Integrand size = 34, antiderivative size = 41 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 \sqrt {c-c \sec (e+f x)} \, dx=-\frac {2 c (a+a \sec (e+f x))^3 \tan (e+f x)}{7 f \sqrt {c-c \sec (e+f x)}} \]
Time = 0.77 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.34 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 \sqrt {c-c \sec (e+f x)} \, dx=\frac {16 a^3 \cos ^6\left (\frac {1}{2} (e+f x)\right ) \cot \left (\frac {1}{2} (e+f x)\right ) \sec ^3(e+f x) \sqrt {c-c \sec (e+f x)}}{7 f} \]
Time = 0.30 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {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.
\(\displaystyle \int \sec (e+f x) (a \sec (e+f x)+a)^3 \sqrt {c-c \sec (e+f x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^3 \sqrt {c-c \csc \left (e+f x+\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 4441 |
\(\displaystyle -\frac {2 c \tan (e+f x) (a \sec (e+f x)+a)^3}{7 f \sqrt {c-c \sec (e+f x)}}\) |
3.1.82.3.1 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)]
Time = 5.64 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.12
method | result | size |
default | \(\frac {2 a^{3} \left (\cos \left (f x +e \right )+1\right )^{4} \sqrt {-c \left (\sec \left (f x +e \right )-1\right )}\, \sec \left (f x +e \right )^{3} \csc \left (f x +e \right )}{7 f}\) | \(46\) |
parts | \(-\frac {2 a^{3} \sqrt {-c \left (\sec \left (f x +e \right )-1\right )}\, \sin \left (f x +e \right )}{f \left (\cos \left (f x +e \right )-1\right )}-\frac {2 a^{3} \sqrt {-c \left (\sec \left (f x +e \right )-1\right )}\, \left (-5+16 \cos \left (f x +e \right )^{4}+8 \cos \left (f x +e \right )^{3}-2 \cos \left (f x +e \right )^{2}+\cos \left (f x +e \right )\right ) \sec \left (f x +e \right )^{3} \csc \left (f x +e \right )}{35 f}-\frac {2 a^{3} \sqrt {-c \left (\sec \left (f x +e \right )-1\right )}\, \left (-\sec \left (f x +e \right ) \csc \left (f x +e \right )+2 \cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{f}+\frac {2 a^{3} \sqrt {-c \left (\sec \left (f x +e \right )-1\right )}\, \left (3+8 \cos \left (f x +e \right )^{3}+4 \cos \left (f x +e \right )^{2}-\cos \left (f x +e \right )\right ) \sec \left (f x +e \right )^{2} \csc \left (f x +e \right )}{5 f}\) | \(227\) |
Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (37) = 74\).
Time = 0.26 (sec) , antiderivative size = 97, normalized size of antiderivative = 2.37 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 \sqrt {c-c \sec (e+f x)} \, dx=\frac {2 \, {\left (a^{3} \cos \left (f x + e\right )^{4} + 4 \, a^{3} \cos \left (f x + e\right )^{3} + 6 \, a^{3} \cos \left (f x + e\right )^{2} + 4 \, a^{3} \cos \left (f x + e\right ) + a^{3}\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{7 \, f \cos \left (f x + e\right )^{3} \sin \left (f x + e\right )} \]
2/7*(a^3*cos(f*x + e)^4 + 4*a^3*cos(f*x + e)^3 + 6*a^3*cos(f*x + e)^2 + 4* a^3*cos(f*x + e) + a^3)*sqrt((c*cos(f*x + e) - c)/cos(f*x + e))/(f*cos(f*x + e)^3*sin(f*x + e))
\[ \int \sec (e+f x) (a+a \sec (e+f x))^3 \sqrt {c-c \sec (e+f x)} \, dx=a^{3} \left (\int \sqrt {- c \sec {\left (e + f x \right )} + c} \sec {\left (e + f x \right )}\, dx + \int 3 \sqrt {- c \sec {\left (e + f x \right )} + c} \sec ^{2}{\left (e + f x \right )}\, dx + \int 3 \sqrt {- c \sec {\left (e + f x \right )} + c} \sec ^{3}{\left (e + f x \right )}\, dx + \int \sqrt {- c \sec {\left (e + f x \right )} + c} \sec ^{4}{\left (e + f x \right )}\, dx\right ) \]
a**3*(Integral(sqrt(-c*sec(e + f*x) + c)*sec(e + f*x), x) + Integral(3*sqr t(-c*sec(e + f*x) + c)*sec(e + f*x)**2, x) + Integral(3*sqrt(-c*sec(e + f* x) + c)*sec(e + f*x)**3, x) + Integral(sqrt(-c*sec(e + f*x) + c)*sec(e + f *x)**4, x))
\[ \int \sec (e+f x) (a+a \sec (e+f x))^3 \sqrt {c-c \sec (e+f x)} \, dx=\int { {\left (a \sec \left (f x + e\right ) + a\right )}^{3} \sqrt {-c \sec \left (f x + e\right ) + c} \sec \left (f x + e\right ) \,d x } \]
2/7*(7*(cos(2*f*x + 2*e)^2 + sin(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1)^ (3/4)*(5*(a^3*f*cos(2*f*x + 2*e)^2 + a^3*f*sin(2*f*x + 2*e)^2 + 2*a^3*f*co s(2*f*x + 2*e) + a^3*f)*integrate((((cos(10*f*x + 10*e)*cos(2*f*x + 2*e) + 4*cos(8*f*x + 8*e)*cos(2*f*x + 2*e) + 6*cos(6*f*x + 6*e)*cos(2*f*x + 2*e) + 4*cos(4*f*x + 4*e)*cos(2*f*x + 2*e) + cos(2*f*x + 2*e)^2 + sin(10*f*x + 10*e)*sin(2*f*x + 2*e) + 4*sin(8*f*x + 8*e)*sin(2*f*x + 2*e) + 6*sin(6*f* x + 6*e)*sin(2*f*x + 2*e) + 4*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + sin(2*f* x + 2*e)^2)*cos(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + (cos(2* f*x + 2*e)*sin(10*f*x + 10*e) + 4*cos(2*f*x + 2*e)*sin(8*f*x + 8*e) + 6*co s(2*f*x + 2*e)*sin(6*f*x + 6*e) + 4*cos(2*f*x + 2*e)*sin(4*f*x + 4*e) - co s(10*f*x + 10*e)*sin(2*f*x + 2*e) - 4*cos(8*f*x + 8*e)*sin(2*f*x + 2*e) - 6*cos(6*f*x + 6*e)*sin(2*f*x + 2*e) - 4*cos(4*f*x + 4*e)*sin(2*f*x + 2*e)) *sin(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*cos(1/2*arctan2(sin (2*f*x + 2*e), cos(2*f*x + 2*e) + 1)) - ((cos(2*f*x + 2*e)*sin(10*f*x + 10 *e) + 4*cos(2*f*x + 2*e)*sin(8*f*x + 8*e) + 6*cos(2*f*x + 2*e)*sin(6*f*x + 6*e) + 4*cos(2*f*x + 2*e)*sin(4*f*x + 4*e) - cos(10*f*x + 10*e)*sin(2*f*x + 2*e) - 4*cos(8*f*x + 8*e)*sin(2*f*x + 2*e) - 6*cos(6*f*x + 6*e)*sin(2*f *x + 2*e) - 4*cos(4*f*x + 4*e)*sin(2*f*x + 2*e))*cos(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - (cos(10*f*x + 10*e)*cos(2*f*x + 2*e) + 4*cos (8*f*x + 8*e)*cos(2*f*x + 2*e) + 6*cos(6*f*x + 6*e)*cos(2*f*x + 2*e) + ...
Time = 0.82 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.80 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 \sqrt {c-c \sec (e+f x)} \, dx=\frac {16 \, \sqrt {2} a^{3} c^{4}}{7 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {7}{2}} f} \]
Time = 17.16 (sec) , antiderivative size = 375, normalized size of antiderivative = 9.15 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 \sqrt {c-c \sec (e+f x)} \, dx=\frac {\sqrt {c-\frac {c}{\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}}{2}}}\,\left (\frac {a^3\,2{}\mathrm {i}}{f}+\frac {a^3\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,2{}\mathrm {i}}{7\,f}\right )}{{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}-1}-\frac {\sqrt {c-\frac {c}{\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}}{2}}}\,\left (\frac {a^3\,8{}\mathrm {i}}{f}+\frac {a^3\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,8{}\mathrm {i}}{7\,f}\right )}{\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}-1\right )\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^2}+\frac {\sqrt {c-\frac {c}{\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}}{2}}}\,\left (\frac {a^3\,4{}\mathrm {i}}{f}+\frac {a^3\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,36{}\mathrm {i}}{7\,f}\right )}{\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}-1\right )\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}+\frac {\sqrt {c-\frac {c}{\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}}{2}}}\,\left (\frac {a^3\,16{}\mathrm {i}}{7\,f}-\frac {a^3\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,16{}\mathrm {i}}{7\,f}\right )}{\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}-1\right )\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^3} \]
((c - c/(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2))^(1/2)*((a^3*2i)/f + (a^3*exp(e*1i + f*x*1i)*2i)/(7*f)))/(exp(e*1i + f*x*1i) - 1) - ((c - c/ (exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2))^(1/2)*((a^3*8i)/f + (a^3* exp(e*1i + f*x*1i)*8i)/(7*f)))/((exp(e*1i + f*x*1i) - 1)*(exp(e*2i + f*x*2 i) + 1)^2) + ((c - c/(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2))^(1/2 )*((a^3*4i)/f + (a^3*exp(e*1i + f*x*1i)*36i)/(7*f)))/((exp(e*1i + f*x*1i) - 1)*(exp(e*2i + f*x*2i) + 1)) + ((c - c/(exp(- e*1i - f*x*1i)/2 + exp(e*1 i + f*x*1i)/2))^(1/2)*((a^3*16i)/(7*f) - (a^3*exp(e*1i + f*x*1i)*16i)/(7*f )))/((exp(e*1i + f*x*1i) - 1)*(exp(e*2i + f*x*2i) + 1)^3)