Integrand size = 25, antiderivative size = 221 \[ \int \frac {(a+b \tan (e+f x))^2}{(c+d \tan (e+f x))^3} \, dx=-\frac {\left (b^2 c \left (c^2-3 d^2\right )-2 a b d \left (3 c^2-d^2\right )-a^2 \left (c^3-3 c d^2\right )\right ) x}{\left (c^2+d^2\right )^3}-\frac {\left (2 a b c \left (c^2-3 d^2\right )+b^2 d \left (3 c^2-d^2\right )-a^2 \left (3 c^2 d-d^3\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{\left (c^2+d^2\right )^3 f}-\frac {(b c-a d)^2}{2 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}+\frac {2 (b c-a d) (a c+b d)}{\left (c^2+d^2\right )^2 f (c+d \tan (e+f x))} \]
-(b^2*c*(c^2-3*d^2)-2*a*b*d*(3*c^2-d^2)-a^2*(c^3-3*c*d^2))*x/(c^2+d^2)^3-( 2*a*b*c*(c^2-3*d^2)+b^2*d*(3*c^2-d^2)-a^2*(3*c^2*d-d^3))*ln(c*cos(f*x+e)+d *sin(f*x+e))/(c^2+d^2)^3/f-1/2*(-a*d+b*c)^2/d/(c^2+d^2)/f/(c+d*tan(f*x+e)) ^2+2*(-a*d+b*c)*(a*c+b*d)/(c^2+d^2)^2/f/(c+d*tan(f*x+e))
Result contains complex when optimal does not.
Time = 4.52 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.32 \[ \int \frac {(a+b \tan (e+f x))^2}{(c+d \tan (e+f x))^3} \, dx=-\frac {\frac {d^2 (a+b \tan (e+f x))^3}{(c+d \tan (e+f x))^2}-\frac {b d (a+b \tan (e+f x))^2}{c+d \tan (e+f x)}+(b c-a d) \left (\frac {(a+i b)^2 (i c+d)^3 \log (i-\tan (e+f x))}{\left (c^2+d^2\right )^2}+\frac {i (a-i b)^2 (c+i d) \log (i+\tan (e+f x))}{(c-i d)^2}+\frac {2 \left (-2 a b c \left (c^2-3 d^2\right )+b^2 d \left (-3 c^2+d^2\right )+a^2 \left (3 c^2 d-d^3\right )\right ) \log (c+d \tan (e+f x))}{\left (c^2+d^2\right )^2}-\frac {2 (-b c+a d) \left (2 a c d+b \left (-c^2+d^2\right )\right )}{d \left (c^2+d^2\right ) (c+d \tan (e+f x))}\right )}{2 (-b c+a d) \left (c^2+d^2\right ) f} \]
-1/2*((d^2*(a + b*Tan[e + f*x])^3)/(c + d*Tan[e + f*x])^2 - (b*d*(a + b*Ta n[e + f*x])^2)/(c + d*Tan[e + f*x]) + (b*c - a*d)*(((a + I*b)^2*(I*c + d)^ 3*Log[I - Tan[e + f*x]])/(c^2 + d^2)^2 + (I*(a - I*b)^2*(c + I*d)*Log[I + Tan[e + f*x]])/(c - I*d)^2 + (2*(-2*a*b*c*(c^2 - 3*d^2) + b^2*d*(-3*c^2 + d^2) + a^2*(3*c^2*d - d^3))*Log[c + d*Tan[e + f*x]])/(c^2 + d^2)^2 - (2*(- (b*c) + a*d)*(2*a*c*d + b*(-c^2 + d^2)))/(d*(c^2 + d^2)*(c + d*Tan[e + f*x ]))))/((-(b*c) + a*d)*(c^2 + d^2)*f)
Time = 0.99 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.10, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3042, 4025, 3042, 4012, 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.
| \(\displaystyle \int \frac {(a+b \tan (e+f x))^2}{(c+d \tan (e+f x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \tan (e+f x))^2}{(c+d \tan (e+f x))^3}dx\) |
| \(\Big \downarrow \) 4025 |
| \(\displaystyle \frac {\int \frac {c a^2+2 b d a-b^2 c+\left (-d a^2+2 b c a+b^2 d\right ) \tan (e+f x)}{(c+d \tan (e+f x))^2}dx}{c^2+d^2}-\frac {(b c-a d)^2}{2 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {c a^2+2 b d a-b^2 c+\left (-d a^2+2 b c a+b^2 d\right ) \tan (e+f x)}{(c+d \tan (e+f x))^2}dx}{c^2+d^2}-\frac {(b c-a d)^2}{2 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^2}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {\frac {\int -\frac {(b (c-d)-a (c+d)) (a (c-d)+b (c+d))-2 (b c-a d) (a c+b d) \tan (e+f x)}{c+d \tan (e+f x)}dx}{c^2+d^2}+\frac {2 (b c-a d) (a c+b d)}{f \left (c^2+d^2\right ) (c+d \tan (e+f x))}}{c^2+d^2}-\frac {(b c-a d)^2}{2 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 (b c-a d) (a c+b d)}{f \left (c^2+d^2\right ) (c+d \tan (e+f x))}-\frac {\int \frac {(b (c-d)-a (c+d)) (a (c-d)+b (c+d))-2 (b c-a d) (a c+b d) \tan (e+f x)}{c+d \tan (e+f x)}dx}{c^2+d^2}}{c^2+d^2}-\frac {(b c-a d)^2}{2 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 (b c-a d) (a c+b d)}{f \left (c^2+d^2\right ) (c+d \tan (e+f x))}-\frac {\int \frac {(b (c-d)-a (c+d)) (a (c-d)+b (c+d))-2 (b c-a d) (a c+b d) \tan (e+f x)}{c+d \tan (e+f x)}dx}{c^2+d^2}}{c^2+d^2}-\frac {(b c-a d)^2}{2 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^2}\) |
| \(\Big \downarrow \) 4014 |
| \(\displaystyle \frac {\frac {2 (b c-a d) (a c+b d)}{f \left (c^2+d^2\right ) (c+d \tan (e+f x))}-\frac {\frac {\left (-\left (a^2 \left (3 c^2 d-d^3\right )\right )+2 a b c \left (c^2-3 d^2\right )+b^2 d \left (3 c^2-d^2\right )\right ) \int \frac {d-c \tan (e+f x)}{c+d \tan (e+f x)}dx}{c^2+d^2}+\frac {x \left (-\left (a^2 \left (c^3-3 c d^2\right )\right )-2 a b d \left (3 c^2-d^2\right )+b^2 c \left (c^2-3 d^2\right )\right )}{c^2+d^2}}{c^2+d^2}}{c^2+d^2}-\frac {(b c-a d)^2}{2 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 (b c-a d) (a c+b d)}{f \left (c^2+d^2\right ) (c+d \tan (e+f x))}-\frac {\frac {\left (-\left (a^2 \left (3 c^2 d-d^3\right )\right )+2 a b c \left (c^2-3 d^2\right )+b^2 d \left (3 c^2-d^2\right )\right ) \int \frac {d-c \tan (e+f x)}{c+d \tan (e+f x)}dx}{c^2+d^2}+\frac {x \left (-\left (a^2 \left (c^3-3 c d^2\right )\right )-2 a b d \left (3 c^2-d^2\right )+b^2 c \left (c^2-3 d^2\right )\right )}{c^2+d^2}}{c^2+d^2}}{c^2+d^2}-\frac {(b c-a d)^2}{2 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^2}\) |
| \(\Big \downarrow \) 4013 |
| \(\displaystyle \frac {\frac {2 (b c-a d) (a c+b d)}{f \left (c^2+d^2\right ) (c+d \tan (e+f x))}-\frac {\frac {\left (-\left (a^2 \left (3 c^2 d-d^3\right )\right )+2 a b c \left (c^2-3 d^2\right )+b^2 d \left (3 c^2-d^2\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left (c^2+d^2\right )}+\frac {x \left (-\left (a^2 \left (c^3-3 c d^2\right )\right )-2 a b d \left (3 c^2-d^2\right )+b^2 c \left (c^2-3 d^2\right )\right )}{c^2+d^2}}{c^2+d^2}}{c^2+d^2}-\frac {(b c-a d)^2}{2 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^2}\) |
-1/2*(b*c - a*d)^2/(d*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^2) + (-((((b^2*c* (c^2 - 3*d^2) - 2*a*b*d*(3*c^2 - d^2) - a^2*(c^3 - 3*c*d^2))*x)/(c^2 + d^2 ) + ((2*a*b*c*(c^2 - 3*d^2) + b^2*d*(3*c^2 - d^2) - a^2*(3*c^2*d - d^3))*L og[c*Cos[e + f*x] + d*Sin[e + f*x]])/((c^2 + d^2)*f))/(c^2 + d^2)) + (2*(b *c - a*d)*(a*c + b*d))/((c^2 + d^2)*f*(c + d*Tan[e + f*x])))/(c^2 + d^2)
3.13.25.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/ (f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x] )^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a , b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1 ]
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)* (x_)]), x_Symbol] :> Simp[(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_)])^2, x_Symbol] :> Simp[(b*c - a*d)^2*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[a*c^2 + 2*b*c*d - a*d^2 - (b*c^2 - 2*a*c*d - b*d^2)*Ta n[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0]
Time = 0.39 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.37
| method | result | size |
| derivativedivides | \(\frac {-\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{2 \left (c^{2}+d^{2}\right ) d \left (c +d \tan \left (f x +e \right )\right )^{2}}+\frac {\left (3 a^{2} c^{2} d -a^{2} d^{3}-2 a b \,c^{3}+6 a b c \,d^{2}-3 b^{2} c^{2} d +b^{2} d^{3}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{\left (c^{2}+d^{2}\right )^{3}}-\frac {2 \left (a^{2} c d -a b \,c^{2}+a b \,d^{2}-b^{2} c d \right )}{\left (c^{2}+d^{2}\right )^{2} \left (c +d \tan \left (f x +e \right )\right )}+\frac {\frac {\left (-3 a^{2} c^{2} d +a^{2} d^{3}+2 a b \,c^{3}-6 a b c \,d^{2}+3 b^{2} c^{2} d -b^{2} d^{3}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (a^{2} c^{3}-3 a^{2} c \,d^{2}+6 a b \,c^{2} d -2 a b \,d^{3}-b^{2} c^{3}+3 b^{2} c \,d^{2}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{\left (c^{2}+d^{2}\right )^{3}}}{f}\) | \(303\) |
| default | \(\frac {-\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{2 \left (c^{2}+d^{2}\right ) d \left (c +d \tan \left (f x +e \right )\right )^{2}}+\frac {\left (3 a^{2} c^{2} d -a^{2} d^{3}-2 a b \,c^{3}+6 a b c \,d^{2}-3 b^{2} c^{2} d +b^{2} d^{3}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{\left (c^{2}+d^{2}\right )^{3}}-\frac {2 \left (a^{2} c d -a b \,c^{2}+a b \,d^{2}-b^{2} c d \right )}{\left (c^{2}+d^{2}\right )^{2} \left (c +d \tan \left (f x +e \right )\right )}+\frac {\frac {\left (-3 a^{2} c^{2} d +a^{2} d^{3}+2 a b \,c^{3}-6 a b c \,d^{2}+3 b^{2} c^{2} d -b^{2} d^{3}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (a^{2} c^{3}-3 a^{2} c \,d^{2}+6 a b \,c^{2} d -2 a b \,d^{3}-b^{2} c^{3}+3 b^{2} c \,d^{2}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{\left (c^{2}+d^{2}\right )^{3}}}{f}\) | \(303\) |
| norman | \(\frac {\frac {\left (a^{2} c^{3}-3 a^{2} c \,d^{2}+6 a b \,c^{2} d -2 a b \,d^{3}-b^{2} c^{3}+3 b^{2} c \,d^{2}\right ) c^{2} x}{\left (c^{4}+2 c^{2} d^{2}+d^{4}\right ) \left (c^{2}+d^{2}\right )}+\frac {d^{2} \left (a^{2} c^{3}-3 a^{2} c \,d^{2}+6 a b \,c^{2} d -2 a b \,d^{3}-b^{2} c^{3}+3 b^{2} c \,d^{2}\right ) x \left (\tan ^{2}\left (f x +e \right )\right )}{\left (c^{4}+2 c^{2} d^{2}+d^{4}\right ) \left (c^{2}+d^{2}\right )}-\frac {5 a^{2} c^{2} d^{3}+a^{2} d^{5}-6 a b \,c^{3} d^{2}+2 a b c \,d^{4}+b^{2} c^{4} d -3 b^{2} c^{2} d^{3}}{2 f \,d^{2} \left (c^{4}+2 c^{2} d^{2}+d^{4}\right )}-\frac {2 \left (a^{2} c \,d^{3}-a b \,c^{2} d^{2}+a b \,d^{4}-b^{2} c \,d^{3}\right ) \tan \left (f x +e \right )}{f d \left (c^{4}+2 c^{2} d^{2}+d^{4}\right )}+\frac {2 d \left (a^{2} c^{3}-3 a^{2} c \,d^{2}+6 a b \,c^{2} d -2 a b \,d^{3}-b^{2} c^{3}+3 b^{2} c \,d^{2}\right ) c x \tan \left (f x +e \right )}{\left (c^{4}+2 c^{2} d^{2}+d^{4}\right ) \left (c^{2}+d^{2}\right )}}{\left (c +d \tan \left (f x +e \right )\right )^{2}}+\frac {\left (3 a^{2} c^{2} d -a^{2} d^{3}-2 a b \,c^{3}+6 a b c \,d^{2}-3 b^{2} c^{2} d +b^{2} d^{3}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{f \left (c^{6}+3 c^{4} d^{2}+3 c^{2} d^{4}+d^{6}\right )}-\frac {\left (3 a^{2} c^{2} d -a^{2} d^{3}-2 a b \,c^{3}+6 a b c \,d^{2}-3 b^{2} c^{2} d +b^{2} d^{3}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f \left (c^{6}+3 c^{4} d^{2}+3 c^{2} d^{4}+d^{6}\right )}\) | \(595\) |
| risch | \(\text {Expression too large to display}\) | \(1243\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1461\) |
1/f*(-1/2*(a^2*d^2-2*a*b*c*d+b^2*c^2)/(c^2+d^2)/d/(c+d*tan(f*x+e))^2+(3*a^ 2*c^2*d-a^2*d^3-2*a*b*c^3+6*a*b*c*d^2-3*b^2*c^2*d+b^2*d^3)/(c^2+d^2)^3*ln( c+d*tan(f*x+e))-2*(a^2*c*d-a*b*c^2+a*b*d^2-b^2*c*d)/(c^2+d^2)^2/(c+d*tan(f *x+e))+1/(c^2+d^2)^3*(1/2*(-3*a^2*c^2*d+a^2*d^3+2*a*b*c^3-6*a*b*c*d^2+3*b^ 2*c^2*d-b^2*d^3)*ln(1+tan(f*x+e)^2)+(a^2*c^3-3*a^2*c*d^2+6*a*b*c^2*d-2*a*b *d^3-b^2*c^3+3*b^2*c*d^2)*arctan(tan(f*x+e))))
Leaf count of result is larger than twice the leaf count of optimal. 663 vs. \(2 (219) = 438\).
Time = 0.28 (sec) , antiderivative size = 663, normalized size of antiderivative = 3.00 \[ \int \frac {(a+b \tan (e+f x))^2}{(c+d \tan (e+f x))^3} \, dx=-\frac {3 \, b^{2} c^{4} d - 10 \, a b c^{3} d^{2} + 2 \, a b c d^{4} + a^{2} d^{5} + {\left (7 \, a^{2} - 3 \, b^{2}\right )} c^{2} d^{3} - 2 \, {\left (6 \, a b c^{4} d - 2 \, a b c^{2} d^{3} + {\left (a^{2} - b^{2}\right )} c^{5} - 3 \, {\left (a^{2} - b^{2}\right )} c^{3} d^{2}\right )} f x - {\left (b^{2} c^{4} d - 6 \, a b c^{3} d^{2} + 6 \, a b c d^{4} - a^{2} d^{5} + 5 \, {\left (a^{2} - b^{2}\right )} c^{2} d^{3} + 2 \, {\left (6 \, a b c^{2} d^{3} - 2 \, a b d^{5} + {\left (a^{2} - b^{2}\right )} c^{3} d^{2} - 3 \, {\left (a^{2} - b^{2}\right )} c d^{4}\right )} f x\right )} \tan \left (f x + e\right )^{2} + {\left (2 \, a b c^{5} - 6 \, a b c^{3} d^{2} - 3 \, {\left (a^{2} - b^{2}\right )} c^{4} d + {\left (a^{2} - b^{2}\right )} c^{2} d^{3} + {\left (2 \, a b c^{3} d^{2} - 6 \, a b c d^{4} - 3 \, {\left (a^{2} - b^{2}\right )} c^{2} d^{3} + {\left (a^{2} - b^{2}\right )} d^{5}\right )} \tan \left (f x + e\right )^{2} + 2 \, {\left (2 \, a b c^{4} d - 6 \, a b c^{2} d^{3} - 3 \, {\left (a^{2} - b^{2}\right )} c^{3} d^{2} + {\left (a^{2} - b^{2}\right )} c d^{4}\right )} \tan \left (f x + e\right )\right )} \log \left (\frac {d^{2} \tan \left (f x + e\right )^{2} + 2 \, c d \tan \left (f x + e\right ) + c^{2}}{\tan \left (f x + e\right )^{2} + 1}\right ) - 2 \, {\left (b^{2} c^{5} - 4 \, a b c^{4} d + 6 \, a b c^{2} d^{3} - 2 \, a b d^{5} + 3 \, {\left (a^{2} - b^{2}\right )} c^{3} d^{2} - {\left (3 \, a^{2} - 2 \, b^{2}\right )} c d^{4} + 2 \, {\left (6 \, a b c^{3} d^{2} - 2 \, a b c d^{4} + {\left (a^{2} - b^{2}\right )} c^{4} d - 3 \, {\left (a^{2} - b^{2}\right )} c^{2} d^{3}\right )} f x\right )} \tan \left (f x + e\right )}{2 \, {\left ({\left (c^{6} d^{2} + 3 \, c^{4} d^{4} + 3 \, c^{2} d^{6} + d^{8}\right )} f \tan \left (f x + e\right )^{2} + 2 \, {\left (c^{7} d + 3 \, c^{5} d^{3} + 3 \, c^{3} d^{5} + c d^{7}\right )} f \tan \left (f x + e\right ) + {\left (c^{8} + 3 \, c^{6} d^{2} + 3 \, c^{4} d^{4} + c^{2} d^{6}\right )} f\right )}} \]
-1/2*(3*b^2*c^4*d - 10*a*b*c^3*d^2 + 2*a*b*c*d^4 + a^2*d^5 + (7*a^2 - 3*b^ 2)*c^2*d^3 - 2*(6*a*b*c^4*d - 2*a*b*c^2*d^3 + (a^2 - b^2)*c^5 - 3*(a^2 - b ^2)*c^3*d^2)*f*x - (b^2*c^4*d - 6*a*b*c^3*d^2 + 6*a*b*c*d^4 - a^2*d^5 + 5* (a^2 - b^2)*c^2*d^3 + 2*(6*a*b*c^2*d^3 - 2*a*b*d^5 + (a^2 - b^2)*c^3*d^2 - 3*(a^2 - b^2)*c*d^4)*f*x)*tan(f*x + e)^2 + (2*a*b*c^5 - 6*a*b*c^3*d^2 - 3 *(a^2 - b^2)*c^4*d + (a^2 - b^2)*c^2*d^3 + (2*a*b*c^3*d^2 - 6*a*b*c*d^4 - 3*(a^2 - b^2)*c^2*d^3 + (a^2 - b^2)*d^5)*tan(f*x + e)^2 + 2*(2*a*b*c^4*d - 6*a*b*c^2*d^3 - 3*(a^2 - b^2)*c^3*d^2 + (a^2 - b^2)*c*d^4)*tan(f*x + e))* log((d^2*tan(f*x + e)^2 + 2*c*d*tan(f*x + e) + c^2)/(tan(f*x + e)^2 + 1)) - 2*(b^2*c^5 - 4*a*b*c^4*d + 6*a*b*c^2*d^3 - 2*a*b*d^5 + 3*(a^2 - b^2)*c^3 *d^2 - (3*a^2 - 2*b^2)*c*d^4 + 2*(6*a*b*c^3*d^2 - 2*a*b*c*d^4 + (a^2 - b^2 )*c^4*d - 3*(a^2 - b^2)*c^2*d^3)*f*x)*tan(f*x + e))/((c^6*d^2 + 3*c^4*d^4 + 3*c^2*d^6 + d^8)*f*tan(f*x + e)^2 + 2*(c^7*d + 3*c^5*d^3 + 3*c^3*d^5 + c *d^7)*f*tan(f*x + e) + (c^8 + 3*c^6*d^2 + 3*c^4*d^4 + c^2*d^6)*f)
Exception generated. \[ \int \frac {(a+b \tan (e+f x))^2}{(c+d \tan (e+f x))^3} \, dx=\text {Exception raised: AttributeError} \]
Time = 0.44 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.88 \[ \int \frac {(a+b \tan (e+f x))^2}{(c+d \tan (e+f x))^3} \, dx=\frac {\frac {2 \, {\left (6 \, a b c^{2} d - 2 \, a b d^{3} + {\left (a^{2} - b^{2}\right )} c^{3} - 3 \, {\left (a^{2} - b^{2}\right )} c d^{2}\right )} {\left (f x + e\right )}}{c^{6} + 3 \, c^{4} d^{2} + 3 \, c^{2} d^{4} + d^{6}} - \frac {2 \, {\left (2 \, a b c^{3} - 6 \, a b c d^{2} - 3 \, {\left (a^{2} - b^{2}\right )} c^{2} d + {\left (a^{2} - b^{2}\right )} d^{3}\right )} \log \left (d \tan \left (f x + e\right ) + c\right )}{c^{6} + 3 \, c^{4} d^{2} + 3 \, c^{2} d^{4} + d^{6}} + \frac {{\left (2 \, a b c^{3} - 6 \, a b c d^{2} - 3 \, {\left (a^{2} - b^{2}\right )} c^{2} d + {\left (a^{2} - b^{2}\right )} d^{3}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{c^{6} + 3 \, c^{4} d^{2} + 3 \, c^{2} d^{4} + d^{6}} - \frac {b^{2} c^{4} - 6 \, a b c^{3} d + 2 \, a b c d^{3} + a^{2} d^{4} + {\left (5 \, a^{2} - 3 \, b^{2}\right )} c^{2} d^{2} - 4 \, {\left (a b c^{2} d^{2} - a b d^{4} - {\left (a^{2} - b^{2}\right )} c d^{3}\right )} \tan \left (f x + e\right )}{c^{6} d + 2 \, c^{4} d^{3} + c^{2} d^{5} + {\left (c^{4} d^{3} + 2 \, c^{2} d^{5} + d^{7}\right )} \tan \left (f x + e\right )^{2} + 2 \, {\left (c^{5} d^{2} + 2 \, c^{3} d^{4} + c d^{6}\right )} \tan \left (f x + e\right )}}{2 \, f} \]
1/2*(2*(6*a*b*c^2*d - 2*a*b*d^3 + (a^2 - b^2)*c^3 - 3*(a^2 - b^2)*c*d^2)*( f*x + e)/(c^6 + 3*c^4*d^2 + 3*c^2*d^4 + d^6) - 2*(2*a*b*c^3 - 6*a*b*c*d^2 - 3*(a^2 - b^2)*c^2*d + (a^2 - b^2)*d^3)*log(d*tan(f*x + e) + c)/(c^6 + 3* c^4*d^2 + 3*c^2*d^4 + d^6) + (2*a*b*c^3 - 6*a*b*c*d^2 - 3*(a^2 - b^2)*c^2* d + (a^2 - b^2)*d^3)*log(tan(f*x + e)^2 + 1)/(c^6 + 3*c^4*d^2 + 3*c^2*d^4 + d^6) - (b^2*c^4 - 6*a*b*c^3*d + 2*a*b*c*d^3 + a^2*d^4 + (5*a^2 - 3*b^2)* c^2*d^2 - 4*(a*b*c^2*d^2 - a*b*d^4 - (a^2 - b^2)*c*d^3)*tan(f*x + e))/(c^6 *d + 2*c^4*d^3 + c^2*d^5 + (c^4*d^3 + 2*c^2*d^5 + d^7)*tan(f*x + e)^2 + 2* (c^5*d^2 + 2*c^3*d^4 + c*d^6)*tan(f*x + e)))/f
Leaf count of result is larger than twice the leaf count of optimal. 597 vs. \(2 (219) = 438\).
Time = 0.66 (sec) , antiderivative size = 597, normalized size of antiderivative = 2.70 \[ \int \frac {(a+b \tan (e+f x))^2}{(c+d \tan (e+f x))^3} \, dx=\frac {\frac {2 \, {\left (a^{2} c^{3} - b^{2} c^{3} + 6 \, a b c^{2} d - 3 \, a^{2} c d^{2} + 3 \, b^{2} c d^{2} - 2 \, a b d^{3}\right )} {\left (f x + e\right )}}{c^{6} + 3 \, c^{4} d^{2} + 3 \, c^{2} d^{4} + d^{6}} + \frac {{\left (2 \, a b c^{3} - 3 \, a^{2} c^{2} d + 3 \, b^{2} c^{2} d - 6 \, a b c d^{2} + a^{2} d^{3} - b^{2} d^{3}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{c^{6} + 3 \, c^{4} d^{2} + 3 \, c^{2} d^{4} + d^{6}} - \frac {2 \, {\left (2 \, a b c^{3} d - 3 \, a^{2} c^{2} d^{2} + 3 \, b^{2} c^{2} d^{2} - 6 \, a b c d^{3} + a^{2} d^{4} - b^{2} d^{4}\right )} \log \left ({\left | d \tan \left (f x + e\right ) + c \right |}\right )}{c^{6} d + 3 \, c^{4} d^{3} + 3 \, c^{2} d^{5} + d^{7}} + \frac {6 \, a b c^{3} d^{3} \tan \left (f x + e\right )^{2} - 9 \, a^{2} c^{2} d^{4} \tan \left (f x + e\right )^{2} + 9 \, b^{2} c^{2} d^{4} \tan \left (f x + e\right )^{2} - 18 \, a b c d^{5} \tan \left (f x + e\right )^{2} + 3 \, a^{2} d^{6} \tan \left (f x + e\right )^{2} - 3 \, b^{2} d^{6} \tan \left (f x + e\right )^{2} + 16 \, a b c^{4} d^{2} \tan \left (f x + e\right ) - 22 \, a^{2} c^{3} d^{3} \tan \left (f x + e\right ) + 22 \, b^{2} c^{3} d^{3} \tan \left (f x + e\right ) - 36 \, a b c^{2} d^{4} \tan \left (f x + e\right ) + 2 \, a^{2} c d^{5} \tan \left (f x + e\right ) - 2 \, b^{2} c d^{5} \tan \left (f x + e\right ) - 4 \, a b d^{6} \tan \left (f x + e\right ) - b^{2} c^{6} + 12 \, a b c^{5} d - 14 \, a^{2} c^{4} d^{2} + 11 \, b^{2} c^{4} d^{2} - 14 \, a b c^{3} d^{3} - 3 \, a^{2} c^{2} d^{4} - 2 \, a b c d^{5} - a^{2} d^{6}}{{\left (c^{6} d + 3 \, c^{4} d^{3} + 3 \, c^{2} d^{5} + d^{7}\right )} {\left (d \tan \left (f x + e\right ) + c\right )}^{2}}}{2 \, f} \]
1/2*(2*(a^2*c^3 - b^2*c^3 + 6*a*b*c^2*d - 3*a^2*c*d^2 + 3*b^2*c*d^2 - 2*a* b*d^3)*(f*x + e)/(c^6 + 3*c^4*d^2 + 3*c^2*d^4 + d^6) + (2*a*b*c^3 - 3*a^2* c^2*d + 3*b^2*c^2*d - 6*a*b*c*d^2 + a^2*d^3 - b^2*d^3)*log(tan(f*x + e)^2 + 1)/(c^6 + 3*c^4*d^2 + 3*c^2*d^4 + d^6) - 2*(2*a*b*c^3*d - 3*a^2*c^2*d^2 + 3*b^2*c^2*d^2 - 6*a*b*c*d^3 + a^2*d^4 - b^2*d^4)*log(abs(d*tan(f*x + e) + c))/(c^6*d + 3*c^4*d^3 + 3*c^2*d^5 + d^7) + (6*a*b*c^3*d^3*tan(f*x + e)^ 2 - 9*a^2*c^2*d^4*tan(f*x + e)^2 + 9*b^2*c^2*d^4*tan(f*x + e)^2 - 18*a*b*c *d^5*tan(f*x + e)^2 + 3*a^2*d^6*tan(f*x + e)^2 - 3*b^2*d^6*tan(f*x + e)^2 + 16*a*b*c^4*d^2*tan(f*x + e) - 22*a^2*c^3*d^3*tan(f*x + e) + 22*b^2*c^3*d ^3*tan(f*x + e) - 36*a*b*c^2*d^4*tan(f*x + e) + 2*a^2*c*d^5*tan(f*x + e) - 2*b^2*c*d^5*tan(f*x + e) - 4*a*b*d^6*tan(f*x + e) - b^2*c^6 + 12*a*b*c^5* d - 14*a^2*c^4*d^2 + 11*b^2*c^4*d^2 - 14*a*b*c^3*d^3 - 3*a^2*c^2*d^4 - 2*a *b*c*d^5 - a^2*d^6)/((c^6*d + 3*c^4*d^3 + 3*c^2*d^5 + d^7)*(d*tan(f*x + e) + c)^2))/f
Time = 9.02 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.66 \[ \int \frac {(a+b \tan (e+f x))^2}{(c+d \tan (e+f x))^3} \, dx=-\frac {\frac {2\,\mathrm {tan}\left (e+f\,x\right )\,\left (a^2\,c\,d^2-a\,b\,c^2\,d+a\,b\,d^3-b^2\,c\,d^2\right )}{c^4+2\,c^2\,d^2+d^4}+\frac {5\,a^2\,c^2\,d^2+a^2\,d^4-6\,a\,b\,c^3\,d+2\,a\,b\,c\,d^3+b^2\,c^4-3\,b^2\,c^2\,d^2}{2\,d\,\left (c^4+2\,c^2\,d^2+d^4\right )}}{f\,\left (c^2+2\,c\,d\,\mathrm {tan}\left (e+f\,x\right )+d^2\,{\mathrm {tan}\left (e+f\,x\right )}^2\right )}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )\,\left (-a^2\,1{}\mathrm {i}+2\,a\,b+b^2\,1{}\mathrm {i}\right )}{2\,f\,\left (-c^3-c^2\,d\,3{}\mathrm {i}+3\,c\,d^2+d^3\,1{}\mathrm {i}\right )}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,\left (-a^2+a\,b\,2{}\mathrm {i}+b^2\right )}{2\,f\,\left (-c^3\,1{}\mathrm {i}-3\,c^2\,d+c\,d^2\,3{}\mathrm {i}+d^3\right )}-\frac {\ln \left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (2\,a\,b\,c^3+\left (3\,b^2-3\,a^2\right )\,c^2\,d-6\,a\,b\,c\,d^2+\left (a^2-b^2\right )\,d^3\right )}{f\,\left (c^6+3\,c^4\,d^2+3\,c^2\,d^4+d^6\right )} \]
- ((2*tan(e + f*x)*(a^2*c*d^2 - b^2*c*d^2 + a*b*d^3 - a*b*c^2*d))/(c^4 + d ^4 + 2*c^2*d^2) + (a^2*d^4 + b^2*c^4 + 5*a^2*c^2*d^2 - 3*b^2*c^2*d^2 + 2*a *b*c*d^3 - 6*a*b*c^3*d)/(2*d*(c^4 + d^4 + 2*c^2*d^2)))/(f*(c^2 + d^2*tan(e + f*x)^2 + 2*c*d*tan(e + f*x))) - (log(tan(e + f*x) - 1i)*(2*a*b - a^2*1i + b^2*1i))/(2*f*(3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)) - (log(tan(e + f*x) + 1i)*(a*b*2i - a^2 + b^2))/(2*f*(c*d^2*3i - 3*c^2*d - c^3*1i + d^3)) - (l og(c + d*tan(e + f*x))*(d^3*(a^2 - b^2) - c^2*d*(3*a^2 - 3*b^2) + 2*a*b*c^ 3 - 6*a*b*c*d^2))/(f*(c^6 + d^6 + 3*c^2*d^4 + 3*c^4*d^2))