right )} f x e^{\left (6 i \, f x + 6 i \, e\right )} - 6 \, {\left (3 \, c^{4} + 10 i \, c^{3} d - 8 \, c^{2} d^{2} + 10 i \, c d^{3} - 11 \, d^{4}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} - 3 \, {\left (3 \, c^{4} + 8 i \, c^{3} d - 2 \, c^{2} d^{2} + 8 i \, c d^{3} - 5 \, d^{4}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{96 \, {\left (-i \, a^{3} c^{5} + 3 \, a^{3} c^{4} d + 2 i \, a^{3} c^{3} d^{2} + 2 \, a^{3} c^{2} d^{3} + 3 i \, a^{3} c d^{4} - a^{3} d^{5}\right )} f} \]
-1/96*(96*d^4*e^(6*I*f*x + 6*I*e)*log(((I*c + d)*e^(2*I*f*x + 2*I*e) + I*c - d)/(I*c + d)) - 2*c^4 - 4*I*c^3*d - 4*I*c*d^3 + 2*d^4 - 12*(-I*c^4 + 4* c^3*d + 6*I*c^2*d^2 - 4*c*d^3 + 15*I*d^4)*f*x*e^(6*I*f*x + 6*I*e) - 6*(3*c ^4 + 10*I*c^3*d - 8*c^2*d^2 + 10*I*c*d^3 - 11*d^4)*e^(4*I*f*x + 4*I*e) - 3 *(3*c^4 + 8*I*c^3*d - 2*c^2*d^2 + 8*I*c*d^3 - 5*d^4)*e^(2*I*f*x + 2*I*e))* e^(-6*I*f*x - 6*I*e)/((-I*a^3*c^5 + 3*a^3*c^4*d + 2*I*a^3*c^3*d^2 + 2*a^3* c^2*d^3 + 3*I*a^3*c*d^4 - a^3*d^5)*f)
Time = 11.75 (sec) , antiderivative size = 1192, normalized size of antiderivative = 5.09 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))} \, dx=\frac {x \left (c^{3} + 5 i c^{2} d - 11 c d^{2} - 15 i d^{3}\right )}{8 a^{3} c^{4} + 32 i a^{3} c^{3} d - 48 a^{3} c^{2} d^{2} - 32 i a^{3} c d^{3} + 8 a^{3} d^{4}} + \begin {cases} \frac {\left (512 i a^{6} c^{5} f^{2} e^{6 i e} - 2560 a^{6} c^{4} d f^{2} e^{6 i e} - 5120 i a^{6} c^{3} d^{2} f^{2} e^{6 i e} + 5120 a^{6} c^{2} d^{3} f^{2} e^{6 i e} + 2560 i a^{6} c d^{4} f^{2} e^{6 i e} - 512 a^{6} d^{5} f^{2} e^{6 i e}\right ) e^{- 6 i f x} + \left (2304 i a^{6} c^{5} f^{2} e^{8 i e} - 13056 a^{6} c^{4} d f^{2} e^{8 i e} - 29184 i a^{6} c^{3} d^{2} f^{2} e^{8 i e} + 32256 a^{6} c^{2} d^{3} f^{2} e^{8 i e} + 17664 i a^{6} c d^{4} f^{2} e^{8 i e} - 3840 a^{6} d^{5} f^{2} e^{8 i e}\right ) e^{- 4 i f x} + \left (4608 i a^{6} c^{5} f^{2} e^{10 i e} - 29184 a^{6} c^{4} d f^{2} e^{10 i e} - 76800 i a^{6} c^{3} d^{2} f^{2} e^{10 i e} + 101376 a^{6} c^{2} d^{3} f^{2} e^{10 i e} + 66048 i a^{6} c d^{4} f^{2} e^{10 i e} - 16896 a^{6} d^{5} f^{2} e^{10 i e}\right ) e^{- 2 i f x}}{24576 a^{9} c^{6} f^{3} e^{12 i e} + 147456 i a^{9} c^{5} d f^{3} e^{12 i e} - 368640 a^{9} c^{4} d^{2} f^{3} e^{12 i e} - 491520 i a^{9} c^{3} d^{3} f^{3} e^{12 i e} + 368640 a^{9} c^{2} d^{4} f^{3} e^{12 i e} + 147456 i a^{9} c d^{5} f^{3} e^{12 i e} - 24576 a^{9} d^{6} f^{3} e^{12 i e}} & \text {for}\: 24576 a^{9} c^{6} f^{3} e^{12 i e} + 147456 i a^{9} c^{5} d f^{3} e^{12 i e} - 368640 a^{9} c^{4} d^{2} f^{3} e^{12 i e} - 491520 i a^{9} c^{3} d^{3} f^{3} e^{12 i e} + 368640 a^{9} c^{2} d^{4} f^{3} e^{12 i e} + 147456 i a^{9} c d^{5} f^{3} e^{12 i e} - 24576 a^{9} d^{6} f^{3} e^{12 i e} \neq 0 \\x \left (- \frac {c^{3} + 5 i c^{2} d - 11 c d^{2} - 15 i d^{3}}{8 a^{3} c^{4} + 32 i a^{3} c^{3} d - 48 a^{3} c^{2} d^{2} - 32 i a^{3} c d^{3} + 8 a^{3} d^{4}} + \frac {c^{3} e^{6 i e} + 3 c^{3} e^{4 i e} + 3 c^{3} e^{2 i e} + c^{3} + 5 i c^{2} d e^{6 i e} + 13 i c^{2} d e^{4 i e} + 11 i c^{2} d e^{2 i e} + 3 i c^{2} d - 11 c d^{2} e^{6 i e} - 21 c d^{2} e^{4 i e} - 13 c d^{2} e^{2 i e} - 3 c d^{2} - 15 i d^{3} e^{6 i e} - 11 i d^{3} e^{4 i e} - 5 i d^{3} e^{2 i e} - i d^{3}}{8 a^{3} c^{4} e^{6 i e} + 32 i a^{3} c^{3} d e^{6 i e} - 48 a^{3} c^{2} d^{2} e^{6 i e} - 32 i a^{3} c d^{3} e^{6 i e} + 8 a^{3} d^{4} e^{6 i e}}\right ) & \text {otherwise} \end {cases} - \frac {i d^{4} \log {\left (\frac {c + i d}{c e^{2 i e} - i d e^{2 i e}} + e^{2 i f x} \right )}}{a^{3} f \left (c - i d\right ) \left (c + i d\right )^{4}} \]
x*(c**3 + 5*I*c**2*d - 11*c*d**2 - 15*I*d**3)/(8*a**3*c**4 + 32*I*a**3*c** 3*d - 48*a**3*c**2*d**2 - 32*I*a**3*c*d**3 + 8*a**3*d**4) + Piecewise((((5 12*I*a**6*c**5*f**2*exp(6*I*e) - 2560*a**6*c**4*d*f**2*exp(6*I*e) - 5120*I *a**6*c**3*d**2*f**2*exp(6*I*e) + 5120*a**6*c**2*d**3*f**2*exp(6*I*e) + 25 60*I*a**6*c*d**4*f**2*exp(6*I*e) - 512*a**6*d**5*f**2*exp(6*I*e))*exp(-6*I *f*x) + (2304*I*a**6*c**5*f**2*exp(8*I*e) - 13056*a**6*c**4*d*f**2*exp(8*I *e) - 29184*I*a**6*c**3*d**2*f**2*exp(8*I*e) + 32256*a**6*c**2*d**3*f**2*e xp(8*I*e) + 17664*I*a**6*c*d**4*f**2*exp(8*I*e) - 3840*a**6*d**5*f**2*exp( 8*I*e))*exp(-4*I*f*x) + (4608*I*a**6*c**5*f**2*exp(10*I*e) - 29184*a**6*c* *4*d*f**2*exp(10*I*e) - 76800*I*a**6*c**3*d**2*f**2*exp(10*I*e) + 101376*a **6*c**2*d**3*f**2*exp(10*I*e) + 66048*I*a**6*c*d**4*f**2*exp(10*I*e) - 16 896*a**6*d**5*f**2*exp(10*I*e))*exp(-2*I*f*x))/(24576*a**9*c**6*f**3*exp(1 2*I*e) + 147456*I*a**9*c**5*d*f**3*exp(12*I*e) - 368640*a**9*c**4*d**2*f** 3*exp(12*I*e) - 491520*I*a**9*c**3*d**3*f**3*exp(12*I*e) + 368640*a**9*c** 2*d**4*f**3*exp(12*I*e) + 147456*I*a**9*c*d**5*f**3*exp(12*I*e) - 24576*a* *9*d**6*f**3*exp(12*I*e)), Ne(24576*a**9*c**6*f**3*exp(12*I*e) + 147456*I* a**9*c**5*d*f**3*exp(12*I*e) - 368640*a**9*c**4*d**2*f**3*exp(12*I*e) - 49 1520*I*a**9*c**3*d**3*f**3*exp(12*I*e) + 368640*a**9*c**2*d**4*f**3*exp(12 *I*e) + 147456*I*a**9*c*d**5*f**3*exp(12*I*e) - 24576*a**9*d**6*f**3*exp(1 2*I*e), 0)), (x*(-(c**3 + 5*I*c**2*d - 11*c*d**2 - 15*I*d**3)/(8*a**3*c...
Exception generated. \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))} \, dx=\text {Exception raised: RuntimeError} \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 424 vs. \(2 (198) = 396\).
Time = 0.60 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.81 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))} \, dx=\frac {\frac {192 \, d^{5} \log \left (d \tan \left (f x + e\right ) + c\right )}{2 i \, a^{3} c^{5} d - 6 \, a^{3} c^{4} d^{2} - 4 i \, a^{3} c^{3} d^{3} - 4 \, a^{3} c^{2} d^{4} - 6 i \, a^{3} c d^{5} + 2 \, a^{3} d^{6}} + \frac {6 \, {\left (-i \, c^{3} + 5 \, c^{2} d + 11 i \, c d^{2} - 15 \, d^{3}\right )} \log \left (\tan \left (f x + e\right ) - i\right )}{a^{3} c^{4} + 4 i \, a^{3} c^{3} d - 6 \, a^{3} c^{2} d^{2} - 4 i \, a^{3} c d^{3} + a^{3} d^{4}} + \frac {192 \, \log \left (\tan \left (f x + e\right ) + i\right )}{-32 i \, a^{3} c - 32 \, a^{3} d} + \frac {11 i \, c^{3} \tan \left (f x + e\right )^{3} - 55 \, c^{2} d \tan \left (f x + e\right )^{3} - 121 i \, c d^{2} \tan \left (f x + e\right )^{3} + 165 \, d^{3} \tan \left (f x + e\right )^{3} + 45 \, c^{3} \tan \left (f x + e\right )^{2} + 225 i \, c^{2} d \tan \left (f x + e\right )^{2} - 495 \, c d^{2} \tan \left (f x + e\right )^{2} - 579 i \, d^{3} \tan \left (f x + e\right )^{2} - 69 i \, c^{3} \tan \left (f x + e\right ) + 345 \, c^{2} d \tan \left (f x + e\right ) + 711 i \, c d^{2} \tan \left (f x + e\right ) - 699 \, d^{3} \tan \left (f x + e\right ) - 51 \, c^{3} - 223 i \, c^{2} d + 385 \, c d^{2} + 301 i \, d^{3}}{{\left (a^{3} c^{4} + 4 i \, a^{3} c^{3} d - 6 \, a^{3} c^{2} d^{2} - 4 i \, a^{3} c d^{3} + a^{3} d^{4}\right )} {\left (\tan \left (f x + e\right ) - i\right )}^{3}}}{96 \, f} \]
1/96*(192*d^5*log(d*tan(f*x + e) + c)/(2*I*a^3*c^5*d - 6*a^3*c^4*d^2 - 4*I *a^3*c^3*d^3 - 4*a^3*c^2*d^4 - 6*I*a^3*c*d^5 + 2*a^3*d^6) + 6*(-I*c^3 + 5* c^2*d + 11*I*c*d^2 - 15*d^3)*log(tan(f*x + e) - I)/(a^3*c^4 + 4*I*a^3*c^3* d - 6*a^3*c^2*d^2 - 4*I*a^3*c*d^3 + a^3*d^4) + 192*log(tan(f*x + e) + I)/( -32*I*a^3*c - 32*a^3*d) + (11*I*c^3*tan(f*x + e)^3 - 55*c^2*d*tan(f*x + e) ^3 - 121*I*c*d^2*tan(f*x + e)^3 + 165*d^3*tan(f*x + e)^3 + 45*c^3*tan(f*x + e)^2 + 225*I*c^2*d*tan(f*x + e)^2 - 495*c*d^2*tan(f*x + e)^2 - 579*I*d^3 *tan(f*x + e)^2 - 69*I*c^3*tan(f*x + e) + 345*c^2*d*tan(f*x + e) + 711*I*c *d^2*tan(f*x + e) - 699*d^3*tan(f*x + e) - 51*c^3 - 223*I*c^2*d + 385*c*d^ 2 + 301*I*d^3)/((a^3*c^4 + 4*I*a^3*c^3*d - 6*a^3*c^2*d^2 - 4*I*a^3*c*d^3 + a^3*d^4)*(tan(f*x + e) - I)^3))/f
Time = 10.89 (sec) , antiderivative size = 1952, normalized size of antiderivative = 8.34 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))} \, dx=\text {Too large to display} \]
symsum(log(- (81*c*d^5 + c^5*d + d^6*56i - c^2*d^4*64i - 30*c^3*d^3 + c^4* d^2*8i)*(a^3*d^8 + a^3*c*d^7*6i - 15*a^3*c^2*d^6 - a^3*c^3*d^5*20i + 15*a^ 3*c^4*d^4 + a^3*c^5*d^3*6i - a^3*c^6*d^2) - root(a^9*c^5*d^5*e^3*7168i + 3 584*a^9*c^6*d^4*e^3 - 3584*a^9*c^4*d^6*e^3 + 3328*a^9*c^8*d^2*e^3 - 3328*a ^9*c^2*d^8*e^3 + a^9*c^7*d^3*e^3*2048i + a^9*c^3*d^7*e^3*2048i - a^9*c^9*d *e^3*1536i - a^9*c*d^9*e^3*1536i + 256*a^9*d^10*e^3 - 256*a^9*c^10*e^3 - a ^3*c*d^7*e*56i - a^3*c^7*d*e*8i - 68*a^3*c^2*d^6*e + a^3*c^5*d^3*e*56i - 5 4*a^3*c^4*d^4*e + 28*a^3*c^6*d^2*e + a^3*c^3*d^5*e*8i - 241*a^3*d^8*e - a^ 3*c^8*e - c^3*d^4*1i + 5*c^2*d^5 + c*d^6*11i - 15*d^7, e, k)*((a^3*d^8 + a ^3*c*d^7*6i - 15*a^3*c^2*d^6 - a^3*c^3*d^5*20i + 15*a^3*c^4*d^4 + a^3*c^5* d^3*6i - a^3*c^6*d^2)*(8*a^3*c^7 - a^3*d^7*56i - 264*a^3*c*d^6 + a^3*c^6*d *56i + a^3*c^2*d^5*520i + 568*a^3*c^3*d^4 - a^3*c^4*d^3*392i - 184*a^3*c^5 *d^2) + root(a^9*c^5*d^5*e^3*7168i + 3584*a^9*c^6*d^4*e^3 - 3584*a^9*c^4*d ^6*e^3 + 3328*a^9*c^8*d^2*e^3 - 3328*a^9*c^2*d^8*e^3 + a^9*c^7*d^3*e^3*204 8i + a^9*c^3*d^7*e^3*2048i - a^9*c^9*d*e^3*1536i - a^9*c*d^9*e^3*1536i + 2 56*a^9*d^10*e^3 - 256*a^9*c^10*e^3 - a^3*c*d^7*e*56i - a^3*c^7*d*e*8i - 68 *a^3*c^2*d^6*e + a^3*c^5*d^3*e*56i - 54*a^3*c^4*d^4*e + 28*a^3*c^6*d^2*e + a^3*c^3*d^5*e*8i - 241*a^3*d^8*e - a^3*c^8*e - c^3*d^4*1i + 5*c^2*d^5 + c *d^6*11i - 15*d^7, e, k)*((512*a^6*c^7*d - 512*a^6*c*d^7 + a^6*c^2*d^6*307 2i + 7680*a^6*c^3*d^5 - a^6*c^4*d^4*10240i - 7680*a^6*c^5*d^3 + a^6*c^6...