Integrand size = 30, antiderivative size = 148 \[ \int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{7/2}} \, dx=-\frac {4 a^3 \tan (e+f x)}{3 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{7/2}}-\frac {a^3 \tan (e+f x)}{c^2 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2}}+\frac {a^3 \log (1-\cos (e+f x)) \tan (e+f x)}{c^3 f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \]
-4/3*a^3*tan(f*x+e)/f/(c-c*sec(f*x+e))^(7/2)/(a+a*sec(f*x+e))^(1/2)-a^3*ta n(f*x+e)/c^2/f/(c-c*sec(f*x+e))^(3/2)/(a+a*sec(f*x+e))^(1/2)+a^3*ln(1-cos( f*x+e))*tan(f*x+e)/c^3/f/(a+a*sec(f*x+e))^(1/2)/(c-c*sec(f*x+e))^(1/2)
Time = 2.35 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.65 \[ \int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{7/2}} \, dx=-\frac {a^3 \left (-3 \log (\cos (e+f x))-3 \log (1-\sec (e+f x))+\frac {-4-3 (-1+\sec (e+f x))^2}{(-1+\sec (e+f x))^3}\right ) \tan (e+f x)}{3 c^3 f \sqrt {a (1+\sec (e+f x))} \sqrt {c-c \sec (e+f x)}} \]
-1/3*(a^3*(-3*Log[Cos[e + f*x]] - 3*Log[1 - Sec[e + f*x]] + (-4 - 3*(-1 + Sec[e + f*x])^2)/(-1 + Sec[e + f*x])^3)*Tan[e + f*x])/(c^3*f*Sqrt[a*(1 + S ec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])
Time = 0.68 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {3042, 4398, 3042, 4395, 3042, 4399, 16}
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.
| \(\displaystyle \int \frac {(a \sec (e+f x)+a)^{5/2}}{(c-c \sec (e+f x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^{5/2}}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{7/2}}dx\) |
| \(\Big \downarrow \) 4398 |
| \(\displaystyle \frac {a^2 \int \frac {\sqrt {\sec (e+f x) a+a}}{(c-c \sec (e+f x))^{3/2}}dx}{c^2}-\frac {4 a^3 \tan (e+f x)}{3 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a^2 \int \frac {\sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{c^2}-\frac {4 a^3 \tan (e+f x)}{3 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 4395 |
| \(\displaystyle \frac {a^2 \left (\frac {\int \frac {\sqrt {\sec (e+f x) a+a}}{\sqrt {c-c \sec (e+f x)}}dx}{c}-\frac {a \tan (e+f x)}{f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}\right )}{c^2}-\frac {4 a^3 \tan (e+f x)}{3 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a^2 \left (\frac {\int \frac {\sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}}{\sqrt {c-c \csc \left (e+f x+\frac {\pi }{2}\right )}}dx}{c}-\frac {a \tan (e+f x)}{f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}\right )}{c^2}-\frac {4 a^3 \tan (e+f x)}{3 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 4399 |
| \(\displaystyle \frac {a^2 \left (\frac {a \tan (e+f x) \int \frac {1}{c \cos (e+f x)-c}d\cos (e+f x)}{f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}-\frac {a \tan (e+f x)}{f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}\right )}{c^2}-\frac {4 a^3 \tan (e+f x)}{3 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {a^2 \left (\frac {a \tan (e+f x) \log (1-\cos (e+f x))}{c f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}-\frac {a \tan (e+f x)}{f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}\right )}{c^2}-\frac {4 a^3 \tan (e+f x)}{3 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}}\) |
(-4*a^3*Tan[e + f*x])/(3*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^( 7/2)) + (a^2*(-((a*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(3/2))) + (a*Log[1 - Cos[e + f*x]]*Tan[e + f*x])/(c*f*Sqrt[a + a*S ec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])))/c^2
3.2.7.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*(csc[(e_.) + (f_.)*(x_)]*(d_ .) + (c_))^(n_), x_Symbol] :> Simp[-2*a*Cot[e + f*x]*((c + d*Csc[e + f*x])^ n/(f*(2*n + 1)*Sqrt[a + b*Csc[e + f*x]])), x] + Simp[1/c Int[Sqrt[a + b*C sc[e + f*x]]*(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] && LtQ[n, -2^(-1)]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(5/2)*(csc[(e_.) + (f_.)*(x_)]*( d_.) + (c_))^(n_.), x_Symbol] :> Simp[-8*a^3*Cot[e + f*x]*((c + d*Csc[e + f *x])^n/(f*(2*n + 1)*Sqrt[a + b*Csc[e + f*x]])), x] + Simp[a^2/c^2 Int[Sqr t[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && LtQ[n, -2^(-1) ]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d _.) + (c_))^(n_), x_Symbol] :> Simp[(-a)*c*(Cot[e + f*x]/(f*Sqrt[a + b*Csc[ e + f*x]]*Sqrt[c + d*Csc[e + f*x]])) Subst[Int[(b + a*x)^(m - 1/2)*((d + c*x)^(n - 1/2)/x^(m + n)), x], x, Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e , f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m - 1/2] && EqQ[m + n, 0]
Time = 2.20 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.77
| method | result | size |
| default | \(\frac {\sqrt {2}\, a^{2} \sqrt {-\frac {2 a}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\, \left (1-\cos \left (f x +e \right )\right ) \left (12 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{6} \csc \left (f x +e \right )^{6}-6 \ln \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}+1\right ) \left (1-\cos \left (f x +e \right )\right )^{6} \csc \left (f x +e \right )^{6}+6 \left (1-\cos \left (f x +e \right )\right )^{4} \csc \left (f x +e \right )^{4}-3 \left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}+1\right ) \csc \left (f x +e \right )}{12 f \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )^{3} \left (\frac {c \left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\right )^{\frac {7}{2}}}\) | \(262\) |
| risch | \(\frac {a^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) x}{c^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}}-\frac {2 a^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) \left (f x +e \right )}{c^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, f}+\frac {2 i a^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left (15 \,{\mathrm e}^{5 i \left (f x +e \right )}-36 \,{\mathrm e}^{4 i \left (f x +e \right )}+58 \,{\mathrm e}^{3 i \left (f x +e \right )}-36 \,{\mathrm e}^{2 i \left (f x +e \right )}+15 \,{\mathrm e}^{i \left (f x +e \right )}\right )}{3 c^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{5} \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, f}-\frac {2 i a^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{c^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, f}\) | \(456\) |
1/12/f*2^(1/2)*a^2*(-2*a/((1-cos(f*x+e))^2*csc(f*x+e)^2-1))^(1/2)/((1-cos( f*x+e))^2*csc(f*x+e)^2-1)^3/(c*(1-cos(f*x+e))^2/((1-cos(f*x+e))^2*csc(f*x+ e)^2-1)*csc(f*x+e)^2)^(7/2)*(1-cos(f*x+e))*(12*ln(-cot(f*x+e)+csc(f*x+e))* (1-cos(f*x+e))^6*csc(f*x+e)^6-6*ln((1-cos(f*x+e))^2*csc(f*x+e)^2+1)*(1-cos (f*x+e))^6*csc(f*x+e)^6+6*(1-cos(f*x+e))^4*csc(f*x+e)^4-3*(1-cos(f*x+e))^2 *csc(f*x+e)^2+1)*csc(f*x+e)
\[ \int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{7/2}} \, dx=\int { \frac {{\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{{\left (-c \sec \left (f x + e\right ) + c\right )}^{\frac {7}{2}}} \,d x } \]
integral((a^2*sec(f*x + e)^2 + 2*a^2*sec(f*x + e) + a^2)*sqrt(a*sec(f*x + e) + a)*sqrt(-c*sec(f*x + e) + c)/(c^4*sec(f*x + e)^4 - 4*c^4*sec(f*x + e) ^3 + 6*c^4*sec(f*x + e)^2 - 4*c^4*sec(f*x + e) + c^4), x)
Timed out. \[ \int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{7/2}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 3738 vs. \(2 (134) = 268\).
Time = 1.85 (sec) , antiderivative size = 3738, normalized size of antiderivative = 25.26 \[ \int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{7/2}} \, dx=\text {Too large to display} \]
-1/3*(3*(f*x + e)*a^2*cos(6*f*x + 6*e)^2 + 675*(f*x + e)*a^2*cos(4*f*x + 4 *e)^2 + 675*(f*x + e)*a^2*cos(2*f*x + 2*e)^2 + 108*(f*x + e)*a^2*cos(5/2*a rctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 1200*(f*x + e)*a^2*cos(3/2 *arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 108*(f*x + e)*a^2*cos(1/ 2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 3*(f*x + e)*a^2*sin(6*f *x + 6*e)^2 + 675*(f*x + e)*a^2*sin(4*f*x + 4*e)^2 + 675*(f*x + e)*a^2*sin (2*f*x + 2*e)^2 + 108*(f*x + e)*a^2*sin(5/2*arctan2(sin(2*f*x + 2*e), cos( 2*f*x + 2*e)))^2 + 1200*(f*x + e)*a^2*sin(3/2*arctan2(sin(2*f*x + 2*e), co s(2*f*x + 2*e)))^2 + 108*(f*x + e)*a^2*sin(1/2*arctan2(sin(2*f*x + 2*e), c os(2*f*x + 2*e)))^2 + 90*(f*x + e)*a^2*cos(2*f*x + 2*e) + 3*(f*x + e)*a^2 - 72*a^2*sin(2*f*x + 2*e) - 6*(a^2*cos(6*f*x + 6*e)^2 + 225*a^2*cos(4*f*x + 4*e)^2 + 225*a^2*cos(2*f*x + 2*e)^2 + 36*a^2*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 400*a^2*cos(3/2*arctan2(sin(2*f*x + 2*e), co s(2*f*x + 2*e)))^2 + 36*a^2*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + a^2*sin(6*f*x + 6*e)^2 + 225*a^2*sin(4*f*x + 4*e)^2 + 450*a^2*s in(4*f*x + 4*e)*sin(2*f*x + 2*e) + 225*a^2*sin(2*f*x + 2*e)^2 + 36*a^2*sin (5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 400*a^2*sin(3/2*arct an2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 36*a^2*sin(1/2*arctan2(sin(2* f*x + 2*e), cos(2*f*x + 2*e)))^2 + 30*a^2*cos(2*f*x + 2*e) + a^2 + 2*(15*a ^2*cos(4*f*x + 4*e) + 15*a^2*cos(2*f*x + 2*e) + a^2)*cos(6*f*x + 6*e) +...
\[ \int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{7/2}} \, dx=\int { \frac {{\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{{\left (-c \sec \left (f x + e\right ) + c\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{7/2}} \, dx=\int \frac {{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{5/2}}{{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^{7/2}} \,d x \]