Integrand size = 35, antiderivative size = 981 \[ \int \frac {\tan ^2(d+e x)}{\left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )^{3/2}} \, dx=-\frac {\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 (a-b+c)^{3/2} e}+\frac {\tan (d+e x) \left (b^2-2 a c-b c+(b-2 c) c \tan ^2(d+e x)\right )}{(a-b+c) \left (b^2-4 a c\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {(b-2 c) \sqrt {c} \tan (d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{(a-b+c) \left (b^2-4 a c\right ) e \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )}+\frac {\sqrt [4]{a} (b-2 c) \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}}}{(a-b+c) \left (b^2-4 a c\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac {\sqrt [4]{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 ) (a-b+c) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\left (\sqrt {a}-\sqrt {c}\right ) \sqrt [4]{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 (b-2 \sqrt {a} \sqrt {c}\right ) (a-b+c) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\left (\sqrt {a}+\sqrt {c}\right ) \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} (a-b+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)^(3/2)/e+tan(e*x+d)*(b^2-2*a*c-b*c+(b-2*c)*c*tan(e*x+d)^2)/(a-b +c)/(-4*a*c+b^2)/e/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)-(b-2*c)*c^(1/2) *(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)*tan(e*x+d)/(a-b+c)/(-4*a*c+b^2)/e /(a^(1/2)+c^(1/2)*tan(e*x+d)^2)+a^(1/4)*(b-2*c)*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)) )*EllipticE(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-b+c)/(-4*a*c+b^2)/e/(a+b*t an(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)+1/2*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)))*Elliptic F(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)/(a-b+c)/e/(a^(1/2)-c^(1/2))/(a+ b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)-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)))*EllipticPi(si n(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)/(a-b+c)/e/(a^(1/2)-c^(1/2))/(a+b*tan(e*...
Result contains complex when optimal does not.
Time = 19.77 (sec) , antiderivative size = 831, normalized size of antiderivative = 0.85 \[ \int \frac {\tan ^2(d+e x)}{\left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )^{3/2}} \, dx=\frac {\sqrt {\frac {3 a+b+3 c+4 a \cos (2 (d+e x))-4 c \cos (2 (d+e x))+a \cos (4 (d+e x))-b \cos (4 (d+e x))+c \cos (4 (d+e x))}{3+4 \cos (2 (d+e x))+\cos (4 (d+e x))}} \left (\frac {(b-2 c) \sin (2 (d+e x))}{2 (-a+b-c) \left (b^2-4 a c\right )}+\frac {2 b^2 \sin (2 (d+e x))-4 a c \sin (2 (d+e x))-4 c^2 \sin (2 (d+e x))+b^2 \sin (4 (d+e x))-2 a c \sin (4 (d+e x))-2 b c \sin (4 (d+e x))+2 c^2 \sin (4 (d+e x))}{(a-b+c) \left (-b^2+4 a c\right ) (-3 a-b-3 c-4 a \cos (2 (d+e x))+4 c \cos (2 (d+e x))-a \cos (4 (d+e x))+b \cos (4 (d+e x))-c \cos (4 (d+e x)))}\right )}{e}+\frac {\frac {i \sqrt {2} \left ((b-2 c) \left (-b+\sqrt {b^2-4 a c}\right ) E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \tan (d+e x)\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+\left (b^2-b \sqrt {b^2-4 a c}+2 c \left (-2 a+\sqrt {b^2-4 a c}\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \tan (d+e x)\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-2 \left (b^2-4 a c\right ) \operatorname {EllipticPi}\left (\frac {b+\sqrt {b^2-4 a c}}{2 c},i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \tan (d+e x)\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c \tan ^2(d+e x)}{b+\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c \tan ^2(d+e x)}{b-\sqrt {b^2-4 a c}}}}{\sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}}}-\frac {4 (b-2 c) \tan (d+e x) \left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )}{1+\tan ^2(d+e x)}}{4 (a-b+c) \left (-b^2+4 a c\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \]
(Sqrt[(3*a + b + 3*c + 4*a*Cos[2*(d + e*x)] - 4*c*Cos[2*(d + e*x)] + a*Cos [4*(d + e*x)] - b*Cos[4*(d + e*x)] + c*Cos[4*(d + e*x)])/(3 + 4*Cos[2*(d + e*x)] + Cos[4*(d + e*x)])]*(((b - 2*c)*Sin[2*(d + e*x)])/(2*(-a + b - c)* (b^2 - 4*a*c)) + (2*b^2*Sin[2*(d + e*x)] - 4*a*c*Sin[2*(d + e*x)] - 4*c^2* Sin[2*(d + e*x)] + b^2*Sin[4*(d + e*x)] - 2*a*c*Sin[4*(d + e*x)] - 2*b*c*S in[4*(d + e*x)] + 2*c^2*Sin[4*(d + e*x)])/((a - b + c)*(-b^2 + 4*a*c)*(-3* a - b - 3*c - 4*a*Cos[2*(d + e*x)] + 4*c*Cos[2*(d + e*x)] - a*Cos[4*(d + e *x)] + b*Cos[4*(d + e*x)] - c*Cos[4*(d + e*x)]))))/e + ((I*Sqrt[2]*((b - 2 *c)*(-b + Sqrt[b^2 - 4*a*c])*EllipticE[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 - Sqrt[b^2 - 4*a* c])] + (b^2 - b*Sqrt[b^2 - 4*a*c] + 2*c*(-2*a + Sqrt[b^2 - 4*a*c]))*Ellipt icF[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 - Sqrt[b^2 - 4*a*c])] - 2*(b^2 - 4*a*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 + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])*Sqr t[(b + Sqrt[b^2 - 4*a*c] + 2*c*Tan[d + e*x]^2)/(b + Sqrt[b^2 - 4*a*c])]*Sq rt[1 + (2*c*Tan[d + e*x]^2)/(b - Sqrt[b^2 - 4*a*c])])/Sqrt[c/(b + Sqrt[b^2 - 4*a*c])] - (4*(b - 2*c)*Tan[d + e*x]*(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4))/(1 + Tan[d + e*x]^2))/(4*(a - b + c)*(-b^2 + 4*a*c)*e*Sqrt[a + b* Tan[d + e*x]^2 + c*Tan[d + e*x]^4])
Time = 1.26 (sec) , antiderivative size = 824, normalized size of antiderivative = 0.84, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.371, Rules used = {3042, 4183, 1638, 25, 27, 2206, 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.
| \(\displaystyle \int \frac {\tan ^2(d+e x)}{\left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan (d+e x)^2}{\left (a+b \tan (d+e x)^2+c \tan (d+e x)^4\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4183 |
| \(\displaystyle \frac {\int \frac {\tan ^2(d+e x)}{\left (\tan ^2(d+e x)+1\right ) \left (c \tan ^4(d+e x)+b \tan ^2(d+e x)+a\right )^{3/2}}d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 1638 |
| \(\displaystyle \frac {-\frac {\int -\frac {\frac {c^{3/2} \tan ^4(d+e x)}{\sqrt {a}}+\frac {\left (b-c+\sqrt {a} \sqrt {c}\right ) \sqrt {c} \tan ^2(d+e x)}{\sqrt {a}}+a}{\left (c \tan ^4(d+e x)+b \tan ^2(d+e x)+a\right )^{3/2}}d\tan (d+e x)}{\left (1-\frac {\sqrt {c}}{\sqrt {a}}\right ) (a-b+c)}-\frac {\int \frac {\sqrt {c} \tan ^2(d+e x)+\sqrt {a}}{\sqrt {a} \left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)}{\left (1-\frac {\sqrt {c}}{\sqrt {a}}\right ) (a-b+c)}}{e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {\frac {c^{3/2} \tan ^4(d+e x)}{\sqrt {a}}+\frac {\left (b-c+\sqrt {a} \sqrt {c}\right ) \sqrt {c} \tan ^2(d+e x)}{\sqrt {a}}+a}{\left (c \tan ^4(d+e x)+b \tan ^2(d+e x)+a\right )^{3/2}}d\tan (d+e x)}{\left (1-\frac {\sqrt {c}}{\sqrt {a}}\right ) (a-b+c)}-\frac {\int \frac {\sqrt {c} \tan ^2(d+e x)+\sqrt {a}}{\sqrt {a} \left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)}{\left (1-\frac {\sqrt {c}}{\sqrt {a}}\right ) (a-b+c)}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\frac {c^{3/2} \tan ^4(d+e x)}{\sqrt {a}}+\frac {\left (b-c+\sqrt {a} \sqrt {c}\right ) \sqrt {c} \tan ^2(d+e x)}{\sqrt {a}}+a}{\left (c \tan ^4(d+e x)+b \tan ^2(d+e x)+a\right )^{3/2}}d\tan (d+e x)}{\left (1-\frac {\sqrt {c}}{\sqrt {a}}\right ) (a-b+c)}-\frac {\int \frac {\sqrt {c} \tan ^2(d+e x)+\sqrt {a}}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)}{\sqrt {a} \left (1-\frac {\sqrt {c}}{\sqrt {a}}\right ) (a-b+c)}}{e}\) |
| \(\Big \downarrow \) 2206 |
| \(\displaystyle \frac {\frac {\frac {\left (\sqrt {a}-\sqrt {c}\right ) \tan (d+e x) \left (-2 a c+b^2+c (b-2 c) \tan ^2(d+e x)-b c\right )}{\sqrt {a} \left (b^2-4 a c\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\int -\frac {\sqrt {a} \sqrt {c} \left (b^2-c b+\sqrt {a} \sqrt {c} b-\left (\sqrt {a}-\sqrt {c}\right ) (b-2 c) \sqrt {c} \tan ^2(d+e x)-2 a c-2 a^{3/2} \sqrt {c}\right )}{\sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)}{a \left (b^2-4 a c\right )}}{\left (1-\frac {\sqrt {c}}{\sqrt {a}}\right ) (a-b+c)}-\frac {\int \frac {\sqrt {c} \tan ^2(d+e x)+\sqrt {a}}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)}{\sqrt {a} \left (1-\frac {\sqrt {c}}{\sqrt {a}}\right ) (a-b+c)}}{e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\sqrt {a} \sqrt {c} \left (b^2-c b+\sqrt {a} \sqrt {c} b-\left (\sqrt {a}-\sqrt {c}\right ) (b-2 c) \sqrt {c} \tan ^2(d+e x)-2 a c-2 a^{3/2} \sqrt {c}\right )}{\sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)}{a \left (b^2-4 a c\right )}+\frac {\left (\sqrt {a}-\sqrt {c}\right ) \tan (d+e x) \left (-2 a c+b^2+c (b-2 c) \tan ^2(d+e x)-b c\right )}{\sqrt {a} \left (b^2-4 a c\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}}{\left (1-\frac {\sqrt {c}}{\sqrt {a}}\right ) (a-b+c)}-\frac {\int \frac {\sqrt {c} \tan ^2(d+e x)+\sqrt {a}}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)}{\sqrt {a} \left (1-\frac {\sqrt {c}}{\sqrt {a}}\right ) (a-b+c)}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\sqrt {c} \int \frac {b^2-c b+\sqrt {a} \sqrt {c} b-\left (\sqrt {a}-\sqrt {c}\right ) (b-2 c) \sqrt {c} \tan ^2(d+e x)-2 a c-2 a^{3/2} \sqrt {c}}{\sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)}{\sqrt {a} \left (b^2-4 a c\right )}+\frac {\left (\sqrt {a}-\sqrt {c}\right ) \tan (d+e x) \left (-2 a c+b^2+c (b-2 c) \tan ^2(d+e x)-b c\right )}{\sqrt {a} \left (b^2-4 a c\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}}{\left (1-\frac {\sqrt {c}}{\sqrt {a}}\right ) (a-b+c)}-\frac {\int \frac {\sqrt {c} \tan ^2(d+e x)+\sqrt {a}}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)}{\sqrt {a} \left (1-\frac {\sqrt {c}}{\sqrt {a}}\right ) (a-b+c)}}{e}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {\frac {\frac {\sqrt {c} \left (\sqrt {a} \left (\sqrt {a}-\sqrt {c}\right ) (b-2 c) \int \frac {\sqrt {a}-\sqrt {c} \tan ^2(d+e x)}{\sqrt {a} \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)-\left (2 \sqrt {a} \sqrt {c}+b\right ) (a-b+c) \int \frac {1}{\sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)\right )}{\sqrt {a} \left (b^2-4 a c\right )}+\frac {\left (\sqrt {a}-\sqrt {c}\right ) \tan (d+e x) \left (-2 a c+b^2+c (b-2 c) \tan ^2(d+e x)-b c\right )}{\sqrt {a} \left (b^2-4 a c\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}}{\left (1-\frac {\sqrt {c}}{\sqrt {a}}\right ) (a-b+c)}-\frac {\int \frac {\sqrt {c} \tan ^2(d+e x)+\sqrt {a}}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)}{\sqrt {a} \left (1-\frac {\sqrt {c}}{\sqrt {a}}\right ) (a-b+c)}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\sqrt {c} \left (\left (\sqrt {a}-\sqrt {c}\right ) (b-2 c) \int \frac {\sqrt {a}-\sqrt {c} \tan ^2(d+e x)}{\sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)-\left (2 \sqrt {a} \sqrt {c}+b\right ) (a-b+c) \int \frac {1}{\sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)\right )}{\sqrt {a} \left (b^2-4 a c\right )}+\frac {\left (\sqrt {a}-\sqrt {c}\right ) \tan (d+e x) \left (-2 a c+b^2+c (b-2 c) \tan ^2(d+e x)-b c\right )}{\sqrt {a} \left (b^2-4 a c\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}}{\left (1-\frac {\sqrt {c}}{\sqrt {a}}\right ) (a-b+c)}-\frac {\int \frac {\sqrt {c} \tan ^2(d+e x)+\sqrt {a}}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)}{\sqrt {a} \left (1-\frac {\sqrt {c}}{\sqrt {a}}\right ) (a-b+c)}}{e}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {\frac {\frac {\sqrt {c} \left (\left (\sqrt {a}-\sqrt {c}\right ) (b-2 c) \int \frac {\sqrt {a}-\sqrt {c} \tan ^2(d+e x)}{\sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)-\frac {\left (2 \sqrt {a} \sqrt {c}+b\right ) (a-b+c) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{\sqrt {a} \left (b^2-4 a c\right )}+\frac {\left (\sqrt {a}-\sqrt {c}\right ) \tan (d+e x) \left (-2 a c+b^2+c (b-2 c) \tan ^2(d+e x)-b c\right )}{\sqrt {a} \left (b^2-4 a c\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}}{\left (1-\frac {\sqrt {c}}{\sqrt {a}}\right ) (a-b+c)}-\frac {\int \frac {\sqrt {c} \tan ^2(d+e x)+\sqrt {a}}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)}{\sqrt {a} \left (1-\frac {\sqrt {c}}{\sqrt {a}}\right ) (a-b+c)}}{e}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {\frac {\frac {\sqrt {c} \left (\left (\sqrt {a}-\sqrt {c}\right ) (b-2 c) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\tan (d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{\sqrt {a}+\sqrt {c} \tan ^2(d+e x)}\right )-\frac {\left (2 \sqrt {a} \sqrt {c}+b\right ) (a-b+c) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{\sqrt {a} \left (b^2-4 a c\right )}+\frac {\left (\sqrt {a}-\sqrt {c}\right ) \tan (d+e x) \left (-2 a c+b^2+c (b-2 c) \tan ^2(d+e x)-b c\right )}{\sqrt {a} \left (b^2-4 a c\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}}{\left (1-\frac {\sqrt {c}}{\sqrt {a}}\right ) (a-b+c)}-\frac {\int \frac {\sqrt {c} \tan ^2(d+e x)+\sqrt {a}}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}d\tan (d+e x)}{\sqrt {a} \left (1-\frac {\sqrt {c}}{\sqrt {a}}\right ) (a-b+c)}}{e}\) |
| \(\Big \downarrow \) 2220 |
| \(\displaystyle \frac {\frac {\frac {\left (\sqrt {a}-\sqrt {c}\right ) \tan (d+e x) \left (b^2-c b+(b-2 c) c \tan ^2(d+e x)-2 a c\right )}{\sqrt {a} \left (b^2-4 a c\right ) \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}+\frac {\sqrt {c} \left (\left (\sqrt {a}-\sqrt {c}\right ) (b-2 c) \left (\frac {\sqrt [4]{a} 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 {c} \tan ^2(d+e x)+\sqrt {a}\right ) \sqrt {\frac {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left (\sqrt {c} \tan ^2(d+e x)+\sqrt {a}\right )^2}}}{\sqrt [4]{c} \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}-\frac {\tan (d+e x) \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}{\sqrt {c} \tan ^2(d+e x)+\sqrt {a}}\right )-\frac {\left (b+2 \sqrt {a} \sqrt {c}\right ) (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 {c} \tan ^2(d+e x)+\sqrt {a}\right ) \sqrt {\frac {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left (\sqrt {c} \tan ^2(d+e x)+\sqrt {a}\right )^2}}}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}\right )}{\sqrt {a} \left (b^2-4 a c\right )}}{\left (1-\frac {\sqrt {c}}{\sqrt {a}}\right ) (a-b+c)}-\frac {\frac {\left (\sqrt {a}-\sqrt {c}\right ) \arctan \left (\frac {\sqrt {a-b+c} \tan (d+e x)}{\sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}\right )}{2 \sqrt {a-b+c}}+\frac {\left (\sqrt {a}+\sqrt {c}\right ) \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 {c} \tan ^2(d+e x)+\sqrt {a}\right ) \sqrt {\frac {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left (\sqrt {c} \tan ^2(d+e x)+\sqrt {a}\right )^2}}}{4 \sqrt [4]{a} \sqrt [4]{c} \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}}{\sqrt {a} \left (1-\frac {\sqrt {c}}{\sqrt {a}}\right ) (a-b+c)}}{e}\) |
(-((((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]) + ((Sqrt[a] + Sqr t[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]*(Sqrt[a] + Sqr t[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])*(a - b + c))) + ( ((Sqrt[a] - Sqrt[c])*Tan[d + e*x]*(b^2 - 2*a*c - b*c + (b - 2*c)*c*Tan[d + e*x]^2))/(Sqrt[a]*(b^2 - 4*a*c)*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x ]^4]) + (Sqrt[c]*(-1/2*((b + 2*Sqrt[a]*Sqrt[c])*(a - b + c)*EllipticF[2*Ar cTan[(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])/(a^(1/4)*c^(1/4)*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) + (Sqrt[a] - Sqrt[c])*(b - 2*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]*( b^2 - 4*a*c)))/((1 - Sqrt[c]/Sqrt[a])*(a - b + c)))/e
3.1.51.3.1 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[((x_)^(m_)*((a_.) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_))/((d_) + (e_.)*(x _)^2), x_Symbol] :> Simp[(-(-d/e)^(m/2))*((c*d^2 - b*d*e + a*e^2)^(p + 1/2) /(e^(2*p)*(Rt[c/a, 2]*d - e))) Int[(1 + Rt[c/a, 2]*x^2)/((d + e*x^2)*Sqrt [a + b*x^2 + c*x^4]), x], x] + Simp[(c*d^2 - b*d*e + a*e^2)^(p + 1/2)/(Rt[c /a, 2]*d - e) Int[(a + b*x^2 + c*x^4)^p*ExpandToSum[((Rt[c/a, 2]*d - e)*( c*d^2 - b*d*e + a*e^2)^(-p - 1/2)*x^m + ((-d/e)^(m/2)*(1 + Rt[c/a, 2]*x^2)* (a + b*x^2 + c*x^4)^(-p - 1/2))/e^(2*p))/(d + e*x^2), x], x], x] /; FreeQ[{ a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && ILtQ[p + 1/2, 0] && IGtQ[m/2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]
Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Poly nomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^ 4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b ^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(a + b*x^2 + c *x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[Px, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4* p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x ^2] && Expon[Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
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[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]
Leaf count of result is larger than twice the leaf count of optimal. \(3597\) vs. \(2(995)=1990\).
Time = 0.60 (sec) , antiderivative size = 3598, normalized size of antiderivative = 3.67
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(3598\) |
| default | \(\text {Expression too large to display}\) | \(3598\) |
1/e*(-2*c*(1/2/a*b/(4*a*c-b^2)*tan(e*x+d)^3-1/2*(2*a*c-b^2)/a/(4*a*c-b^2)/ c*tan(e*x+d))/((tan(e*x+d)^4+b/c*tan(e*x+d)^2+a/c)*c)^(1/2)+1/4*(1/a-(2*a* c-b^2)/a/(4*a*c-b^2))*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2*(-b+( -4*a*c+b^2)^(1/2))/a*tan(e*x+d)^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*tan (e*x+d)^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))-1/2*b/(4*a*c-b^2)*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2) )/a)^(1/2)*(4-2*(-b+(-4*a*c+b^2)^(1/2))/a*tan(e*x+d)^2)^(1/2)*(4+2*(b+(-4* a*c+b^2)^(1/2))/a*tan(e*x+d)^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))-Ellipt icE(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)))+2*c*(1/2*(2*a*c-b^2+b*c)/a/(4*a*c-b^2)/ (a-b+c)*tan(e*x+d)^3+1/2*(3*a*b*c-2*a*c^2-b^3+b^2*c)/a/(4*a*c-b^2)/(a-b+c) /c*tan(e*x+d))/((tan(e*x+d)^4+b/c*tan(e*x+d)^2+a/c)*c)^(1/2)+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*t an(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/(a-b+c)*b-1/4*2^(1/2)/(-1/a*b+1/a*(-4*a*c+b^...
Timed out. \[ \int \frac {\tan ^2(d+e x)}{\left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\tan ^2(d+e x)}{\left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )^{3/2}} \, dx=\int \frac {\tan ^{2}{\left (d + e x \right )}}{\left (a + b \tan ^{2}{\left (d + e x \right )} + c \tan ^{4}{\left (d + e x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\tan ^2(d+e x)}{\left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )^{3/2}} \, dx=\int { \frac {\tan \left (e x + d\right )^{2}}{{\left (c \tan \left (e x + d\right )^{4} + b \tan \left (e x + d\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\tan ^2(d+e x)}{\left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )^{3/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\tan ^2(d+e x)}{\left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )^{3/2}} \, dx=\int \frac {{\mathrm {tan}\left (d+e\,x\right )}^2}{{\left (c\,{\mathrm {tan}\left (d+e\,x\right )}^4+b\,{\mathrm {tan}\left (d+e\,x\right )}^2+a\right )}^{3/2}} \,d x \]