Integrand size = 45, antiderivative size = 415 \[ \int \frac {(c+d \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^2} \, dx=-\frac {\left (a^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-b^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-2 a b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) x}{\left (a^2+b^2\right )^2}-\frac {\left (2 a b \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+a^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )-b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \log (\cos (e+f x))}{\left (a^2+b^2\right )^2 f}-\frac {(b c-a d) \left (a^3 b B d-2 a^4 C d-b^4 (B c+2 A d)-a b^3 (2 A c-2 c C-3 B d)+a^2 b^2 (B c-4 C d)\right ) \log (a+b \tan (e+f x))}{b^3 \left (a^2+b^2\right )^2 f}+\frac {\left (A b^2-a b B+2 a^2 C+b^2 C\right ) d^2 \tan (e+f x)}{b^2 \left (a^2+b^2\right ) f}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^2}{b \left (a^2+b^2\right ) f (a+b \tan (e+f x))} \] Output:
-(a^2*(c^2*C+2*B*c*d-C*d^2-A*(c^2-d^2))-b^2*(c^2*C+2*B*c*d-C*d^2-A*(c^2-d^ 2))-2*a*b*(2*c*(A-C)*d+B*(c^2-d^2)))*x/(a^2+b^2)^2-(2*a*b*(c^2*C+2*B*c*d-C *d^2-A*(c^2-d^2))+a^2*(2*c*(A-C)*d+B*(c^2-d^2))-b^2*(2*c*(A-C)*d+B*(c^2-d^ 2)))*ln(cos(f*x+e))/(a^2+b^2)^2/f-(-a*d+b*c)*(a^3*b*B*d-2*a^4*C*d-b^4*(2*A *d+B*c)-a*b^3*(2*A*c-3*B*d-2*C*c)+a^2*b^2*(B*c-4*C*d))*ln(a+b*tan(f*x+e))/ b^3/(a^2+b^2)^2/f+(A*b^2-B*a*b+2*C*a^2+C*b^2)*d^2*tan(f*x+e)/b^2/(a^2+b^2) /f-(A*b^2-a*(B*b-C*a))*(c+d*tan(f*x+e))^2/b/(a^2+b^2)/f/(a+b*tan(f*x+e))
Result contains complex when optimal does not.
Time = 4.04 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.67 \[ \int \frac {(c+d \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^2} \, dx=\frac {\frac {(-i A+B+i C) (c+i d)^2 \log (i-\tan (e+f x))}{(a+i b)^2}+\frac {(i A+B-i C) (c-i d)^2 \log (i+\tan (e+f x))}{(a-i b)^2}+\frac {2 (b c-a d) \left (-a^3 b B d+2 a^4 C d+b^4 (B c+2 A d)+a b^3 (2 A c-2 c C-3 B d)+a^2 b^2 (-B c+4 C d)\right ) \log (a+b \tan (e+f x))}{b^3 \left (a^2+b^2\right )^2}-\frac {2 \left (A b^2-a b B+2 a^2 C+b^2 C\right ) (b c-a d)^2}{b^3 \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {2 C (c+d \tan (e+f x))^2}{b (a+b \tan (e+f x))}}{2 f} \] Input:
Integrate[((c + d*Tan[e + f*x])^2*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)) /(a + b*Tan[e + f*x])^2,x]
Output:
((((-I)*A + B + I*C)*(c + I*d)^2*Log[I - Tan[e + f*x]])/(a + I*b)^2 + ((I* A + B - I*C)*(c - I*d)^2*Log[I + Tan[e + f*x]])/(a - I*b)^2 + (2*(b*c - a* d)*(-(a^3*b*B*d) + 2*a^4*C*d + b^4*(B*c + 2*A*d) + a*b^3*(2*A*c - 2*c*C - 3*B*d) + a^2*b^2*(-(B*c) + 4*C*d))*Log[a + b*Tan[e + f*x]])/(b^3*(a^2 + b^ 2)^2) - (2*(A*b^2 - a*b*B + 2*a^2*C + b^2*C)*(b*c - a*d)^2)/(b^3*(a^2 + b^ 2)*(a + b*Tan[e + f*x])) + (2*C*(c + d*Tan[e + f*x])^2)/(b*(a + b*Tan[e + f*x])))/(2*f)
Time = 3.26 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.03, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3042, 4128, 3042, 4120, 3042, 4109, 3042, 3956, 4100, 16}
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.
| \(\displaystyle \int \frac {(c+d \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c+d \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan (e+f x)^2\right )}{(a+b \tan (e+f x))^2}dx\) |
| \(\Big \downarrow \) 4128 |
| \(\displaystyle \frac {\int \frac {(c+d \tan (e+f x)) \left (\left (2 C a^2-b B a+A b^2+b^2 C\right ) d \tan ^2(e+f x)-b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)+(b B-a C) (b c-2 a d)+A b (a c+2 b d)\right )}{a+b \tan (e+f x)}dx}{b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^2}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(c+d \tan (e+f x)) \left (\left (2 C a^2-b B a+A b^2+b^2 C\right ) d \tan (e+f x)^2-b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)+(b B-a C) (b c-2 a d)+A b (a c+2 b d)\right )}{a+b \tan (e+f x)}dx}{b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^2}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\) |
| \(\Big \downarrow \) 4120 |
| \(\displaystyle \frac {\frac {d^2 \tan (e+f x) \left (2 a^2 C-a b B+A b^2+b^2 C\right )}{b f}-\frac {\int \frac {-\left (\left (2 a A c d-2 a c C d-A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )+b \left (C c^2+2 B d c-C d^2\right )\right ) \tan (e+f x) b^2\right )-c ((b B-a C) (b c-2 a d)+A b (a c+2 b d)) b+a \left (2 C a^2-b B a+A b^2+b^2 C\right ) d^2-\left (a^2+b^2\right ) d (2 b c C-2 a d C+b B d) \tan ^2(e+f x)}{a+b \tan (e+f x)}dx}{b}}{b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^2}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {d^2 \tan (e+f x) \left (2 a^2 C-a b B+A b^2+b^2 C\right )}{b f}-\frac {\int \frac {-\left (\left (2 a A c d-2 a c C d-A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )+b \left (C c^2+2 B d c-C d^2\right )\right ) \tan (e+f x) b^2\right )-c ((b B-a C) (b c-2 a d)+A b (a c+2 b d)) b+a \left (2 C a^2-b B a+A b^2+b^2 C\right ) d^2-\left (a^2+b^2\right ) d (2 b c C-2 a d C+b B d) \tan (e+f x)^2}{a+b \tan (e+f x)}dx}{b}}{b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^2}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\) |
| \(\Big \downarrow \) 4109 |
| \(\displaystyle \frac {\frac {d^2 \tan (e+f x) \left (2 a^2 C-a b B+A b^2+b^2 C\right )}{b f}-\frac {-\frac {b^2 \left (a^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )+2 a b \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )-b^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right ) \int \tan (e+f x)dx}{a^2+b^2}+\frac {(b c-a d) \left (-2 a^4 C d+a^3 b B d+a^2 b^2 (B c-4 C d)-a b^3 (2 A c-3 B d-2 c C)-b^4 (2 A d+B c)\right ) \int \frac {\tan ^2(e+f x)+1}{a+b \tan (e+f x)}dx}{a^2+b^2}+\frac {b^2 x \left (a^2 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )-2 a b \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )-b^2 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )\right )}{a^2+b^2}}{b}}{b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^2}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {d^2 \tan (e+f x) \left (2 a^2 C-a b B+A b^2+b^2 C\right )}{b f}-\frac {-\frac {b^2 \left (a^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )+2 a b \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )-b^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right ) \int \tan (e+f x)dx}{a^2+b^2}+\frac {(b c-a d) \left (-2 a^4 C d+a^3 b B d+a^2 b^2 (B c-4 C d)-a b^3 (2 A c-3 B d-2 c C)-b^4 (2 A d+B c)\right ) \int \frac {\tan (e+f x)^2+1}{a+b \tan (e+f x)}dx}{a^2+b^2}+\frac {b^2 x \left (a^2 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )-2 a b \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )-b^2 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )\right )}{a^2+b^2}}{b}}{b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^2}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {\frac {d^2 \tan (e+f x) \left (2 a^2 C-a b B+A b^2+b^2 C\right )}{b f}-\frac {\frac {(b c-a d) \left (-2 a^4 C d+a^3 b B d+a^2 b^2 (B c-4 C d)-a b^3 (2 A c-3 B d-2 c C)-b^4 (2 A d+B c)\right ) \int \frac {\tan (e+f x)^2+1}{a+b \tan (e+f x)}dx}{a^2+b^2}+\frac {b^2 \log (\cos (e+f x)) \left (a^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )+2 a b \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )-b^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )}{f \left (a^2+b^2\right )}+\frac {b^2 x \left (a^2 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )-2 a b \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )-b^2 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )\right )}{a^2+b^2}}{b}}{b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^2}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\) |
| \(\Big \downarrow \) 4100 |
| \(\displaystyle \frac {\frac {d^2 \tan (e+f x) \left (2 a^2 C-a b B+A b^2+b^2 C\right )}{b f}-\frac {\frac {(b c-a d) \left (-2 a^4 C d+a^3 b B d+a^2 b^2 (B c-4 C d)-a b^3 (2 A c-3 B d-2 c C)-b^4 (2 A d+B c)\right ) \int \frac {1}{a+b \tan (e+f x)}d(b \tan (e+f x))}{b f \left (a^2+b^2\right )}+\frac {b^2 \log (\cos (e+f x)) \left (a^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )+2 a b \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )-b^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )}{f \left (a^2+b^2\right )}+\frac {b^2 x \left (a^2 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )-2 a b \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )-b^2 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )\right )}{a^2+b^2}}{b}}{b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^2}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {d^2 \tan (e+f x) \left (2 a^2 C-a b B+A b^2+b^2 C\right )}{b f}-\frac {\frac {b^2 \log (\cos (e+f x)) \left (a^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )+2 a b \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )-b^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )}{f \left (a^2+b^2\right )}+\frac {b^2 x \left (a^2 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )-2 a b \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )-b^2 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )\right )}{a^2+b^2}+\frac {(b c-a d) \left (-2 a^4 C d+a^3 b B d+a^2 b^2 (B c-4 C d)-a b^3 (2 A c-3 B d-2 c C)-b^4 (2 A d+B c)\right ) \log (a+b \tan (e+f x))}{b f \left (a^2+b^2\right )}}{b}}{b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^2}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\) |
Input:
Int[((c + d*Tan[e + f*x])^2*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^2,x]
Output:
-(((A*b^2 - a*(b*B - a*C))*(c + d*Tan[e + f*x])^2)/(b*(a^2 + b^2)*f*(a + b *Tan[e + f*x]))) + (-(((b^2*(a^2*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - b^2*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - 2*a*b*(2*c*(A - C)*d + B*(c^2 - d^2)))*x)/(a^2 + b^2) + (b^2*(2*a*b*(c^2*C + 2*B*c*d - C*d^2 - A* (c^2 - d^2)) + a^2*(2*c*(A - C)*d + B*(c^2 - d^2)) - b^2*(2*c*(A - C)*d + B*(c^2 - d^2)))*Log[Cos[e + f*x]])/((a^2 + b^2)*f) + ((b*c - a*d)*(a^3*b*B *d - 2*a^4*C*d - b^4*(B*c + 2*A*d) - a*b^3*(2*A*c - 2*c*C - 3*B*d) + a^2*b ^2*(B*c - 4*C*d))*Log[a + b*Tan[e + f*x]])/(b*(a^2 + b^2)*f))/b) + ((A*b^2 - a*b*B + 2*a^2*C + b^2*C)*d^2*Tan[e + f*x])/(b*f))/(b*(a^2 + b^2))
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, 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_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[A/(b*f) Subst[Int[(a + x)^m, x], x, b* Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[A, C]
Int[((A_) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2 )/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(a*A + b*B - a *C)*(x/(a^2 + b^2)), x] + (Simp[(A*b^2 - a*b*B + a^2*C)/(a^2 + b^2) Int[( 1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x]), x], x] - Simp[(A*b - a*B - b*C)/( a^2 + b^2) Int[Tan[e + f*x], x], x]) /; FreeQ[{a, b, e, f, A, B, C}, x] & & NeQ[A*b^2 - a*b*B + a^2*C, 0] && NeQ[a^2 + b^2, 0] && NeQ[A*b - a*B - b*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*Tan[e + f*x]*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 2))), x] - Simp[1/(d*(n + 2)) Int[(c + d*Tan[e + f*x])^n*Si mp[b*c*C - a*A*d*(n + 2) - (A*b + a*B - b*C)*d*(n + 2)*Tan[e + f*x] - (a*C* d*(n + 2) - b*(c*C - B*d*(n + 2)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[c^2 + d^2, 0] && !LtQ[n, -1]
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[(A*d^2 + c*(c*C - B*d))*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Sim p[1/(d*(n + 1)*(c^2 + d^2)) Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*d*(b*d*m - a*c*(n + 1)) + (c*C - B*d)*(b*c*m + a*d* (n + 1)) - d*(n + 1)*((A - C)*(b*c - a*d) + B*(a*c + b*d))*Tan[e + f*x] - b *(d*(B*c - A*d)*(m + n + 1) - C*(c^2*m - d^2*(n + 1)))*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 [a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Time = 0.28 (sec) , antiderivative size = 552, normalized size of antiderivative = 1.33
| method | result | size |
| derivativedivides | \(\frac {\frac {\tan \left (f x +e \right ) C \,d^{2}}{b^{2}}+\frac {\frac {\left (2 A \,a^{2} c d -2 A a b \,c^{2}+2 A a b \,d^{2}-2 A \,b^{2} c d +B \,a^{2} c^{2}-B \,a^{2} d^{2}+4 B a b c d -B \,b^{2} c^{2}+B \,b^{2} d^{2}-2 C \,a^{2} c d +2 C a b \,c^{2}-2 C a b \,d^{2}+2 C \,b^{2} c d \right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}+\left (A \,a^{2} c^{2}-A \,a^{2} d^{2}+4 A a b c d -A \,b^{2} c^{2}+A \,b^{2} d^{2}-2 B \,a^{2} c d +2 B a b \,c^{2}-2 B a b \,d^{2}+2 B \,b^{2} c d -C \,a^{2} c^{2}+a^{2} C \,d^{2}-4 C a b c d +C \,b^{2} c^{2}-b^{2} d^{2} C \right ) \arctan \left (\tan \left (f x +e \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}-\frac {A \,a^{2} d^{2} b^{2}-2 A a \,b^{3} c d +A \,b^{4} c^{2}-B \,a^{3} d^{2} b +2 B \,a^{2} c d \,b^{2}-B a \,b^{3} c^{2}+a^{4} C \,d^{2}-2 C \,a^{3} c d b +C \,a^{2} c^{2} b^{2}}{b^{3} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (f x +e \right )\right )}+\frac {\left (-2 A \,a^{2} b^{3} c d +2 A a \,b^{4} c^{2}-2 A a \,b^{4} d^{2}+2 A \,b^{5} c d +B \,a^{4} b \,d^{2}-B \,a^{2} b^{3} c^{2}+3 B \,a^{2} b^{3} d^{2}-4 B a \,b^{4} c d +B \,b^{5} c^{2}-2 C \,a^{5} d^{2}+2 C \,a^{4} b c d -4 C \,a^{3} b^{2} d^{2}+6 C \,a^{2} b^{3} c d -2 C a \,b^{4} c^{2}\right ) \ln \left (a +b \tan \left (f x +e \right )\right )}{b^{3} \left (a^{2}+b^{2}\right )^{2}}}{f}\) | \(552\) |
| default | \(\frac {\frac {\tan \left (f x +e \right ) C \,d^{2}}{b^{2}}+\frac {\frac {\left (2 A \,a^{2} c d -2 A a b \,c^{2}+2 A a b \,d^{2}-2 A \,b^{2} c d +B \,a^{2} c^{2}-B \,a^{2} d^{2}+4 B a b c d -B \,b^{2} c^{2}+B \,b^{2} d^{2}-2 C \,a^{2} c d +2 C a b \,c^{2}-2 C a b \,d^{2}+2 C \,b^{2} c d \right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}+\left (A \,a^{2} c^{2}-A \,a^{2} d^{2}+4 A a b c d -A \,b^{2} c^{2}+A \,b^{2} d^{2}-2 B \,a^{2} c d +2 B a b \,c^{2}-2 B a b \,d^{2}+2 B \,b^{2} c d -C \,a^{2} c^{2}+a^{2} C \,d^{2}-4 C a b c d +C \,b^{2} c^{2}-b^{2} d^{2} C \right ) \arctan \left (\tan \left (f x +e \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}-\frac {A \,a^{2} d^{2} b^{2}-2 A a \,b^{3} c d +A \,b^{4} c^{2}-B \,a^{3} d^{2} b +2 B \,a^{2} c d \,b^{2}-B a \,b^{3} c^{2}+a^{4} C \,d^{2}-2 C \,a^{3} c d b +C \,a^{2} c^{2} b^{2}}{b^{3} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (f x +e \right )\right )}+\frac {\left (-2 A \,a^{2} b^{3} c d +2 A a \,b^{4} c^{2}-2 A a \,b^{4} d^{2}+2 A \,b^{5} c d +B \,a^{4} b \,d^{2}-B \,a^{2} b^{3} c^{2}+3 B \,a^{2} b^{3} d^{2}-4 B a \,b^{4} c d +B \,b^{5} c^{2}-2 C \,a^{5} d^{2}+2 C \,a^{4} b c d -4 C \,a^{3} b^{2} d^{2}+6 C \,a^{2} b^{3} c d -2 C a \,b^{4} c^{2}\right ) \ln \left (a +b \tan \left (f x +e \right )\right )}{b^{3} \left (a^{2}+b^{2}\right )^{2}}}{f}\) | \(552\) |
| norman | \(\frac {\frac {a \left (A \,a^{2} c^{2}-A \,a^{2} d^{2}+4 A a b c d -A \,b^{2} c^{2}+A \,b^{2} d^{2}-2 B \,a^{2} c d +2 B a b \,c^{2}-2 B a b \,d^{2}+2 B \,b^{2} c d -C \,a^{2} c^{2}+a^{2} C \,d^{2}-4 C a b c d +C \,b^{2} c^{2}-b^{2} d^{2} C \right ) x}{a^{4}+2 b^{2} a^{2}+b^{4}}+\frac {C \,d^{2} \tan \left (f x +e \right )^{2}}{f b}+\frac {b \left (A \,a^{2} c^{2}-A \,a^{2} d^{2}+4 A a b c d -A \,b^{2} c^{2}+A \,b^{2} d^{2}-2 B \,a^{2} c d +2 B a b \,c^{2}-2 B a b \,d^{2}+2 B \,b^{2} c d -C \,a^{2} c^{2}+a^{2} C \,d^{2}-4 C a b c d +C \,b^{2} c^{2}-b^{2} d^{2} C \right ) x \tan \left (f x +e \right )}{a^{4}+2 b^{2} a^{2}+b^{4}}-\frac {A \,a^{2} d^{2} b^{2}-2 A a \,b^{3} c d +A \,b^{4} c^{2}-B \,a^{3} d^{2} b +2 B \,a^{2} c d \,b^{2}-B a \,b^{3} c^{2}+2 a^{4} C \,d^{2}-2 C \,a^{3} c d b +C \,a^{2} c^{2} b^{2}+C \,a^{2} b^{2} d^{2}}{f \,b^{3} \left (a^{2}+b^{2}\right )}}{a +b \tan \left (f x +e \right )}+\frac {\left (2 A \,a^{2} c d -2 A a b \,c^{2}+2 A a b \,d^{2}-2 A \,b^{2} c d +B \,a^{2} c^{2}-B \,a^{2} d^{2}+4 B a b c d -B \,b^{2} c^{2}+B \,b^{2} d^{2}-2 C \,a^{2} c d +2 C a b \,c^{2}-2 C a b \,d^{2}+2 C \,b^{2} c d \right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2 f \left (a^{4}+2 b^{2} a^{2}+b^{4}\right )}-\frac {\left (2 A \,a^{2} b^{3} c d -2 A a \,b^{4} c^{2}+2 A a \,b^{4} d^{2}-2 A \,b^{5} c d -B \,a^{4} b \,d^{2}+B \,a^{2} b^{3} c^{2}-3 B \,a^{2} b^{3} d^{2}+4 B a \,b^{4} c d -B \,b^{5} c^{2}+2 C \,a^{5} d^{2}-2 C \,a^{4} b c d +4 C \,a^{3} b^{2} d^{2}-6 C \,a^{2} b^{3} c d +2 C a \,b^{4} c^{2}\right ) \ln \left (a +b \tan \left (f x +e \right )\right )}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) b^{3} f}\) | \(745\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(2062\) |
| risch | \(\text {Expression too large to display}\) | \(2406\) |
Input:
int((c+d*tan(f*x+e))^2*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e))^2, x,method=_RETURNVERBOSE)
Output:
1/f*(tan(f*x+e)*C*d^2/b^2+1/(a^2+b^2)^2*(1/2*(2*A*a^2*c*d-2*A*a*b*c^2+2*A* a*b*d^2-2*A*b^2*c*d+B*a^2*c^2-B*a^2*d^2+4*B*a*b*c*d-B*b^2*c^2+B*b^2*d^2-2* C*a^2*c*d+2*C*a*b*c^2-2*C*a*b*d^2+2*C*b^2*c*d)*ln(1+tan(f*x+e)^2)+(A*a^2*c ^2-A*a^2*d^2+4*A*a*b*c*d-A*b^2*c^2+A*b^2*d^2-2*B*a^2*c*d+2*B*a*b*c^2-2*B*a *b*d^2+2*B*b^2*c*d-C*a^2*c^2+C*a^2*d^2-4*C*a*b*c*d+C*b^2*c^2-C*b^2*d^2)*ar ctan(tan(f*x+e)))-1/b^3*(A*a^2*b^2*d^2-2*A*a*b^3*c*d+A*b^4*c^2-B*a^3*b*d^2 +2*B*a^2*b^2*c*d-B*a*b^3*c^2+C*a^4*d^2-2*C*a^3*b*c*d+C*a^2*b^2*c^2)/(a^2+b ^2)/(a+b*tan(f*x+e))+(-2*A*a^2*b^3*c*d+2*A*a*b^4*c^2-2*A*a*b^4*d^2+2*A*b^5 *c*d+B*a^4*b*d^2-B*a^2*b^3*c^2+3*B*a^2*b^3*d^2-4*B*a*b^4*c*d+B*b^5*c^2-2*C *a^5*d^2+2*C*a^4*b*c*d-4*C*a^3*b^2*d^2+6*C*a^2*b^3*c*d-2*C*a*b^4*c^2)/b^3/ (a^2+b^2)^2*ln(a+b*tan(f*x+e)))
Leaf count of result is larger than twice the leaf count of optimal. 964 vs. \(2 (413) = 826\).
Time = 0.39 (sec) , antiderivative size = 964, normalized size of antiderivative = 2.32 \[ \int \frac {(c+d \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^2} \, dx =\text {Too large to display} \] Input:
integrate((c+d*tan(f*x+e))^2*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+ e))^2,x, algorithm="fricas")
Output:
1/2*(2*(C*a^4*b^2 + 2*C*a^2*b^4 + C*b^6)*d^2*tan(f*x + e)^2 - 2*(C*a^2*b^4 - B*a*b^5 + A*b^6)*c^2 + 4*(C*a^3*b^3 - B*a^2*b^4 + A*a*b^5)*c*d - 2*(C*a ^4*b^2 - B*a^3*b^3 + A*a^2*b^4)*d^2 + 2*(((A - C)*a^3*b^3 + 2*B*a^2*b^4 - (A - C)*a*b^5)*c^2 - 2*(B*a^3*b^3 - 2*(A - C)*a^2*b^4 - B*a*b^5)*c*d - ((A - C)*a^3*b^3 + 2*B*a^2*b^4 - (A - C)*a*b^5)*d^2)*f*x - ((B*a^3*b^3 - 2*(A - C)*a^2*b^4 - B*a*b^5)*c^2 - 2*(C*a^5*b - (A - 3*C)*a^3*b^3 - 2*B*a^2*b^ 4 + A*a*b^5)*c*d + (2*C*a^6 - B*a^5*b + 4*C*a^4*b^2 - 3*B*a^3*b^3 + 2*A*a^ 2*b^4)*d^2 + ((B*a^2*b^4 - 2*(A - C)*a*b^5 - B*b^6)*c^2 - 2*(C*a^4*b^2 - ( A - 3*C)*a^2*b^4 - 2*B*a*b^5 + A*b^6)*c*d + (2*C*a^5*b - B*a^4*b^2 + 4*C*a ^3*b^3 - 3*B*a^2*b^4 + 2*A*a*b^5)*d^2)*tan(f*x + e))*log((b^2*tan(f*x + e) ^2 + 2*a*b*tan(f*x + e) + a^2)/(tan(f*x + e)^2 + 1)) - (2*(C*a^5*b + 2*C*a ^3*b^3 + C*a*b^5)*c*d - (2*C*a^6 - B*a^5*b + 4*C*a^4*b^2 - 2*B*a^3*b^3 + 2 *C*a^2*b^4 - B*a*b^5)*d^2 + (2*(C*a^4*b^2 + 2*C*a^2*b^4 + C*b^6)*c*d - (2* C*a^5*b - B*a^4*b^2 + 4*C*a^3*b^3 - 2*B*a^2*b^4 + 2*C*a*b^5 - B*b^6)*d^2)* tan(f*x + e))*log(1/(tan(f*x + e)^2 + 1)) + 2*((C*a^3*b^3 - B*a^2*b^4 + A* a*b^5)*c^2 - 2*(C*a^4*b^2 - B*a^3*b^3 + A*a^2*b^4)*c*d + (2*C*a^5*b - B*a^ 4*b^2 + (A + 2*C)*a^3*b^3 + C*a*b^5)*d^2 + (((A - C)*a^2*b^4 + 2*B*a*b^5 - (A - C)*b^6)*c^2 - 2*(B*a^2*b^4 - 2*(A - C)*a*b^5 - B*b^6)*c*d - ((A - C) *a^2*b^4 + 2*B*a*b^5 - (A - C)*b^6)*d^2)*f*x)*tan(f*x + e))/((a^4*b^4 + 2* a^2*b^6 + b^8)*f*tan(f*x + e) + (a^5*b^3 + 2*a^3*b^5 + a*b^7)*f)
Result contains complex when optimal does not.
Time = 2.67 (sec) , antiderivative size = 16225, normalized size of antiderivative = 39.10 \[ \int \frac {(c+d \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^2} \, dx=\text {Too large to display} \] Input:
integrate((c+d*tan(f*x+e))**2*(A+B*tan(f*x+e)+C*tan(f*x+e)**2)/(a+b*tan(f* x+e))**2,x)
Output:
Piecewise((zoo*x*(c + d*tan(e))**2*(A + B*tan(e) + C*tan(e)**2)/tan(e)**2, Eq(a, 0) & Eq(b, 0) & Eq(f, 0)), ((A*c**2*x + A*c*d*log(tan(e + f*x)**2 + 1)/f - A*d**2*x + A*d**2*tan(e + f*x)/f + B*c**2*log(tan(e + f*x)**2 + 1) /(2*f) - 2*B*c*d*x + 2*B*c*d*tan(e + f*x)/f - B*d**2*log(tan(e + f*x)**2 + 1)/(2*f) + B*d**2*tan(e + f*x)**2/(2*f) - C*c**2*x + C*c**2*tan(e + f*x)/ f - C*c*d*log(tan(e + f*x)**2 + 1)/f + C*c*d*tan(e + f*x)**2/f + C*d**2*x + C*d**2*tan(e + f*x)**3/(3*f) - C*d**2*tan(e + f*x)/f)/a**2, Eq(b, 0)), ( -A*c**2*f*x*tan(e + f*x)**2/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) + 2*I*A*c**2*f*x*tan(e + f*x)/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) + A*c**2*f*x/(4*b**2*f*tan(e + f*x)* *2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) - A*c**2*tan(e + f*x)/(4*b**2*f*t an(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) + 2*I*A*c**2/(4*b**2* f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) + 2*I*A*c*d*f*x*ta n(e + f*x)**2/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2 *f) + 4*A*c*d*f*x*tan(e + f*x)/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan( e + f*x) - 4*b**2*f) - 2*I*A*c*d*f*x/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2* f*tan(e + f*x) - 4*b**2*f) + 2*I*A*c*d*tan(e + f*x)/(4*b**2*f*tan(e + f*x) **2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) + A*d**2*f*x*tan(e + f*x)**2/(4* b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) - 4*b**2*f) - 2*I*A*d**2* f*x*tan(e + f*x)/(4*b**2*f*tan(e + f*x)**2 - 8*I*b**2*f*tan(e + f*x) - ...
Time = 0.12 (sec) , antiderivative size = 496, normalized size of antiderivative = 1.20 \[ \int \frac {(c+d \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^2} \, dx=\frac {\frac {2 \, C d^{2} \tan \left (f x + e\right )}{b^{2}} + \frac {2 \, {\left ({\left ({\left (A - C\right )} a^{2} + 2 \, B a b - {\left (A - C\right )} b^{2}\right )} c^{2} - 2 \, {\left (B a^{2} - 2 \, {\left (A - C\right )} a b - B b^{2}\right )} c d - {\left ({\left (A - C\right )} a^{2} + 2 \, B a b - {\left (A - C\right )} b^{2}\right )} d^{2}\right )} {\left (f x + e\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, {\left ({\left (B a^{2} b^{3} - 2 \, {\left (A - C\right )} a b^{4} - B b^{5}\right )} c^{2} - 2 \, {\left (C a^{4} b - {\left (A - 3 \, C\right )} a^{2} b^{3} - 2 \, B a b^{4} + A b^{5}\right )} c d + {\left (2 \, C a^{5} - B a^{4} b + 4 \, C a^{3} b^{2} - 3 \, B a^{2} b^{3} + 2 \, A a b^{4}\right )} d^{2}\right )} \log \left (b \tan \left (f x + e\right ) + a\right )}{a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}} + \frac {{\left ({\left (B a^{2} - 2 \, {\left (A - C\right )} a b - B b^{2}\right )} c^{2} + 2 \, {\left ({\left (A - C\right )} a^{2} + 2 \, B a b - {\left (A - C\right )} b^{2}\right )} c d - {\left (B a^{2} - 2 \, {\left (A - C\right )} a b - B b^{2}\right )} d^{2}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, {\left ({\left (C a^{2} b^{2} - B a b^{3} + A b^{4}\right )} c^{2} - 2 \, {\left (C a^{3} b - B a^{2} b^{2} + A a b^{3}\right )} c d + {\left (C a^{4} - B a^{3} b + A a^{2} b^{2}\right )} d^{2}\right )}}{a^{3} b^{3} + a b^{5} + {\left (a^{2} b^{4} + b^{6}\right )} \tan \left (f x + e\right )}}{2 \, f} \] Input:
integrate((c+d*tan(f*x+e))^2*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+ e))^2,x, algorithm="maxima")
Output:
1/2*(2*C*d^2*tan(f*x + e)/b^2 + 2*(((A - C)*a^2 + 2*B*a*b - (A - C)*b^2)*c ^2 - 2*(B*a^2 - 2*(A - C)*a*b - B*b^2)*c*d - ((A - C)*a^2 + 2*B*a*b - (A - C)*b^2)*d^2)*(f*x + e)/(a^4 + 2*a^2*b^2 + b^4) - 2*((B*a^2*b^3 - 2*(A - C )*a*b^4 - B*b^5)*c^2 - 2*(C*a^4*b - (A - 3*C)*a^2*b^3 - 2*B*a*b^4 + A*b^5) *c*d + (2*C*a^5 - B*a^4*b + 4*C*a^3*b^2 - 3*B*a^2*b^3 + 2*A*a*b^4)*d^2)*lo g(b*tan(f*x + e) + a)/(a^4*b^3 + 2*a^2*b^5 + b^7) + ((B*a^2 - 2*(A - C)*a* b - B*b^2)*c^2 + 2*((A - C)*a^2 + 2*B*a*b - (A - C)*b^2)*c*d - (B*a^2 - 2* (A - C)*a*b - B*b^2)*d^2)*log(tan(f*x + e)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) - 2*((C*a^2*b^2 - B*a*b^3 + A*b^4)*c^2 - 2*(C*a^3*b - B*a^2*b^2 + A*a*b^3) *c*d + (C*a^4 - B*a^3*b + A*a^2*b^2)*d^2)/(a^3*b^3 + a*b^5 + (a^2*b^4 + b^ 6)*tan(f*x + e)))/f
Time = 0.75 (sec) , antiderivative size = 701, normalized size of antiderivative = 1.69 \[ \int \frac {(c+d \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^2} \, dx=\frac {{\left (A a^{2} c^{2} - C a^{2} c^{2} + 2 \, B a b c^{2} - A b^{2} c^{2} + C b^{2} c^{2} - 2 \, B a^{2} c d + 4 \, A a b c d - 4 \, C a b c d + 2 \, B b^{2} c d - A a^{2} d^{2} + C a^{2} d^{2} - 2 \, B a b d^{2} + A b^{2} d^{2} - C b^{2} d^{2}\right )} {\left (f x + e\right )}}{a^{4} f + 2 \, a^{2} b^{2} f + b^{4} f} + \frac {{\left (B a^{2} c^{2} - 2 \, A a b c^{2} + 2 \, C a b c^{2} - B b^{2} c^{2} + 2 \, A a^{2} c d - 2 \, C a^{2} c d + 4 \, B a b c d - 2 \, A b^{2} c d + 2 \, C b^{2} c d - B a^{2} d^{2} + 2 \, A a b d^{2} - 2 \, C a b d^{2} + B b^{2} d^{2}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{2 \, {\left (a^{4} f + 2 \, a^{2} b^{2} f + b^{4} f\right )}} - \frac {{\left (B a^{2} b^{3} c^{2} - 2 \, A a b^{4} c^{2} + 2 \, C a b^{4} c^{2} - B b^{5} c^{2} - 2 \, C a^{4} b c d + 2 \, A a^{2} b^{3} c d - 6 \, C a^{2} b^{3} c d + 4 \, B a b^{4} c d - 2 \, A b^{5} c d + 2 \, C a^{5} d^{2} - B a^{4} b d^{2} + 4 \, C a^{3} b^{2} d^{2} - 3 \, B a^{2} b^{3} d^{2} + 2 \, A a b^{4} d^{2}\right )} \log \left ({\left | b \tan \left (f x + e\right ) + a \right |}\right )}{a^{4} b^{3} f + 2 \, a^{2} b^{5} f + b^{7} f} + \frac {C d^{2} \tan \left (f x + e\right )}{b^{2} f} - \frac {C a^{4} b^{2} c^{2} - B a^{3} b^{3} c^{2} + A a^{2} b^{4} c^{2} + C a^{2} b^{4} c^{2} - B a b^{5} c^{2} + A b^{6} c^{2} - 2 \, C a^{5} b c d + 2 \, B a^{4} b^{2} c d - 2 \, A a^{3} b^{3} c d - 2 \, C a^{3} b^{3} c d + 2 \, B a^{2} b^{4} c d - 2 \, A a b^{5} c d + C a^{6} d^{2} - B a^{5} b d^{2} + A a^{4} b^{2} d^{2} + C a^{4} b^{2} d^{2} - B a^{3} b^{3} d^{2} + A a^{2} b^{4} d^{2}}{{\left (a^{2} + b^{2}\right )}^{2} {\left (b \tan \left (f x + e\right ) + a\right )} b^{3} f} \] Input:
integrate((c+d*tan(f*x+e))^2*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+ e))^2,x, algorithm="giac")
Output:
(A*a^2*c^2 - C*a^2*c^2 + 2*B*a*b*c^2 - A*b^2*c^2 + C*b^2*c^2 - 2*B*a^2*c*d + 4*A*a*b*c*d - 4*C*a*b*c*d + 2*B*b^2*c*d - A*a^2*d^2 + C*a^2*d^2 - 2*B*a *b*d^2 + A*b^2*d^2 - C*b^2*d^2)*(f*x + e)/(a^4*f + 2*a^2*b^2*f + b^4*f) + 1/2*(B*a^2*c^2 - 2*A*a*b*c^2 + 2*C*a*b*c^2 - B*b^2*c^2 + 2*A*a^2*c*d - 2*C *a^2*c*d + 4*B*a*b*c*d - 2*A*b^2*c*d + 2*C*b^2*c*d - B*a^2*d^2 + 2*A*a*b*d ^2 - 2*C*a*b*d^2 + B*b^2*d^2)*log(tan(f*x + e)^2 + 1)/(a^4*f + 2*a^2*b^2*f + b^4*f) - (B*a^2*b^3*c^2 - 2*A*a*b^4*c^2 + 2*C*a*b^4*c^2 - B*b^5*c^2 - 2 *C*a^4*b*c*d + 2*A*a^2*b^3*c*d - 6*C*a^2*b^3*c*d + 4*B*a*b^4*c*d - 2*A*b^5 *c*d + 2*C*a^5*d^2 - B*a^4*b*d^2 + 4*C*a^3*b^2*d^2 - 3*B*a^2*b^3*d^2 + 2*A *a*b^4*d^2)*log(abs(b*tan(f*x + e) + a))/(a^4*b^3*f + 2*a^2*b^5*f + b^7*f) + C*d^2*tan(f*x + e)/(b^2*f) - (C*a^4*b^2*c^2 - B*a^3*b^3*c^2 + A*a^2*b^4 *c^2 + C*a^2*b^4*c^2 - B*a*b^5*c^2 + A*b^6*c^2 - 2*C*a^5*b*c*d + 2*B*a^4*b ^2*c*d - 2*A*a^3*b^3*c*d - 2*C*a^3*b^3*c*d + 2*B*a^2*b^4*c*d - 2*A*a*b^5*c *d + C*a^6*d^2 - B*a^5*b*d^2 + A*a^4*b^2*d^2 + C*a^4*b^2*d^2 - B*a^3*b^3*d ^2 + A*a^2*b^4*d^2)/((a^2 + b^2)^2*(b*tan(f*x + e) + a)*b^3*f)
Time = 29.77 (sec) , antiderivative size = 3958, normalized size of antiderivative = 9.54 \[ \int \frac {(c+d \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^2} \, dx=\text {Too large to display} \] Input:
int(((c + d*tan(e + f*x))^2*(A + B*tan(e + f*x) + C*tan(e + f*x)^2))/(a + b*tan(e + f*x))^2,x)
Output:
(log((2*C^2*a^5*d^4 + 4*C^2*a^3*b^2*d^4 - 2*C^2*a^5*c^2*d^2 - A*B*b^5*c^4 - 2*A*C*a^5*d^4 + B*C*b^5*c^4 - A^2*a*b^4*c^4 - A^2*a*b^4*d^4 + B^2*a*b^4* c^4 + B^2*a*b^4*d^4 - C^2*a*b^4*c^4 + 2*A^2*b^5*c*d^3 - 2*A^2*b^5*c^3*d + C^2*a*b^4*d^4 + 2*B^2*b^5*c^3*d - 4*C^2*a^3*b^2*c^2*d^2 + A*B*a^2*b^3*c^4 + 3*A*B*a^2*b^3*d^4 - 4*A*C*a^3*b^2*d^4 - B*C*a^2*b^3*c^4 + 5*A*B*b^5*c^2* d^2 + 2*A*C*a^5*c^2*d^2 - 3*B*C*a^2*b^3*d^4 - B*C*b^5*c^2*d^2 + 2*B^2*a^4* b*c*d^3 - 2*C^2*a^4*b*c*d^3 + 2*C^2*a^4*b*c^3*d + 6*A^2*a*b^4*c^2*d^2 - 2* A^2*a^2*b^3*c*d^3 + 2*A^2*a^2*b^3*c^3*d - 6*B^2*a*b^4*c^2*d^2 + 6*B^2*a^2* b^3*c*d^3 - 2*B^2*a^2*b^3*c^3*d + 4*C^2*a*b^4*c^2*d^2 - 6*C^2*a^2*b^3*c*d^ 3 + 6*C^2*a^2*b^3*c^3*d + A*B*a^4*b*d^4 + 2*A*C*a*b^4*c^4 - B*C*a^4*b*d^4 - 2*A*C*b^5*c*d^3 + 2*A*C*b^5*c^3*d - 4*B*C*a^5*c*d^3 - 8*A*B*a*b^4*c*d^3 + 8*A*B*a*b^4*c^3*d + 2*A*C*a^4*b*c*d^3 - 2*A*C*a^4*b*c^3*d + 4*B*C*a*b^4* c*d^3 - 8*B*C*a*b^4*c^3*d - A*B*a^4*b*c^2*d^2 - 10*A*C*a*b^4*c^2*d^2 + 8*A *C*a^2*b^3*c*d^3 - 8*A*C*a^2*b^3*c^3*d - 8*B*C*a^3*b^2*c*d^3 + 5*B*C*a^4*b *c^2*d^2 - 8*A*B*a^2*b^3*c^2*d^2 + 4*A*C*a^3*b^2*c^2*d^2 + 16*B*C*a^2*b^3* c^2*d^2)/(b^2*(a^2 + b^2)^2) + ((c*1i + d)^2*((tan(e + f*x)*(3*B*b^5*c^2 - 5*B*b^5*d^2 - 4*C*a^5*d^2 + 6*A*b^5*c*d - 10*C*b^5*c*d + 4*A*a*b^4*c^2 - 4*A*a*b^4*d^2 + 2*B*a^4*b*d^2 - 4*C*a*b^4*c^2 + 8*C*a*b^4*d^2 - B*a^2*b^3* c^2 + B*a^2*b^3*d^2 - 8*B*a*b^4*c*d + 4*C*a^4*b*c*d - 2*A*a^2*b^3*c*d + 2* C*a^2*b^3*c*d))/(b^2*(a^2 + b^2)) - (A*b^2*d^2 - A*b^2*c^2 - 8*C*a^2*d^...
Time = 0.17 (sec) , antiderivative size = 1586, normalized size of antiderivative = 3.82 \[ \int \frac {(c+d \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^2} \, dx =\text {Too large to display} \] Input:
int((c+d*tan(f*x+e))^2*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e))^2, x)
Output:
(2*log(tan(e + f*x)**2 + 1)*tan(e + f*x)*a**3*b**4*c*d - log(tan(e + f*x)* *2 + 1)*tan(e + f*x)*a**2*b**5*c**2 + log(tan(e + f*x)**2 + 1)*tan(e + f*x )*a**2*b**5*d**2 - 2*log(tan(e + f*x)**2 + 1)*tan(e + f*x)*a**2*b**4*c**2* d + 2*log(tan(e + f*x)**2 + 1)*tan(e + f*x)*a*b**6*c*d + 2*log(tan(e + f*x )**2 + 1)*tan(e + f*x)*a*b**5*c**3 - 2*log(tan(e + f*x)**2 + 1)*tan(e + f* x)*a*b**5*c*d**2 - log(tan(e + f*x)**2 + 1)*tan(e + f*x)*b**7*c**2 + log(t an(e + f*x)**2 + 1)*tan(e + f*x)*b**7*d**2 + 2*log(tan(e + f*x)**2 + 1)*ta n(e + f*x)*b**6*c**2*d + 2*log(tan(e + f*x)**2 + 1)*a**4*b**3*c*d - log(ta n(e + f*x)**2 + 1)*a**3*b**4*c**2 + log(tan(e + f*x)**2 + 1)*a**3*b**4*d** 2 - 2*log(tan(e + f*x)**2 + 1)*a**3*b**3*c**2*d + 2*log(tan(e + f*x)**2 + 1)*a**2*b**5*c*d + 2*log(tan(e + f*x)**2 + 1)*a**2*b**4*c**3 - 2*log(tan(e + f*x)**2 + 1)*a**2*b**4*c*d**2 - log(tan(e + f*x)**2 + 1)*a*b**6*c**2 + log(tan(e + f*x)**2 + 1)*a*b**6*d**2 + 2*log(tan(e + f*x)**2 + 1)*a*b**5*c **2*d - 4*log(tan(e + f*x)*b + a)*tan(e + f*x)*a**5*b*c*d**2 + 2*log(tan(e + f*x)*b + a)*tan(e + f*x)*a**4*b**3*d**2 + 4*log(tan(e + f*x)*b + a)*tan (e + f*x)*a**4*b**2*c**2*d - 4*log(tan(e + f*x)*b + a)*tan(e + f*x)*a**3*b **4*c*d - 8*log(tan(e + f*x)*b + a)*tan(e + f*x)*a**3*b**3*c*d**2 + 2*log( tan(e + f*x)*b + a)*tan(e + f*x)*a**2*b**5*c**2 + 2*log(tan(e + f*x)*b + a )*tan(e + f*x)*a**2*b**5*d**2 + 12*log(tan(e + f*x)*b + a)*tan(e + f*x)*a* *2*b**4*c**2*d - 4*log(tan(e + f*x)*b + a)*tan(e + f*x)*a*b**6*c*d - 4*...