Integrand size = 36, antiderivative size = 160 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{5/2}} \, dx=\frac {3 \csc (e+f x)}{8 a^2 c^2 f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}-\frac {\cot ^2(e+f x) \csc (e+f x)}{4 a^2 c^2 f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}-\frac {3 \text {arctanh}(\cos (e+f x)) \tan (e+f x)}{8 a^2 c^2 f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \] Output:
3/8*csc(f*x+e)/a^2/c^2/f/(a+a*sec(f*x+e))^(1/2)/(c-c*sec(f*x+e))^(1/2)-1/4 *cot(f*x+e)^2*csc(f*x+e)/a^2/c^2/f/(a+a*sec(f*x+e))^(1/2)/(c-c*sec(f*x+e)) ^(1/2)-3/8*arctanh(cos(f*x+e))*tan(f*x+e)/a^2/c^2/f/(a+a*sec(f*x+e))^(1/2) /(c-c*sec(f*x+e))^(1/2)
Time = 0.49 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.48 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{5/2}} \, dx=\frac {\left (3-2 \cot ^2(e+f x)\right ) \csc (e+f x)-3 \text {arctanh}(\sec (e+f x)) \tan (e+f x)}{8 a^2 c^2 f \sqrt {a (1+\sec (e+f x))} \sqrt {c-c \sec (e+f x)}} \] Input:
Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(5 /2)),x]
Output:
((3 - 2*Cot[e + f*x]^2)*Csc[e + f*x] - 3*ArcTanh[Sec[e + f*x]]*Tan[e + f*x ])/(8*a^2*c^2*f*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])
Time = 0.63 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.64, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3042, 4447, 25, 3042, 3091, 3042, 3091, 3042, 4257}
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.
| \(\displaystyle \int \frac {\sec (e+f x)}{(a \sec (e+f x)+a)^{5/2} (c-c \sec (e+f x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right )}{\left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^{5/2} \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4447 |
| \(\displaystyle -\frac {\tan (e+f x) \int -\cot ^4(e+f x) \csc (e+f x)dx}{a^2 c^2 \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\tan (e+f x) \int \cot ^4(e+f x) \csc (e+f x)dx}{a^2 c^2 \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\tan (e+f x) \int \sec \left (e+f x-\frac {\pi }{2}\right ) \tan \left (e+f x-\frac {\pi }{2}\right )^4dx}{a^2 c^2 \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle \frac {\tan (e+f x) \left (-\frac {3}{4} \int \cot ^2(e+f x) \csc (e+f x)dx-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 f}\right )}{a^2 c^2 \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\tan (e+f x) \left (-\frac {3}{4} \int \sec \left (e+f x-\frac {\pi }{2}\right ) \tan \left (e+f x-\frac {\pi }{2}\right )^2dx-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 f}\right )}{a^2 c^2 \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle \frac {\tan (e+f x) \left (-\frac {3}{4} \left (-\frac {1}{2} \int \csc (e+f x)dx-\frac {\cot (e+f x) \csc (e+f x)}{2 f}\right )-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 f}\right )}{a^2 c^2 \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\tan (e+f x) \left (-\frac {3}{4} \left (-\frac {1}{2} \int \csc (e+f x)dx-\frac {\cot (e+f x) \csc (e+f x)}{2 f}\right )-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 f}\right )}{a^2 c^2 \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\tan (e+f x) \left (-\frac {3}{4} \left (\frac {\text {arctanh}(\cos (e+f x))}{2 f}-\frac {\cot (e+f x) \csc (e+f x)}{2 f}\right )-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 f}\right )}{a^2 c^2 \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}\) |
Input:
Int[Sec[e + f*x]/((a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(5/2)),x ]
Output:
((-1/4*(Cot[e + f*x]^3*Csc[e + f*x])/f - (3*(ArcTanh[Cos[e + f*x]]/(2*f) - (Cot[e + f*x]*Csc[e + f*x])/(2*f)))/4)*Tan[e + f*x])/(a^2*c^2*Sqrt[a + a* Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[b*(a*Sec[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Simp[b^2*((n - 1)/(m + n - 1)) Int[(a*Sec[e + f*x])^m*( b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] & & NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(cs c[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(m_), x_Symbol] :> Simp[((-a)*c)^(m + 1 /2)*(Cot[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]])) In t[Csc[e + f*x]*Cot[e + f*x]^(2*m), 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]
Leaf count of result is larger than twice the leaf count of optimal. \(309\) vs. \(2(142)=284\).
Time = 2.37 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.94
| method | result | size |
| default | \(-\frac {\left (192 \sin \left (\frac {f x}{2}+\frac {e}{2}\right )^{4} \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4} \ln \left (-\cot \left (\frac {f x}{2}+\frac {e}{2}\right )+\csc \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )-192 \sin \left (\frac {f x}{2}+\frac {e}{2}\right )^{4} \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4} \ln \left (-\cot \left (\frac {f x}{2}+\frac {e}{2}\right )+\csc \left (\frac {f x}{2}+\frac {e}{2}\right )\right )+192 \sin \left (\frac {f x}{2}+\frac {e}{2}\right )^{4} \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4} \ln \left (-\cot \left (\frac {f x}{2}+\frac {e}{2}\right )+\csc \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )-35 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}+230 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-275 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+96 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-8\right ) \sec \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} \csc \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{512 f \left (2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right ) \sqrt {\frac {a \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1}}\, \sqrt {-\frac {c \sin \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1}}\, a^{2} c^{2}}\) | \(310\) |
| risch | \(\frac {i \left (5 \,{\mathrm e}^{7 i \left (f x +e \right )}+3 \,{\mathrm e}^{5 i \left (f x +e \right )}+3 \,{\mathrm e}^{3 i \left (f x +e \right )}+5 \,{\mathrm e}^{i \left (f x +e \right )}\right )}{4 a^{2} c^{2} \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{3} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{3} \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}-\frac {3 i \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{8 a^{2} c^{2} \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}+\frac {3 i \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{8 a^{2} c^{2} \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\) | \(396\) |
Input:
int(sec(f*x+e)/(a+a*sec(f*x+e))^(5/2)/(c-c*sec(f*x+e))^(5/2),x,method=_RET URNVERBOSE)
Output:
-1/512/f*(192*sin(1/2*f*x+1/2*e)^4*cos(1/2*f*x+1/2*e)^4*ln(-cot(1/2*f*x+1/ 2*e)+csc(1/2*f*x+1/2*e)-1)-192*sin(1/2*f*x+1/2*e)^4*cos(1/2*f*x+1/2*e)^4*l n(-cot(1/2*f*x+1/2*e)+csc(1/2*f*x+1/2*e))+192*sin(1/2*f*x+1/2*e)^4*cos(1/2 *f*x+1/2*e)^4*ln(-cot(1/2*f*x+1/2*e)+csc(1/2*f*x+1/2*e)+1)-35*cos(1/2*f*x+ 1/2*e)^8+230*cos(1/2*f*x+1/2*e)^6-275*cos(1/2*f*x+1/2*e)^4+96*cos(1/2*f*x+ 1/2*e)^2-8)/(2*cos(1/2*f*x+1/2*e)^2-1)/(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)/a^2/c^2*sec(1/2*f*x+1/2*e)^3*csc(1/2*f*x+1/2*e)^3
Time = 0.27 (sec) , antiderivative size = 490, normalized size of antiderivative = 3.06 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{5/2}} \, dx=\left [-\frac {3 \, {\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sqrt {-a c} \log \left (-\frac {4 \, {\left (2 \, \sqrt {-a c} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )^{2} + {\left (a c \cos \left (f x + e\right )^{2} + a c\right )} \sin \left (f x + e\right )\right )}}{{\left (\cos \left (f x + e\right )^{2} - 1\right )} \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) - 2 \, {\left (5 \, \cos \left (f x + e\right )^{4} - 3 \, \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{16 \, {\left (a^{3} c^{3} f \cos \left (f x + e\right )^{4} - 2 \, a^{3} c^{3} f \cos \left (f x + e\right )^{2} + a^{3} c^{3} f\right )} \sin \left (f x + e\right )}, \frac {3 \, {\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )^{2}}{a c \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) + {\left (5 \, \cos \left (f x + e\right )^{4} - 3 \, \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{8 \, {\left (a^{3} c^{3} f \cos \left (f x + e\right )^{4} - 2 \, a^{3} c^{3} f \cos \left (f x + e\right )^{2} + a^{3} c^{3} f\right )} \sin \left (f x + e\right )}\right ] \] Input:
integrate(sec(f*x+e)/(a+a*sec(f*x+e))^(5/2)/(c-c*sec(f*x+e))^(5/2),x, algo rithm="fricas")
Output:
[-1/16*(3*(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)*sqrt(-a*c)*log(-4*(2*sqr t(-a*c)*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)^2 + (a*c*cos(f*x + e)^2 + a*c)*sin(f*x + e))/(( cos(f*x + e)^2 - 1)*sin(f*x + e)))*sin(f*x + e) - 2*(5*cos(f*x + e)^4 - 3* cos(f*x + e)^2)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt((c*cos(f*x + e) - c)/cos(f*x + e)))/((a^3*c^3*f*cos(f*x + e)^4 - 2*a^3*c^3*f*cos(f*x + e)^2 + a^3*c^3*f)*sin(f*x + e)), 1/8*(3*(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)*sqrt(a*c)*arctan(sqrt(a*c)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*s qrt((c*cos(f*x + e) - c)/cos(f*x + e))*cos(f*x + e)^2/(a*c*sin(f*x + e)))* sin(f*x + e) + (5*cos(f*x + e)^4 - 3*cos(f*x + e)^2)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt((c*cos(f*x + e) - c)/cos(f*x + e)))/((a^3*c^3*f*co s(f*x + e)^4 - 2*a^3*c^3*f*cos(f*x + e)^2 + a^3*c^3*f)*sin(f*x + e))]
Timed out. \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(sec(f*x+e)/(a+a*sec(f*x+e))**(5/2)/(c-c*sec(f*x+e))**(5/2),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 1659 vs. \(2 (142) = 284\).
Time = 0.35 (sec) , antiderivative size = 1659, normalized size of antiderivative = 10.37 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{5/2}} \, dx=\text {Too large to display} \] Input:
integrate(sec(f*x+e)/(a+a*sec(f*x+e))^(5/2)/(c-c*sec(f*x+e))^(5/2),x, algo rithm="maxima")
Output:
1/8*(3*(2*(4*cos(6*f*x + 6*e) - 6*cos(4*f*x + 4*e) + 4*cos(2*f*x + 2*e) - 1)*cos(8*f*x + 8*e) - cos(8*f*x + 8*e)^2 + 8*(6*cos(4*f*x + 4*e) - 4*cos(2 *f*x + 2*e) + 1)*cos(6*f*x + 6*e) - 16*cos(6*f*x + 6*e)^2 + 12*(4*cos(2*f* x + 2*e) - 1)*cos(4*f*x + 4*e) - 36*cos(4*f*x + 4*e)^2 - 16*cos(2*f*x + 2* e)^2 + 4*(2*sin(6*f*x + 6*e) - 3*sin(4*f*x + 4*e) + 2*sin(2*f*x + 2*e))*si n(8*f*x + 8*e) - sin(8*f*x + 8*e)^2 + 16*(3*sin(4*f*x + 4*e) - 2*sin(2*f*x + 2*e))*sin(6*f*x + 6*e) - 16*sin(6*f*x + 6*e)^2 - 36*sin(4*f*x + 4*e)^2 + 48*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) - 16*sin(2*f*x + 2*e)^2 + 8*cos(2*f *x + 2*e) - 1)*arctan2(sin(f*x + e), cos(f*x + e) + 1) - 3*(2*(4*cos(6*f*x + 6*e) - 6*cos(4*f*x + 4*e) + 4*cos(2*f*x + 2*e) - 1)*cos(8*f*x + 8*e) - cos(8*f*x + 8*e)^2 + 8*(6*cos(4*f*x + 4*e) - 4*cos(2*f*x + 2*e) + 1)*cos(6 *f*x + 6*e) - 16*cos(6*f*x + 6*e)^2 + 12*(4*cos(2*f*x + 2*e) - 1)*cos(4*f* x + 4*e) - 36*cos(4*f*x + 4*e)^2 - 16*cos(2*f*x + 2*e)^2 + 4*(2*sin(6*f*x + 6*e) - 3*sin(4*f*x + 4*e) + 2*sin(2*f*x + 2*e))*sin(8*f*x + 8*e) - sin(8 *f*x + 8*e)^2 + 16*(3*sin(4*f*x + 4*e) - 2*sin(2*f*x + 2*e))*sin(6*f*x + 6 *e) - 16*sin(6*f*x + 6*e)^2 - 36*sin(4*f*x + 4*e)^2 + 48*sin(4*f*x + 4*e)* sin(2*f*x + 2*e) - 16*sin(2*f*x + 2*e)^2 + 8*cos(2*f*x + 2*e) - 1)*arctan2 (sin(f*x + e), cos(f*x + e) - 1) - 2*(5*sin(7*f*x + 7*e) + 3*sin(5*f*x + 5 *e) + 3*sin(3*f*x + 3*e) + 5*sin(f*x + e))*cos(8*f*x + 8*e) - 20*(2*sin(6* f*x + 6*e) - 3*sin(4*f*x + 4*e) + 2*sin(2*f*x + 2*e))*cos(7*f*x + 7*e) ...
Time = 0.87 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.14 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{5/2}} \, dx=-\frac {\frac {{\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{2} c^{2} - 6 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )} c^{3}}{c^{4}} - \frac {18 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{2} + 28 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )} c + 11 \, c^{2}}{c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4}} + 12 \, \log \left ({\left | c \right |} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}\right ) - 12 \, \log \left ({\left | c \right |}\right ) + 11}{64 \, \sqrt {-a c} a^{2} c f {\left | c \right |} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} \] Input:
integrate(sec(f*x+e)/(a+a*sec(f*x+e))^(5/2)/(c-c*sec(f*x+e))^(5/2),x, algo rithm="giac")
Output:
-1/64*(((c*tan(1/2*f*x + 1/2*e)^2 - c)^2*c^2 - 6*(c*tan(1/2*f*x + 1/2*e)^2 - c)*c^3)/c^4 - (18*(c*tan(1/2*f*x + 1/2*e)^2 - c)^2 + 28*(c*tan(1/2*f*x + 1/2*e)^2 - c)*c + 11*c^2)/(c^2*tan(1/2*f*x + 1/2*e)^4) + 12*log(abs(c)*t an(1/2*f*x + 1/2*e)^2) - 12*log(abs(c)) + 11)/(sqrt(-a*c)*a^2*c*f*abs(c)*s gn(tan(1/2*f*x + 1/2*e)^3 + tan(1/2*f*x + 1/2*e)))
Timed out. \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{5/2}} \, dx=\int \frac {1}{\cos \left (e+f\,x\right )\,{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{5/2}\,{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^{5/2}} \,d x \] Input:
int(1/(cos(e + f*x)*(a + a/cos(e + f*x))^(5/2)*(c - c/cos(e + f*x))^(5/2)) ,x)
Output:
int(1/(cos(e + f*x)*(a + a/cos(e + f*x))^(5/2)*(c - c/cos(e + f*x))^(5/2)) , x)
\[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{5/2}} \, dx=-\frac {\sqrt {c}\, \sqrt {a}\, \left (\int \frac {\sqrt {\sec \left (f x +e \right )+1}\, \sqrt {-\sec \left (f x +e \right )+1}\, \sec \left (f x +e \right )}{\sec \left (f x +e \right )^{6}-3 \sec \left (f x +e \right )^{4}+3 \sec \left (f x +e \right )^{2}-1}d x \right )}{a^{3} c^{3}} \] Input:
int(sec(f*x+e)/(a+a*sec(f*x+e))^(5/2)/(c-c*sec(f*x+e))^(5/2),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)**6 - 3*sec(e + f*x)**4 + 3*sec(e + f*x)**2 - 1 ),x))/(a**3*c**3)