\(\int \frac {1}{(a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{3/2}} \, dx\) [129]
Optimal result
Integrand size = 30, antiderivative size = 345 \[
\int \frac {1}{(a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{3/2}} \, dx=\frac {\log (\cos (e+f x)) \tan (e+f x)}{a^2 c f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}+\frac {5 \log (1-\sec (e+f x)) \tan (e+f x)}{16 a^2 c f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}+\frac {11 \log (1+\sec (e+f x)) \tan (e+f x)}{16 a^2 c f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}-\frac {\tan (e+f x)}{8 a^2 c f (1-\sec (e+f x)) \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}-\frac {\tan (e+f x)}{8 a^2 c f (1+\sec (e+f x))^2 \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}-\frac {\tan (e+f x)}{2 a^2 c f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}
\]
[Out]
ln(cos(f*x+e))*tan(f*x+e)/a^2/c/f/(a+a*sec(f*x+e))^(1/2)/(c-c*sec(f*x+e))^(1/2)+5/16*ln(1-sec(f*x+e))*tan(f*x+
e)/a^2/c/f/(a+a*sec(f*x+e))^(1/2)/(c-c*sec(f*x+e))^(1/2)+11/16*ln(1+sec(f*x+e))*tan(f*x+e)/a^2/c/f/(a+a*sec(f*
x+e))^(1/2)/(c-c*sec(f*x+e))^(1/2)-1/8*tan(f*x+e)/a^2/c/f/(1-sec(f*x+e))/(a+a*sec(f*x+e))^(1/2)/(c-c*sec(f*x+e
))^(1/2)-1/8*tan(f*x+e)/a^2/c/f/(1+sec(f*x+e))^2/(a+a*sec(f*x+e))^(1/2)/(c-c*sec(f*x+e))^(1/2)-1/2*tan(f*x+e)/
a^2/c/f/(1+sec(f*x+e))/(a+a*sec(f*x+e))^(1/2)/(c-c*sec(f*x+e))^(1/2)
Rubi [A] (verified)
Time = 0.21 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {3997, 90}
\[
\int \frac {1}{(a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{3/2}} \, dx=-\frac {\tan (e+f x)}{8 a^2 c f (1-\sec (e+f x)) \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}-\frac {\tan (e+f x)}{2 a^2 c f (\sec (e+f x)+1) \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}-\frac {\tan (e+f x)}{8 a^2 c f (\sec (e+f x)+1)^2 \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}+\frac {5 \tan (e+f x) \log (1-\sec (e+f x))}{16 a^2 c f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}+\frac {11 \tan (e+f x) \log (\sec (e+f x)+1)}{16 a^2 c f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}+\frac {\tan (e+f x) \log (\cos (e+f x))}{a^2 c f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}
\]
[In]
Int[1/((a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(3/2)),x]
[Out]
(Log[Cos[e + f*x]]*Tan[e + f*x])/(a^2*c*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (5*Log[1 - Sec[
e + f*x]]*Tan[e + f*x])/(16*a^2*c*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (11*Log[1 + Sec[e + f
*x]]*Tan[e + f*x])/(16*a^2*c*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - Tan[e + f*x]/(8*a^2*c*f*(1
- Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - Tan[e + f*x]/(8*a^2*c*f*(1 + Sec[e + f*x
])^2*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - Tan[e + f*x]/(2*a^2*c*f*(1 + Sec[e + f*x])*Sqrt[a +
a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])
Rule 90
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))
Rule 3997
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.), x_Symbol] :> Di
st[a*c*(Cot[e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]])), Subst[Int[(a + b*x)^(m - 1/2)*((c
+ d*x)^(n - 1/2)/x), x], x, Csc[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && E
qQ[a^2 - b^2, 0]
Rubi steps \begin{align*}
\text {integral}& = -\frac {(a c \tan (e+f x)) \text {Subst}\left (\int \frac {1}{x (a+a x)^3 (c-c x)^2} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \\ & = -\frac {(a c \tan (e+f x)) \text {Subst}\left (\int \left (\frac {1}{8 a^3 c^2 (-1+x)^2}-\frac {5}{16 a^3 c^2 (-1+x)}+\frac {1}{a^3 c^2 x}-\frac {1}{4 a^3 c^2 (1+x)^3}-\frac {1}{2 a^3 c^2 (1+x)^2}-\frac {11}{16 a^3 c^2 (1+x)}\right ) \, dx,x,\sec (e+f x)\right )}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \\ & = \frac {\log (\cos (e+f x)) \tan (e+f x)}{a^2 c f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}+\frac {5 \log (1-\sec (e+f x)) \tan (e+f x)}{16 a^2 c f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}+\frac {11 \log (1+\sec (e+f x)) \tan (e+f x)}{16 a^2 c f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}-\frac {\tan (e+f x)}{8 a^2 c f (1-\sec (e+f x)) \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}-\frac {\tan (e+f x)}{8 a^2 c f (1+\sec (e+f x))^2 \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}-\frac {\tan (e+f x)}{2 a^2 c f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.89 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.34
\[
\int \frac {1}{(a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{3/2}} \, dx=\frac {\left (16 \log (\cos (e+f x))+5 \log (1-\sec (e+f x))+11 \log (1+\sec (e+f x))+\frac {2}{-1+\sec (e+f x)}-\frac {2}{(1+\sec (e+f x))^2}-\frac {8}{1+\sec (e+f x)}\right ) \tan (e+f x)}{16 a^2 c f \sqrt {a (1+\sec (e+f x))} \sqrt {c-c \sec (e+f x)}}
\]
[In]
Integrate[1/((a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(3/2)),x]
[Out]
((16*Log[Cos[e + f*x]] + 5*Log[1 - Sec[e + f*x]] + 11*Log[1 + Sec[e + f*x]] + 2/(-1 + Sec[e + f*x]) - 2/(1 + S
ec[e + f*x])^2 - 8/(1 + Sec[e + f*x]))*Tan[e + f*x])/(16*a^2*c*f*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e +
f*x]])
Maple [A] (warning: unable to verify)
Time = 2.26 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.76
| | |
| method | result | size |
| | |
| default |
\(\frac {\sqrt {2}\, \sqrt {-\frac {2 a}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\, \left (1-\cos \left (f x +e \right )\right ) \left (-\left (1-\cos \left (f x +e \right )\right )^{6} \csc \left (f x +e \right )^{6}+10 \left (1-\cos \left (f x +e \right )\right )^{4} \csc \left (f x +e \right )^{4}+20 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-32 \ln \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}+1\right ) \left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}+2\right ) \csc \left (f x +e \right )}{64 f \,a^{3} \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right ) \left (\frac {c \left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\right )^{\frac {3}{2}}}\) |
\(262\) |
| risch |
\(\frac {\left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) x}{a^{2} c \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right ) \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 )}}}}-\frac {2 \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) \left (f x +e \right )}{a^{2} c \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right ) \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 )}}}\, f}-\frac {i \left (5 \,{\mathrm e}^{5 i \left (f x +e \right )}-6 \,{\mathrm e}^{4 i \left (f x +e \right )}-14 \,{\mathrm e}^{3 i \left (f x +e \right )}-6 \,{\mathrm e}^{2 i \left (f x +e \right )}+5 \,{\mathrm e}^{i \left (f x +e \right )}\right )}{4 a^{2} c \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\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}}{1+{\mathrm e}^{2 i \left (f x +e \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}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, f}-\frac {5 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 \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right ) \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 )}}}\, f}-\frac {11 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 \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right ) \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 )}}}\, f}\) |
\(621\) |
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[In]
int(1/(a+a*sec(f*x+e))^(5/2)/(c-c*sec(f*x+e))^(3/2),x,method=_RETURNVERBOSE)
[Out]
1/64/f*2^(1/2)/a^3*(-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)/(c*(1-cos(
f*x+e))^2/((1-cos(f*x+e))^2*csc(f*x+e)^2-1)*csc(f*x+e)^2)^(3/2)*(1-cos(f*x+e))*(-(1-cos(f*x+e))^6*csc(f*x+e)^6
+10*(1-cos(f*x+e))^4*csc(f*x+e)^4+20*ln(-cot(f*x+e)+csc(f*x+e))*(1-cos(f*x+e))^2*csc(f*x+e)^2-32*ln((1-cos(f*x
+e))^2*csc(f*x+e)^2+1)*(1-cos(f*x+e))^2*csc(f*x+e)^2+2)*csc(f*x+e)
Fricas [F]
\[
\int \frac {1}{(a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{3/2}} \, dx=\int { \frac {1}{{\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (-c \sec \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x }
\]
[In]
integrate(1/(a+a*sec(f*x+e))^(5/2)/(c-c*sec(f*x+e))^(3/2),x, algorithm="fricas")
[Out]
integral(sqrt(a*sec(f*x + e) + a)*sqrt(-c*sec(f*x + e) + c)/(a^3*c^2*sec(f*x + e)^5 + a^3*c^2*sec(f*x + e)^4 -
2*a^3*c^2*sec(f*x + e)^3 - 2*a^3*c^2*sec(f*x + e)^2 + a^3*c^2*sec(f*x + e) + a^3*c^2), x)
Sympy [F(-1)]
Timed out. \[
\int \frac {1}{(a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{3/2}} \, dx=\text {Timed out}
\]
[In]
integrate(1/(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. 4272 vs. \(2 (309) = 618\).
Time = 1.90 (sec) , antiderivative size = 4272, normalized size of antiderivative = 12.38
\[
\int \frac {1}{(a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{3/2}} \, dx=\text {Too large to display}
\]
[In]
integrate(1/(a+a*sec(f*x+e))^(5/2)/(c-c*sec(f*x+e))^(3/2),x, algorithm="maxima")
[Out]
-1/8*(8*(f*x + e)*cos(6*f*x + 6*e)^2 + 8*(f*x + e)*cos(4*f*x + 4*e)^2 + 8*(f*x + e)*cos(2*f*x + 2*e)^2 + 32*(f
*x + e)*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 128*(f*x + e)*cos(3/2*arctan2(sin(2*f*x + 2*e
), cos(2*f*x + 2*e)))^2 + 32*(f*x + e)*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 8*(f*x + e)*si
n(6*f*x + 6*e)^2 + 8*(f*x + e)*sin(4*f*x + 4*e)^2 + 8*(f*x + e)*sin(2*f*x + 2*e)^2 + 32*(f*x + e)*sin(5/2*arct
an2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 128*(f*x + e)*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))
)^2 + 32*(f*x + e)*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 8*f*x + 11*(2*(cos(4*f*x + 4*e) +
cos(2*f*x + 2*e) - 1)*cos(6*f*x + 6*e) - cos(6*f*x + 6*e)^2 - 2*(cos(2*f*x + 2*e) - 1)*cos(4*f*x + 4*e) - cos(
4*f*x + 4*e)^2 - cos(2*f*x + 2*e)^2 - 4*(cos(6*f*x + 6*e) - cos(4*f*x + 4*e) - cos(2*f*x + 2*e) - 4*cos(3/2*ar
ctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 2*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 1)*cos(5
/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 4*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 8
*(cos(6*f*x + 6*e) - cos(4*f*x + 4*e) - cos(2*f*x + 2*e) + 2*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e
))) + 1)*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 16*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x
+ 2*e)))^2 - 4*(cos(6*f*x + 6*e) - cos(4*f*x + 4*e) - cos(2*f*x + 2*e) + 1)*cos(1/2*arctan2(sin(2*f*x + 2*e),
cos(2*f*x + 2*e))) - 4*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 2*(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 - sin(4*f*x + 4*e)^2 - 2*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) -
sin(2*f*x + 2*e)^2 - 4*(sin(6*f*x + 6*e) - sin(4*f*x + 4*e) - sin(2*f*x + 2*e) - 4*sin(3/2*arctan2(sin(2*f*x +
2*e), cos(2*f*x + 2*e))) + 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))) - 4*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 8*(sin(6*f*x + 6*e) -
sin(4*f*x + 4*e) - sin(2*f*x + 2*e) + 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*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 - 4*(sin(6*f*
x + 6*e) - sin(4*f*x + 4*e) - sin(2*f*x + 2*e))*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 4*sin(1
/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 2*cos(2*f*x + 2*e) - 1)*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) + 5*(2*(cos(4*f*x + 4*e)
+ cos(2*f*x + 2*e) - 1)*cos(6*f*x + 6*e) - cos(6*f*x + 6*e)^2 - 2*(cos(2*f*x + 2*e) - 1)*cos(4*f*x + 4*e) - c
os(4*f*x + 4*e)^2 - cos(2*f*x + 2*e)^2 - 4*(cos(6*f*x + 6*e) - cos(4*f*x + 4*e) - cos(2*f*x + 2*e) - 4*cos(3/2
*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 2*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 1)*co
s(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 4*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2
+ 8*(cos(6*f*x + 6*e) - cos(4*f*x + 4*e) - cos(2*f*x + 2*e) + 2*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x +
2*e))) + 1)*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 16*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*
f*x + 2*e)))^2 - 4*(cos(6*f*x + 6*e) - cos(4*f*x + 4*e) - cos(2*f*x + 2*e) + 1)*cos(1/2*arctan2(sin(2*f*x + 2*
e), cos(2*f*x + 2*e))) - 4*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 2*(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 - sin(4*f*x + 4*e)^2 - 2*sin(4*f*x + 4*e)*sin(2*f*x + 2*e)
- sin(2*f*x + 2*e)^2 - 4*(sin(6*f*x + 6*e) - sin(4*f*x + 4*e) - sin(2*f*x + 2*e) - 4*sin(3/2*arctan2(sin(2*f*
x + 2*e), cos(2*f*x + 2*e))) + 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))) - 4*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 8*(sin(6*f*x + 6*e)
- sin(4*f*x + 4*e) - sin(2*f*x + 2*e) + 2*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*sin(3/2*arcta
n2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 16*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 - 4*(sin(6
*f*x + 6*e) - sin(4*f*x + 4*e) - sin(2*f*x + 2*e))*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 4*si
n(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 2*cos(2*f*x + 2*e) - 1)*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) + 4*(4*f*x - 4*(f*x +
e)*cos(4*f*x + 4*e) - 4*(f*x + e)*cos(2*f*x + 2*e) + 4*e + 3*sin(4*f*x + 4*e) + 3*sin(2*f*x + 2*e))*cos(6*f*x
+ 6*e) - 16*(f*x - (f*x + e)*cos(2*f*x + 2*e) + e)*cos(4*f*x + 4*e) - 16*(f*x + e)*cos(2*f*x + 2*e) + 2*(16*f
*x + 16*(f*x + e)*cos(6*f*x + 6*e) - 16*(f*x + e)*cos(4*f*x + 4*e) - 16*(f*x + e)*cos(2*f*x + 2*e) - 64*(f*x +
e)*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 32*(f*x + e)*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(
2*f*x + 2*e))) + 16*e + 5*sin(6*f*x + 6*e) + 7*sin(4*f*x + 4*e) + 7*sin(2*f*x + 2*e) + 8*sin(3/2*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*(16*f*x + 16*(f*x +
e)*cos(6*f*x + 6*e) - 16*(f*x + e)*cos(4*f*x + 4*e) - 16*(f*x + e)*cos(2*f*x + 2*e) + 32*(f*x + e)*cos(1/2*arc
tan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 16*e + 7*sin(6*f*x + 6*e) + 5*sin(4*f*x + 4*e) + 5*sin(2*f*x + 2*e
) + 4*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)
)) + 2*(16*f*x + 16*(f*x + e)*cos(6*f*x + 6*e) - 16*(f*x + e)*cos(4*f*x + 4*e) - 16*(f*x + e)*cos(2*f*x + 2*e)
+ 16*e + 5*sin(6*f*x + 6*e) + 7*sin(4*f*x + 4*e) + 7*sin(2*f*x + 2*e))*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(
2*f*x + 2*e))) - 4*(4*(f*x + e)*sin(4*f*x + 4*e) + 4*(f*x + e)*sin(2*f*x + 2*e) + 3*cos(4*f*x + 4*e) + 3*cos(2
*f*x + 2*e))*sin(6*f*x + 6*e) + 4*(4*(f*x + e)*sin(2*f*x + 2*e) + 3)*sin(4*f*x + 4*e) + 2*(16*(f*x + e)*sin(6*
f*x + 6*e) - 16*(f*x + e)*sin(4*f*x + 4*e) - 16*(f*x + e)*sin(2*f*x + 2*e) - 64*(f*x + e)*sin(3/2*arctan2(sin(
2*f*x + 2*e), cos(2*f*x + 2*e))) + 32*(f*x + e)*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 5*cos(6
*f*x + 6*e) - 7*cos(4*f*x + 4*e) - 7*cos(2*f*x + 2*e) - 8*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))
- 5)*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 4*(16*(f*x + e)*sin(6*f*x + 6*e) - 16*(f*x + e)*s
in(4*f*x + 4*e) - 16*(f*x + e)*sin(2*f*x + 2*e) + 32*(f*x + e)*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2
*e))) - 7*cos(6*f*x + 6*e) - 5*cos(4*f*x + 4*e) - 5*cos(2*f*x + 2*e) - 4*cos(1/2*arctan2(sin(2*f*x + 2*e), cos
(2*f*x + 2*e))) - 7)*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 2*(16*(f*x + e)*sin(6*f*x + 6*e) -
16*(f*x + e)*sin(4*f*x + 4*e) - 16*(f*x + e)*sin(2*f*x + 2*e) - 5*cos(6*f*x + 6*e) - 7*cos(4*f*x + 4*e) - 7*c
os(2*f*x + 2*e) - 5)*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 8*e + 12*sin(2*f*x + 2*e))/((a^2*c
*cos(6*f*x + 6*e)^2 + a^2*c*cos(4*f*x + 4*e)^2 + a^2*c*cos(2*f*x + 2*e)^2 + 4*a^2*c*cos(5/2*arctan2(sin(2*f*x
+ 2*e), cos(2*f*x + 2*e)))^2 + 16*a^2*c*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 4*a^2*c*cos(1
/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + a^2*c*sin(6*f*x + 6*e)^2 + a^2*c*sin(4*f*x + 4*e)^2 + 2*a^
2*c*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + a^2*c*sin(2*f*x + 2*e)^2 + 4*a^2*c*sin(5/2*arctan2(sin(2*f*x + 2*e), c
os(2*f*x + 2*e)))^2 + 16*a^2*c*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 4*a^2*c*sin(1/2*arctan
2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 - 2*a^2*c*cos(2*f*x + 2*e) + a^2*c - 2*(a^2*c*cos(4*f*x + 4*e) + a^2*
c*cos(2*f*x + 2*e) - a^2*c)*cos(6*f*x + 6*e) + 2*(a^2*c*cos(2*f*x + 2*e) - a^2*c)*cos(4*f*x + 4*e) + 4*(a^2*c*
cos(6*f*x + 6*e) - a^2*c*cos(4*f*x + 4*e) - a^2*c*cos(2*f*x + 2*e) - 4*a^2*c*cos(3/2*arctan2(sin(2*f*x + 2*e),
cos(2*f*x + 2*e))) + 2*a^2*c*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + a^2*c)*cos(5/2*arctan2(si
n(2*f*x + 2*e), cos(2*f*x + 2*e))) - 8*(a^2*c*cos(6*f*x + 6*e) - a^2*c*cos(4*f*x + 4*e) - a^2*c*cos(2*f*x + 2*
e) + 2*a^2*c*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + a^2*c)*cos(3/2*arctan2(sin(2*f*x + 2*e), c
os(2*f*x + 2*e))) + 4*(a^2*c*cos(6*f*x + 6*e) - a^2*c*cos(4*f*x + 4*e) - a^2*c*cos(2*f*x + 2*e) + a^2*c)*cos(1
/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 2*(a^2*c*sin(4*f*x + 4*e) + a^2*c*sin(2*f*x + 2*e))*sin(6*f*
x + 6*e) + 4*(a^2*c*sin(6*f*x + 6*e) - a^2*c*sin(4*f*x + 4*e) - a^2*c*sin(2*f*x + 2*e) - 4*a^2*c*sin(3/2*arcta
n2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 2*a^2*c*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))) - 8*(a^2*c*sin(6*f*x + 6*e) - a^2*c*sin(4*f*x + 4*e) - a^2*c*si
n(2*f*x + 2*e) + 2*a^2*c*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))) + 4*(a^2*c*sin(6*f*x + 6*e) - a^2*c*sin(4*f*x + 4*e) - a^2*c*sin(2*f*x + 2*e))*sin(1/2*a
rctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*sqrt(a)*sqrt(c)*f)
Giac [A] (verification not implemented)
none
Time = 2.00 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.56
\[
\int \frac {1}{(a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{3/2}} \, dx=-\frac {\frac {10 \, \log \left ({\left | c \right |} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}\right )}{\sqrt {-a c} a^{2} {\left | c \right |}} - \frac {32 \, \log \left ({\left | -c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c \right |}\right )}{\sqrt {-a c} a^{2} {\left | c \right |}} - \frac {2 \, {\left (5 \, c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}}{\sqrt {-a c} a^{2} c {\left | c \right |} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}} + \frac {{\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{2} \sqrt {-a c} a^{2} c^{2} {\left | c \right |} - 8 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )} \sqrt {-a c} a^{2} c^{3} {\left | c \right |}}{a^{5} c^{7}}}{32 \, f}
\]
[In]
integrate(1/(a+a*sec(f*x+e))^(5/2)/(c-c*sec(f*x+e))^(3/2),x, algorithm="giac")
[Out]
-1/32*(10*log(abs(c)*tan(1/2*f*x + 1/2*e)^2)/(sqrt(-a*c)*a^2*abs(c)) - 32*log(abs(-c*tan(1/2*f*x + 1/2*e)^2 -
c))/(sqrt(-a*c)*a^2*abs(c)) - 2*(5*c*tan(1/2*f*x + 1/2*e)^2 - c)/(sqrt(-a*c)*a^2*c*abs(c)*tan(1/2*f*x + 1/2*e)
^2) + ((c*tan(1/2*f*x + 1/2*e)^2 - c)^2*sqrt(-a*c)*a^2*c^2*abs(c) - 8*(c*tan(1/2*f*x + 1/2*e)^2 - c)*sqrt(-a*c
)*a^2*c^3*abs(c))/(a^5*c^7))/f
Mupad [F(-1)]
Timed out. \[
\int \frac {1}{(a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{3/2}} \, dx=\int \frac {1}{{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{5/2}\,{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x
\]
[In]
int(1/((a + a/cos(e + f*x))^(5/2)*(c - c/cos(e + f*x))^(3/2)),x)
[Out]
int(1/((a + a/cos(e + f*x))^(5/2)*(c - c/cos(e + f*x))^(3/2)), x)