Integrand size = 36, antiderivative size = 99 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{3/2}} \, dx=-\frac {a \sqrt {a+a \sec (e+f x)} \tan (e+f x)}{f (c-c \sec (e+f x))^{3/2}}-\frac {a^2 \log (1-\sec (e+f x)) \tan (e+f x)}{c f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \] Output:
-a*(a+a*sec(f*x+e))^(1/2)*tan(f*x+e)/f/(c-c*sec(f*x+e))^(3/2)-a^2*ln(1-sec (f*x+e))*tan(f*x+e)/c/f/(a+a*sec(f*x+e))^(1/2)/(c-c*sec(f*x+e))^(1/2)
Time = 0.51 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.71 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{3/2}} \, dx=-\frac {a^2 \left (\log (1-\sec (e+f x))-\frac {2}{-1+\sec (e+f x)}\right ) \tan (e+f x)}{c f \sqrt {a (1+\sec (e+f x))} \sqrt {c-c \sec (e+f x)}} \] Input:
Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^(3/2))/(c - c*Sec[e + f*x])^( 3/2),x]
Output:
-((a^2*(Log[1 - Sec[e + f*x]] - 2/(-1 + Sec[e + f*x]))*Tan[e + f*x])/(c*f* Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]]))
Time = 0.57 (sec) , antiderivative size = 99, 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, 4440}
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.
| \(\displaystyle \int \frac {\sec (e+f x) (a \sec (e+f x)+a)^{3/2}}{(c-c \sec (e+f x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^{3/2}}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4442 |
| \(\displaystyle -\frac {a \int \frac {\sec (e+f x) \sqrt {\sec (e+f x) a+a}}{\sqrt {c-c \sec (e+f x)}}dx}{c}-\frac {a \tan (e+f x) \sqrt {a \sec (e+f x)+a}}{f (c-c \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}}{\sqrt {c-c \csc \left (e+f x+\frac {\pi }{2}\right )}}dx}{c}-\frac {a \tan (e+f x) \sqrt {a \sec (e+f x)+a}}{f (c-c \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 4440 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \log (1-\sec (e+f x))}{c f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}-\frac {a \tan (e+f x) \sqrt {a \sec (e+f x)+a}}{f (c-c \sec (e+f x))^{3/2}}\) |
Input:
Int[(Sec[e + f*x]*(a + a*Sec[e + f*x])^(3/2))/(c - c*Sec[e + f*x])^(3/2),x ]
Output:
-((a*Sqrt[a + a*Sec[e + f*x]]*Tan[e + f*x])/(f*(c - c*Sec[e + f*x])^(3/2)) ) - (a^2*Log[1 - Sec[e + f*x]]*Tan[e + f*x])/(c*f*Sqrt[a + a*Sec[e + f*x]] *Sqrt[c - c*Sec[e + f*x]])
Int[(csc[(e_.) + (f_.)*(x_)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)])/Sq rt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[a*c*Log[1 + (b/ a)*Csc[e + f*x]]*(Cot[e + f*x]/(b*f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc [e + f*x]])), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && Eq Q[a^2 - b^2, 0]
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)]
Result contains complex when optimal does not.
Time = 1.83 (sec) , antiderivative size = 229, normalized size of antiderivative = 2.31
| method | result | size |
| risch | \(\frac {i a \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left (2 \,{\mathrm e}^{2 i \left (f x +e \right )} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )-{\mathrm e}^{2 i \left (f x +e \right )} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )-4 \,{\mathrm e}^{i \left (f x +e \right )} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )+2 \,{\mathrm e}^{i \left (f x +e \right )} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )+4 \,{\mathrm e}^{i \left (f x +e \right )}+2 \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )-\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )\right )}{c \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\) | \(229\) |
| default | \(-\frac {\sqrt {\frac {a \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1}}\, a \left (4 \ln \left (-\cot \left (\frac {f x}{2}+\frac {e}{2}\right )+\csc \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-2 \ln \left (-\frac {2 \left (\cos \left (\frac {f x}{2}+\frac {e}{2}\right )-\sin \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{\cos \left (\frac {f x}{2}+\frac {e}{2}\right )+1}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-2 \ln \left (-\frac {2 \left (\cos \left (\frac {f x}{2}+\frac {e}{2}\right )+\sin \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{\cos \left (\frac {f x}{2}+\frac {e}{2}\right )+1}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-\cot \left (\frac {f x}{2}+\frac {e}{2}\right )-\sec \left (\frac {f x}{2}+\frac {e}{2}\right ) \csc \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f \sqrt {-\frac {c \sin \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1}}\, c}\) | \(234\) |
Input:
int(sec(f*x+e)*(a+a*sec(f*x+e))^(3/2)/(c-c*sec(f*x+e))^(3/2),x,method=_RET URNVERBOSE)
Output:
I*a*(a*(exp(I*(f*x+e))+1)^2/(exp(2*I*(f*x+e))+1))^(1/2)*(2*exp(2*I*(f*x+e) )*ln(exp(I*(f*x+e))-1)-exp(2*I*(f*x+e))*ln(exp(2*I*(f*x+e))+1)-4*exp(I*(f* x+e))*ln(exp(I*(f*x+e))-1)+2*exp(I*(f*x+e))*ln(exp(2*I*(f*x+e))+1)+4*exp(I *(f*x+e))+2*ln(exp(I*(f*x+e))-1)-ln(exp(2*I*(f*x+e))+1))/c/(exp(I*(f*x+e)) +1)/(exp(I*(f*x+e))-1)/(c*(exp(I*(f*x+e))-1)^2/(exp(2*I*(f*x+e))+1))^(1/2) /f
\[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{3/2}} \, dx=\int { \frac {{\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \sec \left (f x + e\right )}{{\left (-c \sec \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(sec(f*x+e)*(a+a*sec(f*x+e))^(3/2)/(c-c*sec(f*x+e))^(3/2),x, algo rithm="fricas")
Output:
integral((a*sec(f*x + e)^2 + a*sec(f*x + e))*sqrt(a*sec(f*x + e) + a)*sqrt (-c*sec(f*x + e) + c)/(c^2*sec(f*x + e)^2 - 2*c^2*sec(f*x + e) + c^2), x)
\[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{3/2}} \, dx=\int \frac {\left (a \left (\sec {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}} \sec {\left (e + f x \right )}}{\left (- c \left (\sec {\left (e + f x \right )} - 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(sec(f*x+e)*(a+a*sec(f*x+e))**(3/2)/(c-c*sec(f*x+e))**(3/2),x)
Output:
Integral((a*(sec(e + f*x) + 1))**(3/2)*sec(e + f*x)/(-c*(sec(e + f*x) - 1) )**(3/2), x)
Time = 0.14 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.23 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{3/2}} \, dx=\frac {\frac {\sqrt {-a} a \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{c^{\frac {3}{2}}} + \frac {\sqrt {-a} a \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{c^{\frac {3}{2}}} - \frac {2 \, \sqrt {-a} a \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{c^{\frac {3}{2}}} + \frac {\sqrt {-a} a {\left (\cos \left (f x + e\right ) + 1\right )}^{2}}{c^{\frac {3}{2}} \sin \left (f x + e\right )^{2}}}{f} \] Input:
integrate(sec(f*x+e)*(a+a*sec(f*x+e))^(3/2)/(c-c*sec(f*x+e))^(3/2),x, algo rithm="maxima")
Output:
(sqrt(-a)*a*log(sin(f*x + e)/(cos(f*x + e) + 1) + 1)/c^(3/2) + sqrt(-a)*a* log(sin(f*x + e)/(cos(f*x + e) + 1) - 1)/c^(3/2) - 2*sqrt(-a)*a*log(sin(f* x + e)/(cos(f*x + e) + 1))/c^(3/2) + sqrt(-a)*a*(cos(f*x + e) + 1)^2/(c^(3 /2)*sin(f*x + e)^2))/f
Exception generated. \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(sec(f*x+e)*(a+a*sec(f*x+e))^(3/2)/(c-c*sec(f*x+e))^(3/2),x, algo rithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value
Timed out. \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{3/2}} \, dx=\int \frac {{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{3/2}}{\cos \left (e+f\,x\right )\,{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \] Input:
int((a + a/cos(e + f*x))^(3/2)/(cos(e + f*x)*(c - c/cos(e + f*x))^(3/2)),x )
Output:
int((a + a/cos(e + f*x))^(3/2)/(cos(e + f*x)*(c - c/cos(e + f*x))^(3/2)), x)
\[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{3/2}} \, dx=\frac {\sqrt {c}\, \sqrt {a}\, a \left (\int \frac {\sqrt {\sec \left (f x +e \right )+1}\, \sqrt {-\sec \left (f x +e \right )+1}\, \sec \left (f x +e \right )^{2}}{\sec \left (f x +e \right )^{2}-2 \sec \left (f x +e \right )+1}d x +\int \frac {\sqrt {\sec \left (f x +e \right )+1}\, \sqrt {-\sec \left (f x +e \right )+1}\, \sec \left (f x +e \right )}{\sec \left (f x +e \right )^{2}-2 \sec \left (f x +e \right )+1}d x \right )}{c^{2}} \] Input:
int(sec(f*x+e)*(a+a*sec(f*x+e))^(3/2)/(c-c*sec(f*x+e))^(3/2),x)
Output:
(sqrt(c)*sqrt(a)*a*(int((sqrt(sec(e + f*x) + 1)*sqrt( - sec(e + f*x) + 1)* sec(e + f*x)**2)/(sec(e + f*x)**2 - 2*sec(e + f*x) + 1),x) + 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