Integrand size = 26, antiderivative size = 829 \[ \int \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx=\frac {\sqrt {a-b+c} \arctan \left (\frac {\sqrt {a-b+c} \tan (d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e}+\frac {\sqrt {c} \tan (d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{e \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )}-\frac {\sqrt [4]{a} \sqrt [4]{c} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}}}{e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac {\left (b+\sqrt {a} \sqrt {c}-c\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}}}{2 \sqrt [4]{a} \sqrt [4]{c} e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\sqrt [4]{c} (a-b+c) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}}}{2 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {c}\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac {\left (\sqrt {a}+\sqrt {c}\right ) (a-b+c) \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {a}-\sqrt {c}\right )^2}{4 \sqrt {a} \sqrt {c}},2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}}}{4 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {c}\right ) \sqrt [4]{c} e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \]
1/2*arctan((a-b+c)^(1/2)*tan(e*x+d)/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2 ))*(a-b+c)^(1/2)/e+c^(1/2)*(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)*tan(e*x +d)/e/(a^(1/2)+c^(1/2)*tan(e*x+d)^2)-a^(1/4)*c^(1/4)*(cos(2*arctan(c^(1/4) *tan(e*x+d)/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*tan(e*x+d)/a^(1/4)))*E llipticE(sin(2*arctan(c^(1/4)*tan(e*x+d)/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/2 ))^(1/2))*((a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)/(a^(1/2)+c^(1/2)*tan(e*x+d)^2 )^2)^(1/2)*(a^(1/2)+c^(1/2)*tan(e*x+d)^2)/e/(a+b*tan(e*x+d)^2+c*tan(e*x+d) ^4)^(1/2)-1/2*c^(1/4)*(a-b+c)*(cos(2*arctan(c^(1/4)*tan(e*x+d)/a^(1/4)))^2 )^(1/2)/cos(2*arctan(c^(1/4)*tan(e*x+d)/a^(1/4)))*EllipticF(sin(2*arctan(c ^(1/4)*tan(e*x+d)/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))*((a+b*tan(e*x +d)^2+c*tan(e*x+d)^4)/(a^(1/2)+c^(1/2)*tan(e*x+d)^2)^2)^(1/2)*(a^(1/2)+c^( 1/2)*tan(e*x+d)^2)/a^(1/4)/e/(a^(1/2)-c^(1/2))/(a+b*tan(e*x+d)^2+c*tan(e*x +d)^4)^(1/2)+1/4*(a-b+c)*(cos(2*arctan(c^(1/4)*tan(e*x+d)/a^(1/4)))^2)^(1/ 2)/cos(2*arctan(c^(1/4)*tan(e*x+d)/a^(1/4)))*EllipticPi(sin(2*arctan(c^(1/ 4)*tan(e*x+d)/a^(1/4))),-1/4*(a^(1/2)-c^(1/2))^2/a^(1/2)/c^(1/2),1/2*(2-b/ a^(1/2)/c^(1/2))^(1/2))*(a^(1/2)+c^(1/2))*((a+b*tan(e*x+d)^2+c*tan(e*x+d)^ 4)/(a^(1/2)+c^(1/2)*tan(e*x+d)^2)^2)^(1/2)*(a^(1/2)+c^(1/2)*tan(e*x+d)^2)/ a^(1/4)/c^(1/4)/e/(a^(1/2)-c^(1/2))/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2 )+1/2*(cos(2*arctan(c^(1/4)*tan(e*x+d)/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^( 1/4)*tan(e*x+d)/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*tan(e*x+d)/a^(...
Result contains complex when optimal does not.
Time = 12.27 (sec) , antiderivative size = 428, normalized size of antiderivative = 0.52 \[ \int \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx=\frac {i \left (\left (-b+\sqrt {b^2-4 a c}\right ) E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \tan (d+e x)\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-\left (b-2 c+\sqrt {b^2-4 a c}\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \tan (d+e x)\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-2 (a-b+c) \operatorname {EllipticPi}\left (\frac {b+\sqrt {b^2-4 a c}}{2 c},i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \tan (d+e x)\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c \tan ^2(d+e x)}{b+\sqrt {b^2-4 a c}}} \sqrt {1-\frac {2 c \tan ^2(d+e x)}{-b+\sqrt {b^2-4 a c}}}}{2 \sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \]
((I/2)*((-b + Sqrt[b^2 - 4*a*c])*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + S qrt[b^2 - 4*a*c])]*Tan[d + e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - (b - 2*c + Sqrt[b^2 - 4*a*c])*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[ c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqr t[b^2 - 4*a*c])] - 2*(a - b + c)*EllipticPi[(b + Sqrt[b^2 - 4*a*c])/(2*c), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Tan[d + e*x]], (b + Sqr t[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2* c*Tan[d + e*x]^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 - (2*c*Tan[d + e*x]^2)/( -b + Sqrt[b^2 - 4*a*c])])/(Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*e*Sqrt[ a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])
Time = 0.83 (sec) , antiderivative size = 694, normalized size of antiderivative = 0.84, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3042, 4853, 1523, 25, 27, 1511, 27, 1416, 1509, 2220}
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.
\(\displaystyle \int \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sqrt {a+b \tan (d+e x)^2+c \tan (d+e x)^4}dx\) |
\(\Big \downarrow \) 4853 |
\(\displaystyle \frac {\int \frac {\sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}{\tan ^2(d+e x)+1}d\tan (d+e x)}{e}\) |
\(\Big \downarrow \) 1523 |
\(\displaystyle \frac {\frac {(a-b+c) \int \frac {\sqrt {c} \tan ^2(d+e x)+\sqrt {a}}{\sqrt {a} \left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)}{1-\frac {\sqrt {c}}{\sqrt {a}}}-\frac {\int -\frac {\left (1-\frac {\sqrt {c}}{\sqrt {a}}\right ) c \tan ^2(d+e x)+b-c-\sqrt {a} \sqrt {c}}{\sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)}{1-\frac {\sqrt {c}}{\sqrt {a}}}}{e}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {(a-b+c) \int \frac {\sqrt {c} \tan ^2(d+e x)+\sqrt {a}}{\sqrt {a} \left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)}{1-\frac {\sqrt {c}}{\sqrt {a}}}+\frac {\int \frac {\left (1-\frac {\sqrt {c}}{\sqrt {a}}\right ) c \tan ^2(d+e x)+b-c-\sqrt {a} \sqrt {c}}{\sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)}{1-\frac {\sqrt {c}}{\sqrt {a}}}}{e}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {(a-b+c) \int \frac {\sqrt {c} \tan ^2(d+e x)+\sqrt {a}}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)}{\sqrt {a} \left (1-\frac {\sqrt {c}}{\sqrt {a}}\right )}+\frac {\int \frac {\left (1-\frac {\sqrt {c}}{\sqrt {a}}\right ) c \tan ^2(d+e x)+b-c-\sqrt {a} \sqrt {c}}{\sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)}{1-\frac {\sqrt {c}}{\sqrt {a}}}}{e}\) |
\(\Big \downarrow \) 1511 |
\(\displaystyle \frac {\frac {(b-2 c) \int \frac {1}{\sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)-\sqrt {c} \left (\sqrt {a}-\sqrt {c}\right ) \int \frac {\sqrt {a}-\sqrt {c} \tan ^2(d+e x)}{\sqrt {a} \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)}{1-\frac {\sqrt {c}}{\sqrt {a}}}+\frac {(a-b+c) \int \frac {\sqrt {c} \tan ^2(d+e x)+\sqrt {a}}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)}{\sqrt {a} \left (1-\frac {\sqrt {c}}{\sqrt {a}}\right )}}{e}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {(b-2 c) \int \frac {1}{\sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)-\frac {\sqrt {c} \left (\sqrt {a}-\sqrt {c}\right ) \int \frac {\sqrt {a}-\sqrt {c} \tan ^2(d+e x)}{\sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)}{\sqrt {a}}}{1-\frac {\sqrt {c}}{\sqrt {a}}}+\frac {(a-b+c) \int \frac {\sqrt {c} \tan ^2(d+e x)+\sqrt {a}}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)}{\sqrt {a} \left (1-\frac {\sqrt {c}}{\sqrt {a}}\right )}}{e}\) |
\(\Big \downarrow \) 1416 |
\(\displaystyle \frac {\frac {\frac {(b-2 c) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\sqrt {c} \left (\sqrt {a}-\sqrt {c}\right ) \int \frac {\sqrt {a}-\sqrt {c} \tan ^2(d+e x)}{\sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)}{\sqrt {a}}}{1-\frac {\sqrt {c}}{\sqrt {a}}}+\frac {(a-b+c) \int \frac {\sqrt {c} \tan ^2(d+e x)+\sqrt {a}}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)}{\sqrt {a} \left (1-\frac {\sqrt {c}}{\sqrt {a}}\right )}}{e}\) |
\(\Big \downarrow \) 1509 |
\(\displaystyle \frac {\frac {(a-b+c) \int \frac {\sqrt {c} \tan ^2(d+e x)+\sqrt {a}}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)}{\sqrt {a} \left (1-\frac {\sqrt {c}}{\sqrt {a}}\right )}+\frac {\frac {(b-2 c) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\sqrt {c} \left (\sqrt {a}-\sqrt {c}\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\tan (d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{\sqrt {a}+\sqrt {c} \tan ^2(d+e x)}\right )}{\sqrt {a}}}{1-\frac {\sqrt {c}}{\sqrt {a}}}}{e}\) |
\(\Big \downarrow \) 2220 |
\(\displaystyle \frac {\frac {(a-b+c) \left (\frac {\left (\sqrt {a}-\sqrt {c}\right ) \arctan \left (\frac {\sqrt {a-b+c} \tan (d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 \sqrt {a-b+c}}+\frac {\left (\sqrt {a}+\sqrt {c}\right ) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {a}-\sqrt {c}\right )^2}{4 \sqrt {a} \sqrt {c}},2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{4 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{\sqrt {a} \left (1-\frac {\sqrt {c}}{\sqrt {a}}\right )}+\frac {\frac {(b-2 c) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\sqrt {c} \left (\sqrt {a}-\sqrt {c}\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\tan (d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{\sqrt {a}+\sqrt {c} \tan ^2(d+e x)}\right )}{\sqrt {a}}}{1-\frac {\sqrt {c}}{\sqrt {a}}}}{e}\) |
(((a - b + c)*(((Sqrt[a] - Sqrt[c])*ArcTan[(Sqrt[a - b + c]*Tan[d + e*x])/ Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]])/(2*Sqrt[a - b + c]) + ((Sq rt[a] + Sqrt[c])*EllipticPi[-1/4*(Sqrt[a] - Sqrt[c])^2/(Sqrt[a]*Sqrt[c]), 2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sq rt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x ]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(4*a^(1/4)*c^(1/4)*Sqrt[a + b* Tan[d + e*x]^2 + c*Tan[d + e*x]^4])))/(Sqrt[a]*(1 - Sqrt[c]/Sqrt[a])) + (( (b - 2*c)*EllipticF[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt [a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e* x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(2*a^(1/4) *c^(1/4)*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - ((Sqrt[a] - Sqrt [c])*Sqrt[c]*(-((Tan[d + e*x]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4 ])/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)) + (a^(1/4)*EllipticE[2*ArcTan[(c^(1 /4)*Tan[d + e*x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]*(Sqrt[a] + Sqrt[c ]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(c^(1/4)*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])))/Sqrt[a])/(1 - Sqrt[c]/Sqrt[a]))/e
3.1.33.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q ^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 /(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + b*x^2 + c*x^ 4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos Q[c/a]
Int[Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]/((d_) + (e_.)*(x_)^2), x_Symbo l] :> With[{q = Rt[c/a, 2]}, Simp[(c*d^2 - b*d*e + a*e^2)/(e*(e - d*q)) I nt[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x] - Simp[1/(e*(e - d*q)) Int[(c*d - b*e + a*e*q - (c*e - a*d*q^3)*x^2)/Sqrt[a + b*x^2 + c *x^4], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c *d^2 - b*d*e + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]
Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(A rcTan[Rt[-b + c*(d/e) + a*(e/d), 2]*(x/Sqrt[a + b*x^2 + c*x^4])]/(2*d*e*Rt[ -b + c*(d/e) + a*(e/d), 2])), x] + Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(4*d*e*q*Sqrt[a + b*x^2 + c*x^4]))*El lipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*ArcTan[q*x], 1/2 - b/(4*a*q^2)], x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] & & EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && PosQ[-b + c*(d/e) + a*(e/d)]
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 ]]
Time = 0.42 (sec) , antiderivative size = 1497, normalized size of antiderivative = 1.81
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1497\) |
default | \(\text {Expression too large to display}\) | \(1497\) |
1/e*(1/4*2^(1/2)/(-1/a*b+1/a*(-4*a*c+b^2)^(1/2))^(1/2)*(4+2/a*b*tan(e*x+d) ^2-2/a*tan(e*x+d)^2*(-4*a*c+b^2)^(1/2))^(1/2)*(4+2/a*b*tan(e*x+d)^2+2/a*ta n(e*x+d)^2*(-4*a*c+b^2)^(1/2))^(1/2)/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/ 2)*EllipticF(1/2*tan(e*x+d)*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2* (-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))*b-1/4*2^(1/2)/(-1/a*b+1/a*(-4*a *c+b^2)^(1/2))^(1/2)*(4+2/a*b*tan(e*x+d)^2-2/a*tan(e*x+d)^2*(-4*a*c+b^2)^( 1/2))^(1/2)*(4+2/a*b*tan(e*x+d)^2+2/a*tan(e*x+d)^2*(-4*a*c+b^2)^(1/2))^(1/ 2)/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)*EllipticF(1/2*tan(e*x+d)*2^(1/2 )*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c )^(1/2))*c-1/2*c*a*2^(1/2)/(-1/a*b+1/a*(-4*a*c+b^2)^(1/2))^(1/2)*(4+2/a*b* tan(e*x+d)^2-2/a*tan(e*x+d)^2*(-4*a*c+b^2)^(1/2))^(1/2)*(4+2/a*b*tan(e*x+d )^2+2/a*tan(e*x+d)^2*(-4*a*c+b^2)^(1/2))^(1/2)/(a+b*tan(e*x+d)^2+c*tan(e*x +d)^4)^(1/2)/(b+(-4*a*c+b^2)^(1/2))*EllipticF(1/2*tan(e*x+d)*2^(1/2)*((-b+ (-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2) )+1/2*c*a*2^(1/2)/(-1/a*b+1/a*(-4*a*c+b^2)^(1/2))^(1/2)*(4+2/a*b*tan(e*x+d )^2-2/a*tan(e*x+d)^2*(-4*a*c+b^2)^(1/2))^(1/2)*(4+2/a*b*tan(e*x+d)^2+2/a*t an(e*x+d)^2*(-4*a*c+b^2)^(1/2))^(1/2)/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1 /2)/(b+(-4*a*c+b^2)^(1/2))*EllipticE(1/2*tan(e*x+d)*2^(1/2)*((-b+(-4*a*c+b ^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))+a*2^(1/ 2)/(-1/a*b+1/a*(-4*a*c+b^2)^(1/2))^(1/2)*(1+1/2/a*b*tan(e*x+d)^2-1/2/a*...
Timed out. \[ \int \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx=\text {Timed out} \]
\[ \int \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx=\int \sqrt {a + b \tan ^{2}{\left (d + e x \right )} + c \tan ^{4}{\left (d + e x \right )}}\, dx \]
\[ \int \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx=\int { \sqrt {c \tan \left (e x + d\right )^{4} + b \tan \left (e x + d\right )^{2} + a} \,d x } \]
\[ \int \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx=\int { \sqrt {c \tan \left (e x + d\right )^{4} + b \tan \left (e x + d\right )^{2} + a} \,d x } \]
Timed out. \[ \int \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx=\int \sqrt {c\,{\mathrm {tan}\left (d+e\,x\right )}^4+b\,{\mathrm {tan}\left (d+e\,x\right )}^2+a} \,d x \]