Integrand size = 34, antiderivative size = 164 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^{7/2}} \, dx=-\frac {a^2 \arctan \left (\frac {\sqrt {c} \tan (e+f x)}{\sqrt {2} \sqrt {c-c \sec (e+f x)}}\right )}{16 \sqrt {2} c^{7/2} f}-\frac {\left (a^2+a^2 \sec (e+f x)\right ) \tan (e+f x)}{3 f (c-c \sec (e+f x))^{7/2}}+\frac {a^2 \tan (e+f x)}{4 c f (c-c \sec (e+f x))^{5/2}}-\frac {a^2 \tan (e+f x)}{16 c^2 f (c-c \sec (e+f x))^{3/2}} \]
-1/32*a^2*arctan(1/2*c^(1/2)*tan(f*x+e)*2^(1/2)/(c-c*sec(f*x+e))^(1/2))/c^ (7/2)/f*2^(1/2)-1/3*(a^2+a^2*sec(f*x+e))*tan(f*x+e)/f/(c-c*sec(f*x+e))^(7/ 2)+1/4*a^2*tan(f*x+e)/c/f/(c-c*sec(f*x+e))^(5/2)-1/16*a^2*tan(f*x+e)/c^2/f /(c-c*sec(f*x+e))^(3/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.61 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.39 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^{7/2}} \, dx=-\frac {a^2 \operatorname {Hypergeometric2F1}\left (\frac {5}{2},4,\frac {7}{2},\frac {1}{2} (1+\sec (e+f x))\right ) (1+\sec (e+f x))^2 \tan (e+f x)}{40 c^3 f \sqrt {c-c \sec (e+f x)}} \]
-1/40*(a^2*Hypergeometric2F1[5/2, 4, 7/2, (1 + Sec[e + f*x])/2]*(1 + Sec[e + f*x])^2*Tan[e + f*x])/(c^3*f*Sqrt[c - c*Sec[e + f*x]])
Time = 0.78 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.265, Rules used = {3042, 4445, 3042, 4445, 3042, 4283, 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.
| \(\displaystyle \int \frac {\sec (e+f x) (a \sec (e+f x)+a)^2}{(c-c \sec (e+f x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^2}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{7/2}}dx\) |
| \(\Big \downarrow \) 4445 |
| \(\displaystyle -\frac {a \int \frac {\sec (e+f x) (\sec (e+f x) a+a)}{(c-c \sec (e+f x))^{5/2}}dx}{2 c}-\frac {\tan (e+f x) \left (a^2 \sec (e+f x)+a^2\right )}{3 f (c-c \sec (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right ) a+a\right )}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{5/2}}dx}{2 c}-\frac {\tan (e+f x) \left (a^2 \sec (e+f x)+a^2\right )}{3 f (c-c \sec (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 4445 |
| \(\displaystyle -\frac {a \left (-\frac {a \int \frac {\sec (e+f x)}{(c-c \sec (e+f x))^{3/2}}dx}{4 c}-\frac {a \tan (e+f x)}{2 f (c-c \sec (e+f x))^{5/2}}\right )}{2 c}-\frac {\tan (e+f x) \left (a^2 \sec (e+f x)+a^2\right )}{3 f (c-c \sec (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \left (-\frac {a \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right )}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{4 c}-\frac {a \tan (e+f x)}{2 f (c-c \sec (e+f x))^{5/2}}\right )}{2 c}-\frac {\tan (e+f x) \left (a^2 \sec (e+f x)+a^2\right )}{3 f (c-c \sec (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 4283 |
| \(\displaystyle -\frac {a \left (-\frac {a \left (\frac {\int \frac {\sec (e+f x)}{\sqrt {c-c \sec (e+f x)}}dx}{4 c}-\frac {\tan (e+f x)}{2 f (c-c \sec (e+f x))^{3/2}}\right )}{4 c}-\frac {a \tan (e+f x)}{2 f (c-c \sec (e+f x))^{5/2}}\right )}{2 c}-\frac {\tan (e+f x) \left (a^2 \sec (e+f x)+a^2\right )}{3 f (c-c \sec (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \left (-\frac {a \left (\frac {\int \frac {\csc \left (e+f x+\frac {\pi }{2}\right )}{\sqrt {c-c \csc \left (e+f x+\frac {\pi }{2}\right )}}dx}{4 c}-\frac {\tan (e+f x)}{2 f (c-c \sec (e+f x))^{3/2}}\right )}{4 c}-\frac {a \tan (e+f x)}{2 f (c-c \sec (e+f x))^{5/2}}\right )}{2 c}-\frac {\tan (e+f x) \left (a^2 \sec (e+f x)+a^2\right )}{3 f (c-c \sec (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 4282 |
| \(\displaystyle -\frac {a \left (-\frac {a \left (-\frac {\int \frac {1}{\frac {c^2 \tan ^2(e+f x)}{c-c \sec (e+f x)}+2 c}d\frac {c \tan (e+f x)}{\sqrt {c-c \sec (e+f x)}}}{2 c f}-\frac {\tan (e+f x)}{2 f (c-c \sec (e+f x))^{3/2}}\right )}{4 c}-\frac {a \tan (e+f x)}{2 f (c-c \sec (e+f x))^{5/2}}\right )}{2 c}-\frac {\tan (e+f x) \left (a^2 \sec (e+f x)+a^2\right )}{3 f (c-c \sec (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {\tan (e+f x) \left (a^2 \sec (e+f x)+a^2\right )}{3 f (c-c \sec (e+f x))^{7/2}}-\frac {a \left (-\frac {a \left (-\frac {\arctan \left (\frac {\sqrt {c} \tan (e+f x)}{\sqrt {2} \sqrt {c-c \sec (e+f x)}}\right )}{2 \sqrt {2} c^{3/2} f}-\frac {\tan (e+f x)}{2 f (c-c \sec (e+f x))^{3/2}}\right )}{4 c}-\frac {a \tan (e+f x)}{2 f (c-c \sec (e+f x))^{5/2}}\right )}{2 c}\) |
-1/3*((a^2 + a^2*Sec[e + f*x])*Tan[e + f*x])/(f*(c - c*Sec[e + f*x])^(7/2) ) - (a*(-1/2*(a*Tan[e + f*x])/(f*(c - c*Sec[e + f*x])^(5/2)) - (a*(-1/2*Ar cTan[(Sqrt[c]*Tan[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sec[e + f*x]])]/(Sqrt[2]*c ^(3/2)*f) - Tan[e + f*x]/(2*f*(c - c*Sec[e + f*x])^(3/2))))/(4*c)))/(2*c)
3.1.78.3.1 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_), x_ Symbol] :> Simp[b*Cot[e + f*x]*((a + b*Csc[e + f*x])^m/(a*f*(2*m + 1))), x] + Simp[(m + 1)/(a*(2*m + 1)) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1) ] && IntegerQ[2*m]
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. \(319\) vs. \(2(141)=282\).
Time = 5.49 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.95
| method | result | size |
| default | \(-\frac {a^{2} \sqrt {2}\, \left (3 \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )^{\frac {3}{2}} \left (1-\cos \left (f x +e \right )\right )^{4} \sin \left (f x +e \right )-3 \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \left (1-\cos \left (f x +e \right )\right )^{6} \csc \left (f x +e \right )-3 \arctan \left (\frac {1}{\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\right ) \left (1-\cos \left (f x +e \right )\right )^{6} \csc \left (f x +e \right )+6 \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )^{\frac {3}{2}} \left (1-\cos \left (f x +e \right )\right )^{2} \sin \left (f x +e \right )^{3}+8 \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )^{\frac {3}{2}} \sin \left (f x +e \right )^{5}\right )}{96 c^{3} f \sqrt {\frac {c \left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\, \left (1-\cos \left (f x +e \right )\right )^{5} \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\) | \(320\) |
| parts | \(\text {Expression too large to display}\) | \(1097\) |
-1/96*a^2/c^3/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))^5/((1-cos(f*x+e))^2*csc(f*x+e)^2-1) ^(1/2)*(3*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(3/2)*(1-cos(f*x+e))^4*sin(f*x +e)-3*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2)*(1-cos(f*x+e))^6*csc(f*x+e)- 3*arctan(1/((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2))*(1-cos(f*x+e))^6*csc(f *x+e)+6*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(3/2)*(1-cos(f*x+e))^2*sin(f*x+e )^3+8*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(3/2)*sin(f*x+e)^5)
Time = 0.35 (sec) , antiderivative size = 517, normalized size of antiderivative = 3.15 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^{7/2}} \, dx=\left [-\frac {3 \, \sqrt {2} {\left (a^{2} \cos \left (f x + e\right )^{3} - 3 \, a^{2} \cos \left (f x + e\right )^{2} + 3 \, 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 (7 \, a^{2} \cos \left (f x + e\right )^{4} + 29 \, a^{2} \cos \left (f x + e\right )^{3} + 25 \, a^{2} \cos \left (f x + e\right )^{2} + 3 \, 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 )}}}{192 \, {\left (c^{4} f \cos \left (f x + e\right )^{3} - 3 \, c^{4} f \cos \left (f x + e\right )^{2} + 3 \, c^{4} f \cos \left (f x + e\right ) - c^{4} f\right )} \sin \left (f x + e\right )}, \frac {3 \, \sqrt {2} {\left (a^{2} \cos \left (f x + e\right )^{3} - 3 \, a^{2} \cos \left (f x + e\right )^{2} + 3 \, 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 (7 \, a^{2} \cos \left (f x + e\right )^{4} + 29 \, a^{2} \cos \left (f x + e\right )^{3} + 25 \, a^{2} \cos \left (f x + e\right )^{2} + 3 \, 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 )}}}{96 \, {\left (c^{4} f \cos \left (f x + e\right )^{3} - 3 \, c^{4} f \cos \left (f x + e\right )^{2} + 3 \, c^{4} f \cos \left (f x + e\right ) - c^{4} f\right )} \sin \left (f x + e\right )}\right ] \]
[-1/192*(3*sqrt(2)*(a^2*cos(f*x + e)^3 - 3*a^2*cos(f*x + e)^2 + 3*a^2*cos( f*x + e) - a^2)*sqrt(-c)*log((2*sqrt(2)*(cos(f*x + e)^2 + cos(f*x + e))*sq rt(-c)*sqrt((c*cos(f*x + e) - c)/cos(f*x + e)) + (3*c*cos(f*x + e) + c)*si n(f*x + e))/((cos(f*x + e) - 1)*sin(f*x + e)))*sin(f*x + e) - 4*(7*a^2*cos (f*x + e)^4 + 29*a^2*cos(f*x + e)^3 + 25*a^2*cos(f*x + e)^2 + 3*a^2*cos(f* x + e))*sqrt((c*cos(f*x + e) - c)/cos(f*x + e)))/((c^4*f*cos(f*x + e)^3 - 3*c^4*f*cos(f*x + e)^2 + 3*c^4*f*cos(f*x + e) - c^4*f)*sin(f*x + e)), 1/96 *(3*sqrt(2)*(a^2*cos(f*x + e)^3 - 3*a^2*cos(f*x + e)^2 + 3*a^2*cos(f*x + e ) - a^2)*sqrt(c)*arctan(sqrt(2)*sqrt((c*cos(f*x + e) - c)/cos(f*x + e))*co s(f*x + e)/(sqrt(c)*sin(f*x + e)))*sin(f*x + e) + 2*(7*a^2*cos(f*x + e)^4 + 29*a^2*cos(f*x + e)^3 + 25*a^2*cos(f*x + e)^2 + 3*a^2*cos(f*x + e))*sqrt ((c*cos(f*x + e) - c)/cos(f*x + e)))/((c^4*f*cos(f*x + e)^3 - 3*c^4*f*cos( f*x + e)^2 + 3*c^4*f*cos(f*x + e) - c^4*f)*sin(f*x + e))]
\[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^{7/2}} \, dx=a^{2} \left (\int \frac {\sec {\left (e + f x \right )}}{- c^{3} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec ^{3}{\left (e + f x \right )} + 3 c^{3} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec ^{2}{\left (e + f x \right )} - 3 c^{3} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec {\left (e + f x \right )} + c^{3} \sqrt {- c \sec {\left (e + f x \right )} + c}}\, dx + \int \frac {2 \sec ^{2}{\left (e + f x \right )}}{- c^{3} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec ^{3}{\left (e + f x \right )} + 3 c^{3} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec ^{2}{\left (e + f x \right )} - 3 c^{3} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec {\left (e + f x \right )} + c^{3} \sqrt {- c \sec {\left (e + f x \right )} + c}}\, dx + \int \frac {\sec ^{3}{\left (e + f x \right )}}{- c^{3} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec ^{3}{\left (e + f x \right )} + 3 c^{3} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec ^{2}{\left (e + f x \right )} - 3 c^{3} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec {\left (e + f x \right )} + c^{3} \sqrt {- c \sec {\left (e + f x \right )} + c}}\, dx\right ) \]
a**2*(Integral(sec(e + f*x)/(-c**3*sqrt(-c*sec(e + f*x) + c)*sec(e + f*x)* *3 + 3*c**3*sqrt(-c*sec(e + f*x) + c)*sec(e + f*x)**2 - 3*c**3*sqrt(-c*sec (e + f*x) + c)*sec(e + f*x) + c**3*sqrt(-c*sec(e + f*x) + c)), x) + Integr al(2*sec(e + f*x)**2/(-c**3*sqrt(-c*sec(e + f*x) + c)*sec(e + f*x)**3 + 3* c**3*sqrt(-c*sec(e + f*x) + c)*sec(e + f*x)**2 - 3*c**3*sqrt(-c*sec(e + f* x) + c)*sec(e + f*x) + c**3*sqrt(-c*sec(e + f*x) + c)), x) + Integral(sec( e + f*x)**3/(-c**3*sqrt(-c*sec(e + f*x) + c)*sec(e + f*x)**3 + 3*c**3*sqrt (-c*sec(e + f*x) + c)*sec(e + f*x)**2 - 3*c**3*sqrt(-c*sec(e + f*x) + c)*s ec(e + f*x) + c**3*sqrt(-c*sec(e + f*x) + c)), x))
Timed out. \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^{7/2}} \, dx=\text {Timed out} \]
Time = 1.11 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.85 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^{7/2}} \, dx=\frac {\sqrt {2} {\left (3 \, a^{2} \sqrt {c} \arctan \left (\frac {\sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c}}{\sqrt {c}}\right ) + \frac {3 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {5}{2}} a^{2} c + 8 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {3}{2}} a^{2} c^{2} - 3 \, \sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c} a^{2} c^{3}}{c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6}}\right )}}{96 \, c^{4} f} \]
1/96*sqrt(2)*(3*a^2*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)^(5/2)*a^2*c + 8*(c*tan(1/2*f*x + 1/2*e)^2 - c)^(3/2)*a^2*c^2 - 3*sqrt(c*tan(1/2*f*x + 1/2*e)^2 - c)*a^2*c^3 )/(c^3*tan(1/2*f*x + 1/2*e)^6))/(c^4*f)
Timed out. \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^{7/2}} \, dx=\int \frac {{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^2}{\cos \left (e+f\,x\right )\,{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^{7/2}} \,d x \]