Integrand size = 30, antiderivative size = 152 \[ \int \frac {(a+a \sec (e+f x))^{5/2}}{\sqrt {c-c \sec (e+f x)}} \, dx=\frac {a^3 \log (\cos (e+f x)) \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}+\frac {4 a^3 \log (1-\sec (e+f x)) \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}+\frac {a^3 \sec (e+f x) \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \]
a^3*ln(cos(f*x+e))*tan(f*x+e)/f/(a+a*sec(f*x+e))^(1/2)/(c-c*sec(f*x+e))^(1 /2)+4*a^3*ln(1-sec(f*x+e))*tan(f*x+e)/f/(a+a*sec(f*x+e))^(1/2)/(c-c*sec(f* x+e))^(1/2)+a^3*sec(f*x+e)*tan(f*x+e)/f/(a+a*sec(f*x+e))^(1/2)/(c-c*sec(f* x+e))^(1/2)
Time = 0.37 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.45 \[ \int \frac {(a+a \sec (e+f x))^{5/2}}{\sqrt {c-c \sec (e+f x)}} \, dx=\frac {a^3 (\log (\cos (e+f x))+4 \log (1-\sec (e+f x))+\sec (e+f x)) \tan (e+f x)}{f \sqrt {a (1+\sec (e+f x))} \sqrt {c-c \sec (e+f x)}} \]
(a^3*(Log[Cos[e + f*x]] + 4*Log[1 - Sec[e + f*x]] + Sec[e + f*x])*Tan[e + f*x])/(f*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])
Time = 0.33 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.47, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3042, 4400, 27, 93, 2009}
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.
| \(\displaystyle \int \frac {(a \sec (e+f x)+a)^{5/2}}{\sqrt {c-c \sec (e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^{5/2}}{\sqrt {c-c \csc \left (e+f x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 4400 |
| \(\displaystyle -\frac {a c \tan (e+f x) \int \frac {a^2 \cos (e+f x) (\sec (e+f x)+1)^2}{c (1-\sec (e+f x))}d\sec (e+f x)}{f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^3 \tan (e+f x) \int \frac {\cos (e+f x) (\sec (e+f x)+1)^2}{1-\sec (e+f x)}d\sec (e+f x)}{f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}\) |
| \(\Big \downarrow \) 93 |
| \(\displaystyle -\frac {a^3 \tan (e+f x) \int \left (\cos (e+f x)-\frac {4}{\sec (e+f x)-1}-1\right )d\sec (e+f x)}{f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^3 \tan (e+f x) (-\sec (e+f x)-4 \log (1-\sec (e+f x))+\log (\sec (e+f x)))}{f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}\) |
-((a^3*(-4*Log[1 - Sec[e + f*x]] + Log[Sec[e + f*x]] - Sec[e + f*x])*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]))
3.2.4.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_] :> Int[ExpandIntegrand[(e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; Fre eQ[{a, b, c, d, e, f}, x] && IntegerQ[p]
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[(a + b*x)^(m - 1/2)*((c + d *x)^(n - 1/2)/x), x], x, Csc[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n }, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0]
Time = 2.29 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.02
| method | result | size |
| default | \(-\frac {a^{2} \sqrt {a \left (\sec \left (f x +e \right )+1\right )}\, \left (\ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right ) \sin \left (f x +e \right )-8 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ) \sin \left (f x +e \right )+3 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )-1\right ) \sin \left (f x +e \right )+3 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right ) \sin \left (f x +e \right )-\sin \left (f x +e \right )-\tan \left (f x +e \right )\right )}{f \left (\cos \left (f x +e \right )+1\right ) \sqrt {-c \left (\sec \left (f x +e \right )-1\right )}}\) | \(155\) |
| risch | \(-\frac {a^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left (-2 i {\mathrm e}^{i \left (f x +e \right )}+3 i {\mathrm e}^{2 i \left (f x +e \right )} \ln \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )+{\mathrm e}^{3 i \left (f x +e \right )} f x +8 i {\mathrm e}^{3 i \left (f x +e \right )} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )+2 i {\mathrm e}^{2 i \left (f x +e \right )}-3 i {\mathrm e}^{i \left (f x +e \right )} \ln \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )+8 i {\mathrm e}^{i \left (f x +e \right )} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )+2 \,{\mathrm e}^{3 i \left (f x +e \right )} e +{\mathrm e}^{i \left (f x +e \right )} f x -{\mathrm e}^{2 i \left (f x +e \right )} f x -3 i \ln \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right ) {\mathrm e}^{3 i \left (f x +e \right )}-8 i {\mathrm e}^{2 i \left (f x +e \right )} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )+3 i \ln \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )-8 i \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )+2 \,{\mathrm e}^{i \left (f x +e \right )} e -2 \,{\mathrm e}^{2 i \left (f x +e \right )} e -f x -2 e \right )}{\left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, f \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )}\) | \(374\) |
-1/f*a^2*(a*(sec(f*x+e)+1))^(1/2)/(cos(f*x+e)+1)/(-c*(sec(f*x+e)-1))^(1/2) *(ln(2/(cos(f*x+e)+1))*sin(f*x+e)-8*ln(-cot(f*x+e)+csc(f*x+e))*sin(f*x+e)+ 3*ln(-cot(f*x+e)+csc(f*x+e)-1)*sin(f*x+e)+3*ln(-cot(f*x+e)+csc(f*x+e)+1)*s in(f*x+e)-sin(f*x+e)-tan(f*x+e))
\[ \int \frac {(a+a \sec (e+f x))^{5/2}}{\sqrt {c-c \sec (e+f x)}} \, dx=\int { \frac {{\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{\sqrt {-c \sec \left (f x + e\right ) + c}} \,d x } \]
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*sec(f*x + e) - c), x)
Timed out. \[ \int \frac {(a+a \sec (e+f x))^{5/2}}{\sqrt {c-c \sec (e+f x)}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(a+a \sec (e+f x))^{5/2}}{\sqrt {c-c \sec (e+f x)}} \, dx=\text {Exception raised: RuntimeError} \]
\[ \int \frac {(a+a \sec (e+f x))^{5/2}}{\sqrt {c-c \sec (e+f x)}} \, dx=\int { \frac {{\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{\sqrt {-c \sec \left (f x + e\right ) + c}} \,d x } \]
Timed out. \[ \int \frac {(a+a \sec (e+f x))^{5/2}}{\sqrt {c-c \sec (e+f x)}} \, dx=\int \frac {{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{5/2}}{\sqrt {c-\frac {c}{\cos \left (e+f\,x\right )}}} \,d x \]