Integrand size = 39, antiderivative size = 230 \[ \int \frac {a+b \sec (d+e x)}{\left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^2} \, dx=\frac {a x}{b^4}-\frac {\left (a^2-2 b^2\right ) \left (2 a^4-a^2 b^2+b^4\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{5/2} b^4 (a+b)^{5/2} e}-\frac {a \left (3 a^2-5 b^2\right ) \tan (d+e x)}{6 b^2 \left (a^2-b^2\right ) e (b+a \sec (d+e x))^2}-\frac {a \left (6 a^4-11 a^2 b^2+11 b^4\right ) \tan (d+e x)}{6 b^3 \left (a^2-b^2\right )^2 e (b+a \sec (d+e x))}-\frac {a^4 \tan (d+e x)}{3 b e \left (a b+a^2 \sec (d+e x)\right )^3} \]
a*x/b^4-(a^2-2*b^2)*(2*a^4-a^2*b^2+b^4)*arctan((a-b)^(1/2)*tan(1/2*e*x+1/2 *d)/(a+b)^(1/2))/(a-b)^(5/2)/b^4/(a+b)^(5/2)/e-1/6*a*(3*a^2-5*b^2)*tan(e*x +d)/b^2/(a^2-b^2)/e/(b+a*sec(e*x+d))^2-1/6*a*(6*a^4-11*a^2*b^2+11*b^4)*tan (e*x+d)/b^3/(a^2-b^2)^2/e/(b+a*sec(e*x+d))-1/3*a^4*tan(e*x+d)/b/e/(a*b+a^2 *sec(e*x+d))^3
Time = 2.39 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.20 \[ \int \frac {a+b \sec (d+e x)}{\left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^2} \, dx=\frac {(a+b \cos (d+e x)) \sec ^3(d+e x) (a+b \sec (d+e x)) \left (6 a (d+e x) (a+b \cos (d+e x))^3+\frac {6 \left (-2 a^6+5 a^4 b^2-3 a^2 b^4+2 b^6\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {-a^2+b^2}}\right ) (a+b \cos (d+e x))^3}{\left (-a^2+b^2\right )^{5/2}}-2 a^3 b \sin (d+e x)+\frac {a^2 b \left (7 a^2-9 b^2\right ) (a+b \cos (d+e x)) \sin (d+e x)}{(a-b) (a+b)}-\frac {a b \left (11 a^4-23 a^2 b^2+18 b^4\right ) (a+b \cos (d+e x))^2 \sin (d+e x)}{(a-b)^2 (a+b)^2}\right )}{6 b^4 e (b+a \cos (d+e x)) (b+a \sec (d+e x))^4} \]
((a + b*Cos[d + e*x])*Sec[d + e*x]^3*(a + b*Sec[d + e*x])*(6*a*(d + e*x)*( a + b*Cos[d + e*x])^3 + (6*(-2*a^6 + 5*a^4*b^2 - 3*a^2*b^4 + 2*b^6)*ArcTan h[((-a + b)*Tan[(d + e*x)/2])/Sqrt[-a^2 + b^2]]*(a + b*Cos[d + e*x])^3)/(- a^2 + b^2)^(5/2) - 2*a^3*b*Sin[d + e*x] + (a^2*b*(7*a^2 - 9*b^2)*(a + b*Co s[d + e*x])*Sin[d + e*x])/((a - b)*(a + b)) - (a*b*(11*a^4 - 23*a^2*b^2 + 18*b^4)*(a + b*Cos[d + e*x])^2*Sin[d + e*x])/((a - b)^2*(a + b)^2)))/(6*b^ 4*e*(b + a*Cos[d + e*x])*(b + a*Sec[d + e*x])^4)
Time = 1.64 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.31, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.436, Rules used = {3042, 4660, 27, 3042, 4411, 3042, 4548, 3042, 4548, 27, 3042, 4407, 3042, 4318, 3042, 3138, 218}
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.
| \(\displaystyle \int \frac {a+b \sec (d+e x)}{\left (a^2 \sec ^2(d+e x)+2 a b \sec (d+e x)+b^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a+b \sec (d+e x)}{\left (a^2 \sec (d+e x)^2+2 a b \sec (d+e x)+b^2\right )^2}dx\) |
| \(\Big \downarrow \) 4660 |
| \(\displaystyle 16 a^4 \int \frac {a+b \sec (d+e x)}{16 \left (\sec (d+e x) a^2+b a\right )^4}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle a^4 \int \frac {a+b \sec (d+e x)}{\left (\sec (d+e x) a^2+b a\right )^4}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^4 \int \frac {a+b \csc \left (d+e x+\frac {\pi }{2}\right )}{\left (\csc \left (d+e x+\frac {\pi }{2}\right ) a^2+b a\right )^4}dx\) |
| \(\Big \downarrow \) 4411 |
| \(\displaystyle a^4 \left (\frac {\int \frac {-2 \left (a^2-b^2\right ) \sec ^2(d+e x) a^3+3 \left (a^2-b^2\right ) a^3+3 b \left (a^2-b^2\right ) \sec (d+e x) a^2}{\left (\sec (d+e x) a^2+b a\right )^3}dx}{3 a^3 b \left (a^2-b^2\right )}-\frac {\tan (d+e x)}{3 b e \left (a^2 \sec (d+e x)+a b\right )^3}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^4 \left (\frac {\int \frac {-2 \left (a^2-b^2\right ) \csc \left (d+e x+\frac {\pi }{2}\right )^2 a^3+3 \left (a^2-b^2\right ) a^3+3 b \left (a^2-b^2\right ) \csc \left (d+e x+\frac {\pi }{2}\right ) a^2}{\left (\csc \left (d+e x+\frac {\pi }{2}\right ) a^2+b a\right )^3}dx}{3 a^3 b \left (a^2-b^2\right )}-\frac {\tan (d+e x)}{3 b e \left (a^2 \sec (d+e x)+a b\right )^3}\right )\) |
| \(\Big \downarrow \) 4548 |
| \(\displaystyle a^4 \left (\frac {\frac {\int \frac {6 \left (a^2-b^2\right )^2 a^5-\left (3 a^2-5 b^2\right ) \left (a^2-b^2\right ) \sec ^2(d+e x) a^5+2 b \left (a^2-3 b^2\right ) \left (a^2-b^2\right ) \sec (d+e x) a^4}{\left (\sec (d+e x) a^2+b a\right )^2}dx}{2 a^3 b \left (a^2-b^2\right )}-\frac {\left (3 a^2-5 b^2\right ) \tan (d+e x)}{2 b e (a \sec (d+e x)+b)^2}}{3 a^3 b \left (a^2-b^2\right )}-\frac {\tan (d+e x)}{3 b e \left (a^2 \sec (d+e x)+a b\right )^3}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^4 \left (\frac {\frac {\int \frac {6 \left (a^2-b^2\right )^2 a^5-\left (3 a^2-5 b^2\right ) \left (a^2-b^2\right ) \csc \left (d+e x+\frac {\pi }{2}\right )^2 a^5+2 b \left (a^2-3 b^2\right ) \left (a^2-b^2\right ) \csc \left (d+e x+\frac {\pi }{2}\right ) a^4}{\left (\csc \left (d+e x+\frac {\pi }{2}\right ) a^2+b a\right )^2}dx}{2 a^3 b \left (a^2-b^2\right )}-\frac {\left (3 a^2-5 b^2\right ) \tan (d+e x)}{2 b e (a \sec (d+e x)+b)^2}}{3 a^3 b \left (a^2-b^2\right )}-\frac {\tan (d+e x)}{3 b e \left (a^2 \sec (d+e x)+a b\right )^3}\right )\) |
| \(\Big \downarrow \) 4548 |
| \(\displaystyle a^4 \left (\frac {\frac {\frac {\int \frac {3 \left (2 \left (a^2-b^2\right )^3 a^7+b \left (a^6-2 b^2 a^4+3 b^4 a^2-2 b^6\right ) \sec (d+e x) a^6\right )}{\sec (d+e x) a^2+b a}dx}{a^3 b \left (a^2-b^2\right )}-\frac {a^4 \left (6 a^4-11 a^2 b^2+11 b^4\right ) \tan (d+e x)}{b e \left (a^2 \sec (d+e x)+a b\right )}}{2 a^3 b \left (a^2-b^2\right )}-\frac {\left (3 a^2-5 b^2\right ) \tan (d+e x)}{2 b e (a \sec (d+e x)+b)^2}}{3 a^3 b \left (a^2-b^2\right )}-\frac {\tan (d+e x)}{3 b e \left (a^2 \sec (d+e x)+a b\right )^3}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle a^4 \left (\frac {\frac {\frac {3 \int \frac {2 \left (a^2-b^2\right )^3 a^7+b \left (a^6-2 b^2 a^4+3 b^4 a^2-2 b^6\right ) \sec (d+e x) a^6}{\sec (d+e x) a^2+b a}dx}{a^3 b \left (a^2-b^2\right )}-\frac {a^4 \left (6 a^4-11 a^2 b^2+11 b^4\right ) \tan (d+e x)}{b e \left (a^2 \sec (d+e x)+a b\right )}}{2 a^3 b \left (a^2-b^2\right )}-\frac {\left (3 a^2-5 b^2\right ) \tan (d+e x)}{2 b e (a \sec (d+e x)+b)^2}}{3 a^3 b \left (a^2-b^2\right )}-\frac {\tan (d+e x)}{3 b e \left (a^2 \sec (d+e x)+a b\right )^3}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^4 \left (\frac {\frac {\frac {3 \int \frac {2 \left (a^2-b^2\right )^3 a^7+b \left (a^6-2 b^2 a^4+3 b^4 a^2-2 b^6\right ) \csc \left (d+e x+\frac {\pi }{2}\right ) a^6}{\csc \left (d+e x+\frac {\pi }{2}\right ) a^2+b a}dx}{a^3 b \left (a^2-b^2\right )}-\frac {a^4 \left (6 a^4-11 a^2 b^2+11 b^4\right ) \tan (d+e x)}{b e \left (a^2 \sec (d+e x)+a b\right )}}{2 a^3 b \left (a^2-b^2\right )}-\frac {\left (3 a^2-5 b^2\right ) \tan (d+e x)}{2 b e (a \sec (d+e x)+b)^2}}{3 a^3 b \left (a^2-b^2\right )}-\frac {\tan (d+e x)}{3 b e \left (a^2 \sec (d+e x)+a b\right )^3}\right )\) |
| \(\Big \downarrow \) 4407 |
| \(\displaystyle a^4 \left (\frac {\frac {\frac {3 \left (\frac {2 a^6 x \left (a^2-b^2\right )^3}{b}-\frac {a^6 \left (a^2-2 b^2\right ) \left (a^2-b^2\right ) \left (2 a^4-a^2 b^2+b^4\right ) \int \frac {\sec (d+e x)}{\sec (d+e x) a^2+b a}dx}{b}\right )}{a^3 b \left (a^2-b^2\right )}-\frac {a^4 \left (6 a^4-11 a^2 b^2+11 b^4\right ) \tan (d+e x)}{b e \left (a^2 \sec (d+e x)+a b\right )}}{2 a^3 b \left (a^2-b^2\right )}-\frac {\left (3 a^2-5 b^2\right ) \tan (d+e x)}{2 b e (a \sec (d+e x)+b)^2}}{3 a^3 b \left (a^2-b^2\right )}-\frac {\tan (d+e x)}{3 b e \left (a^2 \sec (d+e x)+a b\right )^3}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^4 \left (\frac {\frac {\frac {3 \left (\frac {2 a^6 x \left (a^2-b^2\right )^3}{b}-\frac {a^6 \left (a^2-2 b^2\right ) \left (a^2-b^2\right ) \left (2 a^4-a^2 b^2+b^4\right ) \int \frac {\csc \left (d+e x+\frac {\pi }{2}\right )}{\csc \left (d+e x+\frac {\pi }{2}\right ) a^2+b a}dx}{b}\right )}{a^3 b \left (a^2-b^2\right )}-\frac {a^4 \left (6 a^4-11 a^2 b^2+11 b^4\right ) \tan (d+e x)}{b e \left (a^2 \sec (d+e x)+a b\right )}}{2 a^3 b \left (a^2-b^2\right )}-\frac {\left (3 a^2-5 b^2\right ) \tan (d+e x)}{2 b e (a \sec (d+e x)+b)^2}}{3 a^3 b \left (a^2-b^2\right )}-\frac {\tan (d+e x)}{3 b e \left (a^2 \sec (d+e x)+a b\right )^3}\right )\) |
| \(\Big \downarrow \) 4318 |
| \(\displaystyle a^4 \left (\frac {\frac {\frac {3 \left (\frac {2 a^6 x \left (a^2-b^2\right )^3}{b}-\frac {a^4 \left (a^2-2 b^2\right ) \left (a^2-b^2\right ) \left (2 a^4-a^2 b^2+b^4\right ) \int \frac {1}{\frac {b \cos (d+e x)}{a}+1}dx}{b}\right )}{a^3 b \left (a^2-b^2\right )}-\frac {a^4 \left (6 a^4-11 a^2 b^2+11 b^4\right ) \tan (d+e x)}{b e \left (a^2 \sec (d+e x)+a b\right )}}{2 a^3 b \left (a^2-b^2\right )}-\frac {\left (3 a^2-5 b^2\right ) \tan (d+e x)}{2 b e (a \sec (d+e x)+b)^2}}{3 a^3 b \left (a^2-b^2\right )}-\frac {\tan (d+e x)}{3 b e \left (a^2 \sec (d+e x)+a b\right )^3}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^4 \left (\frac {\frac {\frac {3 \left (\frac {2 a^6 x \left (a^2-b^2\right )^3}{b}-\frac {a^4 \left (a^2-2 b^2\right ) \left (a^2-b^2\right ) \left (2 a^4-a^2 b^2+b^4\right ) \int \frac {1}{\frac {b \sin \left (d+e x+\frac {\pi }{2}\right )}{a}+1}dx}{b}\right )}{a^3 b \left (a^2-b^2\right )}-\frac {a^4 \left (6 a^4-11 a^2 b^2+11 b^4\right ) \tan (d+e x)}{b e \left (a^2 \sec (d+e x)+a b\right )}}{2 a^3 b \left (a^2-b^2\right )}-\frac {\left (3 a^2-5 b^2\right ) \tan (d+e x)}{2 b e (a \sec (d+e x)+b)^2}}{3 a^3 b \left (a^2-b^2\right )}-\frac {\tan (d+e x)}{3 b e \left (a^2 \sec (d+e x)+a b\right )^3}\right )\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle a^4 \left (\frac {\frac {\frac {3 \left (\frac {2 a^6 x \left (a^2-b^2\right )^3}{b}-\frac {2 a^4 \left (a^2-2 b^2\right ) \left (a^2-b^2\right ) \left (2 a^4-a^2 b^2+b^4\right ) \int \frac {1}{\frac {(a-b) \tan ^2\left (\frac {1}{2} (d+e x)\right )}{a}+\frac {a+b}{a}}d\tan \left (\frac {1}{2} (d+e x)\right )}{b e}\right )}{a^3 b \left (a^2-b^2\right )}-\frac {a^4 \left (6 a^4-11 a^2 b^2+11 b^4\right ) \tan (d+e x)}{b e \left (a^2 \sec (d+e x)+a b\right )}}{2 a^3 b \left (a^2-b^2\right )}-\frac {\left (3 a^2-5 b^2\right ) \tan (d+e x)}{2 b e (a \sec (d+e x)+b)^2}}{3 a^3 b \left (a^2-b^2\right )}-\frac {\tan (d+e x)}{3 b e \left (a^2 \sec (d+e x)+a b\right )^3}\right )\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle a^4 \left (\frac {\frac {\frac {3 \left (\frac {2 a^6 x \left (a^2-b^2\right )^3}{b}-\frac {2 a^5 \left (a^2-2 b^2\right ) \left (a^2-b^2\right ) \left (2 a^4-a^2 b^2+b^4\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a+b}}\right )}{b e \sqrt {a-b} \sqrt {a+b}}\right )}{a^3 b \left (a^2-b^2\right )}-\frac {a^4 \left (6 a^4-11 a^2 b^2+11 b^4\right ) \tan (d+e x)}{b e \left (a^2 \sec (d+e x)+a b\right )}}{2 a^3 b \left (a^2-b^2\right )}-\frac {\left (3 a^2-5 b^2\right ) \tan (d+e x)}{2 b e (a \sec (d+e x)+b)^2}}{3 a^3 b \left (a^2-b^2\right )}-\frac {\tan (d+e x)}{3 b e \left (a^2 \sec (d+e x)+a b\right )^3}\right )\) |
a^4*(-1/3*Tan[d + e*x]/(b*e*(a*b + a^2*Sec[d + e*x])^3) + (-1/2*((3*a^2 - 5*b^2)*Tan[d + e*x])/(b*e*(b + a*Sec[d + e*x])^2) + ((3*((2*a^6*(a^2 - b^2 )^3*x)/b - (2*a^5*(a^2 - 2*b^2)*(a^2 - b^2)*(2*a^4 - a^2*b^2 + b^4)*ArcTan [(Sqrt[a - b]*Tan[(d + e*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b*Sqrt[a + b]*e )))/(a^3*b*(a^2 - b^2)) - (a^4*(6*a^4 - 11*a^2*b^2 + 11*b^4)*Tan[d + e*x]) /(b*e*(a*b + a^2*Sec[d + e*x])))/(2*a^3*b*(a^2 - b^2)))/(3*a^3*b*(a^2 - b^ 2)))
3.6.21.3.1 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[((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_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbo l] :> Simp[1/b Int[1/(1 + (a/b)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[c*(x/a), x] - Simp[(b*c - a*d)/a Int[Csc[e + f* x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d _.) + (c_)), x_Symbol] :> Simp[b*(b*c - a*d)*Cot[e + f*x]*((a + b*Csc[e + f *x])^(m + 1)/(a*f*(m + 1)*(a^2 - b^2))), x] + Simp[1/(a*(m + 1)*(a^2 - b^2) ) Int[(a + b*Csc[e + f*x])^(m + 1)*Simp[c*(a^2 - b^2)*(m + 1) - (a*(b*c - a*d)*(m + 1))*Csc[e + f*x] + b*(b*c - a*d)*(m + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && N eQ[a^2 - b^2, 0] && IntegerQ[2*m]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(a*f*(m + 1)*(a^2 - b^2))), x] + Simp[1/(a*(m + 1)*(a^2 - b^2)) Int[(a + b*Csc[e + f*x])^( m + 1)*Simp[A*(a^2 - b^2)*(m + 1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x ] + (A*b^2 - a*b*B + a^2*C)*(m + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
Int[((A_) + (B_.)*sec[(d_.) + (e_.)*(x_)])*((a_) + (b_.)*sec[(d_.) + (e_.)* (x_)] + (c_.)*sec[(d_.) + (e_.)*(x_)]^2)^(n_), x_Symbol] :> Simp[1/(4^n*c^n ) Int[(A + B*Sec[d + e*x])*(b + 2*c*Sec[d + e*x])^(2*n), x], x] /; FreeQ[ {a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[n]
Time = 1.52 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.33
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 \left (\frac {\frac {\left (2 a^{4}-a^{3} b -4 a^{2} b^{2}+3 a \,b^{3}+6 b^{4}\right ) a b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{5}}{2 a^{2}+4 a b +2 b^{2}}+\frac {2 \left (3 a^{4}-8 a^{2} b^{2}+9 b^{4}\right ) a b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{3 \left (a +b \right ) \left (a -b \right )}+\frac {\left (2 a^{4}+a^{3} b -4 a^{2} b^{2}-3 a \,b^{3}+6 b^{4}\right ) a b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{2 a^{2}-4 a b +2 b^{2}}}{\left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2} a -\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2} b +a +b \right )^{3}}+\frac {\left (2 a^{6}-5 a^{4} b^{2}+3 a^{2} b^{4}-2 b^{6}\right ) \arctan \left (\frac {\tan \left (\frac {e x}{2}+\frac {d}{2}\right ) \left (a -b \right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{b^{4}}+\frac {2 a \arctan \left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{b^{4}}}{e}\) | \(307\) |
| default | \(\frac {-\frac {2 \left (\frac {\frac {\left (2 a^{4}-a^{3} b -4 a^{2} b^{2}+3 a \,b^{3}+6 b^{4}\right ) a b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{5}}{2 a^{2}+4 a b +2 b^{2}}+\frac {2 \left (3 a^{4}-8 a^{2} b^{2}+9 b^{4}\right ) a b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{3 \left (a +b \right ) \left (a -b \right )}+\frac {\left (2 a^{4}+a^{3} b -4 a^{2} b^{2}-3 a \,b^{3}+6 b^{4}\right ) a b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{2 a^{2}-4 a b +2 b^{2}}}{\left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2} a -\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2} b +a +b \right )^{3}}+\frac {\left (2 a^{6}-5 a^{4} b^{2}+3 a^{2} b^{4}-2 b^{6}\right ) \arctan \left (\frac {\tan \left (\frac {e x}{2}+\frac {d}{2}\right ) \left (a -b \right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{b^{4}}+\frac {2 a \arctan \left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{b^{4}}}{e}\) | \(307\) |
| risch | \(\frac {a x}{b^{4}}-\frac {i a \left (18 a^{5} b^{2} {\mathrm e}^{5 i \left (e x +d \right )}-39 a^{3} b^{4} {\mathrm e}^{5 i \left (e x +d \right )}+27 a \,b^{6} {\mathrm e}^{5 i \left (e x +d \right )}+54 a^{6} b \,{\mathrm e}^{4 i \left (e x +d \right )}-105 a^{4} b^{3} {\mathrm e}^{4 i \left (e x +d \right )}+63 a^{2} b^{5} {\mathrm e}^{4 i \left (e x +d \right )}+18 b^{7} {\mathrm e}^{4 i \left (e x +d \right )}+44 a^{7} {\mathrm e}^{3 i \left (e x +d \right )}-26 a^{5} b^{2} {\mathrm e}^{3 i \left (e x +d \right )}-66 a^{3} b^{4} {\mathrm e}^{3 i \left (e x +d \right )}+108 a \,b^{6} {\mathrm e}^{3 i \left (e x +d \right )}+78 a^{6} b \,{\mathrm e}^{2 i \left (e x +d \right )}-138 a^{4} b^{3} {\mathrm e}^{2 i \left (e x +d \right )}+84 a^{2} b^{5} {\mathrm e}^{2 i \left (e x +d \right )}+36 b^{7} {\mathrm e}^{2 i \left (e x +d \right )}+48 a^{5} b^{2} {\mathrm e}^{i \left (e x +d \right )}-99 a^{3} b^{4} {\mathrm e}^{i \left (e x +d \right )}+81 a \,b^{6} {\mathrm e}^{i \left (e x +d \right )}+11 a^{4} b^{3}-23 a^{2} b^{5}+18 b^{7}\right )}{3 b^{4} \left (a^{2}-b^{2}\right )^{2} e \left (b \,{\mathrm e}^{2 i \left (e x +d \right )}+2 a \,{\mathrm e}^{i \left (e x +d \right )}+b \right )^{3}}-\frac {\ln \left ({\mathrm e}^{i \left (e x +d \right )}+\frac {-i a^{2}+i b^{2}+\sqrt {-a^{2}+b^{2}}\, a}{\sqrt {-a^{2}+b^{2}}\, b}\right ) a^{6}}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} e \,b^{4}}+\frac {5 \ln \left ({\mathrm e}^{i \left (e x +d \right )}+\frac {-i a^{2}+i b^{2}+\sqrt {-a^{2}+b^{2}}\, a}{\sqrt {-a^{2}+b^{2}}\, b}\right ) a^{4}}{2 \sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} e \,b^{2}}-\frac {3 \ln \left ({\mathrm e}^{i \left (e x +d \right )}+\frac {-i a^{2}+i b^{2}+\sqrt {-a^{2}+b^{2}}\, a}{\sqrt {-a^{2}+b^{2}}\, b}\right ) a^{2}}{2 \sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} e}+\frac {b^{2} \ln \left ({\mathrm e}^{i \left (e x +d \right )}+\frac {-i a^{2}+i b^{2}+\sqrt {-a^{2}+b^{2}}\, a}{\sqrt {-a^{2}+b^{2}}\, b}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} e}+\frac {\ln \left ({\mathrm e}^{i \left (e x +d \right )}+\frac {i a^{2}-i b^{2}+\sqrt {-a^{2}+b^{2}}\, a}{\sqrt {-a^{2}+b^{2}}\, b}\right ) a^{6}}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} e \,b^{4}}-\frac {5 \ln \left ({\mathrm e}^{i \left (e x +d \right )}+\frac {i a^{2}-i b^{2}+\sqrt {-a^{2}+b^{2}}\, a}{\sqrt {-a^{2}+b^{2}}\, b}\right ) a^{4}}{2 \sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} e \,b^{2}}+\frac {3 \ln \left ({\mathrm e}^{i \left (e x +d \right )}+\frac {i a^{2}-i b^{2}+\sqrt {-a^{2}+b^{2}}\, a}{\sqrt {-a^{2}+b^{2}}\, b}\right ) a^{2}}{2 \sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} e}-\frac {b^{2} \ln \left ({\mathrm e}^{i \left (e x +d \right )}+\frac {i a^{2}-i b^{2}+\sqrt {-a^{2}+b^{2}}\, a}{\sqrt {-a^{2}+b^{2}}\, b}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} e}\) | \(1039\) |
1/e*(-2/b^4*((1/2*(2*a^4-a^3*b-4*a^2*b^2+3*a*b^3+6*b^4)*a*b/(a^2+2*a*b+b^2 )*tan(1/2*e*x+1/2*d)^5+2/3*(3*a^4-8*a^2*b^2+9*b^4)*a*b/(a+b)/(a-b)*tan(1/2 *e*x+1/2*d)^3+1/2*(2*a^4+a^3*b-4*a^2*b^2-3*a*b^3+6*b^4)*a*b/(a^2-2*a*b+b^2 )*tan(1/2*e*x+1/2*d))/(tan(1/2*e*x+1/2*d)^2*a-tan(1/2*e*x+1/2*d)^2*b+a+b)^ 3+1/2*(2*a^6-5*a^4*b^2+3*a^2*b^4-2*b^6)/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^ (1/2)*arctan(tan(1/2*e*x+1/2*d)*(a-b)/((a+b)*(a-b))^(1/2)))+2*a/b^4*arctan (tan(1/2*e*x+1/2*d)))
Leaf count of result is larger than twice the leaf count of optimal. 633 vs. \(2 (215) = 430\).
Time = 0.33 (sec) , antiderivative size = 1335, normalized size of antiderivative = 5.80 \[ \int \frac {a+b \sec (d+e x)}{\left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^2} \, dx=\text {Too large to display} \]
[1/12*(12*(a^7*b^3 - 3*a^5*b^5 + 3*a^3*b^7 - a*b^9)*e*x*cos(e*x + d)^3 + 3 6*(a^8*b^2 - 3*a^6*b^4 + 3*a^4*b^6 - a^2*b^8)*e*x*cos(e*x + d)^2 + 36*(a^9 *b - 3*a^7*b^3 + 3*a^5*b^5 - a^3*b^7)*e*x*cos(e*x + d) + 12*(a^10 - 3*a^8* b^2 + 3*a^6*b^4 - a^4*b^6)*e*x + 3*(2*a^9 - 5*a^7*b^2 + 3*a^5*b^4 - 2*a^3* b^6 + (2*a^6*b^3 - 5*a^4*b^5 + 3*a^2*b^7 - 2*b^9)*cos(e*x + d)^3 + 3*(2*a^ 7*b^2 - 5*a^5*b^4 + 3*a^3*b^6 - 2*a*b^8)*cos(e*x + d)^2 + 3*(2*a^8*b - 5*a ^6*b^3 + 3*a^4*b^5 - 2*a^2*b^7)*cos(e*x + d))*sqrt(-a^2 + b^2)*log((2*a*b* cos(e*x + d) + (2*a^2 - b^2)*cos(e*x + d)^2 + 2*sqrt(-a^2 + b^2)*(a*cos(e* x + d) + b)*sin(e*x + d) - a^2 + 2*b^2)/(b^2*cos(e*x + d)^2 + 2*a*b*cos(e* x + d) + a^2)) - 2*(6*a^9*b - 17*a^7*b^3 + 22*a^5*b^5 - 11*a^3*b^7 + (11*a ^7*b^3 - 34*a^5*b^5 + 41*a^3*b^7 - 18*a*b^9)*cos(e*x + d)^2 + 3*(5*a^8*b^2 - 15*a^6*b^4 + 19*a^4*b^6 - 9*a^2*b^8)*cos(e*x + d))*sin(e*x + d))/((a^6* b^7 - 3*a^4*b^9 + 3*a^2*b^11 - b^13)*e*cos(e*x + d)^3 + 3*(a^7*b^6 - 3*a^5 *b^8 + 3*a^3*b^10 - a*b^12)*e*cos(e*x + d)^2 + 3*(a^8*b^5 - 3*a^6*b^7 + 3* a^4*b^9 - a^2*b^11)*e*cos(e*x + d) + (a^9*b^4 - 3*a^7*b^6 + 3*a^5*b^8 - a^ 3*b^10)*e), 1/6*(6*(a^7*b^3 - 3*a^5*b^5 + 3*a^3*b^7 - a*b^9)*e*x*cos(e*x + d)^3 + 18*(a^8*b^2 - 3*a^6*b^4 + 3*a^4*b^6 - a^2*b^8)*e*x*cos(e*x + d)^2 + 18*(a^9*b - 3*a^7*b^3 + 3*a^5*b^5 - a^3*b^7)*e*x*cos(e*x + d) + 6*(a^10 - 3*a^8*b^2 + 3*a^6*b^4 - a^4*b^6)*e*x - 3*(2*a^9 - 5*a^7*b^2 + 3*a^5*b^4 - 2*a^3*b^6 + (2*a^6*b^3 - 5*a^4*b^5 + 3*a^2*b^7 - 2*b^9)*cos(e*x + d)^...
\[ \int \frac {a+b \sec (d+e x)}{\left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^2} \, dx=\int \frac {a + b \sec {\left (d + e x \right )}}{\left (a \sec {\left (d + e x \right )} + b\right )^{4}}\, dx \]
Exception generated. \[ \int \frac {a+b \sec (d+e x)}{\left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 469 vs. \(2 (215) = 430\).
Time = 0.39 (sec) , antiderivative size = 469, normalized size of antiderivative = 2.04 \[ \int \frac {a+b \sec (d+e x)}{\left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^2} \, dx=\frac {\frac {3 \, {\left (2 \, a^{6} - 5 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - 2 \, b^{6}\right )} {\left (\pi \left \lfloor \frac {e x + d}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) - b \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right )} \sqrt {a^{2} - b^{2}}} + \frac {3 \, {\left (e x + d\right )} a}{b^{4}} - \frac {6 \, a^{7} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{5} - 15 \, a^{6} b \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{5} + 30 \, a^{4} b^{3} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{5} - 12 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{5} - 27 \, a^{2} b^{5} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{5} + 18 \, a b^{6} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{5} + 12 \, a^{7} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{3} - 44 \, a^{5} b^{2} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{3} + 68 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{3} - 36 \, a b^{6} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{3} + 6 \, a^{7} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + 15 \, a^{6} b \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) - 30 \, a^{4} b^{3} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) - 12 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + 27 \, a^{2} b^{5} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + 18 \, a b^{6} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )}{{\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} {\left (a \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{2} - b \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{2} + a + b\right )}^{3}}}{3 \, e} \]
1/3*(3*(2*a^6 - 5*a^4*b^2 + 3*a^2*b^4 - 2*b^6)*(pi*floor(1/2*(e*x + d)/pi + 1/2)*sgn(-2*a + 2*b) + arctan(-(a*tan(1/2*e*x + 1/2*d) - b*tan(1/2*e*x + 1/2*d))/sqrt(a^2 - b^2)))/((a^4*b^4 - 2*a^2*b^6 + b^8)*sqrt(a^2 - b^2)) + 3*(e*x + d)*a/b^4 - (6*a^7*tan(1/2*e*x + 1/2*d)^5 - 15*a^6*b*tan(1/2*e*x + 1/2*d)^5 + 30*a^4*b^3*tan(1/2*e*x + 1/2*d)^5 - 12*a^3*b^4*tan(1/2*e*x + 1/2*d)^5 - 27*a^2*b^5*tan(1/2*e*x + 1/2*d)^5 + 18*a*b^6*tan(1/2*e*x + 1/2* d)^5 + 12*a^7*tan(1/2*e*x + 1/2*d)^3 - 44*a^5*b^2*tan(1/2*e*x + 1/2*d)^3 + 68*a^3*b^4*tan(1/2*e*x + 1/2*d)^3 - 36*a*b^6*tan(1/2*e*x + 1/2*d)^3 + 6*a ^7*tan(1/2*e*x + 1/2*d) + 15*a^6*b*tan(1/2*e*x + 1/2*d) - 30*a^4*b^3*tan(1 /2*e*x + 1/2*d) - 12*a^3*b^4*tan(1/2*e*x + 1/2*d) + 27*a^2*b^5*tan(1/2*e*x + 1/2*d) + 18*a*b^6*tan(1/2*e*x + 1/2*d))/((a^4*b^3 - 2*a^2*b^5 + b^7)*(a *tan(1/2*e*x + 1/2*d)^2 - b*tan(1/2*e*x + 1/2*d)^2 + a + b)^3))/e
Time = 37.13 (sec) , antiderivative size = 5469, normalized size of antiderivative = 23.78 \[ \int \frac {a+b \sec (d+e x)}{\left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^2} \, dx=\text {Too large to display} \]
- ((tan(d/2 + (e*x)/2)*(6*a*b^4 + a^4*b + 2*a^5 - 3*a^2*b^3 - 4*a^3*b^2))/ (b^5 - 2*a*b^4 + a^2*b^3) + (tan(d/2 + (e*x)/2)^5*(6*a*b^4 - a^4*b + 2*a^5 + 3*a^2*b^3 - 4*a^3*b^2))/(b^3*(a + b)^2) + (4*tan(d/2 + (e*x)/2)^3*(9*a* b^4 + 3*a^5 - 8*a^3*b^2))/(3*(a*b^3 - b^4)*(a + b)))/(e*(3*a*b^2 - tan(d/2 + (e*x)/2)^4*(3*a*b^2 + 3*a^2*b - 3*a^3 - 3*b^3) - tan(d/2 + (e*x)/2)^2*( 3*a*b^2 - 3*a^2*b - 3*a^3 + 3*b^3) + 3*a^2*b + a^3 + b^3 + tan(d/2 + (e*x) /2)^6*(3*a*b^2 - 3*a^2*b + a^3 - b^3))) - (2*a*atan(((a*((a*((8*(4*b^18 - 14*a^2*b^16 - 6*a^3*b^15 + 26*a^4*b^14 + 14*a^5*b^13 - 30*a^6*b^12 - 10*a^ 7*b^11 + 18*a^8*b^10 + 2*a^9*b^9 - 4*a^10*b^8))/(a*b^15 + b^16 - 3*a^2*b^1 4 - 3*a^3*b^13 + 3*a^4*b^12 + 3*a^5*b^11 - a^6*b^10 - a^7*b^9) - (a*tan(d/ 2 + (e*x)/2)*(8*a*b^17 - 8*a^2*b^16 - 32*a^3*b^15 + 32*a^4*b^14 + 48*a^5*b ^13 - 48*a^6*b^12 - 32*a^7*b^11 + 32*a^8*b^10 + 8*a^9*b^9 - 8*a^10*b^8)*8i )/(b^4*(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6)))*1i)/b^4 - (8*tan(d/2 + (e*x)/2)*(8*a^11*b - 8*a^12 - 4*b^12 + 8*a^2*b^10 + 8*a^3*b^9 - 17*a^4*b^8 - 32*a^5*b^7 + 30*a^6*b^6 + 4 8*a^7*b^5 - 45*a^8*b^4 - 32*a^9*b^3 + 32*a^10*b^2))/(a*b^12 + b^13 - 3*a^2 *b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6)))/b^4 - (a *((a*((8*(4*b^18 - 14*a^2*b^16 - 6*a^3*b^15 + 26*a^4*b^14 + 14*a^5*b^13 - 30*a^6*b^12 - 10*a^7*b^11 + 18*a^8*b^10 + 2*a^9*b^9 - 4*a^10*b^8))/(a*b^15 + b^16 - 3*a^2*b^14 - 3*a^3*b^13 + 3*a^4*b^12 + 3*a^5*b^11 - a^6*b^10 ...