\(\int \tan (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx\) [5]
Optimal result
Integrand size = 31, antiderivative size = 601 \[
\int \tan (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\frac {\sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \arctan \left (\frac {b^2+(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )-b \sqrt {a^2+b^2-2 a c+c^2} \tan (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}+\frac {b \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 \sqrt {c} e}-\frac {\sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \text {arctanh}\left (\frac {b^2+(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )+b \sqrt {a^2+b^2-2 a c+c^2} \tan (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}+\frac {\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{e}
\]
[Out]
1/2*b*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*arctanh(1/2*
(b^2+(a-c)*(a-c+(a^2-2*a*c+b^2+c^2)^(1/2))+b*(a^2-2*a*c+b^2+c^2)^(1/2)*tan(e*x+d))/(a^2-2*a*c+b^2+c^2)^(1/4)*2
^(1/2)/(a^2+b^2+c*(c-(a^2-2*a*c+b^2+c^2)^(1/2))-a*(2*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^2+b^2+c*(c-(a^2-2*a*c+b^2+c^2)^(1/2))-a*(2*c-(a^2-2*a*c+b^2+c^2)^(1/2)))^(1/2)/(a^2-2*a*c
+b^2+c^2)^(1/4)/e*2^(1/2)+1/2*arctan(1/2*(b^2+(a-c)*(a-c-(a^2-2*a*c+b^2+c^2)^(1/2))-b*(a^2-2*a*c+b^2+c^2)^(1/2
)*tan(e*x+d))/(a^2-2*a*c+b^2+c^2)^(1/4)*2^(1/2)/(a^2+b^2+c*(c+(a^2-2*a*c+b^2+c^2)^(1/2))-a*(2*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^2+b^2+c*(c+(a^2-2*a*c+b^2+c^2)^(1/2))-a*(2*c+(a^
2-2*a*c+b^2+c^2)^(1/2)))^(1/2)/(a^2-2*a*c+b^2+c^2)^(1/4)/e*2^(1/2)+(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2)/e
Rubi [A] (verified)
Time = 23.87 (sec) , antiderivative size = 601, normalized size of antiderivative = 1.00, number
of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {3781, 1035, 1092, 635, 212,
1050, 1044, 214, 211} \[
\int \tan (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\frac {\sqrt {-a \left (\sqrt {a^2-2 a c+b^2+c^2}+2 c\right )+c \left (\sqrt {a^2-2 a c+b^2+c^2}+c\right )+a^2+b^2} \arctan \left (\frac {-b \sqrt {a^2-2 a c+b^2+c^2} \tan (d+e x)+(a-c) \left (-\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )+b^2}{\sqrt {2} \sqrt [4]{a^2-2 a c+b^2+c^2} \sqrt {-a \left (\sqrt {a^2-2 a c+b^2+c^2}+2 c\right )+c \left (\sqrt {a^2-2 a c+b^2+c^2}+c\right )+a^2+b^2} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} e \sqrt [4]{a^2-2 a c+b^2+c^2}}-\frac {\sqrt {-a \left (2 c-\sqrt {a^2-2 a c+b^2+c^2}\right )+c \left (c-\sqrt {a^2-2 a c+b^2+c^2}\right )+a^2+b^2} \text {arctanh}\left (\frac {b \sqrt {a^2-2 a c+b^2+c^2} \tan (d+e x)+(a-c) \left (\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )+b^2}{\sqrt {2} \sqrt [4]{a^2-2 a c+b^2+c^2} \sqrt {-a \left (2 c-\sqrt {a^2-2 a c+b^2+c^2}\right )+c \left (c-\sqrt {a^2-2 a c+b^2+c^2}\right )+a^2+b^2} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} e \sqrt [4]{a^2-2 a c+b^2+c^2}}+\frac {b \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 \sqrt {c} e}+\frac {\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{e}
\]
[In]
Int[Tan[d + e*x]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]
[Out]
(Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTan[(b^2
+ (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Tan[d + e*x])/(Sqrt[2]*(a
^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*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]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) +
(b*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(2*Sqrt[c]*e) - (
Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTanh[(b^2
+ (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) + b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Tan[d + e*x])/(Sqrt[2]*(a
^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*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]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) +
Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]/e
Rule 211
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]
Rule 212
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] && (GtQ[a, 0] || LtQ[b, 0])
Rule 214
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]
Rule 635
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 1035
Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp
[h*(a + b*x + c*x^2)^p*((d + f*x^2)^(q + 1)/(2*f*(p + q + 1))), x] - Dist[1/(2*f*(p + q + 1)), Int[(a + b*x +
c*x^2)^(p - 1)*(d + f*x^2)^q*Simp[h*p*(b*d) + a*(-2*g*f)*(p + q + 1) + (2*h*p*(c*d - a*f) + b*(-2*g*f)*(p + q
+ 1))*x + (h*p*((-b)*f) + c*(-2*g*f)*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, f, g, h, q}, x] && NeQ
[b^2 - 4*a*c, 0] && GtQ[p, 0] && NeQ[p + q + 1, 0]
Rule 1044
Int[((g_) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[-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] /; F
reeQ[{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]
Rule 1050
Int[((g_.) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q
= Rt[(c*d - a*f)^2 + a*c*e^2, 2]}, Dist[1/(2*q), Int[Simp[(-a)*h*e - g*(c*d - a*f - q) + (h*(c*d - a*f + q) -
g*c*e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] - Dist[1/(2*q), Int[Simp[(-a)*h*e - g*(c*d - a*f + q
) + (h*(c*d - a*f - q) - g*c*e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d, e, f, g,
h}, x] && NeQ[e^2 - 4*d*f, 0] && NegQ[(-a)*c]
Rule 1092
Int[((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Sym
bol] :> Dist[C/c, Int[1/Sqrt[d + e*x + f*x^2], x], x] + Dist[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, A, B, C}, x] && NeQ[e^2 - 4*d*f, 0]
Rule 3781
Int[tan[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*((f_.)*tan[(d_.) + (e_.)*(x_)])^(n_.) + (c_.)*((f_.)*tan[(d_.
) + (e_.)*(x_)])^(n2_.))^(p_), x_Symbol] :> Dist[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[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
Rubi steps \begin{align*}
\text {integral}& = \frac {\text {Subst}\left (\int \frac {x \sqrt {a+b x+c x^2}}{1+x^2} \, dx,x,\tan (d+e x)\right )}{e} \\ & = \frac {\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{e}-\frac {\text {Subst}\left (\int \frac {\frac {b}{2}-(a-c) x-\frac {b x^2}{2}}{\left (1+x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{e} \\ & = \frac {\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{e}-\frac {\text {Subst}\left (\int \frac {b+(-a+c) x}{\left (1+x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{e}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{2 e} \\ & = \frac {\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{e}+\frac {b \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c \tan (d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{e}+\frac {\text {Subst}\left (\int \frac {-b \sqrt {a^2+b^2-2 a c+c^2}+\left (-b^2-(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) x}{\left (1+x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{2 \sqrt {a^2+b^2-2 a c+c^2} e}-\frac {\text {Subst}\left (\int \frac {b \sqrt {a^2+b^2-2 a c+c^2}+\left (-b^2-(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) x}{\left (1+x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{2 \sqrt {a^2+b^2-2 a c+c^2} e} \\ & = \frac {b \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 )}{2 \sqrt {c} e}+\frac {\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{e}-\frac {\left (b \left (b^2+(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )\right )\right ) \text {Subst}\left (\int \frac {1}{2 b \sqrt {a^2+b^2-2 a c+c^2} \left (b^2+(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )\right )+b x^2} \, dx,x,\frac {-b^2-(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )+b \sqrt {a^2+b^2-2 a c+c^2} \tan (d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{e}-\frac {\left (b \left (b^2+(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )\right )\right ) \text {Subst}\left (\int \frac {1}{-2 b \sqrt {a^2+b^2-2 a c+c^2} \left (b^2+(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )\right )+b x^2} \, dx,x,\frac {-b^2-(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )-b \sqrt {a^2+b^2-2 a c+c^2} \tan (d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{e} \\ & = \frac {\sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \arctan \left (\frac {b^2+(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )-b \sqrt {a^2+b^2-2 a c+c^2} \tan (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}+\frac {b \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 )}{2 \sqrt {c} e}-\frac {\sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \text {arctanh}\left (\frac {b^2+(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )+b \sqrt {a^2+b^2-2 a c+c^2} \tan (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}+\frac {\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{e} \\
\end{align*}
Mathematica [C] (verified)
Result contains complex when optimal does not.
Time = 0.43 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.42
\[
\int \tan (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\frac {-\frac {1}{2} \sqrt {a-i b-c} \text {arctanh}\left (\frac {2 a-i b+(b-2 i c) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )-\frac {1}{2} \sqrt {a+i b-c} \text {arctanh}\left (\frac {2 a+i b+(b+2 i c) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )+\frac {b \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 \sqrt {c}}+\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{e}
\]
[In]
Integrate[Tan[d + e*x]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]
[Out]
(-1/2*(Sqrt[a - I*b - c]*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[(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])])/2 + (b*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*S
qrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(2*Sqrt[c]) + Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/e
Maple [B] (warning: unable to verify)
result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 1.92 (sec) , antiderivative size = 17767879, normalized size of antiderivative =
29563.86
\[\text {output too large to display}\]
[In]
int((a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2)*tan(e*x+d),x)
[Out]
result too large to display
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 2330 vs. \(2 (542) = 1084\).
Time = 0.74 (sec) , antiderivative size = 4660, normalized size of antiderivative = 7.75
\[
\int \tan (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\text {Too large to display}
\]
[In]
integrate((a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2)*tan(e*x+d),x, algorithm="fricas")
[Out]
[1/4*(c*e*sqrt(-(e^2*sqrt(-b^2/e^4) - a + c)/e^2)*log(-(2*(4*a^3*b^2 + 2*a*b^4 + 2*b^4*c - 4*a*b^2*c^2 - (4*a^
4*b + 3*a^2*b^3 + b^5 + (4*a^2*b - b^3)*c^2 - 2*(4*a^3*b + 3*a*b^3)*c)*tan(e*x + d) + (2*(2*a^3*b + a*b^3 + b^
3*c - 2*a*b*c^2)*e^2*tan(e*x + d) + (4*a^4 + 3*a^2*b^2 + b^4 + (4*a^2 - b^2)*c^2 - 2*(4*a^3 + 3*a*b^2)*c)*e^2)
*sqrt(-b^2/e^4))*sqrt(c*tan(e*x + d)^2 + b*tan(e*x + d) + a) + ((4*a^3*b^2 + 3*a*b^4 - 8*a*b^2*c^2 - (4*a^2*b^
2 - 3*b^4)*c)*e*tan(e*x + d)^2 + 2*(4*a^4*b + 3*a^2*b^3 + b^5 + (4*a^2*b - b^3)*c^2 - 2*(4*a^3*b + 3*a*b^3)*c)
*e*tan(e*x + d) - (4*a^3*b^2 + a*b^4 + (4*a^2*b^2 + b^4)*c)*e + ((b^4 + 2*(4*a^2 - b^2)*c^2 - 2*(4*a^3 + 5*a*b
^2)*c)*e^3*tan(e*x + d)^2 - 4*(2*a^3*b + a*b^3 + b^3*c - 2*a*b*c^2)*e^3*tan(e*x + d) - (8*a^4 + 6*a^2*b^2 + b^
4 - 2*(4*a^3 + a*b^2)*c)*e^3)*sqrt(-b^2/e^4))*sqrt(-(e^2*sqrt(-b^2/e^4) - a + c)/e^2))/(tan(e*x + d)^2 + 1)) -
c*e*sqrt(-(e^2*sqrt(-b^2/e^4) - a + c)/e^2)*log(-(2*(4*a^3*b^2 + 2*a*b^4 + 2*b^4*c - 4*a*b^2*c^2 - (4*a^4*b +
3*a^2*b^3 + b^5 + (4*a^2*b - b^3)*c^2 - 2*(4*a^3*b + 3*a*b^3)*c)*tan(e*x + d) + (2*(2*a^3*b + a*b^3 + b^3*c -
2*a*b*c^2)*e^2*tan(e*x + d) + (4*a^4 + 3*a^2*b^2 + b^4 + (4*a^2 - b^2)*c^2 - 2*(4*a^3 + 3*a*b^2)*c)*e^2)*sqrt
(-b^2/e^4))*sqrt(c*tan(e*x + d)^2 + b*tan(e*x + d) + a) - ((4*a^3*b^2 + 3*a*b^4 - 8*a*b^2*c^2 - (4*a^2*b^2 - 3
*b^4)*c)*e*tan(e*x + d)^2 + 2*(4*a^4*b + 3*a^2*b^3 + b^5 + (4*a^2*b - b^3)*c^2 - 2*(4*a^3*b + 3*a*b^3)*c)*e*ta
n(e*x + d) - (4*a^3*b^2 + a*b^4 + (4*a^2*b^2 + b^4)*c)*e + ((b^4 + 2*(4*a^2 - b^2)*c^2 - 2*(4*a^3 + 5*a*b^2)*c
)*e^3*tan(e*x + d)^2 - 4*(2*a^3*b + a*b^3 + b^3*c - 2*a*b*c^2)*e^3*tan(e*x + d) - (8*a^4 + 6*a^2*b^2 + b^4 - 2
*(4*a^3 + a*b^2)*c)*e^3)*sqrt(-b^2/e^4))*sqrt(-(e^2*sqrt(-b^2/e^4) - a + c)/e^2))/(tan(e*x + d)^2 + 1)) + c*e*
sqrt((e^2*sqrt(-b^2/e^4) + a - c)/e^2)*log(-(2*(4*a^3*b^2 + 2*a*b^4 + 2*b^4*c - 4*a*b^2*c^2 - (4*a^4*b + 3*a^2
*b^3 + b^5 + (4*a^2*b - b^3)*c^2 - 2*(4*a^3*b + 3*a*b^3)*c)*tan(e*x + d) - (2*(2*a^3*b + a*b^3 + b^3*c - 2*a*b
*c^2)*e^2*tan(e*x + d) + (4*a^4 + 3*a^2*b^2 + b^4 + (4*a^2 - b^2)*c^2 - 2*(4*a^3 + 3*a*b^2)*c)*e^2)*sqrt(-b^2/
e^4))*sqrt(c*tan(e*x + d)^2 + b*tan(e*x + d) + a) + ((4*a^3*b^2 + 3*a*b^4 - 8*a*b^2*c^2 - (4*a^2*b^2 - 3*b^4)*
c)*e*tan(e*x + d)^2 + 2*(4*a^4*b + 3*a^2*b^3 + b^5 + (4*a^2*b - b^3)*c^2 - 2*(4*a^3*b + 3*a*b^3)*c)*e*tan(e*x
+ d) - (4*a^3*b^2 + a*b^4 + (4*a^2*b^2 + b^4)*c)*e - ((b^4 + 2*(4*a^2 - b^2)*c^2 - 2*(4*a^3 + 5*a*b^2)*c)*e^3*
tan(e*x + d)^2 - 4*(2*a^3*b + a*b^3 + b^3*c - 2*a*b*c^2)*e^3*tan(e*x + d) - (8*a^4 + 6*a^2*b^2 + b^4 - 2*(4*a^
3 + a*b^2)*c)*e^3)*sqrt(-b^2/e^4))*sqrt((e^2*sqrt(-b^2/e^4) + a - c)/e^2))/(tan(e*x + d)^2 + 1)) - c*e*sqrt((e
^2*sqrt(-b^2/e^4) + a - c)/e^2)*log(-(2*(4*a^3*b^2 + 2*a*b^4 + 2*b^4*c - 4*a*b^2*c^2 - (4*a^4*b + 3*a^2*b^3 +
b^5 + (4*a^2*b - b^3)*c^2 - 2*(4*a^3*b + 3*a*b^3)*c)*tan(e*x + d) - (2*(2*a^3*b + a*b^3 + b^3*c - 2*a*b*c^2)*e
^2*tan(e*x + d) + (4*a^4 + 3*a^2*b^2 + b^4 + (4*a^2 - b^2)*c^2 - 2*(4*a^3 + 3*a*b^2)*c)*e^2)*sqrt(-b^2/e^4))*s
qrt(c*tan(e*x + d)^2 + b*tan(e*x + d) + a) - ((4*a^3*b^2 + 3*a*b^4 - 8*a*b^2*c^2 - (4*a^2*b^2 - 3*b^4)*c)*e*ta
n(e*x + d)^2 + 2*(4*a^4*b + 3*a^2*b^3 + b^5 + (4*a^2*b - b^3)*c^2 - 2*(4*a^3*b + 3*a*b^3)*c)*e*tan(e*x + d) -
(4*a^3*b^2 + a*b^4 + (4*a^2*b^2 + b^4)*c)*e - ((b^4 + 2*(4*a^2 - b^2)*c^2 - 2*(4*a^3 + 5*a*b^2)*c)*e^3*tan(e*x
+ d)^2 - 4*(2*a^3*b + a*b^3 + b^3*c - 2*a*b*c^2)*e^3*tan(e*x + d) - (8*a^4 + 6*a^2*b^2 + b^4 - 2*(4*a^3 + a*b
^2)*c)*e^3)*sqrt(-b^2/e^4))*sqrt((e^2*sqrt(-b^2/e^4) + a - c)/e^2))/(tan(e*x + d)^2 + 1)) + b*sqrt(c)*log(8*c^
2*tan(e*x + d)^2 + 8*b*c*tan(e*x + d) + b^2 + 4*sqrt(c*tan(e*x + d)^2 + b*tan(e*x + d) + a)*(2*c*tan(e*x + d)
+ b)*sqrt(c) + 4*a*c) + 4*sqrt(c*tan(e*x + d)^2 + b*tan(e*x + d) + a)*c)/(c*e), 1/4*(c*e*sqrt(-(e^2*sqrt(-b^2/
e^4) - a + c)/e^2)*log(-(2*(4*a^3*b^2 + 2*a*b^4 + 2*b^4*c - 4*a*b^2*c^2 - (4*a^4*b + 3*a^2*b^3 + b^5 + (4*a^2*
b - b^3)*c^2 - 2*(4*a^3*b + 3*a*b^3)*c)*tan(e*x + d) + (2*(2*a^3*b + a*b^3 + b^3*c - 2*a*b*c^2)*e^2*tan(e*x +
d) + (4*a^4 + 3*a^2*b^2 + b^4 + (4*a^2 - b^2)*c^2 - 2*(4*a^3 + 3*a*b^2)*c)*e^2)*sqrt(-b^2/e^4))*sqrt(c*tan(e*x
+ d)^2 + b*tan(e*x + d) + a) + ((4*a^3*b^2 + 3*a*b^4 - 8*a*b^2*c^2 - (4*a^2*b^2 - 3*b^4)*c)*e*tan(e*x + d)^2
+ 2*(4*a^4*b + 3*a^2*b^3 + b^5 + (4*a^2*b - b^3)*c^2 - 2*(4*a^3*b + 3*a*b^3)*c)*e*tan(e*x + d) - (4*a^3*b^2 +
a*b^4 + (4*a^2*b^2 + b^4)*c)*e + ((b^4 + 2*(4*a^2 - b^2)*c^2 - 2*(4*a^3 + 5*a*b^2)*c)*e^3*tan(e*x + d)^2 - 4*(
2*a^3*b + a*b^3 + b^3*c - 2*a*b*c^2)*e^3*tan(e*x + d) - (8*a^4 + 6*a^2*b^2 + b^4 - 2*(4*a^3 + a*b^2)*c)*e^3)*s
qrt(-b^2/e^4))*sqrt(-(e^2*sqrt(-b^2/e^4) - a + c)/e^2))/(tan(e*x + d)^2 + 1)) - c*e*sqrt(-(e^2*sqrt(-b^2/e^4)
- a + c)/e^2)*log(-(2*(4*a^3*b^2 + 2*a*b^4 + 2*b^4*c - 4*a*b^2*c^2 - (4*a^4*b + 3*a^2*b^3 + b^5 + (4*a^2*b - b
^3)*c^2 - 2*(4*a^3*b + 3*a*b^3)*c)*tan(e*x + d) + (2*(2*a^3*b + a*b^3 + b^3*c - 2*a*b*c^2)*e^2*tan(e*x + d) +
(4*a^4 + 3*a^2*b^2 + b^4 + (4*a^2 - b^2)*c^2 - 2*(4*a^3 + 3*a*b^2)*c)*e^2)*sqrt(-b^2/e^4))*sqrt(c*tan(e*x + d)
^2 + b*tan(e*x + d) + a) - ((4*a^3*b^2 + 3*a*b^4 - 8*a*b^2*c^2 - (4*a^2*b^2 - 3*b^4)*c)*e*tan(e*x + d)^2 + 2*(
4*a^4*b + 3*a^2*b^3 + b^5 + (4*a^2*b - b^3)*c^2 - 2*(4*a^3*b + 3*a*b^3)*c)*e*tan(e*x + d) - (4*a^3*b^2 + a*b^4
+ (4*a^2*b^2 + b^4)*c)*e + ((b^4 + 2*(4*a^2 - b^2)*c^2 - 2*(4*a^3 + 5*a*b^2)*c)*e^3*tan(e*x + d)^2 - 4*(2*a^3
*b + a*b^3 + b^3*c - 2*a*b*c^2)*e^3*tan(e*x + d) - (8*a^4 + 6*a^2*b^2 + b^4 - 2*(4*a^3 + a*b^2)*c)*e^3)*sqrt(-
b^2/e^4))*sqrt(-(e^2*sqrt(-b^2/e^4) - a + c)/e^2))/(tan(e*x + d)^2 + 1)) + c*e*sqrt((e^2*sqrt(-b^2/e^4) + a -
c)/e^2)*log(-(2*(4*a^3*b^2 + 2*a*b^4 + 2*b^4*c - 4*a*b^2*c^2 - (4*a^4*b + 3*a^2*b^3 + b^5 + (4*a^2*b - b^3)*c^
2 - 2*(4*a^3*b + 3*a*b^3)*c)*tan(e*x + d) - (2*(2*a^3*b + a*b^3 + b^3*c - 2*a*b*c^2)*e^2*tan(e*x + d) + (4*a^4
+ 3*a^2*b^2 + b^4 + (4*a^2 - b^2)*c^2 - 2*(4*a^3 + 3*a*b^2)*c)*e^2)*sqrt(-b^2/e^4))*sqrt(c*tan(e*x + d)^2 + b
*tan(e*x + d) + a) + ((4*a^3*b^2 + 3*a*b^4 - 8*a*b^2*c^2 - (4*a^2*b^2 - 3*b^4)*c)*e*tan(e*x + d)^2 + 2*(4*a^4*
b + 3*a^2*b^3 + b^5 + (4*a^2*b - b^3)*c^2 - 2*(4*a^3*b + 3*a*b^3)*c)*e*tan(e*x + d) - (4*a^3*b^2 + a*b^4 + (4*
a^2*b^2 + b^4)*c)*e - ((b^4 + 2*(4*a^2 - b^2)*c^2 - 2*(4*a^3 + 5*a*b^2)*c)*e^3*tan(e*x + d)^2 - 4*(2*a^3*b + a
*b^3 + b^3*c - 2*a*b*c^2)*e^3*tan(e*x + d) - (8*a^4 + 6*a^2*b^2 + b^4 - 2*(4*a^3 + a*b^2)*c)*e^3)*sqrt(-b^2/e^
4))*sqrt((e^2*sqrt(-b^2/e^4) + a - c)/e^2))/(tan(e*x + d)^2 + 1)) - c*e*sqrt((e^2*sqrt(-b^2/e^4) + a - c)/e^2)
*log(-(2*(4*a^3*b^2 + 2*a*b^4 + 2*b^4*c - 4*a*b^2*c^2 - (4*a^4*b + 3*a^2*b^3 + b^5 + (4*a^2*b - b^3)*c^2 - 2*(
4*a^3*b + 3*a*b^3)*c)*tan(e*x + d) - (2*(2*a^3*b + a*b^3 + b^3*c - 2*a*b*c^2)*e^2*tan(e*x + d) + (4*a^4 + 3*a^
2*b^2 + b^4 + (4*a^2 - b^2)*c^2 - 2*(4*a^3 + 3*a*b^2)*c)*e^2)*sqrt(-b^2/e^4))*sqrt(c*tan(e*x + d)^2 + b*tan(e*
x + d) + a) - ((4*a^3*b^2 + 3*a*b^4 - 8*a*b^2*c^2 - (4*a^2*b^2 - 3*b^4)*c)*e*tan(e*x + d)^2 + 2*(4*a^4*b + 3*a
^2*b^3 + b^5 + (4*a^2*b - b^3)*c^2 - 2*(4*a^3*b + 3*a*b^3)*c)*e*tan(e*x + d) - (4*a^3*b^2 + a*b^4 + (4*a^2*b^2
+ b^4)*c)*e - ((b^4 + 2*(4*a^2 - b^2)*c^2 - 2*(4*a^3 + 5*a*b^2)*c)*e^3*tan(e*x + d)^2 - 4*(2*a^3*b + a*b^3 +
b^3*c - 2*a*b*c^2)*e^3*tan(e*x + d) - (8*a^4 + 6*a^2*b^2 + b^4 - 2*(4*a^3 + a*b^2)*c)*e^3)*sqrt(-b^2/e^4))*sqr
t((e^2*sqrt(-b^2/e^4) + a - c)/e^2))/(tan(e*x + d)^2 + 1)) - 2*b*sqrt(-c)*arctan(1/2*sqrt(c*tan(e*x + d)^2 + b
*tan(e*x + d) + a)*(2*c*tan(e*x + d) + b)*sqrt(-c)/(c^2*tan(e*x + d)^2 + b*c*tan(e*x + d) + a*c)) + 4*sqrt(c*t
an(e*x + d)^2 + b*tan(e*x + d) + a)*c)/(c*e)]
Sympy [F]
\[
\int \tan (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\int \sqrt {a + b \tan {\left (d + e x \right )} + c \tan ^{2}{\left (d + e x \right )}} \tan {\left (d + e x \right )}\, dx
\]
[In]
integrate((a+b*tan(e*x+d)+c*tan(e*x+d)**2)**(1/2)*tan(e*x+d),x)
[Out]
Integral(sqrt(a + b*tan(d + e*x) + c*tan(d + e*x)**2)*tan(d + e*x), x)
Maxima [F]
\[
\int \tan (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\int { \sqrt {c \tan \left (e x + d\right )^{2} + b \tan \left (e x + d\right ) + a} \tan \left (e x + d\right ) \,d x }
\]
[In]
integrate((a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2)*tan(e*x+d),x, algorithm="maxima")
[Out]
integrate(sqrt(c*tan(e*x + d)^2 + b*tan(e*x + d) + a)*tan(e*x + d), x)
Giac [F(-1)]
Timed out. \[
\int \tan (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\text {Timed out}
\]
[In]
integrate((a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2)*tan(e*x+d),x, algorithm="giac")
[Out]
Timed out
Mupad [F(-1)]
Timed out. \[
\int \tan (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\int \mathrm {tan}\left (d+e\,x\right )\,\sqrt {c\,{\mathrm {tan}\left (d+e\,x\right )}^2+b\,\mathrm {tan}\left (d+e\,x\right )+a} \,d x
\]
[In]
int(tan(d + e*x)*(a + b*tan(d + e*x) + c*tan(d + e*x)^2)^(1/2),x)
[Out]
int(tan(d + e*x)*(a + b*tan(d + e*x) + c*tan(d + e*x)^2)^(1/2), x)