Integrand size = 36, antiderivative size = 42 \[ \int \frac {\sec (e+f x) \sqrt {a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{3/2}} \, dx=-\frac {\sqrt {a+a \sec (e+f x)} \tan (e+f x)}{2 f (c-c \sec (e+f x))^{3/2}} \] Output:
-1/2*(a+a*sec(f*x+e))^(1/2)*tan(f*x+e)/f/(c-c*sec(f*x+e))^(3/2)
Time = 0.44 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00 \[ \int \frac {\sec (e+f x) \sqrt {a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{3/2}} \, dx=\frac {\cot (e+f x) \sqrt {a (1+\sec (e+f x))}}{c f \sqrt {c-c \sec (e+f x)}} \] Input:
Integrate[(Sec[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(c - c*Sec[e + f*x])^(3/ 2),x]
Output:
(Cot[e + f*x]*Sqrt[a*(1 + Sec[e + f*x])])/(c*f*Sqrt[c - c*Sec[e + f*x]])
Time = 0.33 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {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.
\(\displaystyle \int \frac {\sec (e+f x) \sqrt {a \sec (e+f x)+a}}{(c-c \sec (e+f x))^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \sqrt {a \csc \left (e+f x+\frac {\pi }{2}\right )+a}}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
\(\Big \downarrow \) 4438 |
\(\displaystyle -\frac {\tan (e+f x) \sqrt {a \sec (e+f x)+a}}{2 f (c-c \sec (e+f x))^{3/2}}\) |
Input:
Int[(Sec[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(c - c*Sec[e + f*x])^(3/2),x]
Output:
-1/2*(Sqrt[a + a*Sec[e + f*x]]*Tan[e + f*x])/(f*(c - c*Sec[e + f*x])^(3/2) )
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]
Leaf count of result is larger than twice the leaf count of optimal. \(102\) vs. \(2(36)=72\).
Time = 1.68 (sec) , antiderivative size = 103, normalized size of antiderivative = 2.45
method | result | size |
default | \(\frac {\sqrt {\frac {a \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1}}\, \left (\cot \left (\frac {f x}{2}+\frac {e}{2}\right )+\sec \left (\frac {f x}{2}+\frac {e}{2}\right ) \csc \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f \sqrt {-\frac {c \sin \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1}}\, c}\) | \(103\) |
risch | \(\frac {2 i \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, {\mathrm e}^{i \left (f x +e \right )}}{c \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\) | \(105\) |
Input:
int(sec(f*x+e)*(a+a*sec(f*x+e))^(1/2)/(c-c*sec(f*x+e))^(3/2),x,method=_RET URNVERBOSE)
Output:
1/4/f*(a/(2*cos(1/2*f*x+1/2*e)^2-1)*cos(1/2*f*x+1/2*e)^2)^(1/2)/(-c/(2*cos (1/2*f*x+1/2*e)^2-1)*sin(1/2*f*x+1/2*e)^2)^(1/2)/c*(cot(1/2*f*x+1/2*e)+sec (1/2*f*x+1/2*e)*csc(1/2*f*x+1/2*e))
Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (36) = 72\).
Time = 0.13 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.88 \[ \int \frac {\sec (e+f x) \sqrt {a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{3/2}} \, dx=\frac {\sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{{\left (c^{2} f \cos \left (f x + e\right ) - c^{2} f\right )} \sin \left (f x + e\right )} \] Input:
integrate(sec(f*x+e)*(a+a*sec(f*x+e))^(1/2)/(c-c*sec(f*x+e))^(3/2),x, algo rithm="fricas")
Output:
sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt((c*cos(f*x + e) - c)/cos(f*x + e))*cos(f*x + e)/((c^2*f*cos(f*x + e) - c^2*f)*sin(f*x + e))
\[ \int \frac {\sec (e+f x) \sqrt {a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{3/2}} \, dx=\int \frac {\sqrt {a \left (\sec {\left (e + f x \right )} + 1\right )} \sec {\left (e + f x \right )}}{\left (- c \left (\sec {\left (e + f x \right )} - 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(sec(f*x+e)*(a+a*sec(f*x+e))**(1/2)/(c-c*sec(f*x+e))**(3/2),x)
Output:
Integral(sqrt(a*(sec(e + f*x) + 1))*sec(e + f*x)/(-c*(sec(e + f*x) - 1))** (3/2), x)
Leaf count of result is larger than twice the leaf count of optimal. 514 vs. \(2 (36) = 72\).
Time = 0.20 (sec) , antiderivative size = 514, normalized size of antiderivative = 12.24 \[ \int \frac {\sec (e+f x) \sqrt {a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{3/2}} \, dx=-\frac {2 \, {\left ({\left (\sin \left (3 \, f x + 3 \, e\right ) + \sin \left (f x + e\right )\right )} \cos \left (4 \, f x + 4 \, e\right ) - {\left (\cos \left (3 \, f x + 3 \, e\right ) + \cos \left (f x + e\right )\right )} \sin \left (4 \, f x + 4 \, e\right ) + {\left (2 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )} \sin \left (3 \, f x + 3 \, e\right ) - 2 \, \cos \left (3 \, f x + 3 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) - 2 \, \cos \left (f x + e\right ) \sin \left (2 \, f x + 2 \, e\right ) + 2 \, \cos \left (2 \, f x + 2 \, e\right ) \sin \left (f x + e\right ) + \sin \left (f x + e\right )\right )} \sqrt {a} \sqrt {c}}{{\left (c^{2} \cos \left (4 \, f x + 4 \, e\right )^{2} + 4 \, c^{2} \cos \left (3 \, f x + 3 \, e\right )^{2} + 4 \, c^{2} \cos \left (2 \, f x + 2 \, e\right )^{2} + 4 \, c^{2} \cos \left (f x + e\right )^{2} + c^{2} \sin \left (4 \, f x + 4 \, e\right )^{2} + 4 \, c^{2} \sin \left (3 \, f x + 3 \, e\right )^{2} + 4 \, c^{2} \sin \left (2 \, f x + 2 \, e\right )^{2} - 8 \, c^{2} \sin \left (2 \, f x + 2 \, e\right ) \sin \left (f x + e\right ) + 4 \, c^{2} \sin \left (f x + e\right )^{2} - 4 \, c^{2} \cos \left (f x + e\right ) + c^{2} - 2 \, {\left (2 \, c^{2} \cos \left (3 \, f x + 3 \, e\right ) - 2 \, c^{2} \cos \left (2 \, f x + 2 \, e\right ) + 2 \, c^{2} \cos \left (f x + e\right ) - c^{2}\right )} \cos \left (4 \, f x + 4 \, e\right ) - 4 \, {\left (2 \, c^{2} \cos \left (2 \, f x + 2 \, e\right ) - 2 \, c^{2} \cos \left (f x + e\right ) + c^{2}\right )} \cos \left (3 \, f x + 3 \, e\right ) - 4 \, {\left (2 \, c^{2} \cos \left (f x + e\right ) - c^{2}\right )} \cos \left (2 \, f x + 2 \, e\right ) - 4 \, {\left (c^{2} \sin \left (3 \, f x + 3 \, e\right ) - c^{2} \sin \left (2 \, f x + 2 \, e\right ) + c^{2} \sin \left (f x + e\right )\right )} \sin \left (4 \, f x + 4 \, e\right ) - 8 \, {\left (c^{2} \sin \left (2 \, f x + 2 \, e\right ) - c^{2} \sin \left (f x + e\right )\right )} \sin \left (3 \, f x + 3 \, e\right )\right )} f} \] Input:
integrate(sec(f*x+e)*(a+a*sec(f*x+e))^(1/2)/(c-c*sec(f*x+e))^(3/2),x, algo rithm="maxima")
Output:
-2*((sin(3*f*x + 3*e) + sin(f*x + e))*cos(4*f*x + 4*e) - (cos(3*f*x + 3*e) + cos(f*x + e))*sin(4*f*x + 4*e) + (2*cos(2*f*x + 2*e) + 1)*sin(3*f*x + 3 *e) - 2*cos(3*f*x + 3*e)*sin(2*f*x + 2*e) - 2*cos(f*x + e)*sin(2*f*x + 2*e ) + 2*cos(2*f*x + 2*e)*sin(f*x + e) + sin(f*x + e))*sqrt(a)*sqrt(c)/((c^2* cos(4*f*x + 4*e)^2 + 4*c^2*cos(3*f*x + 3*e)^2 + 4*c^2*cos(2*f*x + 2*e)^2 + 4*c^2*cos(f*x + e)^2 + c^2*sin(4*f*x + 4*e)^2 + 4*c^2*sin(3*f*x + 3*e)^2 + 4*c^2*sin(2*f*x + 2*e)^2 - 8*c^2*sin(2*f*x + 2*e)*sin(f*x + e) + 4*c^2*s in(f*x + e)^2 - 4*c^2*cos(f*x + e) + c^2 - 2*(2*c^2*cos(3*f*x + 3*e) - 2*c ^2*cos(2*f*x + 2*e) + 2*c^2*cos(f*x + e) - c^2)*cos(4*f*x + 4*e) - 4*(2*c^ 2*cos(2*f*x + 2*e) - 2*c^2*cos(f*x + e) + c^2)*cos(3*f*x + 3*e) - 4*(2*c^2 *cos(f*x + e) - c^2)*cos(2*f*x + 2*e) - 4*(c^2*sin(3*f*x + 3*e) - c^2*sin( 2*f*x + 2*e) + c^2*sin(f*x + e))*sin(4*f*x + 4*e) - 8*(c^2*sin(2*f*x + 2*e ) - c^2*sin(f*x + e))*sin(3*f*x + 3*e))*f)
Time = 0.65 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.38 \[ \int \frac {\sec (e+f x) \sqrt {a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{3/2}} \, dx=\frac {a^{2} {\left (\frac {1}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}} - 1\right )}}{2 \, \sqrt {-a c} c f {\left | a \right |} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} \] Input:
integrate(sec(f*x+e)*(a+a*sec(f*x+e))^(1/2)/(c-c*sec(f*x+e))^(3/2),x, algo rithm="giac")
Output:
1/2*a^2*(1/tan(1/2*f*x + 1/2*e)^2 - 1)/(sqrt(-a*c)*c*f*abs(a)*sgn(tan(1/2* f*x + 1/2*e)^3 + tan(1/2*f*x + 1/2*e)))
Time = 12.01 (sec) , antiderivative size = 118, normalized size of antiderivative = 2.81 \[ \int \frac {\sec (e+f x) \sqrt {a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{3/2}} \, dx=-\frac {2\,\sqrt {\frac {a\,\left (\cos \left (e+f\,x\right )+1\right )}{\cos \left (e+f\,x\right )}}\,\sqrt {\frac {c\,\left (\cos \left (e+f\,x\right )-1\right )}{\cos \left (e+f\,x\right )}}\,\left (\sin \left (e+f\,x\right )-2\,\sin \left (2\,e+2\,f\,x\right )+\sin \left (3\,e+3\,f\,x\right )\right )}{c^2\,f\,\left (4\,\cos \left (e+f\,x\right )+4\,\cos \left (2\,e+2\,f\,x\right )-4\,\cos \left (3\,e+3\,f\,x\right )+\cos \left (4\,e+4\,f\,x\right )-5\right )} \] Input:
int((a + a/cos(e + f*x))^(1/2)/(cos(e + f*x)*(c - c/cos(e + f*x))^(3/2)),x )
Output:
-(2*((a*(cos(e + f*x) + 1))/cos(e + f*x))^(1/2)*((c*(cos(e + f*x) - 1))/co s(e + f*x))^(1/2)*(sin(e + f*x) - 2*sin(2*e + 2*f*x) + sin(3*e + 3*f*x)))/ (c^2*f*(4*cos(e + f*x) + 4*cos(2*e + 2*f*x) - 4*cos(3*e + 3*f*x) + cos(4*e + 4*f*x) - 5))
\[ \int \frac {\sec (e+f x) \sqrt {a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{3/2}} \, dx=\frac {\sqrt {c}\, \sqrt {a}\, \left (\int \frac {\sqrt {\sec \left (f x +e \right )+1}\, \sqrt {-\sec \left (f x +e \right )+1}\, \sec \left (f x +e \right )}{\sec \left (f x +e \right )^{2}-2 \sec \left (f x +e \right )+1}d x \right )}{c^{2}} \] Input:
int(sec(f*x+e)*(a+a*sec(f*x+e))^(1/2)/(c-c*sec(f*x+e))^(3/2),x)
Output:
(sqrt(c)*sqrt(a)*int((sqrt(sec(e + f*x) + 1)*sqrt( - sec(e + f*x) + 1)*sec (e + f*x))/(sec(e + f*x)**2 - 2*sec(e + f*x) + 1),x))/c**2