Integrand size = 40, antiderivative size = 250 \[ \int \frac {\tan ^2(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx=-\frac {\left (3 a^2 b B-b^3 B-a^3 C+3 a b^2 C\right ) x}{\left (a^2+b^2\right )^3}+\frac {\left (a^3 B-3 a b^2 B+3 a^2 b C-b^3 C\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac {a \left (a^2 b^3 B-3 b^5 B+a^5 C+3 a^3 b^2 C+6 a b^4 C\right ) \log (a+b \tan (c+d x))}{b^3 \left (a^2+b^2\right )^3 d}+\frac {a (b B-a C) \tan ^2(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (2 b^3 B-a^3 C-3 a b^2 C\right )}{b^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \]
-(3*B*a^2*b-B*b^3-C*a^3+3*C*a*b^2)*x/(a^2+b^2)^3+(B*a^3-3*B*a*b^2+3*C*a^2* b-C*b^3)*ln(cos(d*x+c))/(a^2+b^2)^3/d+a*(B*a^2*b^3-3*B*b^5+C*a^5+3*C*a^3*b ^2+6*C*a*b^4)*ln(a+b*tan(d*x+c))/b^3/(a^2+b^2)^3/d+1/2*a*(B*b-C*a)*tan(d*x +c)^2/b/(a^2+b^2)/d/(a+b*tan(d*x+c))^2-a^2*(2*B*b^3-C*a^3-3*C*a*b^2)/b^3/( a^2+b^2)^2/d/(a+b*tan(d*x+c))
Result contains complex when optimal does not.
Time = 6.39 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.94 \[ \int \frac {\tan ^2(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx=-\frac {(B+i C) \log (i-\tan (c+d x))}{2 (a+i b)^3 d}-\frac {(B-i C) \log (i+\tan (c+d x))}{2 (a-i b)^3 d}+\frac {a \left (a^2 b^3 B-3 b^5 B+a^5 C+3 a^3 b^2 C+6 a b^4 C\right ) \log (a+b \tan (c+d x))}{b^3 \left (a^2+b^2\right )^3 d}+\frac {a^3 (b B-a C)}{2 b^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (a^2 b B+3 b^3 B-2 a^3 C-4 a b^2 C\right )}{b^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \]
-1/2*((B + I*C)*Log[I - Tan[c + d*x]])/((a + I*b)^3*d) - ((B - I*C)*Log[I + Tan[c + d*x]])/(2*(a - I*b)^3*d) + (a*(a^2*b^3*B - 3*b^5*B + a^5*C + 3*a ^3*b^2*C + 6*a*b^4*C)*Log[a + b*Tan[c + d*x]])/(b^3*(a^2 + b^2)^3*d) + (a^ 3*(b*B - a*C))/(2*b^3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) - (a^2*(a^2*b* B + 3*b^3*B - 2*a^3*C - 4*a*b^2*C))/(b^3*(a^2 + b^2)^2*d*(a + b*Tan[c + d* x]))
Time = 1.42 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.14, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.325, Rules used = {3042, 4115, 3042, 4088, 27, 3042, 4118, 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.
| \(\displaystyle \int \frac {\tan ^2(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan (c+d x)^2 \left (B \tan (c+d x)+C \tan (c+d x)^2\right )}{(a+b \tan (c+d x))^3}dx\) |
| \(\Big \downarrow \) 4115 |
| \(\displaystyle \int \frac {\tan ^3(c+d x) (B+C \tan (c+d x))}{(a+b \tan (c+d x))^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan (c+d x)^3 (B+C \tan (c+d x))}{(a+b \tan (c+d x))^3}dx\) |
| \(\Big \downarrow \) 4088 |
| \(\displaystyle \frac {\int -\frac {2 \tan (c+d x) \left (-\left (\left (a^2+b^2\right ) C \tan ^2(c+d x)\right )-b (b B-a C) \tan (c+d x)+a (b B-a C)\right )}{(a+b \tan (c+d x))^2}dx}{2 b \left (a^2+b^2\right )}+\frac {a (b B-a C) \tan ^2(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a (b B-a C) \tan ^2(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\int \frac {\tan (c+d x) \left (-\left (\left (a^2+b^2\right ) C \tan ^2(c+d x)\right )-b (b B-a C) \tan (c+d x)+a (b B-a C)\right )}{(a+b \tan (c+d x))^2}dx}{b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a (b B-a C) \tan ^2(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\int \frac {\tan (c+d x) \left (-\left (\left (a^2+b^2\right ) C \tan (c+d x)^2\right )-b (b B-a C) \tan (c+d x)+a (b B-a C)\right )}{(a+b \tan (c+d x))^2}dx}{b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 4118 |
| \(\displaystyle \frac {a (b B-a C) \tan ^2(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\int \frac {\left (B a^2+2 b C a-b^2 B\right ) \tan (c+d x) b^2-\left (a^2+b^2\right )^2 C \tan ^2(c+d x)+a \left (-C a^3-3 b^2 C a+2 b^3 B\right )}{a+b \tan (c+d x)}dx}{b \left (a^2+b^2\right )}+\frac {a^2 \left (a^3 (-C)-3 a b^2 C+2 b^3 B\right )}{b^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a (b B-a C) \tan ^2(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\int \frac {\left (B a^2+2 b C a-b^2 B\right ) \tan (c+d x) b^2-\left (a^2+b^2\right )^2 C \tan (c+d x)^2+a \left (-C a^3-3 b^2 C a+2 b^3 B\right )}{a+b \tan (c+d x)}dx}{b \left (a^2+b^2\right )}+\frac {a^2 \left (a^3 (-C)-3 a b^2 C+2 b^3 B\right )}{b^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 4109 |
| \(\displaystyle \frac {a (b B-a C) \tan ^2(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\frac {b^2 \left (a^3 B+3 a^2 b C-3 a b^2 B-b^3 C\right ) \int \tan (c+d x)dx}{a^2+b^2}-\frac {a \left (a^5 C+3 a^3 b^2 C+a^2 b^3 B+6 a b^4 C-3 b^5 B\right ) \int \frac {\tan ^2(c+d x)+1}{a+b \tan (c+d x)}dx}{a^2+b^2}+\frac {b^2 x \left (a^3 (-C)+3 a^2 b B+3 a b^2 C-b^3 B\right )}{a^2+b^2}}{b \left (a^2+b^2\right )}+\frac {a^2 \left (a^3 (-C)-3 a b^2 C+2 b^3 B\right )}{b^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a (b B-a C) \tan ^2(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\frac {b^2 \left (a^3 B+3 a^2 b C-3 a b^2 B-b^3 C\right ) \int \tan (c+d x)dx}{a^2+b^2}-\frac {a \left (a^5 C+3 a^3 b^2 C+a^2 b^3 B+6 a b^4 C-3 b^5 B\right ) \int \frac {\tan (c+d x)^2+1}{a+b \tan (c+d x)}dx}{a^2+b^2}+\frac {b^2 x \left (a^3 (-C)+3 a^2 b B+3 a b^2 C-b^3 B\right )}{a^2+b^2}}{b \left (a^2+b^2\right )}+\frac {a^2 \left (a^3 (-C)-3 a b^2 C+2 b^3 B\right )}{b^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {a (b B-a C) \tan ^2(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {-\frac {a \left (a^5 C+3 a^3 b^2 C+a^2 b^3 B+6 a b^4 C-3 b^5 B\right ) \int \frac {\tan (c+d x)^2+1}{a+b \tan (c+d x)}dx}{a^2+b^2}-\frac {b^2 \left (a^3 B+3 a^2 b C-3 a b^2 B-b^3 C\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )}+\frac {b^2 x \left (a^3 (-C)+3 a^2 b B+3 a b^2 C-b^3 B\right )}{a^2+b^2}}{b \left (a^2+b^2\right )}+\frac {a^2 \left (a^3 (-C)-3 a b^2 C+2 b^3 B\right )}{b^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 4100 |
| \(\displaystyle \frac {a (b B-a C) \tan ^2(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {-\frac {a \left (a^5 C+3 a^3 b^2 C+a^2 b^3 B+6 a b^4 C-3 b^5 B\right ) \int \frac {1}{a+b \tan (c+d x)}d(b \tan (c+d x))}{b d \left (a^2+b^2\right )}-\frac {b^2 \left (a^3 B+3 a^2 b C-3 a b^2 B-b^3 C\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )}+\frac {b^2 x \left (a^3 (-C)+3 a^2 b B+3 a b^2 C-b^3 B\right )}{a^2+b^2}}{b \left (a^2+b^2\right )}+\frac {a^2 \left (a^3 (-C)-3 a b^2 C+2 b^3 B\right )}{b^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {a (b B-a C) \tan ^2(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {a^2 \left (a^3 (-C)-3 a b^2 C+2 b^3 B\right )}{b^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {-\frac {b^2 \left (a^3 B+3 a^2 b C-3 a b^2 B-b^3 C\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )}+\frac {b^2 x \left (a^3 (-C)+3 a^2 b B+3 a b^2 C-b^3 B\right )}{a^2+b^2}-\frac {a \left (a^5 C+3 a^3 b^2 C+a^2 b^3 B+6 a b^4 C-3 b^5 B\right ) \log (a+b \tan (c+d x))}{b d \left (a^2+b^2\right )}}{b \left (a^2+b^2\right )}}{b \left (a^2+b^2\right )}\) |
(a*(b*B - a*C)*Tan[c + d*x]^2)/(2*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) - (((b^2*(3*a^2*b*B - b^3*B - a^3*C + 3*a*b^2*C)*x)/(a^2 + b^2) - (b^2*(a^ 3*B - 3*a*b^2*B + 3*a^2*b*C - b^3*C)*Log[Cos[c + d*x]])/((a^2 + b^2)*d) - (a*(a^2*b^3*B - 3*b^5*B + a^5*C + 3*a^3*b^2*C + 6*a*b^4*C)*Log[a + b*Tan[c + d*x]])/(b*(a^2 + b^2)*d))/(b*(a^2 + b^2)) + (a^2*(2*b^3*B - a^3*C - 3*a *b^2*C))/(b^2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])))/(b*(a^2 + b^2))
3.1.39.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
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_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[(b*c - a*d)*(B*c - A*d)*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e + f*x ])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Simp[1/(d*(n + 1)*(c^2 + d^2)) Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a*A*d* (b*d*(m - 1) - a*c*(n + 1)) + (b*B*c - (A*b + a*B)*d)*(b*c*(m - 1) + a*d*(n + 1)) - d*((a*A - b*B)*(b*c - a*d) + (A*b + a*B)*(a*c + b*d))*(n + 1)*Tan[ e + f*x] - b*(d*(A*b*c + a*B*c - a*A*d)*(m + n) - b*B*(c^2*(m - 1) - d^2*(n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[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_) + (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_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_ .) + (f_.)*(x_)]^2), x_Symbol] :> Simp[1/b^2 Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*(b*B - a*C + b*C*Tan[e + f*x]), x], x] /; FreeQ [{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[A*b^2 - a*b*B + a^2*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 - a*d))*(c^2*C - B*c*d + A*d^2)*((c + d*Tan[e + f*x])^(n + 1)/(d^2*f*(n + 1)*(c^2 + d^2))), x] + Simp[1/(d*(c^2 + d^2)) Int[(c + d*Tan[e + f*x])^(n + 1)*Simp[a*d*(A*c - c*C + B*d) + b* (c^2*C - B*c*d + A*d^2) + d*(A*b*c + a*B*c - b*c*C - a*A*d + b*B*d + a*C*d) *Tan[e + f*x] + b*C*(c^2 + d^2)*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[c^2 + d^2, 0] && LtQ[n , -1]
Time = 0.15 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.97
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\left (-B \,a^{3}+3 B a \,b^{2}-3 C \,a^{2} b +C \,b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-3 B \,a^{2} b +B \,b^{3}+C \,a^{3}-3 C a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}+\frac {a \left (B \,a^{2} b^{3}-3 B \,b^{5}+C \,a^{5}+3 C \,a^{3} b^{2}+6 C a \,b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3} b^{3}}-\frac {a^{2} \left (B \,a^{2} b +3 B \,b^{3}-2 C \,a^{3}-4 C a \,b^{2}\right )}{b^{3} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {a^{3} \left (B b -C a \right )}{2 b^{3} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}}{d}\) | \(242\) |
| default | \(\frac {\frac {\frac {\left (-B \,a^{3}+3 B a \,b^{2}-3 C \,a^{2} b +C \,b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-3 B \,a^{2} b +B \,b^{3}+C \,a^{3}-3 C a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}+\frac {a \left (B \,a^{2} b^{3}-3 B \,b^{5}+C \,a^{5}+3 C \,a^{3} b^{2}+6 C a \,b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3} b^{3}}-\frac {a^{2} \left (B \,a^{2} b +3 B \,b^{3}-2 C \,a^{3}-4 C a \,b^{2}\right )}{b^{3} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {a^{3} \left (B b -C a \right )}{2 b^{3} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}}{d}\) | \(242\) |
| norman | \(\frac {-\frac {a^{2} \left (B \,a^{3} b +5 B a \,b^{3}-3 C \,a^{4}-7 C \,a^{2} b^{2}\right )}{2 d \,b^{3} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (3 B \,a^{2} b -B \,b^{3}-C \,a^{3}+3 C a \,b^{2}\right ) a^{2} x}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}-\frac {b^{2} \left (3 B \,a^{2} b -B \,b^{3}-C \,a^{3}+3 C a \,b^{2}\right ) x \tan \left (d x +c \right )^{2}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}-\frac {a \left (B \,a^{3} b +3 B a \,b^{3}-2 C \,a^{4}-4 C \,a^{2} b^{2}\right ) \tan \left (d x +c \right )}{d \,b^{2} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {2 b \left (3 B \,a^{2} b -B \,b^{3}-C \,a^{3}+3 C a \,b^{2}\right ) a x \tan \left (d x +c \right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a \left (B \,a^{2} b^{3}-3 B \,b^{5}+C \,a^{5}+3 C \,a^{3} b^{2}+6 C a \,b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d \,b^{3}}-\frac {\left (B \,a^{3}-3 B a \,b^{2}+3 C \,a^{2} b -C \,b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}\) | \(471\) |
| risch | \(-\frac {6 i a^{4} C x}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) b}-\frac {x C}{3 i a^{2} b -i b^{3}-a^{3}+3 a \,b^{2}}+\frac {2 i C c}{d \,b^{3}}+\frac {2 i C x}{b^{3}}-\frac {2 i a^{3} B c}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d}-\frac {2 i a^{6} C c}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d \,b^{3}}+\frac {6 i a \,b^{2} B c}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d}-\frac {12 i a^{2} b C c}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d}+\frac {6 i a \,b^{2} B x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {2 i a^{3} B x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {12 i a^{2} b C x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {2 i a^{2} \left (2 i B a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-i C \,a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-3 i C \,a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+B \,a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+3 B \,b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-2 C \,a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}-4 C a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+3 i B a \,b^{3}-i C \,a^{4}-4 i C \,a^{2} b^{2}-3 B \,b^{4}+C \,a^{3} b +4 C a \,b^{3}\right )}{\left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right )^{2} \left (-i a +b \right )^{2} b^{2} d \left (i a +b \right )^{3}}-\frac {i x B}{3 i a^{2} b -i b^{3}-a^{3}+3 a \,b^{2}}-\frac {2 i a^{6} C x}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) b^{3}}-\frac {6 i a^{4} C c}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d b}-\frac {C \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d \,b^{3}}+\frac {a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) B}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d}-\frac {3 a \,b^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) B}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d}+\frac {a^{6} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) C}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d \,b^{3}}+\frac {3 a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) C}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d b}+\frac {6 a^{2} b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) C}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d}\) | \(1010\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1014\) |
1/d*(1/(a^2+b^2)^3*(1/2*(-B*a^3+3*B*a*b^2-3*C*a^2*b+C*b^3)*ln(1+tan(d*x+c) ^2)+(-3*B*a^2*b+B*b^3+C*a^3-3*C*a*b^2)*arctan(tan(d*x+c)))+a*(B*a^2*b^3-3* B*b^5+C*a^5+3*C*a^3*b^2+6*C*a*b^4)/(a^2+b^2)^3/b^3*ln(a+b*tan(d*x+c))-a^2* (B*a^2*b+3*B*b^3-2*C*a^3-4*C*a*b^2)/b^3/(a^2+b^2)^2/(a+b*tan(d*x+c))+1/2*a ^3*(B*b-C*a)/b^3/(a^2+b^2)/(a+b*tan(d*x+c))^2)
Leaf count of result is larger than twice the leaf count of optimal. 666 vs. \(2 (243) = 486\).
Time = 0.31 (sec) , antiderivative size = 666, normalized size of antiderivative = 2.66 \[ \int \frac {\tan ^2(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx=\frac {C a^{6} b^{2} + B a^{5} b^{3} + 7 \, C a^{4} b^{4} - 5 \, B a^{3} b^{5} + 2 \, {\left (C a^{5} b^{3} - 3 \, B a^{4} b^{4} - 3 \, C a^{3} b^{5} + B a^{2} b^{6}\right )} d x - {\left (3 \, C a^{6} b^{2} - B a^{5} b^{3} + 9 \, C a^{4} b^{4} - 7 \, B a^{3} b^{5} - 2 \, {\left (C a^{3} b^{5} - 3 \, B a^{2} b^{6} - 3 \, C a b^{7} + B b^{8}\right )} d x\right )} \tan \left (d x + c\right )^{2} + {\left (C a^{8} + 3 \, C a^{6} b^{2} + B a^{5} b^{3} + 6 \, C a^{4} b^{4} - 3 \, B a^{3} b^{5} + {\left (C a^{6} b^{2} + 3 \, C a^{4} b^{4} + B a^{3} b^{5} + 6 \, C a^{2} b^{6} - 3 \, B a b^{7}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (C a^{7} b + 3 \, C a^{5} b^{3} + B a^{4} b^{4} + 6 \, C a^{3} b^{5} - 3 \, B a^{2} b^{6}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - {\left (C a^{8} + 3 \, C a^{6} b^{2} + 3 \, C a^{4} b^{4} + C a^{2} b^{6} + {\left (C a^{6} b^{2} + 3 \, C a^{4} b^{4} + 3 \, C a^{2} b^{6} + C b^{8}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (C a^{7} b + 3 \, C a^{5} b^{3} + 3 \, C a^{3} b^{5} + C a b^{7}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \, {\left (C a^{7} b + 3 \, C a^{5} b^{3} - 3 \, B a^{4} b^{4} - 4 \, C a^{3} b^{5} + 3 \, B a^{2} b^{6} - 2 \, {\left (C a^{4} b^{4} - 3 \, B a^{3} b^{5} - 3 \, C a^{2} b^{6} + B a b^{7}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{6} b^{5} + 3 \, a^{4} b^{7} + 3 \, a^{2} b^{9} + b^{11}\right )} d \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b^{4} + 3 \, a^{5} b^{6} + 3 \, a^{3} b^{8} + a b^{10}\right )} d \tan \left (d x + c\right ) + {\left (a^{8} b^{3} + 3 \, a^{6} b^{5} + 3 \, a^{4} b^{7} + a^{2} b^{9}\right )} d\right )}} \]
integrate(tan(d*x+c)^2*(B*tan(d*x+c)+C*tan(d*x+c)^2)/(a+b*tan(d*x+c))^3,x, algorithm="fricas")
1/2*(C*a^6*b^2 + B*a^5*b^3 + 7*C*a^4*b^4 - 5*B*a^3*b^5 + 2*(C*a^5*b^3 - 3* B*a^4*b^4 - 3*C*a^3*b^5 + B*a^2*b^6)*d*x - (3*C*a^6*b^2 - B*a^5*b^3 + 9*C* a^4*b^4 - 7*B*a^3*b^5 - 2*(C*a^3*b^5 - 3*B*a^2*b^6 - 3*C*a*b^7 + B*b^8)*d* x)*tan(d*x + c)^2 + (C*a^8 + 3*C*a^6*b^2 + B*a^5*b^3 + 6*C*a^4*b^4 - 3*B*a ^3*b^5 + (C*a^6*b^2 + 3*C*a^4*b^4 + B*a^3*b^5 + 6*C*a^2*b^6 - 3*B*a*b^7)*t an(d*x + c)^2 + 2*(C*a^7*b + 3*C*a^5*b^3 + B*a^4*b^4 + 6*C*a^3*b^5 - 3*B*a ^2*b^6)*tan(d*x + c))*log((b^2*tan(d*x + c)^2 + 2*a*b*tan(d*x + c) + a^2)/ (tan(d*x + c)^2 + 1)) - (C*a^8 + 3*C*a^6*b^2 + 3*C*a^4*b^4 + C*a^2*b^6 + ( C*a^6*b^2 + 3*C*a^4*b^4 + 3*C*a^2*b^6 + C*b^8)*tan(d*x + c)^2 + 2*(C*a^7*b + 3*C*a^5*b^3 + 3*C*a^3*b^5 + C*a*b^7)*tan(d*x + c))*log(1/(tan(d*x + c)^ 2 + 1)) - 2*(C*a^7*b + 3*C*a^5*b^3 - 3*B*a^4*b^4 - 4*C*a^3*b^5 + 3*B*a^2*b ^6 - 2*(C*a^4*b^4 - 3*B*a^3*b^5 - 3*C*a^2*b^6 + B*a*b^7)*d*x)*tan(d*x + c) )/((a^6*b^5 + 3*a^4*b^7 + 3*a^2*b^9 + b^11)*d*tan(d*x + c)^2 + 2*(a^7*b^4 + 3*a^5*b^6 + 3*a^3*b^8 + a*b^10)*d*tan(d*x + c) + (a^8*b^3 + 3*a^6*b^5 + 3*a^4*b^7 + a^2*b^9)*d)
Exception generated. \[ \int \frac {\tan ^2(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx=\text {Exception raised: AttributeError} \]
Time = 0.37 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.46 \[ \int \frac {\tan ^2(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {2 \, {\left (C a^{3} - 3 \, B a^{2} b - 3 \, C a b^{2} + B b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left (C a^{6} + 3 \, C a^{4} b^{2} + B a^{3} b^{3} + 6 \, C a^{2} b^{4} - 3 \, B a b^{5}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{6} b^{3} + 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} + b^{9}} - \frac {{\left (B a^{3} + 3 \, C a^{2} b - 3 \, B a b^{2} - C b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {3 \, C a^{6} - B a^{5} b + 7 \, C a^{4} b^{2} - 5 \, B a^{3} b^{3} + 2 \, {\left (2 \, C a^{5} b - B a^{4} b^{2} + 4 \, C a^{3} b^{3} - 3 \, B a^{2} b^{4}\right )} \tan \left (d x + c\right )}{a^{6} b^{3} + 2 \, a^{4} b^{5} + a^{2} b^{7} + {\left (a^{4} b^{5} + 2 \, a^{2} b^{7} + b^{9}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b^{4} + 2 \, a^{3} b^{6} + a b^{8}\right )} \tan \left (d x + c\right )}}{2 \, d} \]
integrate(tan(d*x+c)^2*(B*tan(d*x+c)+C*tan(d*x+c)^2)/(a+b*tan(d*x+c))^3,x, algorithm="maxima")
1/2*(2*(C*a^3 - 3*B*a^2*b - 3*C*a*b^2 + B*b^3)*(d*x + c)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 2*(C*a^6 + 3*C*a^4*b^2 + B*a^3*b^3 + 6*C*a^2*b^4 - 3* B*a*b^5)*log(b*tan(d*x + c) + a)/(a^6*b^3 + 3*a^4*b^5 + 3*a^2*b^7 + b^9) - (B*a^3 + 3*C*a^2*b - 3*B*a*b^2 - C*b^3)*log(tan(d*x + c)^2 + 1)/(a^6 + 3* a^4*b^2 + 3*a^2*b^4 + b^6) + (3*C*a^6 - B*a^5*b + 7*C*a^4*b^2 - 5*B*a^3*b^ 3 + 2*(2*C*a^5*b - B*a^4*b^2 + 4*C*a^3*b^3 - 3*B*a^2*b^4)*tan(d*x + c))/(a ^6*b^3 + 2*a^4*b^5 + a^2*b^7 + (a^4*b^5 + 2*a^2*b^7 + b^9)*tan(d*x + c)^2 + 2*(a^5*b^4 + 2*a^3*b^6 + a*b^8)*tan(d*x + c)))/d
Time = 0.83 (sec) , antiderivative size = 458, normalized size of antiderivative = 1.83 \[ \int \frac {\tan ^2(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {2 \, {\left (C a^{3} - 3 \, B a^{2} b - 3 \, C a b^{2} + B b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (B a^{3} + 3 \, C a^{2} b - 3 \, B a b^{2} - C b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left (C a^{6} + 3 \, C a^{4} b^{2} + B a^{3} b^{3} + 6 \, C a^{2} b^{4} - 3 \, B a b^{5}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b^{3} + 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} + b^{9}} - \frac {3 \, C a^{6} b \tan \left (d x + c\right )^{2} + 9 \, C a^{4} b^{3} \tan \left (d x + c\right )^{2} + 3 \, B a^{3} b^{4} \tan \left (d x + c\right )^{2} + 18 \, C a^{2} b^{5} \tan \left (d x + c\right )^{2} - 9 \, B a b^{6} \tan \left (d x + c\right )^{2} + 2 \, C a^{7} \tan \left (d x + c\right ) + 2 \, B a^{6} b \tan \left (d x + c\right ) + 6 \, C a^{5} b^{2} \tan \left (d x + c\right ) + 14 \, B a^{4} b^{3} \tan \left (d x + c\right ) + 28 \, C a^{3} b^{4} \tan \left (d x + c\right ) - 12 \, B a^{2} b^{5} \tan \left (d x + c\right ) + B a^{7} - C a^{6} b + 9 \, B a^{5} b^{2} + 11 \, C a^{4} b^{3} - 4 \, B a^{3} b^{4}}{{\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{2}}}{2 \, d} \]
1/2*(2*(C*a^3 - 3*B*a^2*b - 3*C*a*b^2 + B*b^3)*(d*x + c)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - (B*a^3 + 3*C*a^2*b - 3*B*a*b^2 - C*b^3)*log(tan(d*x + c)^2 + 1)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 2*(C*a^6 + 3*C*a^4*b^2 + B*a^3*b^3 + 6*C*a^2*b^4 - 3*B*a*b^5)*log(abs(b*tan(d*x + c) + a))/(a^6*b^3 + 3*a^4*b^5 + 3*a^2*b^7 + b^9) - (3*C*a^6*b*tan(d*x + c)^2 + 9*C*a^4*b^3* tan(d*x + c)^2 + 3*B*a^3*b^4*tan(d*x + c)^2 + 18*C*a^2*b^5*tan(d*x + c)^2 - 9*B*a*b^6*tan(d*x + c)^2 + 2*C*a^7*tan(d*x + c) + 2*B*a^6*b*tan(d*x + c) + 6*C*a^5*b^2*tan(d*x + c) + 14*B*a^4*b^3*tan(d*x + c) + 28*C*a^3*b^4*tan (d*x + c) - 12*B*a^2*b^5*tan(d*x + c) + B*a^7 - C*a^6*b + 9*B*a^5*b^2 + 11 *C*a^4*b^3 - 4*B*a^3*b^4)/((a^6*b^2 + 3*a^4*b^4 + 3*a^2*b^6 + b^8)*(b*tan( d*x + c) + a)^2))/d
Time = 8.52 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.23 \[ \int \frac {\tan ^2(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {3\,C\,a^6-B\,a^5\,b+7\,C\,a^4\,b^2-5\,B\,a^3\,b^3}{2\,b^3\,\left (a^4+2\,a^2\,b^2+b^4\right )}-\frac {a^2\,\mathrm {tan}\left (c+d\,x\right )\,\left (-2\,C\,a^3+B\,a^2\,b-4\,C\,a\,b^2+3\,B\,b^3\right )}{b^2\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{d\,\left (a^2+2\,a\,b\,\mathrm {tan}\left (c+d\,x\right )+b^2\,{\mathrm {tan}\left (c+d\,x\right )}^2\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-C+B\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^3\,1{}\mathrm {i}+3\,a^2\,b+a\,b^2\,3{}\mathrm {i}-b^3\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B-C\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^3+a^2\,b\,3{}\mathrm {i}+3\,a\,b^2-b^3\,1{}\mathrm {i}\right )}+\frac {a\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (C\,a^5+3\,C\,a^3\,b^2+B\,a^2\,b^3+6\,C\,a\,b^4-3\,B\,b^5\right )}{b^3\,d\,{\left (a^2+b^2\right )}^3} \]
((3*C*a^6 - 5*B*a^3*b^3 + 7*C*a^4*b^2 - B*a^5*b)/(2*b^3*(a^4 + b^4 + 2*a^2 *b^2)) - (a^2*tan(c + d*x)*(3*B*b^3 - 2*C*a^3 + B*a^2*b - 4*C*a*b^2))/(b^2 *(a^4 + b^4 + 2*a^2*b^2)))/(d*(a^2 + b^2*tan(c + d*x)^2 + 2*a*b*tan(c + d* x))) + (log(tan(c + d*x) - 1i)*(B*1i - C))/(2*d*(a*b^2*3i + 3*a^2*b - a^3* 1i - b^3)) + (log(tan(c + d*x) + 1i)*(B - C*1i))/(2*d*(3*a*b^2 + a^2*b*3i - a^3 - b^3*1i)) + (a*log(a + b*tan(c + d*x))*(C*a^5 - 3*B*b^5 + B*a^2*b^3 + 3*C*a^3*b^2 + 6*C*a*b^4))/(b^3*d*(a^2 + b^2)^3)