\(\int (a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{3/2} \, dx\) [102]

Optimal result

Integrand size = 30, antiderivative size = 190 \[ \int (a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{3/2} \, dx=\frac {a^3 c^2 \log (\cos (e+f x)) \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}-\frac {a^2 c^2 \sqrt {a+a \sec (e+f x)} \tan (e+f x)}{f \sqrt {c-c \sec (e+f x)}}-\frac {a c^2 (a+a \sec (e+f x))^{3/2} \tan (e+f x)}{2 f \sqrt {c-c \sec (e+f x)}}+\frac {c^2 (a+a \sec (e+f x))^{5/2} \tan (e+f x)}{3 f \sqrt {c-c \sec (e+f x)}} \]

[Out]

-1/2*a*c^2*(a+a*sec(f*x+e))^(3/2)*tan(f*x+e)/f/(c-c*sec(f*x+e))^(1/2)+1/3*c^2*(a+a*sec(f*x+e))^(5/2)*tan(f*x+e
)/f/(c-c*sec(f*x+e))^(1/2)+a^3*c^2*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*(a+a*sec(f*x+e))^(1/2)*tan(f*x+e)/f/(c-c*sec(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3994, 3991, 3990, 3556} \[ \int (a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{3/2} \, dx=\frac {a^3 c^2 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}-\frac {a^2 c^2 \tan (e+f x) \sqrt {a \sec (e+f x)+a}}{f \sqrt {c-c \sec (e+f x)}}-\frac {a c^2 \tan (e+f x) (a \sec (e+f x)+a)^{3/2}}{2 f \sqrt {c-c \sec (e+f x)}}+\frac {c^2 \tan (e+f x) (a \sec (e+f x)+a)^{5/2}}{3 f \sqrt {c-c \sec (e+f x)}} \]

[In]

Int[(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(3/2),x]

[Out]

(a^3*c^2*Log[Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (a^2*c^2*Sqrt
[a + a*Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[c - c*Sec[e + f*x]]) - (a*c^2*(a + a*Sec[e + f*x])^(3/2)*Tan[e + f*
x])/(2*f*Sqrt[c - c*Sec[e + f*x]]) + (c^2*(a + a*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(3*f*Sqrt[c - c*Sec[e + f*x
]])

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3990

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(m_), x_Symbol] :> Dist
[((-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]

Rule 3991

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] + Dist[c, Int[Sq
rt[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]

Rule 3994

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(3/2)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.), x_Symbol] :> Si
mp[-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] + Dist[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)]

Rubi steps \begin{align*} \text {integral}& = \frac {c^2 (a+a \sec (e+f x))^{5/2} \tan (e+f x)}{3 f \sqrt {c-c \sec (e+f x)}}+c \int (a+a \sec (e+f x))^{5/2} \sqrt {c-c \sec (e+f x)} \, dx \\ & = -\frac {a c^2 (a+a \sec (e+f x))^{3/2} \tan (e+f x)}{2 f \sqrt {c-c \sec (e+f x)}}+\frac {c^2 (a+a \sec (e+f x))^{5/2} \tan (e+f x)}{3 f \sqrt {c-c \sec (e+f x)}}+(a c) \int (a+a \sec (e+f x))^{3/2} \sqrt {c-c \sec (e+f x)} \, dx \\ & = -\frac {a^2 c^2 \sqrt {a+a \sec (e+f x)} \tan (e+f x)}{f \sqrt {c-c \sec (e+f x)}}-\frac {a c^2 (a+a \sec (e+f x))^{3/2} \tan (e+f x)}{2 f \sqrt {c-c \sec (e+f x)}}+\frac {c^2 (a+a \sec (e+f x))^{5/2} \tan (e+f x)}{3 f \sqrt {c-c \sec (e+f x)}}+\left (a^2 c\right ) \int \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)} \, dx \\ & = -\frac {a^2 c^2 \sqrt {a+a \sec (e+f x)} \tan (e+f x)}{f \sqrt {c-c \sec (e+f x)}}-\frac {a c^2 (a+a \sec (e+f x))^{3/2} \tan (e+f x)}{2 f \sqrt {c-c \sec (e+f x)}}+\frac {c^2 (a+a \sec (e+f x))^{5/2} \tan (e+f x)}{3 f \sqrt {c-c \sec (e+f x)}}-\frac {\left (a^3 c^2 \tan (e+f x)\right ) \int \tan (e+f x) \, dx}{\sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \\ & = \frac {a^3 c^2 \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 {a+a \sec (e+f x)} \tan (e+f x)}{f \sqrt {c-c \sec (e+f x)}}-\frac {a c^2 (a+a \sec (e+f x))^{3/2} \tan (e+f x)}{2 f \sqrt {c-c \sec (e+f x)}}+\frac {c^2 (a+a \sec (e+f x))^{5/2} \tan (e+f x)}{3 f \sqrt {c-c \sec (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.46 \[ \int (a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{3/2} \, dx=\frac {a^3 c^2 \left (2+6 \log (\cos (e+f x))-6 \sec (e+f x)+3 \sec ^2(e+f x)+2 \sec ^3(e+f x)\right ) \tan (e+f x)}{6 f \sqrt {a (1+\sec (e+f x))} \sqrt {c-c \sec (e+f x)}} \]

[In]

Integrate[(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(3/2),x]

[Out]

(a^3*c^2*(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*S
qrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])

Maple [A] (verified)

Time = 2.31 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.91

method	result	size
default	\(-\frac {a^{2} \left (6 \cos \left (f x +e \right )^{3} \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )-6 \cos \left (f x +e \right )^{3} \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )-1\right )-6 \cos \left (f x +e \right )^{3} \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right )+7 \cos \left (f x +e \right )^{3}+6 \cos \left (f x +e \right )^{2}-3 \cos \left (f x +e \right )-2\right ) \sqrt {-c \left (\sec \left (f x +e \right )-1\right )}\, \left (\sec \left (f x +e \right )-1\right ) c \sqrt {a \left (\sec \left (f x +e \right )+1\right )}\, \sec \left (f x +e \right ) \csc \left (f x +e \right )}{6 f \left (\cos \left (f x +e \right )-1\right )}\)	\(172\)
risch	\(-\frac {a^{2} c \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left (-6 i {\mathrm e}^{5 i \left (f x +e \right )}+3 \,{\mathrm e}^{6 i \left (f x +e \right )} f x +6 \,{\mathrm e}^{6 i \left (f x +e \right )} e -6 i {\mathrm e}^{i \left (f x +e \right )}+9 \,{\mathrm e}^{4 i \left (f x +e \right )} f x +18 \,{\mathrm e}^{4 i \left (f x +e \right )} e +6 i {\mathrm e}^{4 i \left (f x +e \right )}+9 \,{\mathrm e}^{2 i \left (f x +e \right )} f x +6 i {\mathrm e}^{2 i \left (f x +e \right )}+3 i \ln \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right ) {\mathrm e}^{6 i \left (f x +e \right )}+9 i {\mathrm e}^{2 i \left (f x +e \right )} \ln \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )+18 \,{\mathrm e}^{2 i \left (f x +e \right )} e -4 i {\mathrm e}^{3 i \left (f x +e \right )}+9 i {\mathrm e}^{4 i \left (f x +e \right )} \ln \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )+3 i \ln \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )+3 f x +6 e \right )}{3 \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) f}\)	\(339\)

[In]

int((a+a*sec(f*x+e))^(5/2)*(c-c*sec(f*x+e))^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/6/f*a^2*(6*cos(f*x+e)^3*ln(2/(cos(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(
-cot(f*x+e)+csc(f*x+e)+1)+7*cos(f*x+e)^3+6*cos(f*x+e)^2-3*cos(f*x+e)-2)*(-c*(sec(f*x+e)-1))^(1/2)*(sec(f*x+e)-
1)*c*(a*(sec(f*x+e)+1))^(1/2)/(cos(f*x+e)-1)*sec(f*x+e)*csc(f*x+e)

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 467, normalized size of antiderivative = 2.46 \[ \int (a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{3/2} \, dx=\left [\frac {{\left (a^{2} c \cos \left (f x + e\right )^{2} - 5 \, a^{2} c \cos \left (f x + e\right ) - 2 \, a^{2} c\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) + 3 \, {\left (a^{2} c \cos \left (f x + e\right )^{3} + a^{2} c \cos \left (f x + e\right )^{2}\right )} \sqrt {-a c} \log \left (\frac {a c \cos \left (f x + e\right )^{4} - {\left (\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )\right )} \sqrt {-a c} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) + a c}{2 \, \cos \left (f x + e\right )^{2}}\right )}{6 \, {\left (f \cos \left (f x + e\right )^{3} + f \cos \left (f x + e\right )^{2}\right )}}, \frac {{\left (a^{2} c \cos \left (f x + e\right )^{2} - 5 \, a^{2} c \cos \left (f x + e\right ) - 2 \, a^{2} c\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) + 6 \, {\left (a^{2} c \cos \left (f x + e\right )^{3} + a^{2} c \cos \left (f x + e\right )^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right )}{a c \cos \left (f x + e\right )^{2} + a c}\right )}{6 \, {\left (f \cos \left (f x + e\right )^{3} + f \cos \left (f x + e\right )^{2}\right )}}\right ] \]

[In]

integrate((a+a*sec(f*x+e))^(5/2)*(c-c*sec(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

[1/6*((a^2*c*cos(f*x + e)^2 - 5*a^2*c*cos(f*x + e) - 2*a^2*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) + 3*(a^2*c*cos(f*x + e)^3 + a^2*c*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))*s
qrt((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*((a^2*c*cos(f*x + e)^2 - 5*a^2*c*cos(f*x + e) - 2*a^2*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) + 6*(a^2*c*cos(f*x + e)^3 + a^2*c*cos(f*x + e)^2)*sqrt(a*c)*a
rctan(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)*s
in(f*x + e)/(a*c*cos(f*x + e)^2 + a*c)))/(f*cos(f*x + e)^3 + f*cos(f*x + e)^2)]

Sympy [F(-1)]

Timed out. \[ \int (a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{3/2} \, dx=\text {Timed out} \]

[In]

integrate((a+a*sec(f*x+e))**(5/2)*(c-c*sec(f*x+e))**(3/2),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1356 vs. \(2 (170) = 340\).

Time = 0.43 (sec) , antiderivative size = 1356, normalized size of antiderivative = 7.14 \[ \int (a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{3/2} \, dx=\text {Too large to display} \]

[In]

integrate((a+a*sec(f*x+e))^(5/2)*(c-c*sec(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

-1/3*(3*(f*x + e)*a^2*c*cos(6*f*x + 6*e)^2 + 27*(f*x + e)*a^2*c*cos(4*f*x + 4*e)^2 + 27*(f*x + e)*a^2*c*cos(2*
f*x + 2*e)^2 + 3*(f*x + e)*a^2*c*sin(6*f*x + 6*e)^2 + 27*(f*x + e)*a^2*c*sin(4*f*x + 4*e)^2 + 27*(f*x + e)*a^2
*c*sin(2*f*x + 2*e)^2 + 18*(f*x + e)*a^2*c*cos(2*f*x + 2*e) + 3*(f*x + e)*a^2*c - 6*a^2*c*sin(2*f*x + 2*e) - 3
*(a^2*c*cos(6*f*x + 6*e)^2 + 9*a^2*c*cos(4*f*x + 4*e)^2 + 9*a^2*c*cos(2*f*x + 2*e)^2 + a^2*c*sin(6*f*x + 6*e)^
2 + 9*a^2*c*sin(4*f*x + 4*e)^2 + 18*a^2*c*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + 9*a^2*c*sin(2*f*x + 2*e)^2 + 6*a
^2*c*cos(2*f*x + 2*e) + a^2*c + 2*(3*a^2*c*cos(4*f*x + 4*e) + 3*a^2*c*cos(2*f*x + 2*e) + a^2*c)*cos(6*f*x + 6*
e) + 6*(3*a^2*c*cos(2*f*x + 2*e) + a^2*c)*cos(4*f*x + 4*e) + 6*(a^2*c*sin(4*f*x + 4*e) + a^2*c*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^2*c*cos(4*f*x + 4*e) +
3*(f*x + e)*a^2*c*cos(2*f*x + 2*e) + (f*x + e)*a^2*c - a^2*c*sin(4*f*x + 4*e) - a^2*c*sin(2*f*x + 2*e))*cos(6
*f*x + 6*e) + 18*(3*(f*x + e)*a^2*c*cos(2*f*x + 2*e) + (f*x + e)*a^2*c)*cos(4*f*x + 4*e) - 6*(a^2*c*sin(6*f*x
+ 6*e) + 3*a^2*c*sin(4*f*x + 4*e) + 3*a^2*c*sin(2*f*x + 2*e))*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*
e))) - 4*(a^2*c*sin(6*f*x + 6*e) + 3*a^2*c*sin(4*f*x + 4*e) + 3*a^2*c*sin(2*f*x + 2*e))*cos(3/2*arctan2(sin(2*
f*x + 2*e), cos(2*f*x + 2*e))) - 6*(a^2*c*sin(6*f*x + 6*e) + 3*a^2*c*sin(4*f*x + 4*e) + 3*a^2*c*sin(2*f*x + 2*
e))*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 6*(3*(f*x + e)*a^2*c*sin(4*f*x + 4*e) + 3*(f*x + e)
*a^2*c*sin(2*f*x + 2*e) + a^2*c*cos(4*f*x + 4*e) + a^2*c*cos(2*f*x + 2*e))*sin(6*f*x + 6*e) + 6*(9*(f*x + e)*a
^2*c*sin(2*f*x + 2*e) - a^2*c)*sin(4*f*x + 4*e) + 6*(a^2*c*cos(6*f*x + 6*e) + 3*a^2*c*cos(4*f*x + 4*e) + 3*a^2
*c*cos(2*f*x + 2*e) + a^2*c)*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 4*(a^2*c*cos(6*f*x + 6*e)
+ 3*a^2*c*cos(4*f*x + 4*e) + 3*a^2*c*cos(2*f*x + 2*e) + a^2*c)*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2
*e))) + 6*(a^2*c*cos(6*f*x + 6*e) + 3*a^2*c*cos(4*f*x + 4*e) + 3*a^2*c*cos(2*f*x + 2*e) + a^2*c)*sin(1/2*arcta
n2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*sqrt(a)*sqrt(c)/((2*(3*cos(4*f*x + 4*e) + 3*cos(2*f*x + 2*e) + 1)*cos
(6*f*x + 6*e) + cos(6*f*x + 6*e)^2 + 6*(3*cos(2*f*x + 2*e) + 1)*cos(4*f*x + 4*e) + 9*cos(4*f*x + 4*e)^2 + 9*co
s(2*f*x + 2*e)^2 + 6*(sin(4*f*x + 4*e) + sin(2*f*x + 2*e))*sin(6*f*x + 6*e) + sin(6*f*x + 6*e)^2 + 9*sin(4*f*x
+ 4*e)^2 + 18*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + 9*sin(2*f*x + 2*e)^2 + 6*cos(2*f*x + 2*e) + 1)*f)

Giac [F]

\[ \int (a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{3/2} \, dx=\int { {\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (-c \sec \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \,d x } \]

[In]

integrate((a+a*sec(f*x+e))^(5/2)*(c-c*sec(f*x+e))^(3/2),x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int (a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{3/2} \, dx=\int {\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{5/2}\,{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^{3/2} \,d x \]

[In]

int((a + a/cos(e + f*x))^(5/2)*(c - c/cos(e + f*x))^(3/2),x)

[Out]

int((a + a/cos(e + f*x))^(5/2)*(c - c/cos(e + f*x))^(3/2), x)