Integrand size = 33, antiderivative size = 352 \[ \int \frac {\tan ^2(d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=-\frac {\sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \arctan \left (\frac {b-\left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right ) \tan (d+e x)}{\sqrt {2} \sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2+b^2-2 a c+c^2} e}+\frac {\sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \arctan \left (\frac {b-\left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right ) \tan (d+e x)}{\sqrt {2} \sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2+b^2-2 a c+c^2} e}+\frac {\text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {c} e} \]
arctanh(1/2*(b+2*c*tan(e*x+d))/c^(1/2)/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/ 2))/e/c^(1/2)-1/2*arctan(1/2*(b-(a-c-(a^2-2*a*c+b^2+c^2)^(1/2))*tan(e*x+d) )*2^(1/2)/(a-c-(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)/(a+b*tan(e*x+d)+c*tan(e*x+ d)^2)^(1/2))*(a-c-(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)/e*2^(1/2)/(a^2-2*a*c+b^ 2+c^2)^(1/2)+1/2*arctan(1/2*(b-(a-c+(a^2-2*a*c+b^2+c^2)^(1/2))*tan(e*x+d)) *2^(1/2)/(a-c+(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)/(a+b*tan(e*x+d)+c*tan(e*x+d )^2)^(1/2))*(a-c+(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)/e*2^(1/2)/(a^2-2*a*c+b^2 +c^2)^(1/2)
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.65 \[ \int \frac {\tan ^2(d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=\frac {\frac {i \text {arctanh}\left (\frac {2 a-i b+(b-2 i c) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 \sqrt {a-i b-c}}-\frac {i \text {arctanh}\left (\frac {2 a+i b+(b+2 i c) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 \sqrt {a+i b-c}}+\frac {\text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {c}}}{e} \]
(((I/2)*ArcTanh[(2*a - I*b + (b - (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/Sqrt[a - I*b - c] - ((I /2)*ArcTanh[(2*a + I*b + (b + (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]* Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/Sqrt[a + I*b - c] + ArcTanh [(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x ]^2])]/Sqrt[c])/e
Time = 0.67 (sec) , antiderivative size = 347, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3042, 4183, 2144, 25, 1092, 219, 1318, 1363, 218}
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.
\(\displaystyle \int \frac {\tan ^2(d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\tan (d+e x)^2}{\sqrt {a+b \tan (d+e x)+c \tan (d+e x)^2}}dx\) |
\(\Big \downarrow \) 4183 |
\(\displaystyle \frac {\int \frac {\tan ^2(d+e x)}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\tan (d+e x)}{e}\) |
\(\Big \downarrow \) 2144 |
\(\displaystyle \frac {\int \frac {1}{\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\tan (d+e x)+\int -\frac {1}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\tan (d+e x)}{e}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {1}{\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\tan (d+e x)-\int \frac {1}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\tan (d+e x)}{e}\) |
\(\Big \downarrow \) 1092 |
\(\displaystyle \frac {2 \int \frac {1}{4 c-\frac {(b+2 c \tan (d+e x))^2}{c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\frac {b+2 c \tan (d+e x)}{\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}-\int \frac {1}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\tan (d+e x)}{e}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\frac {\text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {c}}-\int \frac {1}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\tan (d+e x)}{e}\) |
\(\Big \downarrow \) 1318 |
\(\displaystyle \frac {\frac {\int \frac {a-c+b \tan (d+e x)-\sqrt {a^2-2 c a+b^2+c^2}}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\tan (d+e x)}{2 \sqrt {a^2-2 a c+b^2+c^2}}-\frac {\int \frac {a-c+b \tan (d+e x)+\sqrt {a^2-2 c a+b^2+c^2}}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\tan (d+e x)}{2 \sqrt {a^2-2 a c+b^2+c^2}}+\frac {\text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {c}}}{e}\) |
\(\Big \downarrow \) 1363 |
\(\displaystyle \frac {-\frac {b \left (-\sqrt {a^2-2 a c+b^2+c^2}+a-c\right ) \int \frac {1}{\frac {b \left (b-\left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right ) \tan (d+e x)\right )^2}{c \tan ^2(d+e x)+b \tan (d+e x)+a}+2 b \left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right )}d\frac {b-\left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right ) \tan (d+e x)}{\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}}{\sqrt {a^2-2 a c+b^2+c^2}}+\frac {b \left (\sqrt {a^2-2 a c+b^2+c^2}+a-c\right ) \int \frac {1}{\frac {b \left (b-\left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right ) \tan (d+e x)\right )^2}{c \tan ^2(d+e x)+b \tan (d+e x)+a}+2 b \left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right )}d\frac {b-\left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right ) \tan (d+e x)}{\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}}{\sqrt {a^2-2 a c+b^2+c^2}}+\frac {\text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {c}}}{e}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {-\frac {\sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+a-c} \arctan \left (\frac {b-\left (-\sqrt {a^2-2 a c+b^2+c^2}+a-c\right ) \tan (d+e x)}{\sqrt {2} \sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+a-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2-2 a c+b^2+c^2}}+\frac {\sqrt {\sqrt {a^2-2 a c+b^2+c^2}+a-c} \arctan \left (\frac {b-\left (\sqrt {a^2-2 a c+b^2+c^2}+a-c\right ) \tan (d+e x)}{\sqrt {2} \sqrt {\sqrt {a^2-2 a c+b^2+c^2}+a-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2-2 a c+b^2+c^2}}+\frac {\text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {c}}}{e}\) |
(-((Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTan[(b - (a - c - Sqrt[ a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[a - c - Sqrt[a^2 + b ^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2] *Sqrt[a^2 + b^2 - 2*a*c + c^2])) + (Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTan[(b - (a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(S qrt[2]*Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]) + ArcTanh[ (b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x] ^2])]/Sqrt[c])/e
3.1.13.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[1/(((a_.) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_S ymbol] :> With[{q = Rt[(c*d - a*f)^2 + a*c*e^2, 2]}, Simp[1/(2*q) Int[(c* d - a*f + q + c*e*x)/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] - Simp[1/( 2*q) Int[(c*d - a*f - q + c*e*x)/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d, e, f}, x] && NeQ[e^2 - 4*d*f, 0] && NegQ[(-a)*c]
Int[((g_) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f _.)*(x_)^2]), x_Symbol] :> Simp[-2*a*g*h Subst[Int[1/Simp[2*a^2*g*h*c + a *e*x^2, x], x], x, Simp[a*h - g*c*x, x]/Sqrt[d + e*x + f*x^2]], x] /; FreeQ [{a, c, d, e, f, g, h}, x] && EqQ[a*h^2*e + 2*g*h*(c*d - a*f) - g^2*c*e, 0]
Int[(Px_)/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2]}, Simp[C/c Int[1/Sqrt[d + e*x + f*x^2], x], x] + Simp[1/c Int[(A* c - a*C + B*c*x)/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d, e, f}, x] && PolyQ[Px, x, 2]
Int[tan[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*((f_.)*tan[(d_.) + (e_.)*( x_)])^(n_.) + (c_.)*((f_.)*tan[(d_.) + (e_.)*(x_)])^(n2_.))^(p_), x_Symbol] :> Simp[f/e Subst[Int[(x/f)^m*((a + b*x^n + c*x^(2*n))^p/(f^2 + x^2)), x ], x, f*Tan[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[n 2, 2*n] && NeQ[b^2 - 4*a*c, 0]
result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 1.02 (sec) , antiderivative size = 7491751, normalized size of antiderivative = 21283.38
\[\text {output too large to display}\]
Leaf count of result is larger than twice the leaf count of optimal. 4980 vs. \(2 (313) = 626\).
Time = 0.99 (sec) , antiderivative size = 9961, normalized size of antiderivative = 28.30 \[ \int \frac {\tan ^2(d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=\text {Too large to display} \]
\[ \int \frac {\tan ^2(d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=\int \frac {\tan ^{2}{\left (d + e x \right )}}{\sqrt {a + b \tan {\left (d + e x \right )} + c \tan ^{2}{\left (d + e x \right )}}}\, dx \]
\[ \int \frac {\tan ^2(d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=\int { \frac {\tan \left (e x + d\right )^{2}}{\sqrt {c \tan \left (e x + d\right )^{2} + b \tan \left (e x + d\right ) + a}} \,d x } \]
Timed out. \[ \int \frac {\tan ^2(d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\tan ^2(d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=\int \frac {{\mathrm {tan}\left (d+e\,x\right )}^2}{\sqrt {c\,{\mathrm {tan}\left (d+e\,x\right )}^2+b\,\mathrm {tan}\left (d+e\,x\right )+a}} \,d x \]