Integrand size = 34, antiderivative size = 117 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^{5/2}} \, dx=-\frac {3 a^2 \arctan \left (\frac {\sqrt {c} \tan (e+f x)}{\sqrt {2} \sqrt {c-c \sec (e+f x)}}\right )}{4 \sqrt {2} c^{5/2} f}-\frac {a^2 \tan (e+f x)}{f (c-c \sec (e+f x))^{5/2}}+\frac {5 a^2 \tan (e+f x)}{4 c f (c-c \sec (e+f x))^{3/2}} \]
-3/8*a^2*arctan(1/2*c^(1/2)*tan(f*x+e)*2^(1/2)/(c-c*sec(f*x+e))^(1/2))/c^( 5/2)/f*2^(1/2)-a^2*tan(f*x+e)/f/(c-c*sec(f*x+e))^(5/2)+5/4*a^2*tan(f*x+e)/ c/f/(c-c*sec(f*x+e))^(3/2)
Time = 1.13 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.28 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^{5/2}} \, dx=-\frac {a^{3/2} \left (\sqrt {a} \left (-1+4 \sec (e+f x)+5 \sec ^2(e+f x)\right )+6 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a (1+\sec (e+f x))}}{\sqrt {2} \sqrt {a}}\right ) \sec ^2(e+f x) \sqrt {a (1+\sec (e+f x))} \sin ^4\left (\frac {1}{2} (e+f x)\right )\right ) \tan (e+f x)}{4 c^2 f (-1+\sec (e+f x))^2 (1+\sec (e+f x)) \sqrt {c-c \sec (e+f x)}} \]
-1/4*(a^(3/2)*(Sqrt[a]*(-1 + 4*Sec[e + f*x] + 5*Sec[e + f*x]^2) + 6*Sqrt[2 ]*ArcTanh[Sqrt[a*(1 + Sec[e + f*x])]/(Sqrt[2]*Sqrt[a])]*Sec[e + f*x]^2*Sqr t[a*(1 + Sec[e + f*x])]*Sin[(e + f*x)/2]^4)*Tan[e + f*x])/(c^2*f*(-1 + Sec [e + f*x])^2*(1 + Sec[e + f*x])*Sqrt[c - c*Sec[e + f*x]])
Time = 0.62 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.206, Rules used = {3042, 4445, 3042, 4445, 3042, 4282, 216}
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.
| \(\displaystyle \int \frac {\sec (e+f x) (a \sec (e+f x)+a)^2}{(c-c \sec (e+f x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^2}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4445 |
| \(\displaystyle -\frac {3 a \int \frac {\sec (e+f x) (\sec (e+f x) a+a)}{(c-c \sec (e+f x))^{3/2}}dx}{4 c}-\frac {\tan (e+f x) \left (a^2 \sec (e+f x)+a^2\right )}{2 f (c-c \sec (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 a \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right ) a+a\right )}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{4 c}-\frac {\tan (e+f x) \left (a^2 \sec (e+f x)+a^2\right )}{2 f (c-c \sec (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 4445 |
| \(\displaystyle -\frac {3 a \left (-\frac {a \int \frac {\sec (e+f x)}{\sqrt {c-c \sec (e+f x)}}dx}{2 c}-\frac {a \tan (e+f x)}{f (c-c \sec (e+f x))^{3/2}}\right )}{4 c}-\frac {\tan (e+f x) \left (a^2 \sec (e+f x)+a^2\right )}{2 f (c-c \sec (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 a \left (-\frac {a \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right )}{\sqrt {c-c \csc \left (e+f x+\frac {\pi }{2}\right )}}dx}{2 c}-\frac {a \tan (e+f x)}{f (c-c \sec (e+f x))^{3/2}}\right )}{4 c}-\frac {\tan (e+f x) \left (a^2 \sec (e+f x)+a^2\right )}{2 f (c-c \sec (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 4282 |
| \(\displaystyle -\frac {3 a \left (\frac {a \int \frac {1}{\frac {c^2 \tan ^2(e+f x)}{c-c \sec (e+f x)}+2 c}d\frac {c \tan (e+f x)}{\sqrt {c-c \sec (e+f x)}}}{c f}-\frac {a \tan (e+f x)}{f (c-c \sec (e+f x))^{3/2}}\right )}{4 c}-\frac {\tan (e+f x) \left (a^2 \sec (e+f x)+a^2\right )}{2 f (c-c \sec (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {\tan (e+f x) \left (a^2 \sec (e+f x)+a^2\right )}{2 f (c-c \sec (e+f x))^{5/2}}-\frac {3 a \left (\frac {a \arctan \left (\frac {\sqrt {c} \tan (e+f x)}{\sqrt {2} \sqrt {c-c \sec (e+f x)}}\right )}{\sqrt {2} c^{3/2} f}-\frac {a \tan (e+f x)}{f (c-c \sec (e+f x))^{3/2}}\right )}{4 c}\) |
-1/2*((a^2 + a^2*Sec[e + f*x])*Tan[e + f*x])/(f*(c - c*Sec[e + f*x])^(5/2) ) - (3*a*((a*ArcTan[(Sqrt[c]*Tan[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sec[e + f*x ]])])/(Sqrt[2]*c^(3/2)*f) - (a*Tan[e + f*x])/(f*(c - c*Sec[e + f*x])^(3/2) )))/(4*c)
3.1.77.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S ymbol] :> Simp[-2/f Subst[Int[1/(2*a + x^2), x], x, b*(Cot[e + f*x]/Sqrt[ a + b*Csc[e + f*x]])], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]
Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(cs c[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.), x_Symbol] :> Simp[2*a*c*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((c + d*Csc[e + f*x])^(n - 1)/(b*f*(2*m + 1))), x] - Simp[d*((2*n - 1)/(b*(2*m + 1))) Int[Csc[e + f*x]*(a + b*Csc[e + f* x])^(m + 1)*(c + d*Csc[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f }, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[n, 0] && LtQ[m, -2^ (-1)] && IntegerQ[2*m]
Leaf count of result is larger than twice the leaf count of optimal. \(227\) vs. \(2(100)=200\).
Time = 4.58 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.95
| method | result | size |
| default | \(\frac {a^{2} \sqrt {2}\, \left (3 \arctan \left (\frac {1}{\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\right ) \left (1-\cos \left (f x +e \right )\right )^{4} \csc \left (f x +e \right )-3 \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \left (1-\cos \left (f x +e \right )\right )^{2} \sin \left (f x +e \right )-2 \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \sin \left (f x +e \right )^{3}\right )}{8 c^{2} f \sqrt {\frac {c \left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\, \left (1-\cos \left (f x +e \right )\right )^{3} \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\) | \(228\) |
| parts | \(\text {Expression too large to display}\) | \(814\) |
1/8*a^2/c^2/f*2^(1/2)/(c*(1-cos(f*x+e))^2/((1-cos(f*x+e))^2*csc(f*x+e)^2-1 )*csc(f*x+e)^2)^(1/2)/(1-cos(f*x+e))^3/((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^( 1/2)*(3*arctan(1/((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2))*(1-cos(f*x+e))^4 *csc(f*x+e)-3*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2)*(1-cos(f*x+e))^2*sin (f*x+e)-2*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2)*sin(f*x+e)^3)
Time = 0.33 (sec) , antiderivative size = 429, normalized size of antiderivative = 3.67 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^{5/2}} \, dx=\left [-\frac {3 \, \sqrt {2} {\left (a^{2} \cos \left (f x + e\right )^{2} - 2 \, a^{2} \cos \left (f x + e\right ) + a^{2}\right )} \sqrt {-c} \log \left (\frac {2 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right )\right )} \sqrt {-c} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} + {\left (3 \, c \cos \left (f x + e\right ) + c\right )} \sin \left (f x + e\right )}{{\left (\cos \left (f x + e\right ) - 1\right )} \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) + 4 \, {\left (a^{2} \cos \left (f x + e\right )^{3} - 4 \, a^{2} \cos \left (f x + e\right )^{2} - 5 \, a^{2} \cos \left (f x + e\right )\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{16 \, {\left (c^{3} f \cos \left (f x + e\right )^{2} - 2 \, c^{3} f \cos \left (f x + e\right ) + c^{3} f\right )} \sin \left (f x + e\right )}, \frac {3 \, \sqrt {2} {\left (a^{2} \cos \left (f x + e\right )^{2} - 2 \, a^{2} \cos \left (f x + e\right ) + a^{2}\right )} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt {c} \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) - 2 \, {\left (a^{2} \cos \left (f x + e\right )^{3} - 4 \, a^{2} \cos \left (f x + e\right )^{2} - 5 \, a^{2} \cos \left (f x + e\right )\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{8 \, {\left (c^{3} f \cos \left (f x + e\right )^{2} - 2 \, c^{3} f \cos \left (f x + e\right ) + c^{3} f\right )} \sin \left (f x + e\right )}\right ] \]
[-1/16*(3*sqrt(2)*(a^2*cos(f*x + e)^2 - 2*a^2*cos(f*x + e) + a^2)*sqrt(-c) *log((2*sqrt(2)*(cos(f*x + e)^2 + cos(f*x + e))*sqrt(-c)*sqrt((c*cos(f*x + e) - c)/cos(f*x + e)) + (3*c*cos(f*x + e) + c)*sin(f*x + e))/((cos(f*x + e) - 1)*sin(f*x + e)))*sin(f*x + e) + 4*(a^2*cos(f*x + e)^3 - 4*a^2*cos(f* x + e)^2 - 5*a^2*cos(f*x + e))*sqrt((c*cos(f*x + e) - c)/cos(f*x + e)))/(( c^3*f*cos(f*x + e)^2 - 2*c^3*f*cos(f*x + e) + c^3*f)*sin(f*x + e)), 1/8*(3 *sqrt(2)*(a^2*cos(f*x + e)^2 - 2*a^2*cos(f*x + e) + a^2)*sqrt(c)*arctan(sq rt(2)*sqrt((c*cos(f*x + e) - c)/cos(f*x + e))*cos(f*x + e)/(sqrt(c)*sin(f* x + e)))*sin(f*x + e) - 2*(a^2*cos(f*x + e)^3 - 4*a^2*cos(f*x + e)^2 - 5*a ^2*cos(f*x + e))*sqrt((c*cos(f*x + e) - c)/cos(f*x + e)))/((c^3*f*cos(f*x + e)^2 - 2*c^3*f*cos(f*x + e) + c^3*f)*sin(f*x + e))]
\[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^{5/2}} \, dx=a^{2} \left (\int \frac {\sec {\left (e + f x \right )}}{c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec ^{2}{\left (e + f x \right )} - 2 c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec {\left (e + f x \right )} + c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c}}\, dx + \int \frac {2 \sec ^{2}{\left (e + f x \right )}}{c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec ^{2}{\left (e + f x \right )} - 2 c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec {\left (e + f x \right )} + c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c}}\, dx + \int \frac {\sec ^{3}{\left (e + f x \right )}}{c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec ^{2}{\left (e + f x \right )} - 2 c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec {\left (e + f x \right )} + c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c}}\, dx\right ) \]
a**2*(Integral(sec(e + f*x)/(c**2*sqrt(-c*sec(e + f*x) + c)*sec(e + f*x)** 2 - 2*c**2*sqrt(-c*sec(e + f*x) + c)*sec(e + f*x) + c**2*sqrt(-c*sec(e + f *x) + c)), x) + Integral(2*sec(e + f*x)**2/(c**2*sqrt(-c*sec(e + f*x) + c) *sec(e + f*x)**2 - 2*c**2*sqrt(-c*sec(e + f*x) + c)*sec(e + f*x) + c**2*sq rt(-c*sec(e + f*x) + c)), x) + Integral(sec(e + f*x)**3/(c**2*sqrt(-c*sec( e + f*x) + c)*sec(e + f*x)**2 - 2*c**2*sqrt(-c*sec(e + f*x) + c)*sec(e + f *x) + c**2*sqrt(-c*sec(e + f*x) + c)), x))
\[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^{5/2}} \, dx=\int { \frac {{\left (a \sec \left (f x + e\right ) + a\right )}^{2} \sec \left (f x + e\right )}{{\left (-c \sec \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
Time = 1.30 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.91 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^{5/2}} \, dx=\frac {\sqrt {2} {\left (3 \, \sqrt {c} \arctan \left (\frac {\sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c}}{\sqrt {c}}\right ) + \frac {3 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {3}{2}} c + 5 \, \sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c} c^{2}}{c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4}}\right )} a^{2}}{8 \, c^{3} f} \]
1/8*sqrt(2)*(3*sqrt(c)*arctan(sqrt(c*tan(1/2*f*x + 1/2*e)^2 - c)/sqrt(c)) + (3*(c*tan(1/2*f*x + 1/2*e)^2 - c)^(3/2)*c + 5*sqrt(c*tan(1/2*f*x + 1/2*e )^2 - c)*c^2)/(c^2*tan(1/2*f*x + 1/2*e)^4))*a^2/(c^3*f)
Timed out. \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^{5/2}} \, dx=\int \frac {{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^2}{\cos \left (e+f\,x\right )\,{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^{5/2}} \,d x \]