Integrand size = 27, antiderivative size = 287 \[ \int \frac {\sqrt {a+a \sec (e+f x)}}{(c+d \sec (e+f x))^3} \, dx=\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right ) \tan (e+f x)}{c^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {a^{3/2} \sqrt {d} \left (15 c^2+20 c d+8 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a-a \sec (e+f x)}}{\sqrt {a} \sqrt {c+d}}\right ) \tan (e+f x)}{4 c^3 (c+d)^{5/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {a d \tan (e+f x)}{2 c (c+d) f \sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))^2}-\frac {a d (7 c+4 d) \tan (e+f x)}{4 c^2 (c+d)^2 f \sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))} \]
-1/2*a*d*tan(f*x+e)/c/(c+d)/f/(c+d*sec(f*x+e))^2/(a+a*sec(f*x+e))^(1/2)-1/ 4*a*d*(7*c+4*d)*tan(f*x+e)/c^2/(c+d)^2/f/(c+d*sec(f*x+e))/(a+a*sec(f*x+e)) ^(1/2)+2*a^(3/2)*arctanh((a-a*sec(f*x+e))^(1/2)/a^(1/2))*tan(f*x+e)/c^3/f/ (a-a*sec(f*x+e))^(1/2)/(a+a*sec(f*x+e))^(1/2)-1/4*a^(3/2)*(15*c^2+20*c*d+8 *d^2)*arctanh(d^(1/2)*(a-a*sec(f*x+e))^(1/2)/a^(1/2)/(c+d)^(1/2))*d^(1/2)* tan(f*x+e)/c^3/(c+d)^(5/2)/f/(a-a*sec(f*x+e))^(1/2)/(a+a*sec(f*x+e))^(1/2)
Time = 7.57 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {a+a \sec (e+f x)}}{(c+d \sec (e+f x))^3} \, dx=\frac {(d+c \cos (e+f x))^3 \sec \left (\frac {1}{2} (e+f x)\right ) \sec ^{\frac {5}{2}}(e+f x) \sqrt {a (1+\sec (e+f x))} \left (\frac {\left (8 (c+d)^{5/2} \arctan \left (\frac {\tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}}}\right )-\sqrt {d} \left (15 c^2+20 c d+8 d^2\right ) \arctan \left (\frac {\sqrt {d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d} \sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}}}\right )\right ) \sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}} \sec \left (\frac {1}{2} (e+f x)\right ) \sqrt {1+\sec (e+f x)}}{(c+d)^{5/2} \sqrt {\frac {1}{1+\cos (e+f x)}}}-\frac {2 c d (d (7 c+4 d)+3 c (3 c+2 d) \cos (e+f x)) \sec ^{\frac {3}{2}}(e+f x) \sin \left (\frac {1}{2} (e+f x)\right )}{(c+d)^2 (c+d \sec (e+f x))^2}\right )}{8 c^3 f (c+d \sec (e+f x))^3} \]
((d + c*Cos[e + f*x])^3*Sec[(e + f*x)/2]*Sec[e + f*x]^(5/2)*Sqrt[a*(1 + Se c[e + f*x])]*(((8*(c + d)^(5/2)*ArcTan[Tan[(e + f*x)/2]/Sqrt[Cos[e + f*x]/ (1 + Cos[e + f*x])]] - Sqrt[d]*(15*c^2 + 20*c*d + 8*d^2)*ArcTan[(Sqrt[d]*T an[(e + f*x)/2])/(Sqrt[c + d]*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])])])*Sqr t[Cos[e + f*x]/(1 + Cos[e + f*x])]*Sec[(e + f*x)/2]*Sqrt[1 + Sec[e + f*x]] )/((c + d)^(5/2)*Sqrt[(1 + Cos[e + f*x])^(-1)]) - (2*c*d*(d*(7*c + 4*d) + 3*c*(3*c + 2*d)*Cos[e + f*x])*Sec[e + f*x]^(3/2)*Sin[(e + f*x)/2])/((c + d )^2*(c + d*Sec[e + f*x])^2)))/(8*c^3*f*(c + d*Sec[e + f*x])^3)
Time = 0.45 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.93, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {3042, 4428, 114, 27, 168, 27, 174, 73, 219, 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.
| \(\displaystyle \int \frac {\sqrt {a \sec (e+f x)+a}}{(c+d \sec (e+f x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {a \csc \left (e+f x+\frac {\pi }{2}\right )+a}}{\left (c+d \csc \left (e+f x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 4428 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \int \frac {\cos (e+f x)}{\sqrt {a-a \sec (e+f x)} (c+d \sec (e+f x))^3}d\sec (e+f x)}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\int \frac {a \cos (e+f x) (4 (c+d)-3 d \sec (e+f x))}{2 \sqrt {a-a \sec (e+f x)} (c+d \sec (e+f x))^2}d\sec (e+f x)}{2 a c (c+d)}+\frac {d \sqrt {a-a \sec (e+f x)}}{2 a c (c+d) (c+d \sec (e+f x))^2}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\int \frac {\cos (e+f x) (4 (c+d)-3 d \sec (e+f x))}{\sqrt {a-a \sec (e+f x)} (c+d \sec (e+f x))^2}d\sec (e+f x)}{4 c (c+d)}+\frac {d \sqrt {a-a \sec (e+f x)}}{2 a c (c+d) (c+d \sec (e+f x))^2}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\frac {\int \frac {a \cos (e+f x) \left (8 (c+d)^2-d (7 c+4 d) \sec (e+f x)\right )}{2 \sqrt {a-a \sec (e+f x)} (c+d \sec (e+f x))}d\sec (e+f x)}{a c (c+d)}+\frac {d (7 c+4 d) \sqrt {a-a \sec (e+f x)}}{a c (c+d) (c+d \sec (e+f x))}}{4 c (c+d)}+\frac {d \sqrt {a-a \sec (e+f x)}}{2 a c (c+d) (c+d \sec (e+f x))^2}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\frac {\int \frac {\cos (e+f x) \left (8 (c+d)^2-d (7 c+4 d) \sec (e+f x)\right )}{\sqrt {a-a \sec (e+f x)} (c+d \sec (e+f x))}d\sec (e+f x)}{2 c (c+d)}+\frac {d (7 c+4 d) \sqrt {a-a \sec (e+f x)}}{a c (c+d) (c+d \sec (e+f x))}}{4 c (c+d)}+\frac {d \sqrt {a-a \sec (e+f x)}}{2 a c (c+d) (c+d \sec (e+f x))^2}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\frac {\frac {8 (c+d)^2 \int \frac {\cos (e+f x)}{\sqrt {a-a \sec (e+f x)}}d\sec (e+f x)}{c}-\frac {d \left (15 c^2+20 c d+8 d^2\right ) \int \frac {1}{\sqrt {a-a \sec (e+f x)} (c+d \sec (e+f x))}d\sec (e+f x)}{c}}{2 c (c+d)}+\frac {d (7 c+4 d) \sqrt {a-a \sec (e+f x)}}{a c (c+d) (c+d \sec (e+f x))}}{4 c (c+d)}+\frac {d \sqrt {a-a \sec (e+f x)}}{2 a c (c+d) (c+d \sec (e+f x))^2}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\frac {\frac {2 d \left (15 c^2+20 c d+8 d^2\right ) \int \frac {1}{c+d-\frac {d (a-a \sec (e+f x))}{a}}d\sqrt {a-a \sec (e+f x)}}{a c}-\frac {16 (c+d)^2 \int \frac {1}{1-\frac {a-a \sec (e+f x)}{a}}d\sqrt {a-a \sec (e+f x)}}{a c}}{2 c (c+d)}+\frac {d (7 c+4 d) \sqrt {a-a \sec (e+f x)}}{a c (c+d) (c+d \sec (e+f x))}}{4 c (c+d)}+\frac {d \sqrt {a-a \sec (e+f x)}}{2 a c (c+d) (c+d \sec (e+f x))^2}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\frac {\frac {2 d \left (15 c^2+20 c d+8 d^2\right ) \int \frac {1}{c+d-\frac {d (a-a \sec (e+f x))}{a}}d\sqrt {a-a \sec (e+f x)}}{a c}-\frac {16 (c+d)^2 \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right )}{\sqrt {a} c}}{2 c (c+d)}+\frac {d (7 c+4 d) \sqrt {a-a \sec (e+f x)}}{a c (c+d) (c+d \sec (e+f x))}}{4 c (c+d)}+\frac {d \sqrt {a-a \sec (e+f x)}}{2 a c (c+d) (c+d \sec (e+f x))^2}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\frac {\frac {2 \sqrt {d} \left (15 c^2+20 c d+8 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a-a \sec (e+f x)}}{\sqrt {a} \sqrt {c+d}}\right )}{\sqrt {a} c \sqrt {c+d}}-\frac {16 (c+d)^2 \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right )}{\sqrt {a} c}}{2 c (c+d)}+\frac {d (7 c+4 d) \sqrt {a-a \sec (e+f x)}}{a c (c+d) (c+d \sec (e+f x))}}{4 c (c+d)}+\frac {d \sqrt {a-a \sec (e+f x)}}{2 a c (c+d) (c+d \sec (e+f x))^2}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
-((a^2*((d*Sqrt[a - a*Sec[e + f*x]])/(2*a*c*(c + d)*(c + d*Sec[e + f*x])^2 ) + (((-16*(c + d)^2*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]])/(Sqrt[a]*c ) + (2*Sqrt[d]*(15*c^2 + 20*c*d + 8*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])])/(Sqrt[a]*c*Sqrt[c + d]))/(2*c*(c + d)) + (d*(7*c + 4*d)*Sqrt[a - a*Sec[e + f*x]])/(a*c*(c + d)*(c + d*Sec[e + f*x] )))/(4*c*(c + d)))*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Se c[e + f*x]]))
3.2.53.3.1 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
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[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*(csc[(e_.) + (f_.)*(x_)]*( d_.) + (c_))^(n_.), x_Symbol] :> Simp[a^2*(Cot[e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[a - b*Csc[e + f*x]])) Subst[Int[(a + b*x)^(m - 1/2)*((c + d *x)^n/(x*Sqrt[a - b*x])), x], x, Csc[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0 ] && IntegerQ[m - 1/2]
Leaf count of result is larger than twice the leaf count of optimal. \(76268\) vs. \(2(249)=498\).
Time = 17.66 (sec) , antiderivative size = 76269, normalized size of antiderivative = 265.75
Leaf count of result is larger than twice the leaf count of optimal. 545 vs. \(2 (249) = 498\).
Time = 4.36 (sec) , antiderivative size = 2368, normalized size of antiderivative = 8.25 \[ \int \frac {\sqrt {a+a \sec (e+f x)}}{(c+d \sec (e+f x))^3} \, dx=\text {Too large to display} \]
[1/8*((15*c^2*d^2 + 20*c*d^3 + 8*d^4 + (15*c^4 + 20*c^3*d + 8*c^2*d^2)*cos (f*x + e)^3 + (15*c^4 + 50*c^3*d + 48*c^2*d^2 + 16*c*d^3)*cos(f*x + e)^2 + (30*c^3*d + 55*c^2*d^2 + 36*c*d^3 + 8*d^4)*cos(f*x + e))*sqrt(-a*d/(c + d ))*log((2*(c + d)*sqrt(-a*d/(c + d))*sqrt((a*cos(f*x + e) + a)/cos(f*x + e ))*cos(f*x + e)*sin(f*x + e) + (a*c + 2*a*d)*cos(f*x + e)^2 - a*d + (a*c + a*d)*cos(f*x + e))/(c*cos(f*x + e)^2 + (c + d)*cos(f*x + e) + d)) + 8*(c^ 2*d^2 + 2*c*d^3 + d^4 + (c^4 + 2*c^3*d + c^2*d^2)*cos(f*x + e)^3 + (c^4 + 4*c^3*d + 5*c^2*d^2 + 2*c*d^3)*cos(f*x + e)^2 + (2*c^3*d + 5*c^2*d^2 + 4*c *d^3 + d^4)*cos(f*x + e))*sqrt(-a)*log((2*a*cos(f*x + e)^2 - 2*sqrt(-a)*sq rt((a*cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)*sin(f*x + e) + a*cos(f* x + e) - a)/(cos(f*x + e) + 1)) - 2*(3*(3*c^3*d + 2*c^2*d^2)*cos(f*x + e)^ 2 + (7*c^2*d^2 + 4*c*d^3)*cos(f*x + e))*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sin(f*x + e))/((c^7 + 2*c^6*d + c^5*d^2)*f*cos(f*x + e)^3 + (c^7 + 4 *c^6*d + 5*c^5*d^2 + 2*c^4*d^3)*f*cos(f*x + e)^2 + (2*c^6*d + 5*c^5*d^2 + 4*c^4*d^3 + c^3*d^4)*f*cos(f*x + e) + (c^5*d^2 + 2*c^4*d^3 + c^3*d^4)*f), -1/8*(16*(c^2*d^2 + 2*c*d^3 + d^4 + (c^4 + 2*c^3*d + c^2*d^2)*cos(f*x + e) ^3 + (c^4 + 4*c^3*d + 5*c^2*d^2 + 2*c*d^3)*cos(f*x + e)^2 + (2*c^3*d + 5*c ^2*d^2 + 4*c*d^3 + d^4)*cos(f*x + e))*sqrt(a)*arctan(sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)/(sqrt(a)*sin(f*x + e))) - (15*c^2*d^2 + 20 *c*d^3 + 8*d^4 + (15*c^4 + 20*c^3*d + 8*c^2*d^2)*cos(f*x + e)^3 + (15*c...
\[ \int \frac {\sqrt {a+a \sec (e+f x)}}{(c+d \sec (e+f x))^3} \, dx=\int \frac {\sqrt {a \left (\sec {\left (e + f x \right )} + 1\right )}}{\left (c + d \sec {\left (e + f x \right )}\right )^{3}}\, dx \]
Timed out. \[ \int \frac {\sqrt {a+a \sec (e+f x)}}{(c+d \sec (e+f x))^3} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {a+a \sec (e+f x)}}{(c+d \sec (e+f x))^3} \, dx=\int { \frac {\sqrt {a \sec \left (f x + e\right ) + a}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {a+a \sec (e+f x)}}{(c+d \sec (e+f x))^3} \, dx=\int \frac {\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}}{{\left (c+\frac {d}{\cos \left (e+f\,x\right )}\right )}^3} \,d x \]