Integrand size = 26, antiderivative size = 110 \[ \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) \, dx=4 a^3 (A-i B) x-\frac {4 a^3 (i A+B) \log (\cos (e+f x))}{f}-\frac {2 a^3 (A-i B) \tan (e+f x)}{f}+\frac {a (i A+B) (a+i a \tan (e+f x))^2}{2 f}+\frac {B (a+i a \tan (e+f x))^3}{3 f} \]
4*a^3*(A-I*B)*x-4*a^3*(I*A+B)*ln(cos(f*x+e))/f-2*a^3*(A-I*B)*tan(f*x+e)/f+ 1/2*a*(I*A+B)*(a+I*a*tan(f*x+e))^2/f+1/3*B*(a+I*a*tan(f*x+e))^3/f
Time = 0.92 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.66 \[ \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) \, dx=\frac {B (a+i a \tan (e+f x))^3+\frac {3}{2} a^3 (i A+B) \left (8 \log (i+\tan (e+f x))+6 i \tan (e+f x)-\tan ^2(e+f x)\right )}{3 f} \]
(B*(a + I*a*Tan[e + f*x])^3 + (3*a^3*(I*A + B)*(8*Log[I + Tan[e + f*x]] + (6*I)*Tan[e + f*x] - Tan[e + f*x]^2))/2)/(3*f)
Time = 0.46 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.90, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3042, 4010, 3042, 3959, 3042, 3958, 3042, 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.
| \(\displaystyle \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x))dx\) |
| \(\Big \downarrow \) 4010 |
| \(\displaystyle (A-i B) \int (i \tan (e+f x) a+a)^3dx+\frac {B (a+i a \tan (e+f x))^3}{3 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle (A-i B) \int (i \tan (e+f x) a+a)^3dx+\frac {B (a+i a \tan (e+f x))^3}{3 f}\) |
| \(\Big \downarrow \) 3959 |
| \(\displaystyle (A-i B) \left (2 a \int (i \tan (e+f x) a+a)^2dx+\frac {i a (a+i a \tan (e+f x))^2}{2 f}\right )+\frac {B (a+i a \tan (e+f x))^3}{3 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle (A-i B) \left (2 a \int (i \tan (e+f x) a+a)^2dx+\frac {i a (a+i a \tan (e+f x))^2}{2 f}\right )+\frac {B (a+i a \tan (e+f x))^3}{3 f}\) |
| \(\Big \downarrow \) 3958 |
| \(\displaystyle (A-i B) \left (2 a \left (2 i a^2 \int \tan (e+f x)dx-\frac {a^2 \tan (e+f x)}{f}+2 a^2 x\right )+\frac {i a (a+i a \tan (e+f x))^2}{2 f}\right )+\frac {B (a+i a \tan (e+f x))^3}{3 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle (A-i B) \left (2 a \left (2 i a^2 \int \tan (e+f x)dx-\frac {a^2 \tan (e+f x)}{f}+2 a^2 x\right )+\frac {i a (a+i a \tan (e+f x))^2}{2 f}\right )+\frac {B (a+i a \tan (e+f x))^3}{3 f}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle (A-i B) \left (2 a \left (-\frac {a^2 \tan (e+f x)}{f}-\frac {2 i a^2 \log (\cos (e+f x))}{f}+2 a^2 x\right )+\frac {i a (a+i a \tan (e+f x))^2}{2 f}\right )+\frac {B (a+i a \tan (e+f x))^3}{3 f}\) |
(B*(a + I*a*Tan[e + f*x])^3)/(3*f) + (A - I*B)*(((I/2)*a*(a + I*a*Tan[e + f*x])^2)/f + 2*a*(2*a^2*x - ((2*I)*a^2*Log[Cos[e + f*x]])/f - (a^2*Tan[e + f*x])/f))
3.7.96.3.1 Defintions of rubi rules used
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[(c_.) + (d_.)*(x_)])^2, x_Symbol] :> Simp[(a^2 - b^2) *x, x] + (Simp[b^2*(Tan[c + d*x]/d), x] + Simp[2*a*b Int[Tan[c + d*x], x] , x]) /; FreeQ[{a, b, c, d}, x]
Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((a + b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[2*a Int[(a + b*Tan[c + d* x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0] && GtQ[n , 1]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*((a + b*Tan[e + f*x])^m/(f*m)), x] + Simp [(b*c + a*d)/b Int[(a + b*Tan[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e , f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && !LtQ[m, 0]
Time = 0.13 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.91
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (-\frac {i B \tan \left (f x +e \right )^{3}}{3}-\frac {i A \tan \left (f x +e \right )^{2}}{2}+4 i \tan \left (f x +e \right ) B -\frac {3 B \tan \left (f x +e \right )^{2}}{2}-3 A \tan \left (f x +e \right )+\frac {\left (4 i A +4 B \right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}+\left (-4 i B +4 A \right ) \arctan \left (\tan \left (f x +e \right )\right )\right )}{f}\) | \(100\) |
| default | \(\frac {a^{3} \left (-\frac {i B \tan \left (f x +e \right )^{3}}{3}-\frac {i A \tan \left (f x +e \right )^{2}}{2}+4 i \tan \left (f x +e \right ) B -\frac {3 B \tan \left (f x +e \right )^{2}}{2}-3 A \tan \left (f x +e \right )+\frac {\left (4 i A +4 B \right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}+\left (-4 i B +4 A \right ) \arctan \left (\tan \left (f x +e \right )\right )\right )}{f}\) | \(100\) |
| norman | \(\left (-4 i B \,a^{3}+4 a^{3} A \right ) x -\frac {\left (i A \,a^{3}+3 B \,a^{3}\right ) \tan \left (f x +e \right )^{2}}{2 f}-\frac {\left (-4 i B \,a^{3}+3 a^{3} A \right ) \tan \left (f x +e \right )}{f}-\frac {i B \,a^{3} \tan \left (f x +e \right )^{3}}{3 f}+\frac {2 \left (i A \,a^{3}+B \,a^{3}\right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{f}\) | \(117\) |
| parallelrisch | \(\frac {-2 i B \,a^{3} \tan \left (f x +e \right )^{3}-3 i A \tan \left (f x +e \right )^{2} a^{3}-24 i B x \,a^{3} f +12 i A \ln \left (1+\tan \left (f x +e \right )^{2}\right ) a^{3}+24 A x \,a^{3} f +24 i B \tan \left (f x +e \right ) a^{3}-9 B \tan \left (f x +e \right )^{2} a^{3}-18 A \tan \left (f x +e \right ) a^{3}+12 B \ln \left (1+\tan \left (f x +e \right )^{2}\right ) a^{3}}{6 f}\) | \(128\) |
| risch | \(\frac {8 i a^{3} B e}{f}-\frac {8 a^{3} A e}{f}-\frac {2 a^{3} \left (12 i A \,{\mathrm e}^{4 i \left (f x +e \right )}+24 B \,{\mathrm e}^{4 i \left (f x +e \right )}+21 i A \,{\mathrm e}^{2 i \left (f x +e \right )}+33 B \,{\mathrm e}^{2 i \left (f x +e \right )}+9 i A +13 B \right )}{3 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{3}}-\frac {4 a^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) B}{f}-\frac {4 i a^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) A}{f}\) | \(145\) |
| parts | \(A \,a^{3} x +\frac {\left (-i A \,a^{3}-3 B \,a^{3}\right ) \left (\frac {\tan \left (f x +e \right )^{2}}{2}-\frac {\ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}\right )}{f}+\frac {\left (3 i A \,a^{3}+B \,a^{3}\right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2 f}+\frac {\left (3 i B \,a^{3}-3 a^{3} A \right ) \left (\tan \left (f x +e \right )-\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}-\frac {i B \,a^{3} \left (\frac {\tan \left (f x +e \right )^{3}}{3}-\tan \left (f x +e \right )+\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}\) | \(149\) |
1/f*a^3*(-1/3*I*B*tan(f*x+e)^3-1/2*I*A*tan(f*x+e)^2+4*I*B*tan(f*x+e)-3/2*B *tan(f*x+e)^2-3*A*tan(f*x+e)+1/2*(4*I*A+4*B)*ln(1+tan(f*x+e)^2)+(-4*I*B+4* A)*arctan(tan(f*x+e)))
Time = 0.26 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.59 \[ \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) \, dx=-\frac {2 \, {\left (12 \, {\left (i \, A + 2 \, B\right )} a^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, {\left (7 i \, A + 11 \, B\right )} a^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (9 i \, A + 13 \, B\right )} a^{3} + 6 \, {\left ({\left (i \, A + B\right )} a^{3} e^{\left (6 i \, f x + 6 i \, e\right )} + 3 \, {\left (i \, A + B\right )} a^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, {\left (i \, A + B\right )} a^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (i \, A + B\right )} a^{3}\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )\right )}}{3 \, {\left (f e^{\left (6 i \, f x + 6 i \, e\right )} + 3 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
-2/3*(12*(I*A + 2*B)*a^3*e^(4*I*f*x + 4*I*e) + 3*(7*I*A + 11*B)*a^3*e^(2*I *f*x + 2*I*e) + (9*I*A + 13*B)*a^3 + 6*((I*A + B)*a^3*e^(6*I*f*x + 6*I*e) + 3*(I*A + B)*a^3*e^(4*I*f*x + 4*I*e) + 3*(I*A + B)*a^3*e^(2*I*f*x + 2*I*e ) + (I*A + B)*a^3)*log(e^(2*I*f*x + 2*I*e) + 1))/(f*e^(6*I*f*x + 6*I*e) + 3*f*e^(4*I*f*x + 4*I*e) + 3*f*e^(2*I*f*x + 2*I*e) + f)
Time = 0.37 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.67 \[ \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) \, dx=- \frac {4 i a^{3} \left (A - i B\right ) \log {\left (e^{2 i f x} + e^{- 2 i e} \right )}}{f} + \frac {- 18 i A a^{3} - 26 B a^{3} + \left (- 42 i A a^{3} e^{2 i e} - 66 B a^{3} e^{2 i e}\right ) e^{2 i f x} + \left (- 24 i A a^{3} e^{4 i e} - 48 B a^{3} e^{4 i e}\right ) e^{4 i f x}}{3 f e^{6 i e} e^{6 i f x} + 9 f e^{4 i e} e^{4 i f x} + 9 f e^{2 i e} e^{2 i f x} + 3 f} \]
-4*I*a**3*(A - I*B)*log(exp(2*I*f*x) + exp(-2*I*e))/f + (-18*I*A*a**3 - 26 *B*a**3 + (-42*I*A*a**3*exp(2*I*e) - 66*B*a**3*exp(2*I*e))*exp(2*I*f*x) + (-24*I*A*a**3*exp(4*I*e) - 48*B*a**3*exp(4*I*e))*exp(4*I*f*x))/(3*f*exp(6* I*e)*exp(6*I*f*x) + 9*f*exp(4*I*e)*exp(4*I*f*x) + 9*f*exp(2*I*e)*exp(2*I*f *x) + 3*f)
Time = 0.31 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.87 \[ \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) \, dx=-\frac {2 i \, B a^{3} \tan \left (f x + e\right )^{3} + 3 \, {\left (i \, A + 3 \, B\right )} a^{3} \tan \left (f x + e\right )^{2} - 24 \, {\left (f x + e\right )} {\left (A - i \, B\right )} a^{3} + 12 \, {\left (-i \, A - B\right )} a^{3} \log \left (\tan \left (f x + e\right )^{2} + 1\right ) + 6 \, {\left (3 \, A - 4 i \, B\right )} a^{3} \tan \left (f x + e\right )}{6 \, f} \]
-1/6*(2*I*B*a^3*tan(f*x + e)^3 + 3*(I*A + 3*B)*a^3*tan(f*x + e)^2 - 24*(f* x + e)*(A - I*B)*a^3 + 12*(-I*A - B)*a^3*log(tan(f*x + e)^2 + 1) + 6*(3*A - 4*I*B)*a^3*tan(f*x + e))/f
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 312 vs. \(2 (94) = 188\).
Time = 0.49 (sec) , antiderivative size = 312, normalized size of antiderivative = 2.84 \[ \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) \, dx=-\frac {2 \, {\left (6 i \, A a^{3} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 6 \, B a^{3} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 18 i \, A a^{3} e^{\left (4 i \, f x + 4 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 18 \, B a^{3} e^{\left (4 i \, f x + 4 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 18 i \, A a^{3} e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 18 \, B a^{3} e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 12 i \, A a^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 24 \, B a^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 21 i \, A a^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 33 \, B a^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 6 i \, A a^{3} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 6 \, B a^{3} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 9 i \, A a^{3} + 13 \, B a^{3}\right )}}{3 \, {\left (f e^{\left (6 i \, f x + 6 i \, e\right )} + 3 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
-2/3*(6*I*A*a^3*e^(6*I*f*x + 6*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) + 6*B*a^3 *e^(6*I*f*x + 6*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) + 18*I*A*a^3*e^(4*I*f*x + 4*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) + 18*B*a^3*e^(4*I*f*x + 4*I*e)*log(e ^(2*I*f*x + 2*I*e) + 1) + 18*I*A*a^3*e^(2*I*f*x + 2*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) + 18*B*a^3*e^(2*I*f*x + 2*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) + 12*I*A*a^3*e^(4*I*f*x + 4*I*e) + 24*B*a^3*e^(4*I*f*x + 4*I*e) + 21*I*A*a^3 *e^(2*I*f*x + 2*I*e) + 33*B*a^3*e^(2*I*f*x + 2*I*e) + 6*I*A*a^3*log(e^(2*I *f*x + 2*I*e) + 1) + 6*B*a^3*log(e^(2*I*f*x + 2*I*e) + 1) + 9*I*A*a^3 + 13 *B*a^3)/(f*e^(6*I*f*x + 6*I*e) + 3*f*e^(4*I*f*x + 4*I*e) + 3*f*e^(2*I*f*x + 2*I*e) + f)
Time = 8.48 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.14 \[ \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) \, dx=-\frac {{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (\frac {B\,a^3}{2}+\frac {a^3\,\left (2\,B+A\,1{}\mathrm {i}\right )}{2}\right )}{f}+\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,\left (4\,B\,a^3+A\,a^3\,4{}\mathrm {i}\right )}{f}+\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (B\,a^3\,1{}\mathrm {i}-a^3\,\left (2\,A-B\,1{}\mathrm {i}\right )+a^3\,\left (2\,B+A\,1{}\mathrm {i}\right )\,1{}\mathrm {i}\right )}{f}-\frac {B\,a^3\,{\mathrm {tan}\left (e+f\,x\right )}^3\,1{}\mathrm {i}}{3\,f} \]