Integrand size = 41, antiderivative size = 316 \[ \int \frac {a+b \tan (d+e x)}{\left (b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right )^{3/2}} \, dx=-\frac {\left (a^2-b^2\right ) (b+a \tan (d+e x))}{2 \left (a^2+b^2\right ) e \left (b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right )^{3/2}}-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \log (b \cos (d+e x)+a \sin (d+e x)) (b+a \tan (d+e x))^3}{\left (a^2+b^2\right )^3 e \left (b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right )^{3/2}}-\frac {4 b \left (a^2-b^2\right ) x \left (a b+a^2 \tan (d+e x)\right )^3}{a^2 \left (a^2+b^2\right )^3 \left (b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right )^{3/2}}-\frac {b \left (3 a^2-b^2\right ) \left (a b+a^2 \tan (d+e x)\right )^3}{\left (a^2+b^2\right )^2 e \left (a^3 b+a^4 \tan (d+e x)\right ) \left (b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right )^{3/2}} \]
-1/2*(a^2-b^2)*(b+a*tan(e*x+d))/(a^2+b^2)/e/(b^2+2*a*b*tan(e*x+d)+a^2*tan( e*x+d)^2)^(3/2)-(a^4-6*a^2*b^2+b^4)*ln(b*cos(e*x+d)+a*sin(e*x+d))*(b+a*tan (e*x+d))^3/(a^2+b^2)^3/e/(b^2+2*a*b*tan(e*x+d)+a^2*tan(e*x+d)^2)^(3/2)-4*b *(a^2-b^2)*x*(a*b+a^2*tan(e*x+d))^3/a^2/(a^2+b^2)^3/(b^2+2*a*b*tan(e*x+d)+ a^2*tan(e*x+d)^2)^(3/2)-b*(3*a^2-b^2)*(a*b+a^2*tan(e*x+d))^3/(a^2+b^2)^2/e /(a^3*b+a^4*tan(e*x+d))/(b^2+2*a*b*tan(e*x+d)+a^2*tan(e*x+d)^2)^(3/2)
Result contains complex when optimal does not.
Time = 3.60 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.85 \[ \int \frac {a+b \tan (d+e x)}{\left (b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right )^{3/2}} \, dx=\frac {(b+a \tan (d+e x))^3 \left (b \left (\frac {i \log (i-\tan (d+e x))}{(a-i b)^2}-\frac {i \log (i+\tan (d+e x))}{(a+i b)^2}+\frac {2 a \left (2 b \log (b+a \tan (d+e x))-\frac {a^2+b^2}{b+a \tan (d+e x)}\right )}{\left (a^2+b^2\right )^2}\right )+(a-b) (a+b) \left (\frac {\log (i-\tan (d+e x))}{(a-i b)^3}+\frac {\log (i+\tan (d+e x))}{(a+i b)^3}+\frac {a \left (-2 \left (a^2-3 b^2\right ) \log (b+a \tan (d+e x))-\frac {\left (a^2+b^2\right ) \left (a^2+5 b^2+4 a b \tan (d+e x)\right )}{(b+a \tan (d+e x))^2}\right )}{\left (a^2+b^2\right )^3}\right )\right )}{2 a e \left ((b+a \tan (d+e x))^2\right )^{3/2}} \]
((b + a*Tan[d + e*x])^3*(b*((I*Log[I - Tan[d + e*x]])/(a - I*b)^2 - (I*Log [I + Tan[d + e*x]])/(a + I*b)^2 + (2*a*(2*b*Log[b + a*Tan[d + e*x]] - (a^2 + b^2)/(b + a*Tan[d + e*x])))/(a^2 + b^2)^2) + (a - b)*(a + b)*(Log[I - T an[d + e*x]]/(a - I*b)^3 + Log[I + Tan[d + e*x]]/(a + I*b)^3 + (a*(-2*(a^2 - 3*b^2)*Log[b + a*Tan[d + e*x]] - ((a^2 + b^2)*(a^2 + 5*b^2 + 4*a*b*Tan[ d + e*x]))/(b + a*Tan[d + e*x])^2))/(a^2 + b^2)^3)))/(2*a*e*((b + a*Tan[d + e*x])^2)^(3/2))
Time = 1.06 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.74, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.293, Rules used = {3042, 4193, 27, 3042, 4012, 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.
| \(\displaystyle \int \frac {a+b \tan (d+e x)}{\left (a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a+b \tan (d+e x)}{\left (a^2 \tan (d+e x)^2+2 a b \tan (d+e x)+b^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4193 |
| \(\displaystyle \frac {8 \left (a^2 \tan (d+e x)+a b\right )^3 \int \frac {a+b \tan (d+e x)}{8 \left (\tan (d+e x) a^2+b a\right )^3}dx}{\left (a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a^2 \tan (d+e x)+a b\right )^3 \int \frac {a+b \tan (d+e x)}{\left (\tan (d+e x) a^2+b a\right )^3}dx}{\left (a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 \tan (d+e x)+a b\right )^3 \int \frac {a+b \tan (d+e x)}{\left (\tan (d+e x) a^2+b a\right )^3}dx}{\left (a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {\left (a^2 \tan (d+e x)+a b\right )^3 \left (\frac {\int \frac {2 a^2 b-a \left (a^2-b^2\right ) \tan (d+e x)}{\left (\tan (d+e x) a^2+b a\right )^2}dx}{a^2 \left (a^2+b^2\right )}-\frac {a^2-b^2}{2 a e \left (a^2+b^2\right ) \left (a^2 \tan (d+e x)+a b\right )^2}\right )}{\left (a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 \tan (d+e x)+a b\right )^3 \left (\frac {\int \frac {2 a^2 b-a \left (a^2-b^2\right ) \tan (d+e x)}{\left (\tan (d+e x) a^2+b a\right )^2}dx}{a^2 \left (a^2+b^2\right )}-\frac {a^2-b^2}{2 a e \left (a^2+b^2\right ) \left (a^2 \tan (d+e x)+a b\right )^2}\right )}{\left (a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {\left (a^2 \tan (d+e x)+a b\right )^3 \left (\frac {\frac {\int -\frac {\left (a^2-3 b^2\right ) a^3+b \left (3 a^2-b^2\right ) \tan (d+e x) a^2}{\tan (d+e x) a^2+b a}dx}{a^2 \left (a^2+b^2\right )}-\frac {b \left (3 a^2-b^2\right )}{e \left (a^2+b^2\right ) \left (a^2 \tan (d+e x)+a b\right )}}{a^2 \left (a^2+b^2\right )}-\frac {a^2-b^2}{2 a e \left (a^2+b^2\right ) \left (a^2 \tan (d+e x)+a b\right )^2}\right )}{\left (a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (a^2 \tan (d+e x)+a b\right )^3 \left (\frac {-\frac {\int \frac {\left (a^2-3 b^2\right ) a^3+b \left (3 a^2-b^2\right ) \tan (d+e x) a^2}{\tan (d+e x) a^2+b a}dx}{a^2 \left (a^2+b^2\right )}-\frac {b \left (3 a^2-b^2\right )}{e \left (a^2+b^2\right ) \left (a^2 \tan (d+e x)+a b\right )}}{a^2 \left (a^2+b^2\right )}-\frac {a^2-b^2}{2 a e \left (a^2+b^2\right ) \left (a^2 \tan (d+e x)+a b\right )^2}\right )}{\left (a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 \tan (d+e x)+a b\right )^3 \left (\frac {-\frac {\int \frac {\left (a^2-3 b^2\right ) a^3+b \left (3 a^2-b^2\right ) \tan (d+e x) a^2}{\tan (d+e x) a^2+b a}dx}{a^2 \left (a^2+b^2\right )}-\frac {b \left (3 a^2-b^2\right )}{e \left (a^2+b^2\right ) \left (a^2 \tan (d+e x)+a b\right )}}{a^2 \left (a^2+b^2\right )}-\frac {a^2-b^2}{2 a e \left (a^2+b^2\right ) \left (a^2 \tan (d+e x)+a b\right )^2}\right )}{\left (a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 4014 |
| \(\displaystyle \frac {\left (a^2 \tan (d+e x)+a b\right )^3 \left (\frac {-\frac {\frac {a \left (a^4-6 a^2 b^2+b^4\right ) \int \frac {a^2-a b \tan (d+e x)}{\tan (d+e x) a^2+b a}dx}{a^2+b^2}+\frac {4 a^2 b x \left (a^2-b^2\right )}{a^2+b^2}}{a^2 \left (a^2+b^2\right )}-\frac {b \left (3 a^2-b^2\right )}{e \left (a^2+b^2\right ) \left (a^2 \tan (d+e x)+a b\right )}}{a^2 \left (a^2+b^2\right )}-\frac {a^2-b^2}{2 a e \left (a^2+b^2\right ) \left (a^2 \tan (d+e x)+a b\right )^2}\right )}{\left (a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 \tan (d+e x)+a b\right )^3 \left (\frac {-\frac {\frac {a \left (a^4-6 a^2 b^2+b^4\right ) \int \frac {a^2-a b \tan (d+e x)}{\tan (d+e x) a^2+b a}dx}{a^2+b^2}+\frac {4 a^2 b x \left (a^2-b^2\right )}{a^2+b^2}}{a^2 \left (a^2+b^2\right )}-\frac {b \left (3 a^2-b^2\right )}{e \left (a^2+b^2\right ) \left (a^2 \tan (d+e x)+a b\right )}}{a^2 \left (a^2+b^2\right )}-\frac {a^2-b^2}{2 a e \left (a^2+b^2\right ) \left (a^2 \tan (d+e x)+a b\right )^2}\right )}{\left (a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 4013 |
| \(\displaystyle \frac {\left (a^2 \tan (d+e x)+a b\right )^3 \left (\frac {-\frac {b \left (3 a^2-b^2\right )}{e \left (a^2+b^2\right ) \left (a^2 \tan (d+e x)+a b\right )}-\frac {\frac {4 a^2 b x \left (a^2-b^2\right )}{a^2+b^2}+\frac {a \left (a^4-6 a^2 b^2+b^4\right ) \log (a \sin (d+e x)+b \cos (d+e x))}{e \left (a^2+b^2\right )}}{a^2 \left (a^2+b^2\right )}}{a^2 \left (a^2+b^2\right )}-\frac {a^2-b^2}{2 a e \left (a^2+b^2\right ) \left (a^2 \tan (d+e x)+a b\right )^2}\right )}{\left (a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right )^{3/2}}\) |
((a*b + a^2*Tan[d + e*x])^3*(-1/2*(a^2 - b^2)/(a*(a^2 + b^2)*e*(a*b + a^2* Tan[d + e*x])^2) + (-(((4*a^2*b*(a^2 - b^2)*x)/(a^2 + b^2) + (a*(a^4 - 6*a ^2*b^2 + b^4)*Log[b*Cos[d + e*x] + a*Sin[d + e*x]])/((a^2 + b^2)*e))/(a^2* (a^2 + b^2))) - (b*(3*a^2 - b^2))/((a^2 + b^2)*e*(a*b + a^2*Tan[d + e*x])) )/(a^2*(a^2 + b^2))))/(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(3/2 )
3.6.17.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((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[(d_.) + (e_.)*(x_)])*((a_) + (b_.)*tan[(d_.) + (e_.)* (x_)] + (c_.)*tan[(d_.) + (e_.)*(x_)]^2)^(n_), x_Symbol] :> Simp[(a + b*Tan [d + e*x] + c*Tan[d + e*x]^2)^n/(b + 2*c*Tan[d + e*x])^(2*n) Int[(A + B*T an[d + e*x])*(b + 2*c*Tan[d + e*x])^(2*n), x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[n]
Leaf count of result is larger than twice the leaf count of optimal. \(621\) vs. \(2(306)=612\).
Time = 0.70 (sec) , antiderivative size = 622, normalized size of antiderivative = 1.97
| method | result | size |
| derivativedivides | \(-\frac {\left (-2 \ln \left (1+\tan \left (e x +d \right )^{2}\right ) a^{5} b \tan \left (e x +d \right )+12 \ln \left (1+\tan \left (e x +d \right )^{2}\right ) a^{3} b^{3} \tan \left (e x +d \right )-2 \ln \left (1+\tan \left (e x +d \right )^{2}\right ) a \,b^{5} \tan \left (e x +d \right )+16 \arctan \left (\tan \left (e x +d \right )\right ) a^{4} b^{2} \tan \left (e x +d \right )-16 \arctan \left (\tan \left (e x +d \right )\right ) a^{2} b^{4} \tan \left (e x +d \right )-12 \ln \left (b +a \tan \left (e x +d \right )\right ) a^{4} b^{2} \tan \left (e x +d \right )^{2}+2 \ln \left (b +a \tan \left (e x +d \right )\right ) a^{2} b^{4} \tan \left (e x +d \right )^{2}+6 \ln \left (1+\tan \left (e x +d \right )^{2}\right ) a^{4} b^{2} \tan \left (e x +d \right )^{2}-\ln \left (1+\tan \left (e x +d \right )^{2}\right ) a^{2} b^{4} \tan \left (e x +d \right )^{2}+8 \arctan \left (\tan \left (e x +d \right )\right ) a^{5} b \tan \left (e x +d \right )^{2}-8 \arctan \left (\tan \left (e x +d \right )\right ) a^{3} b^{3} \tan \left (e x +d \right )^{2}+4 \ln \left (b +a \tan \left (e x +d \right )\right ) a^{5} b \tan \left (e x +d \right )-24 \ln \left (b +a \tan \left (e x +d \right )\right ) a^{3} b^{3} \tan \left (e x +d \right )+a^{6}-3 b^{6}+6 a^{5} b \tan \left (e x +d \right )+4 a^{3} b^{3} \tan \left (e x +d \right )-2 a \,b^{5} \tan \left (e x +d \right )-\ln \left (1+\tan \left (e x +d \right )^{2}\right ) a^{6} \tan \left (e x +d \right )^{2}+2 \ln \left (b +a \tan \left (e x +d \right )\right ) a^{4} b^{2}-12 \ln \left (b +a \tan \left (e x +d \right )\right ) a^{2} b^{4}-\ln \left (1+\tan \left (e x +d \right )^{2}\right ) a^{4} b^{2}+6 \ln \left (1+\tan \left (e x +d \right )^{2}\right ) a^{2} b^{4}+8 \arctan \left (\tan \left (e x +d \right )\right ) a^{3} b^{3}-8 \arctan \left (\tan \left (e x +d \right )\right ) a \,b^{5}+2 \ln \left (b +a \tan \left (e x +d \right )\right ) a^{6} \tan \left (e x +d \right )^{2}+7 a^{4} b^{2}+4 \ln \left (b +a \tan \left (e x +d \right )\right ) a \,b^{5} \tan \left (e x +d \right )+3 a^{2} b^{4}+2 \ln \left (b +a \tan \left (e x +d \right )\right ) b^{6}-\ln \left (1+\tan \left (e x +d \right )^{2}\right ) b^{6}\right ) \left (b +a \tan \left (e x +d \right )\right )}{2 e \left (a^{2}+b^{2}\right )^{3} \left (b^{2}+2 a b \tan \left (e x +d \right )+a^{2} \tan \left (e x +d \right )^{2}\right )^{\frac {3}{2}}}\) | \(622\) |
| default | \(-\frac {\left (-2 \ln \left (1+\tan \left (e x +d \right )^{2}\right ) a^{5} b \tan \left (e x +d \right )+12 \ln \left (1+\tan \left (e x +d \right )^{2}\right ) a^{3} b^{3} \tan \left (e x +d \right )-2 \ln \left (1+\tan \left (e x +d \right )^{2}\right ) a \,b^{5} \tan \left (e x +d \right )+16 \arctan \left (\tan \left (e x +d \right )\right ) a^{4} b^{2} \tan \left (e x +d \right )-16 \arctan \left (\tan \left (e x +d \right )\right ) a^{2} b^{4} \tan \left (e x +d \right )-12 \ln \left (b +a \tan \left (e x +d \right )\right ) a^{4} b^{2} \tan \left (e x +d \right )^{2}+2 \ln \left (b +a \tan \left (e x +d \right )\right ) a^{2} b^{4} \tan \left (e x +d \right )^{2}+6 \ln \left (1+\tan \left (e x +d \right )^{2}\right ) a^{4} b^{2} \tan \left (e x +d \right )^{2}-\ln \left (1+\tan \left (e x +d \right )^{2}\right ) a^{2} b^{4} \tan \left (e x +d \right )^{2}+8 \arctan \left (\tan \left (e x +d \right )\right ) a^{5} b \tan \left (e x +d \right )^{2}-8 \arctan \left (\tan \left (e x +d \right )\right ) a^{3} b^{3} \tan \left (e x +d \right )^{2}+4 \ln \left (b +a \tan \left (e x +d \right )\right ) a^{5} b \tan \left (e x +d \right )-24 \ln \left (b +a \tan \left (e x +d \right )\right ) a^{3} b^{3} \tan \left (e x +d \right )+a^{6}-3 b^{6}+6 a^{5} b \tan \left (e x +d \right )+4 a^{3} b^{3} \tan \left (e x +d \right )-2 a \,b^{5} \tan \left (e x +d \right )-\ln \left (1+\tan \left (e x +d \right )^{2}\right ) a^{6} \tan \left (e x +d \right )^{2}+2 \ln \left (b +a \tan \left (e x +d \right )\right ) a^{4} b^{2}-12 \ln \left (b +a \tan \left (e x +d \right )\right ) a^{2} b^{4}-\ln \left (1+\tan \left (e x +d \right )^{2}\right ) a^{4} b^{2}+6 \ln \left (1+\tan \left (e x +d \right )^{2}\right ) a^{2} b^{4}+8 \arctan \left (\tan \left (e x +d \right )\right ) a^{3} b^{3}-8 \arctan \left (\tan \left (e x +d \right )\right ) a \,b^{5}+2 \ln \left (b +a \tan \left (e x +d \right )\right ) a^{6} \tan \left (e x +d \right )^{2}+7 a^{4} b^{2}+4 \ln \left (b +a \tan \left (e x +d \right )\right ) a \,b^{5} \tan \left (e x +d \right )+3 a^{2} b^{4}+2 \ln \left (b +a \tan \left (e x +d \right )\right ) b^{6}-\ln \left (1+\tan \left (e x +d \right )^{2}\right ) b^{6}\right ) \left (b +a \tan \left (e x +d \right )\right )}{2 e \left (a^{2}+b^{2}\right )^{3} \left (b^{2}+2 a b \tan \left (e x +d \right )+a^{2} \tan \left (e x +d \right )^{2}\right )^{\frac {3}{2}}}\) | \(622\) |
| parts | \(\text {Expression too large to display}\) | \(927\) |
-1/2/e*(-2*ln(1+tan(e*x+d)^2)*a^5*b*tan(e*x+d)+12*ln(1+tan(e*x+d)^2)*a^3*b ^3*tan(e*x+d)-2*ln(1+tan(e*x+d)^2)*a*b^5*tan(e*x+d)+16*arctan(tan(e*x+d))* a^4*b^2*tan(e*x+d)-16*arctan(tan(e*x+d))*a^2*b^4*tan(e*x+d)-12*ln(b+a*tan( e*x+d))*a^4*b^2*tan(e*x+d)^2+2*ln(b+a*tan(e*x+d))*a^2*b^4*tan(e*x+d)^2+6*l n(1+tan(e*x+d)^2)*a^4*b^2*tan(e*x+d)^2-ln(1+tan(e*x+d)^2)*a^2*b^4*tan(e*x+ d)^2+8*arctan(tan(e*x+d))*a^5*b*tan(e*x+d)^2-8*arctan(tan(e*x+d))*a^3*b^3* tan(e*x+d)^2+4*ln(b+a*tan(e*x+d))*a^5*b*tan(e*x+d)-24*ln(b+a*tan(e*x+d))*a ^3*b^3*tan(e*x+d)+a^6-3*b^6+6*a^5*b*tan(e*x+d)+4*a^3*b^3*tan(e*x+d)-2*a*b^ 5*tan(e*x+d)-ln(1+tan(e*x+d)^2)*a^6*tan(e*x+d)^2+2*ln(b+a*tan(e*x+d))*a^4* b^2-12*ln(b+a*tan(e*x+d))*a^2*b^4-ln(1+tan(e*x+d)^2)*a^4*b^2+6*ln(1+tan(e* x+d)^2)*a^2*b^4+8*arctan(tan(e*x+d))*a^3*b^3-8*arctan(tan(e*x+d))*a*b^5+2* ln(b+a*tan(e*x+d))*a^6*tan(e*x+d)^2+7*a^4*b^2+4*ln(b+a*tan(e*x+d))*a*b^5*t an(e*x+d)+3*a^2*b^4+2*ln(b+a*tan(e*x+d))*b^6-ln(1+tan(e*x+d)^2)*b^6)*(b+a* tan(e*x+d))/(a^2+b^2)^3/((b+a*tan(e*x+d))^2)^(3/2)
Time = 0.27 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.12 \[ \int \frac {a+b \tan (d+e x)}{\left (b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right )^{3/2}} \, dx=-\frac {a^{6} + 8 \, a^{4} b^{2} - 5 \, a^{2} b^{4} + 8 \, {\left (a^{3} b^{3} - a b^{5}\right )} e x + {\left (a^{6} - 8 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + 8 \, {\left (a^{5} b - a^{3} b^{3}\right )} e x\right )} \tan \left (e x + d\right )^{2} + {\left (a^{4} b^{2} - 6 \, a^{2} b^{4} + b^{6} + {\left (a^{6} - 6 \, a^{4} b^{2} + a^{2} b^{4}\right )} \tan \left (e x + d\right )^{2} + 2 \, {\left (a^{5} b - 6 \, a^{3} b^{3} + a b^{5}\right )} \tan \left (e x + d\right )\right )} \log \left (\frac {a^{2} \tan \left (e x + d\right )^{2} + 2 \, a b \tan \left (e x + d\right ) + b^{2}}{\tan \left (e x + d\right )^{2} + 1}\right ) + 4 \, {\left (2 \, a^{5} b - 3 \, a^{3} b^{3} + a b^{5} + 4 \, {\left (a^{4} b^{2} - a^{2} b^{4}\right )} e x\right )} \tan \left (e x + d\right )}{2 \, {\left ({\left (a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6}\right )} e \tan \left (e x + d\right )^{2} + 2 \, {\left (a^{7} b + 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + a b^{7}\right )} e \tan \left (e x + d\right ) + {\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )} e\right )}} \]
integrate((a+b*tan(e*x+d))/(b^2+2*a*b*tan(e*x+d)+a^2*tan(e*x+d)^2)^(3/2),x , algorithm="fricas")
-1/2*(a^6 + 8*a^4*b^2 - 5*a^2*b^4 + 8*(a^3*b^3 - a*b^5)*e*x + (a^6 - 8*a^4 *b^2 + 3*a^2*b^4 + 8*(a^5*b - a^3*b^3)*e*x)*tan(e*x + d)^2 + (a^4*b^2 - 6* a^2*b^4 + b^6 + (a^6 - 6*a^4*b^2 + a^2*b^4)*tan(e*x + d)^2 + 2*(a^5*b - 6* a^3*b^3 + a*b^5)*tan(e*x + d))*log((a^2*tan(e*x + d)^2 + 2*a*b*tan(e*x + d ) + b^2)/(tan(e*x + d)^2 + 1)) + 4*(2*a^5*b - 3*a^3*b^3 + a*b^5 + 4*(a^4*b ^2 - a^2*b^4)*e*x)*tan(e*x + d))/((a^8 + 3*a^6*b^2 + 3*a^4*b^4 + a^2*b^6)* e*tan(e*x + d)^2 + 2*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*e*tan(e*x + d ) + (a^6*b^2 + 3*a^4*b^4 + 3*a^2*b^6 + b^8)*e)
\[ \int \frac {a+b \tan (d+e x)}{\left (b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right )^{3/2}} \, dx=\int \frac {a + b \tan {\left (d + e x \right )}}{\left (\left (a \tan {\left (d + e x \right )} + b\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
Time = 0.33 (sec) , antiderivative size = 498, normalized size of antiderivative = 1.58 \[ \int \frac {a+b \tan (d+e x)}{\left (b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right )^{3/2}} \, dx=-\frac {{\left (\frac {2 \, {\left (3 \, a^{2} b - b^{3}\right )} {\left (e x + d\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left (a^{3} - 3 \, a b^{2}\right )} \log \left (a \tan \left (e x + d\right ) + b\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (a^{3} - 3 \, a b^{2}\right )} \log \left (\tan \left (e x + d\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {4 \, a^{2} b \tan \left (e x + d\right ) + a^{3} + 5 \, a b^{2}}{a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6} + {\left (a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \tan \left (e x + d\right )^{2} + 2 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} \tan \left (e x + d\right )}\right )} a + {\left (\frac {2 \, {\left (a^{3} - 3 \, a b^{2}\right )} {\left (e x + d\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {2 \, {\left (3 \, a^{2} b - b^{3}\right )} \log \left (a \tan \left (e x + d\right ) + b\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (e x + d\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {a^{2} b - 3 \, b^{3} + 2 \, {\left (a^{3} - a b^{2}\right )} \tan \left (e x + d\right )}{a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6} + {\left (a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \tan \left (e x + d\right )^{2} + 2 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} \tan \left (e x + d\right )}\right )} b}{2 \, e} \]
integrate((a+b*tan(e*x+d))/(b^2+2*a*b*tan(e*x+d)+a^2*tan(e*x+d)^2)^(3/2),x , algorithm="maxima")
-1/2*((2*(3*a^2*b - b^3)*(e*x + d)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 2 *(a^3 - 3*a*b^2)*log(a*tan(e*x + d) + b)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^ 6) - (a^3 - 3*a*b^2)*log(tan(e*x + d)^2 + 1)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + (4*a^2*b*tan(e*x + d) + a^3 + 5*a*b^2)/(a^4*b^2 + 2*a^2*b^4 + b^6 + (a^6 + 2*a^4*b^2 + a^2*b^4)*tan(e*x + d)^2 + 2*(a^5*b + 2*a^3*b^3 + a*b ^5)*tan(e*x + d)))*a + (2*(a^3 - 3*a*b^2)*(e*x + d)/(a^6 + 3*a^4*b^2 + 3*a ^2*b^4 + b^6) - 2*(3*a^2*b - b^3)*log(a*tan(e*x + d) + b)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + (3*a^2*b - b^3)*log(tan(e*x + d)^2 + 1)/(a^6 + 3*a^4 *b^2 + 3*a^2*b^4 + b^6) + (a^2*b - 3*b^3 + 2*(a^3 - a*b^2)*tan(e*x + d))/( a^4*b^2 + 2*a^2*b^4 + b^6 + (a^6 + 2*a^4*b^2 + a^2*b^4)*tan(e*x + d)^2 + 2 *(a^5*b + 2*a^3*b^3 + a*b^5)*tan(e*x + d)))*b)/e
Time = 0.55 (sec) , antiderivative size = 481, normalized size of antiderivative = 1.52 \[ \int \frac {a+b \tan (d+e x)}{\left (b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right )^{3/2}} \, dx=-\frac {\frac {8 \, {\left (a^{3} b - a b^{3}\right )} {\left (e x + d\right )}}{a^{6} \mathrm {sgn}\left (a \tan \left (e x + d\right ) + b\right ) + 3 \, a^{4} b^{2} \mathrm {sgn}\left (a \tan \left (e x + d\right ) + b\right ) + 3 \, a^{2} b^{4} \mathrm {sgn}\left (a \tan \left (e x + d\right ) + b\right ) + b^{6} \mathrm {sgn}\left (a \tan \left (e x + d\right ) + b\right )} - \frac {{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (e x + d\right )^{2} + 1\right )}{a^{6} \mathrm {sgn}\left (a \tan \left (e x + d\right ) + b\right ) + 3 \, a^{4} b^{2} \mathrm {sgn}\left (a \tan \left (e x + d\right ) + b\right ) + 3 \, a^{2} b^{4} \mathrm {sgn}\left (a \tan \left (e x + d\right ) + b\right ) + b^{6} \mathrm {sgn}\left (a \tan \left (e x + d\right ) + b\right )} + \frac {2 \, {\left (a^{5} - 6 \, a^{3} b^{2} + a b^{4}\right )} \log \left ({\left | a \tan \left (e x + d\right ) + b \right |}\right )}{a^{7} \mathrm {sgn}\left (a \tan \left (e x + d\right ) + b\right ) + 3 \, a^{5} b^{2} \mathrm {sgn}\left (a \tan \left (e x + d\right ) + b\right ) + 3 \, a^{3} b^{4} \mathrm {sgn}\left (a \tan \left (e x + d\right ) + b\right ) + a b^{6} \mathrm {sgn}\left (a \tan \left (e x + d\right ) + b\right )} - \frac {3 \, a^{6} \tan \left (e x + d\right )^{2} - 18 \, a^{4} b^{2} \tan \left (e x + d\right )^{2} + 3 \, a^{2} b^{4} \tan \left (e x + d\right )^{2} - 40 \, a^{3} b^{3} \tan \left (e x + d\right ) + 8 \, a b^{5} \tan \left (e x + d\right ) - a^{6} - 4 \, a^{4} b^{2} - 21 \, a^{2} b^{4} + 6 \, b^{6}}{{\left (a^{6} \mathrm {sgn}\left (a \tan \left (e x + d\right ) + b\right ) + 3 \, a^{4} b^{2} \mathrm {sgn}\left (a \tan \left (e x + d\right ) + b\right ) + 3 \, a^{2} b^{4} \mathrm {sgn}\left (a \tan \left (e x + d\right ) + b\right ) + b^{6} \mathrm {sgn}\left (a \tan \left (e x + d\right ) + b\right )\right )} {\left (a \tan \left (e x + d\right ) + b\right )}^{2}}}{2 \, e} \]
integrate((a+b*tan(e*x+d))/(b^2+2*a*b*tan(e*x+d)+a^2*tan(e*x+d)^2)^(3/2),x , algorithm="giac")
-1/2*(8*(a^3*b - a*b^3)*(e*x + d)/(a^6*sgn(a*tan(e*x + d) + b) + 3*a^4*b^2 *sgn(a*tan(e*x + d) + b) + 3*a^2*b^4*sgn(a*tan(e*x + d) + b) + b^6*sgn(a*t an(e*x + d) + b)) - (a^4 - 6*a^2*b^2 + b^4)*log(tan(e*x + d)^2 + 1)/(a^6*s gn(a*tan(e*x + d) + b) + 3*a^4*b^2*sgn(a*tan(e*x + d) + b) + 3*a^2*b^4*sgn (a*tan(e*x + d) + b) + b^6*sgn(a*tan(e*x + d) + b)) + 2*(a^5 - 6*a^3*b^2 + a*b^4)*log(abs(a*tan(e*x + d) + b))/(a^7*sgn(a*tan(e*x + d) + b) + 3*a^5* b^2*sgn(a*tan(e*x + d) + b) + 3*a^3*b^4*sgn(a*tan(e*x + d) + b) + a*b^6*sg n(a*tan(e*x + d) + b)) - (3*a^6*tan(e*x + d)^2 - 18*a^4*b^2*tan(e*x + d)^2 + 3*a^2*b^4*tan(e*x + d)^2 - 40*a^3*b^3*tan(e*x + d) + 8*a*b^5*tan(e*x + d) - a^6 - 4*a^4*b^2 - 21*a^2*b^4 + 6*b^6)/((a^6*sgn(a*tan(e*x + d) + b) + 3*a^4*b^2*sgn(a*tan(e*x + d) + b) + 3*a^2*b^4*sgn(a*tan(e*x + d) + b) + b ^6*sgn(a*tan(e*x + d) + b))*(a*tan(e*x + d) + b)^2))/e
Timed out. \[ \int \frac {a+b \tan (d+e x)}{\left (b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right )^{3/2}} \, dx=\int \frac {a+b\,\mathrm {tan}\left (d+e\,x\right )}{{\left (a^2\,{\mathrm {tan}\left (d+e\,x\right )}^2+2\,a\,b\,\mathrm {tan}\left (d+e\,x\right )+b^2\right )}^{3/2}} \,d x \]