\(\int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{9/2}} \, dx\) [108]
Optimal result
Integrand size = 30, antiderivative size = 194 \[
\int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{9/2}} \, dx=-\frac {a^3 \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{9/2}}-\frac {a^3 \tan (e+f x)}{2 c^2 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2}}-\frac {a^3 \tan (e+f x)}{c^3 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2}}+\frac {a^3 \log (1-\cos (e+f x)) \tan (e+f x)}{c^4 f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}
\]
[Out]
-a^3*tan(f*x+e)/f/(c-c*sec(f*x+e))^(9/2)/(a+a*sec(f*x+e))^(1/2)-1/2*a^3*tan(f*x+e)/c^2/f/(c-c*sec(f*x+e))^(5/2
)/(a+a*sec(f*x+e))^(1/2)-a^3*tan(f*x+e)/c^3/f/(c-c*sec(f*x+e))^(3/2)/(a+a*sec(f*x+e))^(1/2)+a^3*ln(1-cos(f*x+e
))*tan(f*x+e)/c^4/f/(a+a*sec(f*x+e))^(1/2)/(c-c*sec(f*x+e))^(1/2)
Rubi [A] (verified)
Time = 0.49 (sec) , antiderivative size = 194, 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 = {3995, 3992, 3996, 31}
\[
\int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{9/2}} \, dx=\frac {a^3 \tan (e+f x) \log (1-\cos (e+f x))}{c^4 f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}-\frac {a^3 \tan (e+f x)}{c^3 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}-\frac {a^3 \tan (e+f x)}{2 c^2 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}}-\frac {a^3 \tan (e+f x)}{f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{9/2}}
\]
[In]
Int[(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x])^(9/2),x]
[Out]
-((a^3*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(9/2))) - (a^3*Tan[e + f*x])/(2*c^2*f*Sq
rt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(5/2)) - (a^3*Tan[e + f*x])/(c^3*f*Sqrt[a + a*Sec[e + f*x]]*(c - c
*Sec[e + f*x])^(3/2)) + (a^3*Log[1 - Cos[e + f*x]]*Tan[e + f*x])/(c^4*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Se
c[e + f*x]])
Rule 31
Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]
Rule 3992
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_), x_Symbol] :> Simp[
-2*a*Cot[e + f*x]*((c + d*Csc[e + f*x])^n/(f*(2*n + 1)*Sqrt[a + b*Csc[e + f*x]])), x] + Dist[1/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] && LtQ[n, -2^(-1)]
Rule 3995
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(5/2)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.), x_Symbol] :> Si
mp[-8*a^3*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^2/c^2, Int
[Sqrt[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*
d, 0] && EqQ[a^2 - b^2, 0] && LtQ[n, -2^(-1)]
Rule 3996
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_), x_Symbol] :> Dist
[(-a)*c*(Cot[e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]])), Subst[Int[(b + a*x)^(m - 1/2)*((
d + c*x)^(n - 1/2)/x^(m + n)), x], x, Sin[e + f*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] && EqQ[m + n, 0]
Rubi steps \begin{align*}
\text {integral}& = -\frac {a^3 \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{9/2}}+\frac {a^2 \int \frac {\sqrt {a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{5/2}} \, dx}{c^2} \\ & = -\frac {a^3 \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{9/2}}-\frac {a^3 \tan (e+f x)}{2 c^2 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2}}+\frac {a^2 \int \frac {\sqrt {a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{3/2}} \, dx}{c^3} \\ & = -\frac {a^3 \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{9/2}}-\frac {a^3 \tan (e+f x)}{2 c^2 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2}}-\frac {a^3 \tan (e+f x)}{c^3 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2}}+\frac {a^2 \int \frac {\sqrt {a+a \sec (e+f x)}}{\sqrt {c-c \sec (e+f x)}} \, dx}{c^4} \\ & = -\frac {a^3 \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{9/2}}-\frac {a^3 \tan (e+f x)}{2 c^2 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2}}-\frac {a^3 \tan (e+f x)}{c^3 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2}}+\frac {\left (a^3 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{-c+c x} \, dx,x,\cos (e+f x)\right )}{c^3 f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \\ & = -\frac {a^3 \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{9/2}}-\frac {a^3 \tan (e+f x)}{2 c^2 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2}}-\frac {a^3 \tan (e+f x)}{c^3 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2}}+\frac {a^3 \log (1-\cos (e+f x)) \tan (e+f x)}{c^4 f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \\
\end{align*}
Mathematica [A] (verified)
Time = 5.09 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.55
\[
\int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{9/2}} \, dx=-\frac {a^3 \left (-2 \log (\cos (e+f x))-2 \log (1-\sec (e+f x))+\frac {2+(-1+\sec (e+f x))^2-2 (-1+\sec (e+f x))^3}{(-1+\sec (e+f x))^4}\right ) \tan (e+f x)}{2 c^4 f \sqrt {a (1+\sec (e+f x))} \sqrt {c-c \sec (e+f x)}}
\]
[In]
Integrate[(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x])^(9/2),x]
[Out]
-1/2*(a^3*(-2*Log[Cos[e + f*x]] - 2*Log[1 - Sec[e + f*x]] + (2 + (-1 + Sec[e + f*x])^2 - 2*(-1 + Sec[e + f*x])
^3)/(-1 + Sec[e + f*x])^4)*Tan[e + f*x])/(c^4*f*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])
Maple [A] (warning: unable to verify)
Time = 2.45 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.46
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| method | result | size |
| | |
| default |
\(-\frac {\sqrt {2}\, a^{2} \sqrt {-\frac {2 a}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\, \left (1-\cos \left (f x +e \right )\right ) \left (16 \ln \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}+1\right ) \left (1-\cos \left (f x +e \right )\right )^{8} \csc \left (f x +e \right )^{8}-32 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{8} \csc \left (f x +e \right )^{8}-16 \left (1-\cos \left (f x +e \right )\right )^{6} \csc \left (f x +e \right )^{6}+8 \left (1-\cos \left (f x +e \right )\right )^{4} \csc \left (f x +e \right )^{4}-4 \left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}+1\right ) \csc \left (f x +e \right )}{32 f \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )^{4} \left (\frac {c \left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\right )^{\frac {9}{2}}}\) |
\(284\) |
| risch |
\(\frac {a^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) x}{c^{4} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}}-\frac {2 a^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) \left (f x +e \right )}{c^{4} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, f}+\frac {2 i a^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left (6 \,{\mathrm e}^{7 i \left (f x +e \right )}-23 \,{\mathrm e}^{6 i \left (f x +e \right )}+54 \,{\mathrm e}^{5 i \left (f x +e \right )}-66 \,{\mathrm e}^{4 i \left (f x +e \right )}+54 \,{\mathrm e}^{3 i \left (f x +e \right )}-23 \,{\mathrm e}^{2 i \left (f x +e \right )}+6 \,{\mathrm e}^{i \left (f x +e \right )}\right )}{c^{4} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{7} \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, f}-\frac {2 i a^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{c^{4} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, f}\) |
\(478\) |
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[In]
int((a+a*sec(f*x+e))^(5/2)/(c-c*sec(f*x+e))^(9/2),x,method=_RETURNVERBOSE)
[Out]
-1/32/f*2^(1/2)*a^2*(-2*a/((1-cos(f*x+e))^2*csc(f*x+e)^2-1))^(1/2)/((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^4/(c*(1-c
os(f*x+e))^2/((1-cos(f*x+e))^2*csc(f*x+e)^2-1)*csc(f*x+e)^2)^(9/2)*(1-cos(f*x+e))*(16*ln((1-cos(f*x+e))^2*csc(
f*x+e)^2+1)*(1-cos(f*x+e))^8*csc(f*x+e)^8-32*ln(-cot(f*x+e)+csc(f*x+e))*(1-cos(f*x+e))^8*csc(f*x+e)^8-16*(1-co
s(f*x+e))^6*csc(f*x+e)^6+8*(1-cos(f*x+e))^4*csc(f*x+e)^4-4*(1-cos(f*x+e))^2*csc(f*x+e)^2+1)*csc(f*x+e)
Fricas [F]
\[
\int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{9/2}} \, dx=\int { \frac {{\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{{\left (-c \sec \left (f x + e\right ) + c\right )}^{\frac {9}{2}}} \,d x }
\]
[In]
integrate((a+a*sec(f*x+e))^(5/2)/(c-c*sec(f*x+e))^(9/2),x, algorithm="fricas")
[Out]
integral(-(a^2*sec(f*x + e)^2 + 2*a^2*sec(f*x + e) + a^2)*sqrt(a*sec(f*x + e) + a)*sqrt(-c*sec(f*x + e) + c)/(
c^5*sec(f*x + e)^5 - 5*c^5*sec(f*x + e)^4 + 10*c^5*sec(f*x + e)^3 - 10*c^5*sec(f*x + e)^2 + 5*c^5*sec(f*x + e)
- c^5), x)
Sympy [F(-1)]
Timed out. \[
\int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{9/2}} \, dx=\text {Timed out}
\]
[In]
integrate((a+a*sec(f*x+e))**(5/2)/(c-c*sec(f*x+e))**(9/2),x)
[Out]
Timed out
Maxima [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 6134 vs. \(2 (176) = 352\).
Time = 11.57 (sec) , antiderivative size = 6134, normalized size of antiderivative = 31.62
\[
\int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{9/2}} \, dx=\text {Too large to display}
\]
[In]
integrate((a+a*sec(f*x+e))^(5/2)/(c-c*sec(f*x+e))^(9/2),x, algorithm="maxima")
[Out]
-((f*x + e)*a^2*cos(8*f*x + 8*e)^2 + 784*(f*x + e)*a^2*cos(6*f*x + 6*e)^2 + 4900*(f*x + e)*a^2*cos(4*f*x + 4*e
)^2 + 784*(f*x + e)*a^2*cos(2*f*x + 2*e)^2 + 64*(f*x + e)*a^2*cos(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*
e)))^2 + 3136*(f*x + e)*a^2*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 3136*(f*x + e)*a^2*cos(3/
2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 64*(f*x + e)*a^2*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*
x + 2*e)))^2 + (f*x + e)*a^2*sin(8*f*x + 8*e)^2 + 784*(f*x + e)*a^2*sin(6*f*x + 6*e)^2 + 4900*(f*x + e)*a^2*si
n(4*f*x + 4*e)^2 + 784*(f*x + e)*a^2*sin(2*f*x + 2*e)^2 + 64*(f*x + e)*a^2*sin(7/2*arctan2(sin(2*f*x + 2*e), c
os(2*f*x + 2*e)))^2 + 3136*(f*x + e)*a^2*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 3136*(f*x +
e)*a^2*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 64*(f*x + e)*a^2*sin(1/2*arctan2(sin(2*f*x + 2
*e), cos(2*f*x + 2*e)))^2 + 56*(f*x + e)*a^2*cos(2*f*x + 2*e) + (f*x + e)*a^2 - 46*a^2*sin(2*f*x + 2*e) - 2*(a
^2*cos(8*f*x + 8*e)^2 + 784*a^2*cos(6*f*x + 6*e)^2 + 4900*a^2*cos(4*f*x + 4*e)^2 + 784*a^2*cos(2*f*x + 2*e)^2
+ 64*a^2*cos(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 3136*a^2*cos(5/2*arctan2(sin(2*f*x + 2*e), c
os(2*f*x + 2*e)))^2 + 3136*a^2*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 64*a^2*cos(1/2*arctan2
(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + a^2*sin(8*f*x + 8*e)^2 + 784*a^2*sin(6*f*x + 6*e)^2 + 4900*a^2*sin(4
*f*x + 4*e)^2 + 3920*a^2*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + 784*a^2*sin(2*f*x + 2*e)^2 + 64*a^2*sin(7/2*arcta
n2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 3136*a^2*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 +
3136*a^2*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 64*a^2*sin(1/2*arctan2(sin(2*f*x + 2*e), cos
(2*f*x + 2*e)))^2 + 56*a^2*cos(2*f*x + 2*e) + a^2 + 2*(28*a^2*cos(6*f*x + 6*e) + 70*a^2*cos(4*f*x + 4*e) + 28*
a^2*cos(2*f*x + 2*e) + a^2)*cos(8*f*x + 8*e) + 56*(70*a^2*cos(4*f*x + 4*e) + 28*a^2*cos(2*f*x + 2*e) + a^2)*co
s(6*f*x + 6*e) + 140*(28*a^2*cos(2*f*x + 2*e) + a^2)*cos(4*f*x + 4*e) - 16*(a^2*cos(8*f*x + 8*e) + 28*a^2*cos(
6*f*x + 6*e) + 70*a^2*cos(4*f*x + 4*e) + 28*a^2*cos(2*f*x + 2*e) - 56*a^2*cos(5/2*arctan2(sin(2*f*x + 2*e), co
s(2*f*x + 2*e))) - 56*a^2*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 8*a^2*cos(1/2*arctan2(sin(2*f
*x + 2*e), cos(2*f*x + 2*e))) + a^2)*cos(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 112*(a^2*cos(8*f*x
+ 8*e) + 28*a^2*cos(6*f*x + 6*e) + 70*a^2*cos(4*f*x + 4*e) + 28*a^2*cos(2*f*x + 2*e) - 56*a^2*cos(3/2*arctan2
(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 8*a^2*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + a^2)*cos(
5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 112*(a^2*cos(8*f*x + 8*e) + 28*a^2*cos(6*f*x + 6*e) + 70*a^
2*cos(4*f*x + 4*e) + 28*a^2*cos(2*f*x + 2*e) - 8*a^2*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + a^
2)*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 16*(a^2*cos(8*f*x + 8*e) + 28*a^2*cos(6*f*x + 6*e) +
70*a^2*cos(4*f*x + 4*e) + 28*a^2*cos(2*f*x + 2*e) + a^2)*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))
+ 28*(2*a^2*sin(6*f*x + 6*e) + 5*a^2*sin(4*f*x + 4*e) + 2*a^2*sin(2*f*x + 2*e))*sin(8*f*x + 8*e) + 784*(5*a^2
*sin(4*f*x + 4*e) + 2*a^2*sin(2*f*x + 2*e))*sin(6*f*x + 6*e) - 16*(a^2*sin(8*f*x + 8*e) + 28*a^2*sin(6*f*x + 6
*e) + 70*a^2*sin(4*f*x + 4*e) + 28*a^2*sin(2*f*x + 2*e) - 56*a^2*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x +
2*e))) - 56*a^2*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 8*a^2*sin(1/2*arctan2(sin(2*f*x + 2*e)
, cos(2*f*x + 2*e))))*sin(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 112*(a^2*sin(8*f*x + 8*e) + 28*a^
2*sin(6*f*x + 6*e) + 70*a^2*sin(4*f*x + 4*e) + 28*a^2*sin(2*f*x + 2*e) - 56*a^2*sin(3/2*arctan2(sin(2*f*x + 2*
e), cos(2*f*x + 2*e))) - 8*a^2*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*sin(5/2*arctan2(sin(2*f*x
+ 2*e), cos(2*f*x + 2*e))) - 112*(a^2*sin(8*f*x + 8*e) + 28*a^2*sin(6*f*x + 6*e) + 70*a^2*sin(4*f*x + 4*e) +
28*a^2*sin(2*f*x + 2*e) - 8*a^2*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*sin(3/2*arctan2(sin(2*f*
x + 2*e), cos(2*f*x + 2*e))) - 16*(a^2*sin(8*f*x + 8*e) + 28*a^2*sin(6*f*x + 6*e) + 70*a^2*sin(4*f*x + 4*e) +
28*a^2*sin(2*f*x + 2*e))*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*arctan2(sin(1/2*arctan2(sin(2*f
*x + 2*e), cos(2*f*x + 2*e))), cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 1) + 2*(28*(f*x + e)*a^2
*cos(6*f*x + 6*e) + 70*(f*x + e)*a^2*cos(4*f*x + 4*e) + 28*(f*x + e)*a^2*cos(2*f*x + 2*e) + (f*x + e)*a^2 - 23
*a^2*sin(6*f*x + 6*e) - 66*a^2*sin(4*f*x + 4*e) - 23*a^2*sin(2*f*x + 2*e))*cos(8*f*x + 8*e) + 28*(140*(f*x + e
)*a^2*cos(4*f*x + 4*e) + 56*(f*x + e)*a^2*cos(2*f*x + 2*e) + 2*(f*x + e)*a^2 - 17*a^2*sin(4*f*x + 4*e))*cos(6*
f*x + 6*e) + 28*(140*(f*x + e)*a^2*cos(2*f*x + 2*e) + 5*(f*x + e)*a^2 + 17*a^2*sin(2*f*x + 2*e))*cos(4*f*x + 4
*e) - 4*(4*(f*x + e)*a^2*cos(8*f*x + 8*e) + 112*(f*x + e)*a^2*cos(6*f*x + 6*e) + 280*(f*x + e)*a^2*cos(4*f*x +
4*e) + 112*(f*x + e)*a^2*cos(2*f*x + 2*e) - 224*(f*x + e)*a^2*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2
*e))) - 224*(f*x + e)*a^2*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 32*(f*x + e)*a^2*cos(1/2*arct
an2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 4*(f*x + e)*a^2 + 3*a^2*sin(8*f*x + 8*e) - 8*a^2*sin(6*f*x + 6*e) -
54*a^2*sin(4*f*x + 4*e) - 8*a^2*sin(2*f*x + 2*e) + 48*a^2*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))
) + 48*a^2*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*cos(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x +
2*e))) - 4*(28*(f*x + e)*a^2*cos(8*f*x + 8*e) + 784*(f*x + e)*a^2*cos(6*f*x + 6*e) + 1960*(f*x + e)*a^2*cos(4
*f*x + 4*e) + 784*(f*x + e)*a^2*cos(2*f*x + 2*e) - 1568*(f*x + e)*a^2*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*
f*x + 2*e))) - 224*(f*x + e)*a^2*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 28*(f*x + e)*a^2 + 27*
a^2*sin(8*f*x + 8*e) + 112*a^2*sin(6*f*x + 6*e) + 42*a^2*sin(4*f*x + 4*e) + 112*a^2*sin(2*f*x + 2*e) - 48*a^2*
sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 4
*(28*(f*x + e)*a^2*cos(8*f*x + 8*e) + 784*(f*x + e)*a^2*cos(6*f*x + 6*e) + 1960*(f*x + e)*a^2*cos(4*f*x + 4*e)
+ 784*(f*x + e)*a^2*cos(2*f*x + 2*e) - 224*(f*x + e)*a^2*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))
+ 28*(f*x + e)*a^2 + 27*a^2*sin(8*f*x + 8*e) + 112*a^2*sin(6*f*x + 6*e) + 42*a^2*sin(4*f*x + 4*e) + 112*a^2*s
in(2*f*x + 2*e) - 48*a^2*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*cos(3/2*arctan2(sin(2*f*x + 2*e
), cos(2*f*x + 2*e))) - 4*(4*(f*x + e)*a^2*cos(8*f*x + 8*e) + 112*(f*x + e)*a^2*cos(6*f*x + 6*e) + 280*(f*x +
e)*a^2*cos(4*f*x + 4*e) + 112*(f*x + e)*a^2*cos(2*f*x + 2*e) + 4*(f*x + e)*a^2 + 3*a^2*sin(8*f*x + 8*e) - 8*a^
2*sin(6*f*x + 6*e) - 54*a^2*sin(4*f*x + 4*e) - 8*a^2*sin(2*f*x + 2*e))*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2
*f*x + 2*e))) + 2*(28*(f*x + e)*a^2*sin(6*f*x + 6*e) + 70*(f*x + e)*a^2*sin(4*f*x + 4*e) + 28*(f*x + e)*a^2*si
n(2*f*x + 2*e) + 23*a^2*cos(6*f*x + 6*e) + 66*a^2*cos(4*f*x + 4*e) + 23*a^2*cos(2*f*x + 2*e))*sin(8*f*x + 8*e)
+ 2*(1960*(f*x + e)*a^2*sin(4*f*x + 4*e) + 784*(f*x + e)*a^2*sin(2*f*x + 2*e) + 238*a^2*cos(4*f*x + 4*e) - 23
*a^2)*sin(6*f*x + 6*e) + 4*(980*(f*x + e)*a^2*sin(2*f*x + 2*e) - 119*a^2*cos(2*f*x + 2*e) - 33*a^2)*sin(4*f*x
+ 4*e) - 4*(4*(f*x + e)*a^2*sin(8*f*x + 8*e) + 112*(f*x + e)*a^2*sin(6*f*x + 6*e) + 280*(f*x + e)*a^2*sin(4*f*
x + 4*e) + 112*(f*x + e)*a^2*sin(2*f*x + 2*e) - 224*(f*x + e)*a^2*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x
+ 2*e))) - 224*(f*x + e)*a^2*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 32*(f*x + e)*a^2*sin(1/2*a
rctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 3*a^2*cos(8*f*x + 8*e) + 8*a^2*cos(6*f*x + 6*e) + 54*a^2*cos(4*f
*x + 4*e) + 8*a^2*cos(2*f*x + 2*e) - 48*a^2*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 48*a^2*cos(
3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 3*a^2)*sin(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))
- 4*(28*(f*x + e)*a^2*sin(8*f*x + 8*e) + 784*(f*x + e)*a^2*sin(6*f*x + 6*e) + 1960*(f*x + e)*a^2*sin(4*f*x +
4*e) + 784*(f*x + e)*a^2*sin(2*f*x + 2*e) - 1568*(f*x + e)*a^2*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2
*e))) - 224*(f*x + e)*a^2*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 27*a^2*cos(8*f*x + 8*e) - 112
*a^2*cos(6*f*x + 6*e) - 42*a^2*cos(4*f*x + 4*e) - 112*a^2*cos(2*f*x + 2*e) + 48*a^2*cos(1/2*arctan2(sin(2*f*x
+ 2*e), cos(2*f*x + 2*e))) - 27*a^2)*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 4*(28*(f*x + e)*a^
2*sin(8*f*x + 8*e) + 784*(f*x + e)*a^2*sin(6*f*x + 6*e) + 1960*(f*x + e)*a^2*sin(4*f*x + 4*e) + 784*(f*x + e)*
a^2*sin(2*f*x + 2*e) - 224*(f*x + e)*a^2*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 27*a^2*cos(8*f
*x + 8*e) - 112*a^2*cos(6*f*x + 6*e) - 42*a^2*cos(4*f*x + 4*e) - 112*a^2*cos(2*f*x + 2*e) + 48*a^2*cos(1/2*arc
tan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 27*a^2)*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 4*(
4*(f*x + e)*a^2*sin(8*f*x + 8*e) + 112*(f*x + e)*a^2*sin(6*f*x + 6*e) + 280*(f*x + e)*a^2*sin(4*f*x + 4*e) + 1
12*(f*x + e)*a^2*sin(2*f*x + 2*e) - 3*a^2*cos(8*f*x + 8*e) + 8*a^2*cos(6*f*x + 6*e) + 54*a^2*cos(4*f*x + 4*e)
+ 8*a^2*cos(2*f*x + 2*e) - 3*a^2)*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*sqrt(a)*sqrt(c)/((c^5*
cos(8*f*x + 8*e)^2 + 784*c^5*cos(6*f*x + 6*e)^2 + 4900*c^5*cos(4*f*x + 4*e)^2 + 784*c^5*cos(2*f*x + 2*e)^2 + 6
4*c^5*cos(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 3136*c^5*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(
2*f*x + 2*e)))^2 + 3136*c^5*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 64*c^5*cos(1/2*arctan2(si
n(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + c^5*sin(8*f*x + 8*e)^2 + 784*c^5*sin(6*f*x + 6*e)^2 + 4900*c^5*sin(4*f*
x + 4*e)^2 + 3920*c^5*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + 784*c^5*sin(2*f*x + 2*e)^2 + 64*c^5*sin(7/2*arctan2(
sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 3136*c^5*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 313
6*c^5*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 64*c^5*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*
f*x + 2*e)))^2 + 56*c^5*cos(2*f*x + 2*e) + c^5 + 2*(28*c^5*cos(6*f*x + 6*e) + 70*c^5*cos(4*f*x + 4*e) + 28*c^5
*cos(2*f*x + 2*e) + c^5)*cos(8*f*x + 8*e) + 56*(70*c^5*cos(4*f*x + 4*e) + 28*c^5*cos(2*f*x + 2*e) + c^5)*cos(6
*f*x + 6*e) + 140*(28*c^5*cos(2*f*x + 2*e) + c^5)*cos(4*f*x + 4*e) - 16*(c^5*cos(8*f*x + 8*e) + 28*c^5*cos(6*f
*x + 6*e) + 70*c^5*cos(4*f*x + 4*e) + 28*c^5*cos(2*f*x + 2*e) - 56*c^5*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(2
*f*x + 2*e))) - 56*c^5*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 8*c^5*cos(1/2*arctan2(sin(2*f*x
+ 2*e), cos(2*f*x + 2*e))) + c^5)*cos(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 112*(c^5*cos(8*f*x +
8*e) + 28*c^5*cos(6*f*x + 6*e) + 70*c^5*cos(4*f*x + 4*e) + 28*c^5*cos(2*f*x + 2*e) - 56*c^5*cos(3/2*arctan2(si
n(2*f*x + 2*e), cos(2*f*x + 2*e))) - 8*c^5*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + c^5)*cos(5/2
*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 112*(c^5*cos(8*f*x + 8*e) + 28*c^5*cos(6*f*x + 6*e) + 70*c^5*c
os(4*f*x + 4*e) + 28*c^5*cos(2*f*x + 2*e) - 8*c^5*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + c^5)*
cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 16*(c^5*cos(8*f*x + 8*e) + 28*c^5*cos(6*f*x + 6*e) + 70
*c^5*cos(4*f*x + 4*e) + 28*c^5*cos(2*f*x + 2*e) + c^5)*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) +
28*(2*c^5*sin(6*f*x + 6*e) + 5*c^5*sin(4*f*x + 4*e) + 2*c^5*sin(2*f*x + 2*e))*sin(8*f*x + 8*e) + 784*(5*c^5*si
n(4*f*x + 4*e) + 2*c^5*sin(2*f*x + 2*e))*sin(6*f*x + 6*e) - 16*(c^5*sin(8*f*x + 8*e) + 28*c^5*sin(6*f*x + 6*e)
+ 70*c^5*sin(4*f*x + 4*e) + 28*c^5*sin(2*f*x + 2*e) - 56*c^5*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*
e))) - 56*c^5*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 8*c^5*sin(1/2*arctan2(sin(2*f*x + 2*e), c
os(2*f*x + 2*e))))*sin(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 112*(c^5*sin(8*f*x + 8*e) + 28*c^5*s
in(6*f*x + 6*e) + 70*c^5*sin(4*f*x + 4*e) + 28*c^5*sin(2*f*x + 2*e) - 56*c^5*sin(3/2*arctan2(sin(2*f*x + 2*e),
cos(2*f*x + 2*e))) - 8*c^5*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*sin(5/2*arctan2(sin(2*f*x +
2*e), cos(2*f*x + 2*e))) - 112*(c^5*sin(8*f*x + 8*e) + 28*c^5*sin(6*f*x + 6*e) + 70*c^5*sin(4*f*x + 4*e) + 28*
c^5*sin(2*f*x + 2*e) - 8*c^5*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*sin(3/2*arctan2(sin(2*f*x +
2*e), cos(2*f*x + 2*e))) - 16*(c^5*sin(8*f*x + 8*e) + 28*c^5*sin(6*f*x + 6*e) + 70*c^5*sin(4*f*x + 4*e) + 28*
c^5*sin(2*f*x + 2*e))*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*f)
Giac [F]
\[
\int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{9/2}} \, dx=\int { \frac {{\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{{\left (-c \sec \left (f x + e\right ) + c\right )}^{\frac {9}{2}}} \,d x }
\]
[In]
integrate((a+a*sec(f*x+e))^(5/2)/(c-c*sec(f*x+e))^(9/2),x, algorithm="giac")
[Out]
sage0*x
Mupad [F(-1)]
Timed out. \[
\int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{9/2}} \, dx=\int \frac {{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{5/2}}{{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^{9/2}} \,d x
\]
[In]
int((a + a/cos(e + f*x))^(5/2)/(c - c/cos(e + f*x))^(9/2),x)
[Out]
int((a + a/cos(e + f*x))^(5/2)/(c - c/cos(e + f*x))^(9/2), x)