\(\int \frac {1}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx\) [15]
Optimal result
Integrand size = 24, antiderivative size = 298 \[
\int \frac {1}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=\frac {\sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \arctan \left (\frac {b-\left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right ) \tan (d+e x)}{\sqrt {2} \sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2+b^2-2 a c+c^2} e}-\frac {\sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \arctan \left (\frac {b-\left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right ) \tan (d+e x)}{\sqrt {2} \sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2+b^2-2 a c+c^2} e}
\]
[Out]
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
)
Rubi [A] (verified)
Time = 0.35 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.00, number of
steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1001, 1044, 211}
\[
\int \frac {1}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=\frac {\sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+a-c} \arctan \left (\frac {b-\left (-\sqrt {a^2-2 a c+b^2+c^2}+a-c\right ) \tan (d+e x)}{\sqrt {2} \sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+a-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} e \sqrt {a^2-2 a c+b^2+c^2}}-\frac {\sqrt {\sqrt {a^2-2 a c+b^2+c^2}+a-c} \arctan \left (\frac {b-\left (\sqrt {a^2-2 a c+b^2+c^2}+a-c\right ) \tan (d+e x)}{\sqrt {2} \sqrt {\sqrt {a^2-2 a c+b^2+c^2}+a-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} e \sqrt {a^2-2 a c+b^2+c^2}}
\]
[In]
Int[1/Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]
[Out]
(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]*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]*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 1001
Int[1/(((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[(c*d - a*f + q + c*e*x)/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] - Dist
[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]
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]
Rubi steps \begin{align*}
\text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{e} \\ & = -\frac {\text {Subst}\left (\int \frac {a-c-\sqrt {a^2+b^2-2 a c+c^2}+b 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 {a-c+\sqrt {a^2+b^2-2 a c+c^2}+b 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 {\left (b \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) \text {Subst}\left (\int \frac {1}{2 b \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )+b x^2} \, dx,x,\frac {b-\left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right ) \tan (d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {a^2+b^2-2 a c+c^2} e}-\frac {\left (b \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) \text {Subst}\left (\int \frac {1}{2 b \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )+b x^2} \, dx,x,\frac {b-\left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right ) \tan (d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\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 {\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} \\
\end{align*}
Mathematica [C] (verified)
Result contains complex when optimal does not.
Time = 0.19 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.58
\[
\int \frac {1}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=-\frac {i \left (\frac {\text {arctanh}\left (\frac {2 a-i b+(b-2 i c) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {a-i b-c}}-\frac {\text {arctanh}\left (\frac {2 a+i b+(b+2 i c) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {a+i b-c}}\right )}{2 e}
\]
[In]
Integrate[1/Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]
[Out]
((-1/2*I)*(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*T
an[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]*Sqr
t[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])]/Sqrt[a + I*b - c]))/e
Maple [B] (warning: unable to verify)
result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 0.71 (sec) , antiderivative size = 7300729, normalized size of antiderivative =
24499.09
\[\text {output too large to display}\]
[In]
int(1/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2),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. 4891 vs. \(2 (267) = 534\).
Time = 0.76 (sec) , antiderivative size = 4891, normalized size of antiderivative = 16.41
\[
\int \frac {1}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=\text {Too large to display}
\]
[In]
integrate(1/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2),x, algorithm="fricas")
[Out]
-1/4*sqrt(-((a^2 + b^2 - 2*a*c + c^2)*e^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4 - 4*a*c^3 + c^4 + 2*(3*a^2 + b^2)*
c^2 - 4*(a^3 + a*b^2)*c)*e^4)) + a - c)/((a^2 + b^2 - 2*a*c + c^2)*e^2))*log(-1/2*(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^5*b + 3*a^3*b^3 + a*b^5 - 3*a*b^3*c^2 - 2*a*b*c^4 + (4*a^2*b + b^3)*c^3 - (4*a^4*b + a^2*b^3 - b^5)
*c)*e^2*tan(e*x + d) + (4*a^6 + 7*a^4*b^2 + 4*a^2*b^4 + b^6 + (4*a^2 - b^2)*c^4 - 4*(4*a^3 + a*b^2)*c^3 + 6*(4
*a^4 + 3*a^2*b^2)*c^2 - 4*(4*a^5 + 5*a^3*b^2 + 2*a*b^4)*c)*e^2)*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4 - 4*a*c^3 +
c^4 + 2*(3*a^2 + b^2)*c^2 - 4*(a^3 + a*b^2)*c)*e^4)))*sqrt(c*tan(e*x + d)^2 + b*tan(e*x + d) + a) + (2*(2*a^3*
b^3 + a*b^5 + (4*a^2*b - b^3)*c^3 - 8*(a^3*b + a*b^3)*c^2 + (4*a^4*b + 3*a^2*b^3 + 2*b^5)*c)*e*tan(e*x + d)^2
+ 2*(8*a^4*b^2 + 5*a^2*b^4 + b^6 - 3*b^4*c^2 + 4*a*b^2*c^3 - 6*(2*a^3*b^2 + a*b^4)*c)*e*tan(e*x + d) + 2*(4*a^
5*b + a^3*b^3 + (4*a^3*b + a*b^3)*c^2 - (8*a^4*b + 6*a^2*b^3 + b^5)*c)*e + ((4*a^6*b + 7*a^4*b^3 + 4*a^2*b^5 +
b^7 + 8*a*b*c^5 - (12*a^2*b + 5*b^3)*c^4 - 8*(2*a^3*b - a*b^3)*c^3 + 2*(20*a^4*b + 11*a^2*b^3 - 2*b^5)*c^2 -
4*(6*a^5*b + 8*a^3*b^3 + 3*a*b^5)*c)*e^3*tan(e*x + d)^2 + 2*(4*a^7 + 3*a^5*b^2 - 2*a^3*b^4 - a*b^6 - (4*a^2 -
b^2)*c^5 + (20*a^3 + 7*a*b^2)*c^4 - 2*(20*a^4 + 15*a^2*b^2 + b^4)*c^3 + 2*(20*a^5 + 19*a^3*b^2 + 7*a*b^4)*c^2
- (20*a^6 + 19*a^4*b^2 + 10*a^2*b^4 + 3*b^6)*c)*e^3*tan(e*x + d) - (12*a^6*b + 19*a^4*b^3 + 8*a^2*b^5 + b^7 -
(4*a^2*b + b^3)*c^4 + 6*(4*a^4*b + a^2*b^3)*c^2 - 4*(8*a^5*b + 6*a^3*b^3 + a*b^5)*c)*e^3)*sqrt(-b^2/((a^4 + 2*
a^2*b^2 + b^4 - 4*a*c^3 + c^4 + 2*(3*a^2 + b^2)*c^2 - 4*(a^3 + a*b^2)*c)*e^4)))*sqrt(-((a^2 + b^2 - 2*a*c + c^
2)*e^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4 - 4*a*c^3 + c^4 + 2*(3*a^2 + b^2)*c^2 - 4*(a^3 + a*b^2)*c)*e^4)) + a
- c)/((a^2 + b^2 - 2*a*c + c^2)*e^2)))/(tan(e*x + d)^2 + 1)) + 1/4*sqrt(-((a^2 + b^2 - 2*a*c + c^2)*e^2*sqrt(-
b^2/((a^4 + 2*a^2*b^2 + b^4 - 4*a*c^3 + c^4 + 2*(3*a^2 + b^2)*c^2 - 4*(a^3 + a*b^2)*c)*e^4)) + a - c)/((a^2 +
b^2 - 2*a*c + c^2)*e^2))*log(-1/2*(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^5*b + 3*a^3*b^3 + a*b^5 - 3*a*b^3*c^2
- 2*a*b*c^4 + (4*a^2*b + b^3)*c^3 - (4*a^4*b + a^2*b^3 - b^5)*c)*e^2*tan(e*x + d) + (4*a^6 + 7*a^4*b^2 + 4*a^
2*b^4 + b^6 + (4*a^2 - b^2)*c^4 - 4*(4*a^3 + a*b^2)*c^3 + 6*(4*a^4 + 3*a^2*b^2)*c^2 - 4*(4*a^5 + 5*a^3*b^2 + 2
*a*b^4)*c)*e^2)*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4 - 4*a*c^3 + c^4 + 2*(3*a^2 + b^2)*c^2 - 4*(a^3 + a*b^2)*c)*e
^4)))*sqrt(c*tan(e*x + d)^2 + b*tan(e*x + d) + a) - (2*(2*a^3*b^3 + a*b^5 + (4*a^2*b - b^3)*c^3 - 8*(a^3*b + a
*b^3)*c^2 + (4*a^4*b + 3*a^2*b^3 + 2*b^5)*c)*e*tan(e*x + d)^2 + 2*(8*a^4*b^2 + 5*a^2*b^4 + b^6 - 3*b^4*c^2 + 4
*a*b^2*c^3 - 6*(2*a^3*b^2 + a*b^4)*c)*e*tan(e*x + d) + 2*(4*a^5*b + a^3*b^3 + (4*a^3*b + a*b^3)*c^2 - (8*a^4*b
+ 6*a^2*b^3 + b^5)*c)*e + ((4*a^6*b + 7*a^4*b^3 + 4*a^2*b^5 + b^7 + 8*a*b*c^5 - (12*a^2*b + 5*b^3)*c^4 - 8*(2
*a^3*b - a*b^3)*c^3 + 2*(20*a^4*b + 11*a^2*b^3 - 2*b^5)*c^2 - 4*(6*a^5*b + 8*a^3*b^3 + 3*a*b^5)*c)*e^3*tan(e*x
+ d)^2 + 2*(4*a^7 + 3*a^5*b^2 - 2*a^3*b^4 - a*b^6 - (4*a^2 - b^2)*c^5 + (20*a^3 + 7*a*b^2)*c^4 - 2*(20*a^4 +
15*a^2*b^2 + b^4)*c^3 + 2*(20*a^5 + 19*a^3*b^2 + 7*a*b^4)*c^2 - (20*a^6 + 19*a^4*b^2 + 10*a^2*b^4 + 3*b^6)*c)*
e^3*tan(e*x + d) - (12*a^6*b + 19*a^4*b^3 + 8*a^2*b^5 + b^7 - (4*a^2*b + b^3)*c^4 + 6*(4*a^4*b + a^2*b^3)*c^2
- 4*(8*a^5*b + 6*a^3*b^3 + a*b^5)*c)*e^3)*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4 - 4*a*c^3 + c^4 + 2*(3*a^2 + b^2)*
c^2 - 4*(a^3 + a*b^2)*c)*e^4)))*sqrt(-((a^2 + b^2 - 2*a*c + c^2)*e^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4 - 4*a*c
^3 + c^4 + 2*(3*a^2 + b^2)*c^2 - 4*(a^3 + a*b^2)*c)*e^4)) + a - c)/((a^2 + b^2 - 2*a*c + c^2)*e^2)))/(tan(e*x
+ d)^2 + 1)) - 1/4*sqrt(((a^2 + b^2 - 2*a*c + c^2)*e^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4 - 4*a*c^3 + c^4 + 2*(
3*a^2 + b^2)*c^2 - 4*(a^3 + a*b^2)*c)*e^4)) - a + c)/((a^2 + b^2 - 2*a*c + c^2)*e^2))*log(-1/2*(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^5*b + 3*a^3*b^3 + a*b^5 - 3*a*b^3*c^2 - 2*a*b*c^4 + (4*a^2*b + b^3)*c^3 - (4*a^4*b + a
^2*b^3 - b^5)*c)*e^2*tan(e*x + d) + (4*a^6 + 7*a^4*b^2 + 4*a^2*b^4 + b^6 + (4*a^2 - b^2)*c^4 - 4*(4*a^3 + a*b^
2)*c^3 + 6*(4*a^4 + 3*a^2*b^2)*c^2 - 4*(4*a^5 + 5*a^3*b^2 + 2*a*b^4)*c)*e^2)*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4
- 4*a*c^3 + c^4 + 2*(3*a^2 + b^2)*c^2 - 4*(a^3 + a*b^2)*c)*e^4)))*sqrt(c*tan(e*x + d)^2 + b*tan(e*x + d) + a)
+ (2*(2*a^3*b^3 + a*b^5 + (4*a^2*b - b^3)*c^3 - 8*(a^3*b + a*b^3)*c^2 + (4*a^4*b + 3*a^2*b^3 + 2*b^5)*c)*e*ta
n(e*x + d)^2 + 2*(8*a^4*b^2 + 5*a^2*b^4 + b^6 - 3*b^4*c^2 + 4*a*b^2*c^3 - 6*(2*a^3*b^2 + a*b^4)*c)*e*tan(e*x +
d) + 2*(4*a^5*b + a^3*b^3 + (4*a^3*b + a*b^3)*c^2 - (8*a^4*b + 6*a^2*b^3 + b^5)*c)*e - ((4*a^6*b + 7*a^4*b^3
+ 4*a^2*b^5 + b^7 + 8*a*b*c^5 - (12*a^2*b + 5*b^3)*c^4 - 8*(2*a^3*b - a*b^3)*c^3 + 2*(20*a^4*b + 11*a^2*b^3 -
2*b^5)*c^2 - 4*(6*a^5*b + 8*a^3*b^3 + 3*a*b^5)*c)*e^3*tan(e*x + d)^2 + 2*(4*a^7 + 3*a^5*b^2 - 2*a^3*b^4 - a*b^
6 - (4*a^2 - b^2)*c^5 + (20*a^3 + 7*a*b^2)*c^4 - 2*(20*a^4 + 15*a^2*b^2 + b^4)*c^3 + 2*(20*a^5 + 19*a^3*b^2 +
7*a*b^4)*c^2 - (20*a^6 + 19*a^4*b^2 + 10*a^2*b^4 + 3*b^6)*c)*e^3*tan(e*x + d) - (12*a^6*b + 19*a^4*b^3 + 8*a^2
*b^5 + b^7 - (4*a^2*b + b^3)*c^4 + 6*(4*a^4*b + a^2*b^3)*c^2 - 4*(8*a^5*b + 6*a^3*b^3 + a*b^5)*c)*e^3)*sqrt(-b
^2/((a^4 + 2*a^2*b^2 + b^4 - 4*a*c^3 + c^4 + 2*(3*a^2 + b^2)*c^2 - 4*(a^3 + a*b^2)*c)*e^4)))*sqrt(((a^2 + b^2
- 2*a*c + c^2)*e^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4 - 4*a*c^3 + c^4 + 2*(3*a^2 + b^2)*c^2 - 4*(a^3 + a*b^2)*c
)*e^4)) - a + c)/((a^2 + b^2 - 2*a*c + c^2)*e^2)))/(tan(e*x + d)^2 + 1)) + 1/4*sqrt(((a^2 + b^2 - 2*a*c + c^2)
*e^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4 - 4*a*c^3 + c^4 + 2*(3*a^2 + b^2)*c^2 - 4*(a^3 + a*b^2)*c)*e^4)) - a +
c)/((a^2 + b^2 - 2*a*c + c^2)*e^2))*log(-1/2*(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^5*b + 3*a^3*b^3 + a*b^5 -
3*a*b^3*c^2 - 2*a*b*c^4 + (4*a^2*b + b^3)*c^3 - (4*a^4*b + a^2*b^3 - b^5)*c)*e^2*tan(e*x + d) + (4*a^6 + 7*a^4
*b^2 + 4*a^2*b^4 + b^6 + (4*a^2 - b^2)*c^4 - 4*(4*a^3 + a*b^2)*c^3 + 6*(4*a^4 + 3*a^2*b^2)*c^2 - 4*(4*a^5 + 5*
a^3*b^2 + 2*a*b^4)*c)*e^2)*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4 - 4*a*c^3 + c^4 + 2*(3*a^2 + b^2)*c^2 - 4*(a^3 +
a*b^2)*c)*e^4)))*sqrt(c*tan(e*x + d)^2 + b*tan(e*x + d) + a) - (2*(2*a^3*b^3 + a*b^5 + (4*a^2*b - b^3)*c^3 - 8
*(a^3*b + a*b^3)*c^2 + (4*a^4*b + 3*a^2*b^3 + 2*b^5)*c)*e*tan(e*x + d)^2 + 2*(8*a^4*b^2 + 5*a^2*b^4 + b^6 - 3*
b^4*c^2 + 4*a*b^2*c^3 - 6*(2*a^3*b^2 + a*b^4)*c)*e*tan(e*x + d) + 2*(4*a^5*b + a^3*b^3 + (4*a^3*b + a*b^3)*c^2
- (8*a^4*b + 6*a^2*b^3 + b^5)*c)*e - ((4*a^6*b + 7*a^4*b^3 + 4*a^2*b^5 + b^7 + 8*a*b*c^5 - (12*a^2*b + 5*b^3)
*c^4 - 8*(2*a^3*b - a*b^3)*c^3 + 2*(20*a^4*b + 11*a^2*b^3 - 2*b^5)*c^2 - 4*(6*a^5*b + 8*a^3*b^3 + 3*a*b^5)*c)*
e^3*tan(e*x + d)^2 + 2*(4*a^7 + 3*a^5*b^2 - 2*a^3*b^4 - a*b^6 - (4*a^2 - b^2)*c^5 + (20*a^3 + 7*a*b^2)*c^4 - 2
*(20*a^4 + 15*a^2*b^2 + b^4)*c^3 + 2*(20*a^5 + 19*a^3*b^2 + 7*a*b^4)*c^2 - (20*a^6 + 19*a^4*b^2 + 10*a^2*b^4 +
3*b^6)*c)*e^3*tan(e*x + d) - (12*a^6*b + 19*a^4*b^3 + 8*a^2*b^5 + b^7 - (4*a^2*b + b^3)*c^4 + 6*(4*a^4*b + a^
2*b^3)*c^2 - 4*(8*a^5*b + 6*a^3*b^3 + a*b^5)*c)*e^3)*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4 - 4*a*c^3 + c^4 + 2*(3*
a^2 + b^2)*c^2 - 4*(a^3 + a*b^2)*c)*e^4)))*sqrt(((a^2 + b^2 - 2*a*c + c^2)*e^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b
^4 - 4*a*c^3 + c^4 + 2*(3*a^2 + b^2)*c^2 - 4*(a^3 + a*b^2)*c)*e^4)) - a + c)/((a^2 + b^2 - 2*a*c + c^2)*e^2)))
/(tan(e*x + d)^2 + 1))
Sympy [F]
\[
\int \frac {1}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=\int \frac {1}{\sqrt {a + b \tan {\left (d + e x \right )} + c \tan ^{2}{\left (d + e x \right )}}}\, dx
\]
[In]
integrate(1/(a+b*tan(e*x+d)+c*tan(e*x+d)**2)**(1/2),x)
[Out]
Integral(1/sqrt(a + b*tan(d + e*x) + c*tan(d + e*x)**2), x)
Maxima [F(-2)]
Exception generated. \[
\int \frac {1}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=\text {Exception raised: ValueError}
\]
[In]
integrate(1/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta
Giac [F]
\[
\int \frac {1}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=\int { \frac {1}{\sqrt {c \tan \left (e x + d\right )^{2} + b \tan \left (e x + d\right ) + a}} \,d x }
\]
[In]
integrate(1/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2),x, algorithm="giac")
[Out]
integrate(1/sqrt(c*tan(e*x + d)^2 + b*tan(e*x + d) + a), x)
Mupad [F(-1)]
Timed out. \[
\int \frac {1}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=\int \frac {1}{\sqrt {c\,{\mathrm {tan}\left (d+e\,x\right )}^2+b\,\mathrm {tan}\left (d+e\,x\right )+a}} \,d x
\]
[In]
int(1/(a + b*tan(d + e*x) + c*tan(d + e*x)^2)^(1/2),x)
[Out]
int(1/(a + b*tan(d + e*x) + c*tan(d + e*x)^2)^(1/2), x)