\(\int \tan (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx\) [3]

Optimal result

Integrand size = 31, antiderivative size = 601 \[ \int \tan (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\frac {\sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \arctan \left (\frac {b^2+(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )-b \sqrt {a^2+b^2-2 a c+c^2} \tan (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}+\frac {b \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 \sqrt {c} e}-\frac {\sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \text {arctanh}\left (\frac {b^2+(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )+b \sqrt {a^2+b^2-2 a c+c^2} \tan (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}+\frac {\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{e} \] 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^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))*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 
+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))/c^(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^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))*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+(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2)/e
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.19 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.42 \[ \int \tan (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\frac {-\frac {1}{2} \sqrt {a-i b-c} \text {arctanh}\left (\frac {2 a-i b+(b-2 i c) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )-\frac {1}{2} \sqrt {a+i b-c} \text {arctanh}\left (\frac {2 a+i b+(b+2 i c) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )+\frac {b \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 \sqrt {c}}+\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{e} \] Input:

Integrate[Tan[d + e*x]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]
 

Output:

(-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])]) - (Sqr 
t[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]*Sqrt[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
 

Rubi [A] (verified)

Time = 28.20 (sec) , antiderivative size = 665, normalized size of antiderivative = 1.11, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {3042, 4183, 1354, 27, 2144, 27, 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.

\(\displaystyle \int \tan (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \tan (d+e x) \sqrt {a+b \tan (d+e x)+c \tan (d+e x)^2}dx\)

\(\Big \downarrow \) 4183

\(\displaystyle \frac {\int \frac {\tan (d+e x) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{\tan ^2(d+e x)+1}d\tan (d+e x)}{e}\)

\(\Big \downarrow \) 1354

\(\displaystyle \frac {\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}-\int \frac {-b \tan ^2(d+e x)-2 (a-c) \tan (d+e x)+b}{2 \left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\tan (d+e x)}{e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}-\frac {1}{2} \int \frac {-b \tan ^2(d+e x)-2 (a-c) \tan (d+e x)+b}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\tan (d+e x)}{e}\)

\(\Big \downarrow \) 2144

\(\displaystyle \frac {\frac {1}{2} \left (b \int \frac {1}{\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\tan (d+e x)-\int \frac {2 (b-(a-c) \tan (d+e x))}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\tan (d+e x)\right )+\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{2} \left (b \int \frac {1}{\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\tan (d+e x)-2 \int \frac {b-(a-c) \tan (d+e x)}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\tan (d+e x)\right )+\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{e}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\frac {1}{2} \left (2 b \int \frac {1}{4 c-\frac {(b+2 c \tan (d+e x))^2}{c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\frac {b+2 c \tan (d+e x)}{\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}-2 \int \frac {b-(a-c) \tan (d+e x)}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\tan (d+e x)\right )+\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{e}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {1}{2} \left (\frac {b \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {c}}-2 \int \frac {b-(a-c) \tan (d+e x)}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\tan (d+e x)\right )+\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{e}\)

\(\Big \downarrow \) 1369

\(\displaystyle \frac {\frac {1}{2} \left (\frac {b \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {c}}-2 \left (\frac {\int \frac {b \sqrt {a^2-2 c a+b^2+c^2}-\left (b^2+(a-c) \left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right )\right ) \tan (d+e x)}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\tan (d+e x)}{2 \sqrt {a^2-2 a c+b^2+c^2}}-\frac {\int -\frac {\sqrt {a^2-2 c a+b^2+c^2} b+\left (b^2+(a-c) \left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right )\right ) \tan (d+e x)}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\tan (d+e x)}{2 \sqrt {a^2-2 a c+b^2+c^2}}\right )\right )+\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{e}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {1}{2} \left (\frac {b \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {c}}-2 \left (\frac {\int \frac {\sqrt {a^2-2 c a+b^2+c^2} b+\left (b^2+(a-c) \left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right )\right ) \tan (d+e x)}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\tan (d+e x)}{2 \sqrt {a^2-2 a c+b^2+c^2}}+\frac {\int \frac {b \sqrt {a^2-2 c a+b^2+c^2}-\left (b^2+(a-c) \left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right )\right ) \tan (d+e x)}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\tan (d+e x)}{2 \sqrt {a^2-2 a c+b^2+c^2}}\right )\right )+\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{e}\)

\(\Big \downarrow \) 1363

\(\displaystyle \frac {\frac {1}{2} \left (\frac {b \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {c}}-2 \left (b \left ((a-c) \left (\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )+b^2\right ) \int \frac {1}{\frac {b \left (b^2+\sqrt {a^2-2 c a+b^2+c^2} \tan (d+e x) b+(a-c) \left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right )\right )^2}{c \tan ^2(d+e x)+b \tan (d+e x)+a}-2 b \sqrt {a^2-2 c a+b^2+c^2} \left (b^2+(a-c) \left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right )\right )}d\left (-\frac {b^2+\sqrt {a^2-2 c a+b^2+c^2} \tan (d+e x) b+(a-c) \left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right )}{\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )-b \left ((a-c) \left (-\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )+b^2\right ) \int \frac {1}{\frac {b \left (b^2-\sqrt {a^2-2 c a+b^2+c^2} \tan (d+e x) b+(a-c) \left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right )\right )^2}{c \tan ^2(d+e x)+b \tan (d+e x)+a}+2 b \sqrt {a^2-2 c a+b^2+c^2} \left (b^2+(a-c) \left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right )\right )}d\frac {b^2-\sqrt {a^2-2 c a+b^2+c^2} \tan (d+e x) b+(a-c) \left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right )}{\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )\right )+\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{e}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {1}{2} \left (\frac {b \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {c}}-2 \left (b \left ((a-c) \left (\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )+b^2\right ) \int \frac {1}{\frac {b \left (b^2+\sqrt {a^2-2 c a+b^2+c^2} \tan (d+e x) b+(a-c) \left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right )\right )^2}{c \tan ^2(d+e x)+b \tan (d+e x)+a}-2 b \sqrt {a^2-2 c a+b^2+c^2} \left (b^2+(a-c) \left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right )\right )}d\left (-\frac {b^2+\sqrt {a^2-2 c a+b^2+c^2} \tan (d+e x) b+(a-c) \left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right )}{\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )-\frac {\left ((a-c) \left (-\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )+b^2\right ) \arctan \left (\frac {-b \sqrt {a^2-2 a c+b^2+c^2} \tan (d+e x)+(a-c) \left (-\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )+b^2}{\sqrt {2} \sqrt [4]{a^2-2 a c+b^2+c^2} \sqrt {-a \left (\sqrt {a^2-2 a c+b^2+c^2}+2 c\right )+c \left (\sqrt {a^2-2 a c+b^2+c^2}+c\right )+a^2+b^2} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2-2 a c+b^2+c^2} \sqrt {-a \left (\sqrt {a^2-2 a c+b^2+c^2}+2 c\right )+c \left (\sqrt {a^2-2 a c+b^2+c^2}+c\right )+a^2+b^2}}\right )\right )+\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{e}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {1}{2} \left (\frac {b \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {c}}-2 \left (\frac {\left ((a-c) \left (\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )+b^2\right ) \text {arctanh}\left (\frac {b \sqrt {a^2-2 a c+b^2+c^2} \tan (d+e x)+(a-c) \left (\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )+b^2}{\sqrt {2} \sqrt [4]{a^2-2 a c+b^2+c^2} \sqrt {-a \left (2 c-\sqrt {a^2-2 a c+b^2+c^2}\right )+c \left (c-\sqrt {a^2-2 a c+b^2+c^2}\right )+a^2+b^2} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2-2 a c+b^2+c^2} \sqrt {-a \left (2 c-\sqrt {a^2-2 a c+b^2+c^2}\right )+c \left (c-\sqrt {a^2-2 a c+b^2+c^2}\right )+a^2+b^2}}-\frac {\left ((a-c) \left (-\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )+b^2\right ) \arctan \left (\frac {-b \sqrt {a^2-2 a c+b^2+c^2} \tan (d+e x)+(a-c) \left (-\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )+b^2}{\sqrt {2} \sqrt [4]{a^2-2 a c+b^2+c^2} \sqrt {-a \left (\sqrt {a^2-2 a c+b^2+c^2}+2 c\right )+c \left (\sqrt {a^2-2 a c+b^2+c^2}+c\right )+a^2+b^2} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2-2 a c+b^2+c^2} \sqrt {-a \left (\sqrt {a^2-2 a c+b^2+c^2}+2 c\right )+c \left (\sqrt {a^2-2 a c+b^2+c^2}+c\right )+a^2+b^2}}\right )\right )+\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{e}\)

Input:

Int[Tan[d + e*x]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]
 

Output:

(((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])])/Sqrt[c] - 2*(-(((b^2 + (a - c)*(a - 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*T 
an[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])])) + ((b^2 + (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]))*ArcT 
anh[(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])])/(Sq 
rt[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])])))/2 + Sqrt[ 
a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/e
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 218

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]
 

rule 219

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])
 

rule 221

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 1092

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]
 

rule 1354

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] - Simp[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 1363

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]
 

rule 1369

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]
 

rule 2144

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4183

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]
 

Maple [B] (warning: unable to verify)

result has leaf size over 500,000. Avoiding possible recursion issues.

Time = 2.31 (sec) , antiderivative size = 17767879, normalized size of antiderivative = 29563.86

\[\text {output too large to display}\]

Input:

int(tan(e*x+d)*(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2),x)
 

Output:

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 = 0.59 (sec) , antiderivative size = 4660, normalized size of antiderivative = 7.75 \[ \int \tan (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\text {Too large to display} \] Input:

integrate(tan(e*x+d)*(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2),x, algorithm="f 
ricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \tan (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\int \sqrt {a + b \tan {\left (d + e x \right )} + c \tan ^{2}{\left (d + e x \right )}} \tan {\left (d + e x \right )}\, dx \] Input:

integrate(tan(e*x+d)*(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)*tan(d + e*x), x)
 

Maxima [F]

\[ \int \tan (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\int { \sqrt {c \tan \left (e x + d\right )^{2} + b \tan \left (e x + d\right ) + a} \tan \left (e x + d\right ) \,d x } \] Input:

integrate(tan(e*x+d)*(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2),x, algorithm="m 
axima")
 

Output:

integrate(sqrt(c*tan(e*x + d)^2 + b*tan(e*x + d) + a)*tan(e*x + d), x)
 

Giac [F(-2)]

Exception generated. \[ \int \tan (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(tan(e*x+d)*(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2),x, algorithm="g 
iac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \tan (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\int \mathrm {tan}\left (d+e\,x\right )\,\sqrt {c\,{\mathrm {tan}\left (d+e\,x\right )}^2+b\,\mathrm {tan}\left (d+e\,x\right )+a} \,d x \] Input:

int(tan(d + e*x)*(a + b*tan(d + e*x) + c*tan(d + e*x)^2)^(1/2),x)
 

Output:

int(tan(d + e*x)*(a + b*tan(d + e*x) + c*tan(d + e*x)^2)^(1/2), x)
 

Reduce [F]

\[ \int \tan (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\int \sqrt {\tan \left (e x +d \right )^{2} c +\tan \left (e x +d \right ) b +a}\, \tan \left (e x +d \right )d x \] Input:

int(tan(e*x+d)*(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)*tan(d + e*x),x)