Integrand size = 36, antiderivative size = 133 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{11/2}} \, dx=-\frac {(a+a \sec (e+f x))^{5/2} \tan (e+f x)}{10 f (c-c \sec (e+f x))^{11/2}}-\frac {(a+a \sec (e+f x))^{5/2} \tan (e+f x)}{40 c f (c-c \sec (e+f x))^{9/2}}-\frac {(a+a \sec (e+f x))^{5/2} \tan (e+f x)}{240 c^2 f (c-c \sec (e+f x))^{7/2}} \]
-1/10*(a+a*sec(f*x+e))^(5/2)*tan(f*x+e)/f/(c-c*sec(f*x+e))^(11/2)-1/40*(a+ a*sec(f*x+e))^(5/2)*tan(f*x+e)/c/f/(c-c*sec(f*x+e))^(9/2)-1/240*(a+a*sec(f *x+e))^(5/2)*tan(f*x+e)/c^2/f/(c-c*sec(f*x+e))^(7/2)
Time = 5.43 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.59 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{11/2}} \, dx=\frac {a^3 \left (2+5 \sec (e+f x)+5 \sec ^2(e+f x)\right ) \tan (e+f x)}{15 c^5 f (-1+\sec (e+f x))^5 \sqrt {a (1+\sec (e+f x))} \sqrt {c-c \sec (e+f x)}} \]
(a^3*(2 + 5*Sec[e + f*x] + 5*Sec[e + f*x]^2)*Tan[e + f*x])/(15*c^5*f*(-1 + Sec[e + f*x])^5*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])
Time = 0.84 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3042, 4439, 3042, 4439, 3042, 4438}
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.
| \(\displaystyle \int \frac {\sec (e+f x) (a \sec (e+f x)+a)^{5/2}}{(c-c \sec (e+f x))^{11/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^{5/2}}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{11/2}}dx\) |
| \(\Big \downarrow \) 4439 |
| \(\displaystyle \frac {\int \frac {\sec (e+f x) (\sec (e+f x) a+a)^{5/2}}{(c-c \sec (e+f x))^{9/2}}dx}{5 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^{5/2}}{10 f (c-c \sec (e+f x))^{11/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right ) a+a\right )^{5/2}}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{9/2}}dx}{5 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^{5/2}}{10 f (c-c \sec (e+f x))^{11/2}}\) |
| \(\Big \downarrow \) 4439 |
| \(\displaystyle \frac {\frac {\int \frac {\sec (e+f x) (\sec (e+f x) a+a)^{5/2}}{(c-c \sec (e+f x))^{7/2}}dx}{8 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^{5/2}}{8 f (c-c \sec (e+f x))^{9/2}}}{5 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^{5/2}}{10 f (c-c \sec (e+f x))^{11/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right ) a+a\right )^{5/2}}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{7/2}}dx}{8 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^{5/2}}{8 f (c-c \sec (e+f x))^{9/2}}}{5 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^{5/2}}{10 f (c-c \sec (e+f x))^{11/2}}\) |
| \(\Big \downarrow \) 4438 |
| \(\displaystyle \frac {-\frac {\tan (e+f x) (a \sec (e+f x)+a)^{5/2}}{48 c f (c-c \sec (e+f x))^{7/2}}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^{5/2}}{8 f (c-c \sec (e+f x))^{9/2}}}{5 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^{5/2}}{10 f (c-c \sec (e+f x))^{11/2}}\) |
-1/10*((a + a*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(f*(c - c*Sec[e + f*x])^(1 1/2)) + (-1/8*((a + a*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(f*(c - c*Sec[e + f*x])^(9/2)) - ((a + a*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(48*c*f*(c - c*Se c[e + f*x])^(7/2)))/(5*c)
3.2.32.3.1 Defintions of rubi rules used
Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(cs c[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.), x_Symbol] :> Simp[b*Cot[e + f*x] *(a + b*Csc[e + f*x])^m*((c + d*Csc[e + f*x])^n/(a*f*(2*m + 1))), x] /; Fre eQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] & & EqQ[m + n + 1, 0] && NeQ[2*m + 1, 0]
Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(cs c[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.), x_Symbol] :> Simp[b*Cot[e + f*x] *(a + b*Csc[e + f*x])^m*((c + d*Csc[e + f*x])^n/(a*f*(2*m + 1))), x] + Simp [(m + n + 1)/(a*(2*m + 1)) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)* (c + d*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ [b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && ILtQ[m + n + 1, 0] && NeQ[2*m + 1, 0 ] && !LtQ[n, 0] && !(IGtQ[n + 1/2, 0] && LtQ[n + 1/2, -(m + n)])
Time = 3.62 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.68
| method | result | size |
| default | \(\frac {a^{2} \left (31 \cos \left (f x +e \right )^{2}-8 \cos \left (f x +e \right )+1\right ) \sqrt {a \left (\sec \left (f x +e \right )+1\right )}\, \left (\cos \left (f x +e \right )+1\right )^{2} \tan \left (f x +e \right ) \sec \left (f x +e \right )^{4}}{240 f \left (\sec \left (f x +e \right )-1\right )^{5} \sqrt {-c \left (\sec \left (f x +e \right )-1\right )}\, c^{5}}\) | \(91\) |
| risch | \(\frac {2 i a^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left (15 \,{\mathrm e}^{9 i \left (f x +e \right )}-30 \,{\mathrm e}^{8 i \left (f x +e \right )}+140 \,{\mathrm e}^{7 i \left (f x +e \right )}-170 \,{\mathrm e}^{6 i \left (f x +e \right )}+282 \,{\mathrm e}^{5 i \left (f x +e \right )}-170 \,{\mathrm e}^{4 i \left (f x +e \right )}+140 \,{\mathrm e}^{3 i \left (f x +e \right )}-30 \,{\mathrm e}^{2 i \left (f x +e \right )}+15 \,{\mathrm e}^{i \left (f x +e \right )}\right )}{15 c^{5} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{9} \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, f}\) | \(199\) |
1/240/f*a^2*(31*cos(f*x+e)^2-8*cos(f*x+e)+1)*(a*(sec(f*x+e)+1))^(1/2)*(cos (f*x+e)+1)^2/(sec(f*x+e)-1)^5/(-c*(sec(f*x+e)-1))^(1/2)/c^5*tan(f*x+e)*sec (f*x+e)^4
Time = 0.28 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.46 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{11/2}} \, dx=\frac {{\left (15 \, a^{2} \cos \left (f x + e\right )^{5} - 15 \, a^{2} \cos \left (f x + e\right )^{4} + 20 \, a^{2} \cos \left (f x + e\right )^{3} - 10 \, a^{2} \cos \left (f x + e\right )^{2} + 2 \, a^{2} \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{15 \, {\left (c^{6} f \cos \left (f x + e\right )^{5} - 5 \, c^{6} f \cos \left (f x + e\right )^{4} + 10 \, c^{6} f \cos \left (f x + e\right )^{3} - 10 \, c^{6} f \cos \left (f x + e\right )^{2} + 5 \, c^{6} f \cos \left (f x + e\right ) - c^{6} f\right )} \sin \left (f x + e\right )} \]
1/15*(15*a^2*cos(f*x + e)^5 - 15*a^2*cos(f*x + e)^4 + 20*a^2*cos(f*x + e)^ 3 - 10*a^2*cos(f*x + e)^2 + 2*a^2*cos(f*x + e))*sqrt((a*cos(f*x + e) + a)/ cos(f*x + e))*sqrt((c*cos(f*x + e) - c)/cos(f*x + e))/((c^6*f*cos(f*x + e) ^5 - 5*c^6*f*cos(f*x + e)^4 + 10*c^6*f*cos(f*x + e)^3 - 10*c^6*f*cos(f*x + e)^2 + 5*c^6*f*cos(f*x + e) - c^6*f)*sin(f*x + e))
Timed out. \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{11/2}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 4108 vs. \(2 (115) = 230\).
Time = 17.08 (sec) , antiderivative size = 4108, normalized size of antiderivative = 30.89 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{11/2}} \, dx=\text {Too large to display} \]
-2/15*(1350*a^2*cos(6*f*x + 6*e)*sin(2*f*x + 2*e) + 1350*a^2*cos(4*f*x + 4 *e)*sin(2*f*x + 2*e) - 30*a^2*sin(2*f*x + 2*e) - 10*(3*a^2*sin(8*f*x + 8*e ) + 17*a^2*sin(6*f*x + 6*e) + 17*a^2*sin(4*f*x + 4*e) + 3*a^2*sin(2*f*x + 2*e))*cos(10*f*x + 10*e) - 1350*(a^2*sin(6*f*x + 6*e) + a^2*sin(4*f*x + 4* e))*cos(8*f*x + 8*e) - 5*(3*a^2*sin(10*f*x + 10*e) + 75*a^2*sin(8*f*x + 8* e) + 290*a^2*sin(6*f*x + 6*e) + 290*a^2*sin(4*f*x + 4*e) + 75*a^2*sin(2*f* x + 2*e) - 80*a^2*sin(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 1 92*a^2*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 80*a^2*sin(3 /2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*cos(9/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 20*(7*a^2*sin(10*f*x + 10*e) + 135*a^2*sin(8 *f*x + 8*e) + 450*a^2*sin(6*f*x + 6*e) + 450*a^2*sin(4*f*x + 4*e) + 135*a^ 2*sin(2*f*x + 2*e) - 72*a^2*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 20*a^2*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*cos( 7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 6*(47*a^2*sin(10*f*x + 10*e) + 855*a^2*sin(8*f*x + 8*e) + 2730*a^2*sin(6*f*x + 6*e) + 2730*a^2*si n(4*f*x + 4*e) + 855*a^2*sin(2*f*x + 2*e) + 240*a^2*sin(3/2*arctan2(sin(2* f*x + 2*e), cos(2*f*x + 2*e))) + 160*a^2*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 20*(7*a^2*sin(10*f*x + 10*e) + 135*a^2*sin(8*f*x + 8*e) + 450*a^2*sin(6* f*x + 6*e) + 450*a^2*sin(4*f*x + 4*e) + 135*a^2*sin(2*f*x + 2*e) + 20*a...
\[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{11/2}} \, dx=\int { \frac {{\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {5}{2}} \sec \left (f x + e\right )}{{\left (-c \sec \left (f x + e\right ) + c\right )}^{\frac {11}{2}}} \,d x } \]
Time = 18.62 (sec) , antiderivative size = 419, normalized size of antiderivative = 3.15 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{11/2}} \, dx=\frac {\sqrt {c-\frac {c}{\cos \left (e+f\,x\right )}}\,\left (\frac {a^2\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,136{}\mathrm {i}}{3\,c^6\,f}-\frac {a^2\,\cos \left (e+f\,x\right )\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,1688{}\mathrm {i}}{15\,c^6\,f}+\frac {a^2\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\cos \left (2\,e+2\,f\,x\right )\,\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,160{}\mathrm {i}}{3\,c^6\,f}-\frac {a^2\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\cos \left (3\,e+3\,f\,x\right )\,\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,124{}\mathrm {i}}{3\,c^6\,f}+\frac {a^2\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\cos \left (4\,e+4\,f\,x\right )\,\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,8{}\mathrm {i}}{c^6\,f}-\frac {a^2\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\cos \left (5\,e+5\,f\,x\right )\,\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,4{}\mathrm {i}}{c^6\,f}\right )}{{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sin \left (e+f\,x\right )\,264{}\mathrm {i}-{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sin \left (2\,e+2\,f\,x\right )\,330{}\mathrm {i}+{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sin \left (3\,e+3\,f\,x\right )\,220{}\mathrm {i}-{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sin \left (4\,e+4\,f\,x\right )\,88{}\mathrm {i}+{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sin \left (5\,e+5\,f\,x\right )\,20{}\mathrm {i}-{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sin \left (6\,e+6\,f\,x\right )\,2{}\mathrm {i}} \]
((c - c/cos(e + f*x))^(1/2)*((a^2*exp(e*6i + f*x*6i)*(a + a/cos(e + f*x))^ (1/2)*136i)/(3*c^6*f) - (a^2*cos(e + f*x)*exp(e*6i + f*x*6i)*(a + a/cos(e + f*x))^(1/2)*1688i)/(15*c^6*f) + (a^2*exp(e*6i + f*x*6i)*cos(2*e + 2*f*x) *(a + a/cos(e + f*x))^(1/2)*160i)/(3*c^6*f) - (a^2*exp(e*6i + f*x*6i)*cos( 3*e + 3*f*x)*(a + a/cos(e + f*x))^(1/2)*124i)/(3*c^6*f) + (a^2*exp(e*6i + f*x*6i)*cos(4*e + 4*f*x)*(a + a/cos(e + f*x))^(1/2)*8i)/(c^6*f) - (a^2*exp (e*6i + f*x*6i)*cos(5*e + 5*f*x)*(a + a/cos(e + f*x))^(1/2)*4i)/(c^6*f)))/ (exp(e*6i + f*x*6i)*sin(e + f*x)*264i - exp(e*6i + f*x*6i)*sin(2*e + 2*f*x )*330i + exp(e*6i + f*x*6i)*sin(3*e + 3*f*x)*220i - exp(e*6i + f*x*6i)*sin (4*e + 4*f*x)*88i + exp(e*6i + f*x*6i)*sin(5*e + 5*f*x)*20i - exp(e*6i + f *x*6i)*sin(6*e + 6*f*x)*2i)