Integrand size = 30, antiderivative size = 190 \[ \int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{5/2} \, dx=\frac {a^2 c^3 \log (\cos (e+f x)) \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}-\frac {a^2 c^2 \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)}}-\frac {a^2 c (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{2 f \sqrt {a+a \sec (e+f x)}}+\frac {a^2 (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{3 f \sqrt {a+a \sec (e+f x)}} \]
-1/2*a^2*c*(c-c*sec(f*x+e))^(3/2)*tan(f*x+e)/f/(a+a*sec(f*x+e))^(1/2)+1/3* a^2*(c-c*sec(f*x+e))^(5/2)*tan(f*x+e)/f/(a+a*sec(f*x+e))^(1/2)+a^2*c^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)-a^2 *c^2*(c-c*sec(f*x+e))^(1/2)*tan(f*x+e)/f/(a+a*sec(f*x+e))^(1/2)
Time = 0.83 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.46 \[ \int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{5/2} \, dx=\frac {a^2 c^3 \left (2+6 \log (\cos (e+f x))+6 \sec (e+f x)+3 \sec ^2(e+f x)-2 \sec ^3(e+f x)\right ) \tan (e+f x)}{6 f \sqrt {a (1+\sec (e+f x))} \sqrt {c-c \sec (e+f x)}} \]
(a^2*c^3*(2 + 6*Log[Cos[e + f*x]] + 6*Sec[e + f*x] + 3*Sec[e + f*x]^2 - 2* Sec[e + f*x]^3)*Tan[e + f*x])/(6*f*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*S ec[e + f*x]])
Time = 1.00 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {3042, 4397, 3042, 4394, 3042, 4394, 3042, 4393, 25, 3042, 3956}
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.
| \(\displaystyle \int (a \sec (e+f x)+a)^{3/2} (c-c \sec (e+f x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^{3/2} \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{5/2}dx\) |
| \(\Big \downarrow \) 4397 |
| \(\displaystyle a \int \sqrt {\sec (e+f x) a+a} (c-c \sec (e+f x))^{5/2}dx+\frac {a^2 \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{3 f \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int \sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a} \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{5/2}dx+\frac {a^2 \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{3 f \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 4394 |
| \(\displaystyle a \left (c \int \sqrt {\sec (e+f x) a+a} (c-c \sec (e+f x))^{3/2}dx-\frac {a c \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{2 f \sqrt {a \sec (e+f x)+a}}\right )+\frac {a^2 \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{3 f \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (c \int \sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a} \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}dx-\frac {a c \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{2 f \sqrt {a \sec (e+f x)+a}}\right )+\frac {a^2 \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{3 f \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 4394 |
| \(\displaystyle a \left (c \left (c \int \sqrt {\sec (e+f x) a+a} \sqrt {c-c \sec (e+f x)}dx-\frac {a c \tan (e+f x) \sqrt {c-c \sec (e+f x)}}{f \sqrt {a \sec (e+f x)+a}}\right )-\frac {a c \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{2 f \sqrt {a \sec (e+f x)+a}}\right )+\frac {a^2 \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{3 f \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (c \left (c \int \sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a} \sqrt {c-c \csc \left (e+f x+\frac {\pi }{2}\right )}dx-\frac {a c \tan (e+f x) \sqrt {c-c \sec (e+f x)}}{f \sqrt {a \sec (e+f x)+a}}\right )-\frac {a c \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{2 f \sqrt {a \sec (e+f x)+a}}\right )+\frac {a^2 \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{3 f \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 4393 |
| \(\displaystyle a \left (c \left (\frac {a c^2 \tan (e+f x) \int -\tan (e+f x)dx}{\sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}-\frac {a c \tan (e+f x) \sqrt {c-c \sec (e+f x)}}{f \sqrt {a \sec (e+f x)+a}}\right )-\frac {a c \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{2 f \sqrt {a \sec (e+f x)+a}}\right )+\frac {a^2 \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{3 f \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle a \left (c \left (-\frac {a c^2 \tan (e+f x) \int \tan (e+f x)dx}{\sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}-\frac {a c \tan (e+f x) \sqrt {c-c \sec (e+f x)}}{f \sqrt {a \sec (e+f x)+a}}\right )-\frac {a c \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{2 f \sqrt {a \sec (e+f x)+a}}\right )+\frac {a^2 \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{3 f \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (c \left (-\frac {a c^2 \tan (e+f x) \int \tan (e+f x)dx}{\sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}-\frac {a c \tan (e+f x) \sqrt {c-c \sec (e+f x)}}{f \sqrt {a \sec (e+f x)+a}}\right )-\frac {a c \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{2 f \sqrt {a \sec (e+f x)+a}}\right )+\frac {a^2 \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{3 f \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {a^2 \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{3 f \sqrt {a \sec (e+f x)+a}}+a \left (c \left (\frac {a c^2 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}-\frac {a c \tan (e+f x) \sqrt {c-c \sec (e+f x)}}{f \sqrt {a \sec (e+f x)+a}}\right )-\frac {a c \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{2 f \sqrt {a \sec (e+f x)+a}}\right )\) |
(a^2*(c - c*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(3*f*Sqrt[a + a*Sec[e + f*x] ]) + a*(-1/2*(a*c*(c - c*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(f*Sqrt[a + a*S ec[e + f*x]]) + c*((a*c^2*Log[Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Se c[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (a*c*Sqrt[c - c*Sec[e + f*x]]*Tan[ e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]])))
3.1.94.3.1 Defintions of rubi rules used
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(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]])) Int[Cot[e + f*x]^(2*m), x] , x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b ^2, 0] && IntegerQ[m + 1/2]
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*(csc[(e_.) + (f_.)*(x_)]*(d_ .) + (c_))^(n_.), x_Symbol] :> Simp[2*a*c*Cot[e + f*x]*((c + d*Csc[e + f*x] )^(n - 1)/(f*(2*n - 1)*Sqrt[a + b*Csc[e + f*x]])), x] + Simp[c Int[Sqrt[a + b*Csc[e + f*x]]*(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] && GtQ[n, 1/2]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(3/2)*(csc[(e_.) + (f_.)*(x_)]*( d_.) + (c_))^(n_.), x_Symbol] :> Simp[-2*a^2*Cot[e + f*x]*((c + d*Csc[e + f *x])^n/(f*(2*n + 1)*Sqrt[a + b*Csc[e + f*x]])), x] + Simp[a Int[Sqrt[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && !LeQ[n, -2^(-1)]
Time = 2.10 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.87
| method | result | size |
| default | \(-\frac {a \sqrt {-c \left (\sec \left (f x +e \right )-1\right )}\, \left (\sec \left (f x +e \right )-1\right )^{2} c^{2} \sqrt {a \left (\sec \left (f x +e \right )+1\right )}\, \left (6 \cos \left (f x +e \right )^{3} \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right )+6 \cos \left (f x +e \right )^{3} \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )-1\right )-6 \cos \left (f x +e \right )^{3} \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )+\cos \left (f x +e \right )^{3}+6 \cos \left (f x +e \right )^{2}+3 \cos \left (f x +e \right )-2\right ) \csc \left (f x +e \right )}{6 f \left (\cos \left (f x +e \right )-1\right )^{2}}\) | \(166\) |
| risch | \(-\frac {a \,c^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left (6 i {\mathrm e}^{5 i \left (f x +e \right )}+3 \,{\mathrm e}^{6 i \left (f x +e \right )} f x +6 i {\mathrm e}^{i \left (f x +e \right )}+6 \,{\mathrm e}^{6 i \left (f x +e \right )} e +9 \,{\mathrm e}^{4 i \left (f x +e \right )} f x +6 i {\mathrm e}^{4 i \left (f x +e \right )}+18 \,{\mathrm e}^{4 i \left (f x +e \right )} e +9 \,{\mathrm e}^{2 i \left (f x +e \right )} f x +9 i {\mathrm e}^{4 i \left (f x +e \right )} \ln \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )+6 i {\mathrm e}^{2 i \left (f x +e \right )}+9 i {\mathrm e}^{2 i \left (f x +e \right )} \ln \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )+3 i \ln \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right ) {\mathrm e}^{6 i \left (f x +e \right )}+4 i {\mathrm e}^{3 i \left (f x +e \right )}+3 i \ln \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )+18 \,{\mathrm e}^{2 i \left (f x +e \right )} e +3 f x +6 e \right )}{3 \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) f}\) | \(339\) |
-1/6/f*a*(-c*(sec(f*x+e)-1))^(1/2)*(sec(f*x+e)-1)^2*c^2*(a*(sec(f*x+e)+1)) ^(1/2)*(6*cos(f*x+e)^3*ln(-cot(f*x+e)+csc(f*x+e)+1)+6*cos(f*x+e)^3*ln(-cot (f*x+e)+csc(f*x+e)-1)-6*cos(f*x+e)^3*ln(2/(cos(f*x+e)+1))+cos(f*x+e)^3+6*c os(f*x+e)^2+3*cos(f*x+e)-2)/(cos(f*x+e)-1)^2*csc(f*x+e)
Time = 0.33 (sec) , antiderivative size = 467, normalized size of antiderivative = 2.46 \[ \int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{5/2} \, dx=\left [-\frac {{\left (7 \, a c^{2} \cos \left (f x + e\right )^{2} + a c^{2} \cos \left (f x + e\right ) - 2 \, a c^{2}\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) - 3 \, {\left (a c^{2} \cos \left (f x + e\right )^{3} + a c^{2} \cos \left (f x + e\right )^{2}\right )} \sqrt {-a c} \log \left (\frac {a c \cos \left (f x + e\right )^{4} - {\left (\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )\right )} \sqrt {-a c} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) + a c}{2 \, \cos \left (f x + e\right )^{2}}\right )}{6 \, {\left (f \cos \left (f x + e\right )^{3} + f \cos \left (f x + e\right )^{2}\right )}}, -\frac {{\left (7 \, a c^{2} \cos \left (f x + e\right )^{2} + a c^{2} \cos \left (f x + e\right ) - 2 \, a c^{2}\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) - 6 \, {\left (a c^{2} \cos \left (f x + e\right )^{3} + a c^{2} \cos \left (f x + e\right )^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right )}{a c \cos \left (f x + e\right )^{2} + a c}\right )}{6 \, {\left (f \cos \left (f x + e\right )^{3} + f \cos \left (f x + e\right )^{2}\right )}}\right ] \]
[-1/6*((7*a*c^2*cos(f*x + e)^2 + a*c^2*cos(f*x + e) - 2*a*c^2)*sqrt((a*cos (f*x + e) + a)/cos(f*x + e))*sqrt((c*cos(f*x + e) - c)/cos(f*x + e))*sin(f *x + e) - 3*(a*c^2*cos(f*x + e)^3 + a*c^2*cos(f*x + e)^2)*sqrt(-a*c)*log(1 /2*(a*c*cos(f*x + e)^4 - (cos(f*x + e)^3 + cos(f*x + e))*sqrt(-a*c)*sqrt(( a*cos(f*x + e) + a)/cos(f*x + e))*sqrt((c*cos(f*x + e) - c)/cos(f*x + e))* sin(f*x + e) + a*c)/cos(f*x + e)^2))/(f*cos(f*x + e)^3 + f*cos(f*x + e)^2) , -1/6*((7*a*c^2*cos(f*x + e)^2 + a*c^2*cos(f*x + e) - 2*a*c^2)*sqrt((a*co s(f*x + e) + a)/cos(f*x + e))*sqrt((c*cos(f*x + e) - c)/cos(f*x + e))*sin( f*x + e) - 6*(a*c^2*cos(f*x + e)^3 + a*c^2*cos(f*x + e)^2)*sqrt(a*c)*arcta n(sqrt(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)*sin(f*x + e)/(a*c*cos(f*x + e)^2 + a*c)))/( f*cos(f*x + e)^3 + f*cos(f*x + e)^2)]
Timed out. \[ \int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{5/2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 1356 vs. \(2 (170) = 340\).
Time = 0.48 (sec) , antiderivative size = 1356, normalized size of antiderivative = 7.14 \[ \int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{5/2} \, dx=\text {Too large to display} \]
-1/3*(3*(f*x + e)*a*c^2*cos(6*f*x + 6*e)^2 + 27*(f*x + e)*a*c^2*cos(4*f*x + 4*e)^2 + 27*(f*x + e)*a*c^2*cos(2*f*x + 2*e)^2 + 3*(f*x + e)*a*c^2*sin(6 *f*x + 6*e)^2 + 27*(f*x + e)*a*c^2*sin(4*f*x + 4*e)^2 + 27*(f*x + e)*a*c^2 *sin(2*f*x + 2*e)^2 + 18*(f*x + e)*a*c^2*cos(2*f*x + 2*e) + 3*(f*x + e)*a* c^2 - 6*a*c^2*sin(2*f*x + 2*e) - 3*(a*c^2*cos(6*f*x + 6*e)^2 + 9*a*c^2*cos (4*f*x + 4*e)^2 + 9*a*c^2*cos(2*f*x + 2*e)^2 + a*c^2*sin(6*f*x + 6*e)^2 + 9*a*c^2*sin(4*f*x + 4*e)^2 + 18*a*c^2*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + 9*a*c^2*sin(2*f*x + 2*e)^2 + 6*a*c^2*cos(2*f*x + 2*e) + a*c^2 + 2*(3*a*c^2 *cos(4*f*x + 4*e) + 3*a*c^2*cos(2*f*x + 2*e) + a*c^2)*cos(6*f*x + 6*e) + 6 *(3*a*c^2*cos(2*f*x + 2*e) + a*c^2)*cos(4*f*x + 4*e) + 6*(a*c^2*sin(4*f*x + 4*e) + a*c^2*sin(2*f*x + 2*e))*sin(6*f*x + 6*e))*arctan2(sin(2*f*x + 2*e ), cos(2*f*x + 2*e) + 1) + 6*(3*(f*x + e)*a*c^2*cos(4*f*x + 4*e) + 3*(f*x + e)*a*c^2*cos(2*f*x + 2*e) + (f*x + e)*a*c^2 - a*c^2*sin(4*f*x + 4*e) - a *c^2*sin(2*f*x + 2*e))*cos(6*f*x + 6*e) + 18*(3*(f*x + e)*a*c^2*cos(2*f*x + 2*e) + (f*x + e)*a*c^2)*cos(4*f*x + 4*e) + 6*(a*c^2*sin(6*f*x + 6*e) + 3 *a*c^2*sin(4*f*x + 4*e) + 3*a*c^2*sin(2*f*x + 2*e))*cos(5/2*arctan2(sin(2* f*x + 2*e), cos(2*f*x + 2*e))) + 4*(a*c^2*sin(6*f*x + 6*e) + 3*a*c^2*sin(4 *f*x + 4*e) + 3*a*c^2*sin(2*f*x + 2*e))*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 6*(a*c^2*sin(6*f*x + 6*e) + 3*a*c^2*sin(4*f*x + 4*e) + 3*a*c^2*sin(2*f*x + 2*e))*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x...
\[ \int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{5/2} \, dx=\int { {\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {3}{2}} {\left (-c \sec \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{5/2} \, dx=\int {\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{3/2}\,{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^{5/2} \,d x \]