Integrand size = 24, antiderivative size = 574 \[ \int \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=-\frac {\sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \arctan \left (\frac {b \sqrt {a^2+b^2-2 a c+c^2}-\left (b^2+(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) \tan (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}+\frac {\sqrt {c} \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{e}-\frac {\sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \text {arctanh}\left (\frac {b \sqrt {a^2+b^2-2 a c+c^2}+\left (b^2+(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) \tan (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e} \] Output:
-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)*arctan(1/2*(b*(a^2-2*a*c+b^2+c^2)^(1/2)-(b^2+(a-c)*(a-c+(a^2- 2*a*c+b^2+c^2)^(1/2)))*tan(e*x+d))*2^(1/2)/(a^2-2*a*c+b^2+c^2)^(1/4)/(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))*2^(1/2)/(a^2-2*a*c+b^2+c^2)^(1/4 )/e+c^(1/2)*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-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)*arctanh(1/2*(b*(a^2-2*a*c+b^2+c^2)^(1/2)+(b^ 2+(a-c)*(a-c-(a^2-2*a*c+b^2+c^2)^(1/2)))*tan(e*x+d))*2^(1/2)/(a^2-2*a*c+b^ 2+c^2)^(1/4)/(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))*2^(1/2)/(a^2-2 *a*c+b^2+c^2)^(1/4)/e
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.40 \[ \int \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\frac {-i \sqrt {a-i b-c} \text {arctanh}\left (\frac {2 a-i b+(b-2 i c) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )+i \sqrt {a+i b-c} \text {arctanh}\left (\frac {2 a+i b+(b+2 i c) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )+2 \sqrt {c} \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 e} \] Input:
Integrate[Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]
Output:
((-I)*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])] + I*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*Sqrt[c]*ArcTan h[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e* x]^2])])/(2*e)
Time = 27.95 (sec) , antiderivative size = 637, normalized size of antiderivative = 1.11, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {3042, 4853, 1321, 25, 1092, 219, 1369, 25, 1363, 218, 221}
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.
\(\displaystyle \int \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sqrt {a+b \tan (d+e x)+c \tan (d+e x)^2}dx\) |
\(\Big \downarrow \) 4853 |
\(\displaystyle \frac {\int \frac {\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}d\tan (d+e x)}{e}\) |
\(\Big \downarrow \) 1321 |
\(\displaystyle \frac {c \int \frac {1}{\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\tan (d+e x)-\int -\frac {a-c+b \tan (d+e x)}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\tan (d+e x)}{e}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {c \int \frac {1}{\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\tan (d+e x)+\int \frac {a-c+b \tan (d+e x)}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\tan (d+e x)}{e}\) |
\(\Big \downarrow \) 1092 |
\(\displaystyle \frac {\int \frac {a-c+b \tan (d+e x)}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\tan (d+e x)+2 c \int \frac {1}{4 c-\frac {(b+2 c \tan (d+e x))^2}{c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\frac {b+2 c \tan (d+e x)}{\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}}{e}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\int \frac {a-c+b \tan (d+e x)}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\tan (d+e x)+\sqrt {c} \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{e}\) |
\(\Big \downarrow \) 1369 |
\(\displaystyle \frac {\frac {\int -\frac {b^2-\sqrt {a^2-2 c a+b^2+c^2} \tan (d+e x) b+(a-c) \left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right )}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\tan (d+e x)}{2 \sqrt {a^2-2 a c+b^2+c^2}}-\frac {\int -\frac {b^2+\sqrt {a^2-2 c a+b^2+c^2} \tan (d+e x) b+(a-c) \left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right )}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\tan (d+e x)}{2 \sqrt {a^2-2 a c+b^2+c^2}}+\sqrt {c} \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{e}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {-\frac {\int \frac {b^2-\sqrt {a^2-2 c a+b^2+c^2} \tan (d+e x) b+(a-c) \left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right )}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\tan (d+e x)}{2 \sqrt {a^2-2 a c+b^2+c^2}}+\frac {\int \frac {b^2+\sqrt {a^2-2 c a+b^2+c^2} \tan (d+e x) b+(a-c) \left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right )}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\tan (d+e x)}{2 \sqrt {a^2-2 a c+b^2+c^2}}+\sqrt {c} \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{e}\) |
\(\Big \downarrow \) 1363 |
\(\displaystyle \frac {-b \left ((a-c) \left (-\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )+b^2\right ) \int \frac {1}{\frac {b \left (\sqrt {a^2-2 c a+b^2+c^2} b+\left (b^2+(a-c) \left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right )\right ) \tan (d+e x)\right )^2}{c \tan ^2(d+e x)+b \tan (d+e x)+a}-2 b \sqrt {a^2-2 c a+b^2+c^2} \left (b^2+(a-c) \left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right )\right )}d\left (-\frac {\sqrt {a^2-2 c a+b^2+c^2} b+\left (b^2+(a-c) \left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right )\right ) \tan (d+e x)}{\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )-b \left ((a-c) \left (\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )+b^2\right ) \int \frac {1}{\frac {b \left (b \sqrt {a^2-2 c a+b^2+c^2}-\left (b^2+(a-c) \left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right )\right ) \tan (d+e x)\right )^2}{c \tan ^2(d+e x)+b \tan (d+e x)+a}+2 b \sqrt {a^2-2 c a+b^2+c^2} \left (b^2+(a-c) \left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right )\right )}d\frac {b \sqrt {a^2-2 c a+b^2+c^2}-\left (b^2+(a-c) \left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right )\right ) \tan (d+e x)}{\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}+\sqrt {c} \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{e}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {-b \left ((a-c) \left (-\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )+b^2\right ) \int \frac {1}{\frac {b \left (\sqrt {a^2-2 c a+b^2+c^2} b+\left (b^2+(a-c) \left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right )\right ) \tan (d+e x)\right )^2}{c \tan ^2(d+e x)+b \tan (d+e x)+a}-2 b \sqrt {a^2-2 c a+b^2+c^2} \left (b^2+(a-c) \left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right )\right )}d\left (-\frac {\sqrt {a^2-2 c a+b^2+c^2} b+\left (b^2+(a-c) \left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right )\right ) \tan (d+e x)}{\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )-\frac {\left ((a-c) \left (\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )+b^2\right ) \arctan \left (\frac {b \sqrt {a^2-2 a c+b^2+c^2}-\left ((a-c) \left (\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )+b^2\right ) \tan (d+e x)}{\sqrt {2} \sqrt [4]{a^2-2 a c+b^2+c^2} \sqrt {-a \left (2 c-\sqrt {a^2-2 a c+b^2+c^2}\right )+c \left (c-\sqrt {a^2-2 a c+b^2+c^2}\right )+a^2+b^2} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2-2 a c+b^2+c^2} \sqrt {-a \left (2 c-\sqrt {a^2-2 a c+b^2+c^2}\right )+c \left (c-\sqrt {a^2-2 a c+b^2+c^2}\right )+a^2+b^2}}+\sqrt {c} \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{e}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {-\frac {\left ((a-c) \left (\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )+b^2\right ) \arctan \left (\frac {b \sqrt {a^2-2 a c+b^2+c^2}-\left ((a-c) \left (\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )+b^2\right ) \tan (d+e x)}{\sqrt {2} \sqrt [4]{a^2-2 a c+b^2+c^2} \sqrt {-a \left (2 c-\sqrt {a^2-2 a c+b^2+c^2}\right )+c \left (c-\sqrt {a^2-2 a c+b^2+c^2}\right )+a^2+b^2} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2-2 a c+b^2+c^2} \sqrt {-a \left (2 c-\sqrt {a^2-2 a c+b^2+c^2}\right )+c \left (c-\sqrt {a^2-2 a c+b^2+c^2}\right )+a^2+b^2}}-\frac {\left ((a-c) \left (-\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )+b^2\right ) \text {arctanh}\left (\frac {\left ((a-c) \left (-\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )+b^2\right ) \tan (d+e x)+b \sqrt {a^2-2 a c+b^2+c^2}}{\sqrt {2} \sqrt [4]{a^2-2 a c+b^2+c^2} \sqrt {-a \left (\sqrt {a^2-2 a c+b^2+c^2}+2 c\right )+c \left (\sqrt {a^2-2 a c+b^2+c^2}+c\right )+a^2+b^2} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2-2 a c+b^2+c^2} \sqrt {-a \left (\sqrt {a^2-2 a c+b^2+c^2}+2 c\right )+c \left (\sqrt {a^2-2 a c+b^2+c^2}+c\right )+a^2+b^2}}+\sqrt {c} \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{e}\) |
Input:
Int[Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]
Output:
(-(((b^2 + (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]))*ArcTan[(b*Sqrt [a^2 + b^2 - 2*a*c + c^2] - (b^2 + (a - c)*(a - c + 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)*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[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])] - ((b^2 + (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]))*ArcTanh[(b *Sqrt[a^2 + b^2 - 2*a*c + c^2] + (b^2 + (a - c)*(a - c - 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)*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])]))/e
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[((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]
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[Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2]/((d_) + (f_.)*(x_)^2), x_Symbol] :> Simp[c/f Int[1/Sqrt[a + b*x + c*x^2], x], x] - Simp[1/f Int[(c*d - a*f - b*f*x)/(Sqrt[a + b*x + c*x^2]*(d + f*x^2)), x], x] /; FreeQ[{a, b, c, d, f}, x] && NeQ[b^2 - 4*a*c, 0]
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[((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]}, Simp [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] - Simp[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]
Int[u_, x_Symbol] :> With[{v = FunctionOfTrig[u, x]}, Simp[With[{d = FreeFa ctors[Tan[v], x]}, d/Coefficient[v, x, 1] Subst[Int[SubstFor[1/(1 + d^2*x ^2), Tan[v]/d, u, x], x], x, Tan[v]/d]], x] /; !FalseQ[v] && FunctionOfQ[N onfreeFactors[Tan[v], x], u, x, True] && TryPureTanSubst[ActivateTrig[u], x ]]
result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 1.79 (sec) , antiderivative size = 17246911, normalized size of antiderivative = 30046.88
\[\text {output too large to display}\]
Input:
int((a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2),x)
Output:
result too large to display
Leaf count of result is larger than twice the leaf count of optimal. 2273 vs. \(2 (516) = 1032\).
Time = 0.36 (sec) , antiderivative size = 4547, normalized size of antiderivative = 7.92 \[ \int \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\text {Too large to display} \] Input:
integrate((a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2),x, algorithm="fricas")
Output:
Too large to include
\[ \int \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\int \sqrt {a + b \tan {\left (d + e x \right )} + c \tan ^{2}{\left (d + e x \right )}}\, dx \] Input:
integrate((a+b*tan(e*x+d)+c*tan(e*x+d)**2)**(1/2),x)
Output:
Integral(sqrt(a + b*tan(d + e*x) + c*tan(d + e*x)**2), x)
Exception generated. \[ \int \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume((c-b-a)*(c+b-a)>0)', see `assume ?` for mor
\[ \int \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\int { \sqrt {c \tan \left (e x + d\right )^{2} + b \tan \left (e x + d\right ) + a} \,d x } \] Input:
integrate((a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(c*tan(e*x + d)^2 + b*tan(e*x + d) + a), x)
Timed out. \[ \int \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\int \sqrt {c\,{\mathrm {tan}\left (d+e\,x\right )}^2+b\,\mathrm {tan}\left (d+e\,x\right )+a} \,d x \] Input:
int((a + b*tan(d + e*x) + c*tan(d + e*x)^2)^(1/2),x)
Output:
int((a + b*tan(d + e*x) + c*tan(d + e*x)^2)^(1/2), x)
\[ \int \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\int \sqrt {\tan \left (e x +d \right )^{2} c +\tan \left (e x +d \right ) b +a}d x \] Input:
int((a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2),x)
Output:
int(sqrt(tan(d + e*x)**2*c + tan(d + e*x)*b + a),x)