Integrand size = 36, antiderivative size = 92 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{9/2}} \, dx=-\frac {a \sqrt {a+a \sec (e+f x)} \tan (e+f x)}{4 f (c-c \sec (e+f x))^{9/2}}+\frac {a^2 \tan (e+f x)}{12 c f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{7/2}} \]
1/12*a^2*tan(f*x+e)/c/f/(c-c*sec(f*x+e))^(7/2)/(a+a*sec(f*x+e))^(1/2)-1/4* a*(a+a*sec(f*x+e))^(1/2)*tan(f*x+e)/f/(c-c*sec(f*x+e))^(9/2)
Time = 4.36 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.74 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{9/2}} \, dx=-\frac {a^2 (1+2 \sec (e+f x)) \tan (e+f x)}{6 c^4 f (-1+\sec (e+f x))^4 \sqrt {a (1+\sec (e+f x))} \sqrt {c-c \sec (e+f x)}} \]
-1/6*(a^2*(1 + 2*Sec[e + f*x])*Tan[e + f*x])/(c^4*f*(-1 + Sec[e + f*x])^4* Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])
Time = 0.57 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3042, 4442, 3042, 4441}
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.
\(\displaystyle \int \frac {\sec (e+f x) (a \sec (e+f x)+a)^{3/2}}{(c-c \sec (e+f x))^{9/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^{3/2}}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{9/2}}dx\) |
\(\Big \downarrow \) 4442 |
\(\displaystyle -\frac {a \int \frac {\sec (e+f x) \sqrt {\sec (e+f x) a+a}}{(c-c \sec (e+f x))^{7/2}}dx}{4 c}-\frac {a \tan (e+f x) \sqrt {a \sec (e+f x)+a}}{4 f (c-c \sec (e+f x))^{9/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{7/2}}dx}{4 c}-\frac {a \tan (e+f x) \sqrt {a \sec (e+f x)+a}}{4 f (c-c \sec (e+f x))^{9/2}}\) |
\(\Big \downarrow \) 4441 |
\(\displaystyle \frac {a^2 \tan (e+f x)}{12 c f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}}-\frac {a \tan (e+f x) \sqrt {a \sec (e+f x)+a}}{4 f (c-c \sec (e+f x))^{9/2}}\) |
-1/4*(a*Sqrt[a + a*Sec[e + f*x]]*Tan[e + f*x])/(f*(c - c*Sec[e + f*x])^(9/ 2)) + (a^2*Tan[e + f*x])/(12*c*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f *x])^(7/2))
3.2.21.3.1 Defintions of rubi rules used
Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*Sq rt[csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)], x_Symbol] :> Simp[2*a*c*Cot[e + f *x]*((a + b*Csc[e + f*x])^m/(b*f*(2*m + 1)*Sqrt[c + d*Csc[e + f*x]])), x] / ; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[m, -2^(-1)]
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 - 1/2, 0] && LtQ[ m, -2^(-1)]
Time = 3.34 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.95
method | result | size |
default | \(-\frac {a \left (17 \cos \left (f x +e \right )^{2}-6 \cos \left (f x +e \right )+1\right ) \sqrt {a \left (\sec \left (f x +e \right )+1\right )}\, \left (\cos \left (f x +e \right )+1\right ) \tan \left (f x +e \right ) \sec \left (f x +e \right )^{3}}{96 f \left (\sec \left (f x +e \right )-1\right )^{4} \sqrt {-c \left (\sec \left (f x +e \right )-1\right )}\, c^{4}}\) | \(87\) |
risch | \(\frac {2 i a \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left (3 \,{\mathrm e}^{7 i \left (f x +e \right )}-6 \,{\mathrm e}^{6 i \left (f x +e \right )}+17 \,{\mathrm e}^{5 i \left (f x +e \right )}-16 \,{\mathrm e}^{4 i \left (f x +e \right )}+17 \,{\mathrm e}^{3 i \left (f x +e \right )}-6 \,{\mathrm e}^{2 i \left (f x +e \right )}+3 \,{\mathrm e}^{i \left (f x +e \right )}\right )}{3 c^{4} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{7} \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, f}\) | \(175\) |
-1/96/f*a*(17*cos(f*x+e)^2-6*cos(f*x+e)+1)*(a*(sec(f*x+e)+1))^(1/2)*(cos(f *x+e)+1)/(sec(f*x+e)-1)^4/(-c*(sec(f*x+e)-1))^(1/2)/c^4*tan(f*x+e)*sec(f*x +e)^3
Time = 0.28 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.72 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{9/2}} \, dx=\frac {{\left (6 \, a \cos \left (f x + e\right )^{4} - 6 \, a \cos \left (f x + e\right )^{3} + 4 \, a \cos \left (f x + e\right )^{2} - a \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{6 \, {\left (c^{5} f \cos \left (f x + e\right )^{4} - 4 \, c^{5} f \cos \left (f x + e\right )^{3} + 6 \, c^{5} f \cos \left (f x + e\right )^{2} - 4 \, c^{5} f \cos \left (f x + e\right ) + c^{5} f\right )} \sin \left (f x + e\right )} \]
1/6*(6*a*cos(f*x + e)^4 - 6*a*cos(f*x + e)^3 + 4*a*cos(f*x + e)^2 - a*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^5*f*cos(f*x + e)^4 - 4*c^5*f*cos(f*x + e)^3 + 6*c^5*f* cos(f*x + e)^2 - 4*c^5*f*cos(f*x + e) + c^5*f)*sin(f*x + e))
Timed out. \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{9/2}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 2608 vs. \(2 (80) = 160\).
Time = 3.32 (sec) , antiderivative size = 2608, normalized size of antiderivative = 28.35 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{9/2}} \, dx=\text {Too large to display} \]
2/3*(28*a*cos(6*f*x + 6*e)*sin(4*f*x + 4*e) - 28*a*cos(4*f*x + 4*e)*sin(2* f*x + 2*e) + 2*(3*a*sin(6*f*x + 6*e) + 8*a*sin(4*f*x + 4*e) + 3*a*sin(2*f* x + 2*e))*cos(8*f*x + 8*e) + (3*a*sin(8*f*x + 8*e) + 36*a*sin(6*f*x + 6*e) + 82*a*sin(4*f*x + 4*e) + 36*a*sin(2*f*x + 2*e) - 32*a*sin(5/2*arctan2(si n(2*f*x + 2*e), cos(2*f*x + 2*e))) - 32*a*sin(3/2*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))) + (17*a*sin(8*f*x + 8*e) + 140*a*sin(6*f*x + 6*e) + 294*a*sin(4*f*x + 4*e ) + 140*a*sin(2*f*x + 2*e) + 32*a*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))) + (17*a *sin(8*f*x + 8*e) + 140*a*sin(6*f*x + 6*e) + 294*a*sin(4*f*x + 4*e) + 140* a*sin(2*f*x + 2*e) + 32*a*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2* e))))*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + (3*a*sin(8*f* x + 8*e) + 36*a*sin(6*f*x + 6*e) + 82*a*sin(4*f*x + 4*e) + 36*a*sin(2*f*x + 2*e))*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 2*(3*a*cos( 6*f*x + 6*e) + 8*a*cos(4*f*x + 4*e) + 3*a*cos(2*f*x + 2*e))*sin(8*f*x + 8* e) - 2*(14*a*cos(4*f*x + 4*e) - 3*a)*sin(6*f*x + 6*e) + 4*(7*a*cos(2*f*x + 2*e) + 4*a)*sin(4*f*x + 4*e) + 6*a*sin(2*f*x + 2*e) - (3*a*cos(8*f*x + 8* e) + 36*a*cos(6*f*x + 6*e) + 82*a*cos(4*f*x + 4*e) + 36*a*cos(2*f*x + 2*e) - 32*a*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 32*a*cos(3/ 2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 3*a)*sin(7/2*arctan2(s...
\[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{9/2}} \, dx=\int { \frac {{\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \sec \left (f x + e\right )}{{\left (-c \sec \left (f x + e\right ) + c\right )}^{\frac {9}{2}}} \,d x } \]
Time = 19.89 (sec) , antiderivative size = 340, normalized size of antiderivative = 3.70 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{9/2}} \, dx=\frac {\sqrt {c-\frac {c}{\cos \left (e+f\,x\right )}}\,\left (\frac {a\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,68{}\mathrm {i}}{3\,c^5\,f}-\frac {a\,\cos \left (e+f\,x\right )\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,88{}\mathrm {i}}{3\,c^5\,f}+\frac {a\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\cos \left (2\,e+2\,f\,x\right )\,\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,80{}\mathrm {i}}{3\,c^5\,f}-\frac {a\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\cos \left (3\,e+3\,f\,x\right )\,\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,8{}\mathrm {i}}{c^5\,f}+\frac {a\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\cos \left (4\,e+4\,f\,x\right )\,\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,4{}\mathrm {i}}{c^5\,f}\right )}{{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\sin \left (e+f\,x\right )\,84{}\mathrm {i}-{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\sin \left (2\,e+2\,f\,x\right )\,96{}\mathrm {i}+{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\sin \left (3\,e+3\,f\,x\right )\,54{}\mathrm {i}-{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\sin \left (4\,e+4\,f\,x\right )\,16{}\mathrm {i}+{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\sin \left (5\,e+5\,f\,x\right )\,2{}\mathrm {i}} \]
((c - c/cos(e + f*x))^(1/2)*((a*exp(e*5i + f*x*5i)*(a + a/cos(e + f*x))^(1 /2)*68i)/(3*c^5*f) - (a*cos(e + f*x)*exp(e*5i + f*x*5i)*(a + a/cos(e + f*x ))^(1/2)*88i)/(3*c^5*f) + (a*exp(e*5i + f*x*5i)*cos(2*e + 2*f*x)*(a + a/co s(e + f*x))^(1/2)*80i)/(3*c^5*f) - (a*exp(e*5i + f*x*5i)*cos(3*e + 3*f*x)* (a + a/cos(e + f*x))^(1/2)*8i)/(c^5*f) + (a*exp(e*5i + f*x*5i)*cos(4*e + 4 *f*x)*(a + a/cos(e + f*x))^(1/2)*4i)/(c^5*f)))/(exp(e*5i + f*x*5i)*sin(e + f*x)*84i - exp(e*5i + f*x*5i)*sin(2*e + 2*f*x)*96i + exp(e*5i + f*x*5i)*s in(3*e + 3*f*x)*54i - exp(e*5i + f*x*5i)*sin(4*e + 4*f*x)*16i + exp(e*5i + f*x*5i)*sin(5*e + 5*f*x)*2i)