3.123 \(\int \sec (e+f x) (a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{7/2} \, dx\)
Optimal. Leaf size=134 \[ \frac {a^3 \tan (e+f x) (c-c \sec (e+f x))^{7/2}}{15 f \sqrt {a \sec (e+f x)+a}}+\frac {2 a^2 \tan (e+f x) \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}}{15 f}+\frac {a \tan (e+f x) (a \sec (e+f x)+a)^{3/2} (c-c \sec (e+f x))^{7/2}}{6 f} \]
[Out]
1/6*a*(a+a*sec(f*x+e))^(3/2)*(c-c*sec(f*x+e))^(7/2)*tan(f*x+e)/f+1/15*a^3*(c-c*sec(f*x+e))^(7/2)*tan(f*x+e)/f/
(a+a*sec(f*x+e))^(1/2)+2/15*a^2*(c-c*sec(f*x+e))^(7/2)*(a+a*sec(f*x+e))^(1/2)*tan(f*x+e)/f
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Rubi [A] time = 0.42, antiderivative size = 134, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 2, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used =
{3955, 3953} \[ \frac {2 a^2 \tan (e+f x) \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}}{15 f}+\frac {a^3 \tan (e+f x) (c-c \sec (e+f x))^{7/2}}{15 f \sqrt {a \sec (e+f x)+a}}+\frac {a \tan (e+f x) (a \sec (e+f x)+a)^{3/2} (c-c \sec (e+f x))^{7/2}}{6 f} \]
Antiderivative was successfully verified.
[In]
Int[Sec[e + f*x]*(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(7/2),x]
[Out]
(a^3*(c - c*Sec[e + f*x])^(7/2)*Tan[e + f*x])/(15*f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^2*Sqrt[a + a*Sec[e + f*x]
]*(c - c*Sec[e + f*x])^(7/2)*Tan[e + f*x])/(15*f) + (a*(a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^(7/2)*T
an[e + f*x])/(6*f)
Rule 3953
Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.) +
(c_)], x_Symbol] :> Simp[(2*a*c*Cot[e + f*x]*(a + b*Csc[e + f*x])^m)/(b*f*(2*m + 1)*Sqrt[c + d*Csc[e + f*x]]),
x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[m, -2^(-1)]
Rule 3955
Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)
)^(n_), x_Symbol] :> -Simp[(d*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^(n - 1))/(f*(m + n)), x
] + Dist[(c*(2*n - 1))/(m + n), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^(n - 1), x], x] /
; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[n - 1/2, 0] && !LtQ[m, -2
^(-1)] && !(IGtQ[m - 1/2, 0] && LtQ[m, n])
Rubi steps
\begin {align*} \int \sec (e+f x) (a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{7/2} \, dx &=\frac {a (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{7/2} \tan (e+f x)}{6 f}+\frac {1}{3} (2 a) \int \sec (e+f x) (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{7/2} \, dx\\ &=\frac {2 a^2 \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{7/2} \tan (e+f x)}{15 f}+\frac {a (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{7/2} \tan (e+f x)}{6 f}+\frac {1}{15} \left (4 a^2\right ) \int \sec (e+f x) \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{7/2} \, dx\\ &=\frac {a^3 (c-c \sec (e+f x))^{7/2} \tan (e+f x)}{15 f \sqrt {a+a \sec (e+f x)}}+\frac {2 a^2 \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{7/2} \tan (e+f x)}{15 f}+\frac {a (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{7/2} \tan (e+f x)}{6 f}\\ \end {align*}
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Mathematica [A] time = 1.30, size = 113, normalized size = 0.84 \[ \frac {a^2 c^3 (78 \cos (e+f x)+5 (7 \cos (3 (e+f x))-3 \cos (4 (e+f x))+3 \cos (5 (e+f x))-5)) \csc \left (\frac {1}{2} (e+f x)\right ) \sec \left (\frac {1}{2} (e+f x)\right ) \sec ^5(e+f x) \sqrt {a (\sec (e+f x)+1)} \sqrt {c-c \sec (e+f x)}}{480 f} \]
Antiderivative was successfully verified.
[In]
Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(7/2),x]
[Out]
(a^2*c^3*(78*Cos[e + f*x] + 5*(-5 + 7*Cos[3*(e + f*x)] - 3*Cos[4*(e + f*x)] + 3*Cos[5*(e + f*x)]))*Csc[(e + f*
x)/2]*Sec[(e + f*x)/2]*Sec[e + f*x]^5*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])/(480*f)
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fricas [A] time = 0.48, size = 152, normalized size = 1.13 \[ \frac {{\left (30 \, a^{2} c^{3} \cos \left (f x + e\right )^{5} - 15 \, a^{2} c^{3} \cos \left (f x + e\right )^{4} - 20 \, a^{2} c^{3} \cos \left (f x + e\right )^{3} + 15 \, a^{2} c^{3} \cos \left (f x + e\right )^{2} + 6 \, a^{2} c^{3} \cos \left (f x + e\right ) - 5 \, a^{2} c^{3}\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{30 \, f \cos \left (f x + e\right )^{5} \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(f*x+e)*(a+a*sec(f*x+e))^(5/2)*(c-c*sec(f*x+e))^(7/2),x, algorithm="fricas")
[Out]
1/30*(30*a^2*c^3*cos(f*x + e)^5 - 15*a^2*c^3*cos(f*x + e)^4 - 20*a^2*c^3*cos(f*x + e)^3 + 15*a^2*c^3*cos(f*x +
e)^2 + 6*a^2*c^3*cos(f*x + e) - 5*a^2*c^3)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt((c*cos(f*x + e) - c)/
cos(f*x + e))/(f*cos(f*x + e)^5*sin(f*x + e))
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(f*x+e)*(a+a*sec(f*x+e))^(5/2)*(c-c*sec(f*x+e))^(7/2),x, algorithm="giac")
[Out]
Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)-16/15*a^2*c*sqrt(-a*c)*(-36*c^6*(c*tan(1/2*(f*x+exp(1)))^2-c)-10*c^7-20*c^4*(c*tan(1/2*(f*x+exp(1)))^2-c)
^3-45*c^5*(c*tan(1/2*(f*x+exp(1)))^2-c)^2)*abs(c)*sign(tan(1/2*(f*x+exp(1)))^3+tan(1/2*(f*x+exp(1))))/(c*tan(1
/2*(f*x+exp(1)))^2-c)^6/f
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maple [A] time = 2.43, size = 105, normalized size = 0.78 \[ -\frac {\left (\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}\right )^{\frac {7}{2}} \sqrt {\frac {a \left (1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}}\, \left (21 \left (\cos ^{3}\left (f x +e \right )\right )-33 \left (\cos ^{2}\left (f x +e \right )\right )+21 \cos \left (f x +e \right )-5\right ) \left (\sin ^{5}\left (f x +e \right )\right ) a^{2}}{30 f \left (-1+\cos \left (f x +e \right )\right )^{6} \cos \left (f x +e \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sec(f*x+e)*(a+a*sec(f*x+e))^(5/2)*(c-c*sec(f*x+e))^(7/2),x)
[Out]
-1/30/f*(c*(-1+cos(f*x+e))/cos(f*x+e))^(7/2)*(a*(1+cos(f*x+e))/cos(f*x+e))^(1/2)*(21*cos(f*x+e)^3-33*cos(f*x+e
)^2+21*cos(f*x+e)-5)*sin(f*x+e)^5/(-1+cos(f*x+e))^6/cos(f*x+e)^2*a^2
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maxima [B] time = 0.61, size = 2454, normalized size = 18.31 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(f*x+e)*(a+a*sec(f*x+e))^(5/2)*(c-c*sec(f*x+e))^(7/2),x, algorithm="maxima")
[Out]
2/15*(210*a^2*c^3*cos(3*f*x + 3*e)*sin(2*f*x + 2*e) - 90*a^2*c^3*cos(2*f*x + 2*e)*sin(f*x + e) - 15*a^2*c^3*si
n(f*x + e) - (15*a^2*c^3*sin(11*f*x + 11*e) - 15*a^2*c^3*sin(10*f*x + 10*e) + 35*a^2*c^3*sin(9*f*x + 9*e) + 78
*a^2*c^3*sin(7*f*x + 7*e) - 50*a^2*c^3*sin(6*f*x + 6*e) + 78*a^2*c^3*sin(5*f*x + 5*e) + 35*a^2*c^3*sin(3*f*x +
3*e) - 15*a^2*c^3*sin(2*f*x + 2*e) + 15*a^2*c^3*sin(f*x + e))*cos(12*f*x + 12*e) + 15*(6*a^2*c^3*sin(10*f*x +
10*e) + 15*a^2*c^3*sin(8*f*x + 8*e) + 20*a^2*c^3*sin(6*f*x + 6*e) + 15*a^2*c^3*sin(4*f*x + 4*e) + 6*a^2*c^3*s
in(2*f*x + 2*e))*cos(11*f*x + 11*e) - 3*(70*a^2*c^3*sin(9*f*x + 9*e) + 75*a^2*c^3*sin(8*f*x + 8*e) + 156*a^2*c
^3*sin(7*f*x + 7*e) + 156*a^2*c^3*sin(5*f*x + 5*e) + 75*a^2*c^3*sin(4*f*x + 4*e) + 70*a^2*c^3*sin(3*f*x + 3*e)
+ 30*a^2*c^3*sin(f*x + e))*cos(10*f*x + 10*e) + 35*(15*a^2*c^3*sin(8*f*x + 8*e) + 20*a^2*c^3*sin(6*f*x + 6*e)
+ 15*a^2*c^3*sin(4*f*x + 4*e) + 6*a^2*c^3*sin(2*f*x + 2*e))*cos(9*f*x + 9*e) - 15*(78*a^2*c^3*sin(7*f*x + 7*e
) - 50*a^2*c^3*sin(6*f*x + 6*e) + 78*a^2*c^3*sin(5*f*x + 5*e) + 35*a^2*c^3*sin(3*f*x + 3*e) - 15*a^2*c^3*sin(2
*f*x + 2*e) + 15*a^2*c^3*sin(f*x + e))*cos(8*f*x + 8*e) + 78*(20*a^2*c^3*sin(6*f*x + 6*e) + 15*a^2*c^3*sin(4*f
*x + 4*e) + 6*a^2*c^3*sin(2*f*x + 2*e))*cos(7*f*x + 7*e) - 10*(156*a^2*c^3*sin(5*f*x + 5*e) + 75*a^2*c^3*sin(4
*f*x + 4*e) + 70*a^2*c^3*sin(3*f*x + 3*e) + 30*a^2*c^3*sin(f*x + e))*cos(6*f*x + 6*e) + 234*(5*a^2*c^3*sin(4*f
*x + 4*e) + 2*a^2*c^3*sin(2*f*x + 2*e))*cos(5*f*x + 5*e) - 75*(7*a^2*c^3*sin(3*f*x + 3*e) - 3*a^2*c^3*sin(2*f*
x + 2*e) + 3*a^2*c^3*sin(f*x + e))*cos(4*f*x + 4*e) + (15*a^2*c^3*cos(11*f*x + 11*e) - 15*a^2*c^3*cos(10*f*x +
10*e) + 35*a^2*c^3*cos(9*f*x + 9*e) + 78*a^2*c^3*cos(7*f*x + 7*e) - 50*a^2*c^3*cos(6*f*x + 6*e) + 78*a^2*c^3*
cos(5*f*x + 5*e) + 35*a^2*c^3*cos(3*f*x + 3*e) - 15*a^2*c^3*cos(2*f*x + 2*e) + 15*a^2*c^3*cos(f*x + e))*sin(12
*f*x + 12*e) - 15*(6*a^2*c^3*cos(10*f*x + 10*e) + 15*a^2*c^3*cos(8*f*x + 8*e) + 20*a^2*c^3*cos(6*f*x + 6*e) +
15*a^2*c^3*cos(4*f*x + 4*e) + 6*a^2*c^3*cos(2*f*x + 2*e) + a^2*c^3)*sin(11*f*x + 11*e) + 3*(70*a^2*c^3*cos(9*f
*x + 9*e) + 75*a^2*c^3*cos(8*f*x + 8*e) + 156*a^2*c^3*cos(7*f*x + 7*e) + 156*a^2*c^3*cos(5*f*x + 5*e) + 75*a^2
*c^3*cos(4*f*x + 4*e) + 70*a^2*c^3*cos(3*f*x + 3*e) + 30*a^2*c^3*cos(f*x + e) + 5*a^2*c^3)*sin(10*f*x + 10*e)
- 35*(15*a^2*c^3*cos(8*f*x + 8*e) + 20*a^2*c^3*cos(6*f*x + 6*e) + 15*a^2*c^3*cos(4*f*x + 4*e) + 6*a^2*c^3*cos(
2*f*x + 2*e) + a^2*c^3)*sin(9*f*x + 9*e) + 15*(78*a^2*c^3*cos(7*f*x + 7*e) - 50*a^2*c^3*cos(6*f*x + 6*e) + 78*
a^2*c^3*cos(5*f*x + 5*e) + 35*a^2*c^3*cos(3*f*x + 3*e) - 15*a^2*c^3*cos(2*f*x + 2*e) + 15*a^2*c^3*cos(f*x + e)
)*sin(8*f*x + 8*e) - 78*(20*a^2*c^3*cos(6*f*x + 6*e) + 15*a^2*c^3*cos(4*f*x + 4*e) + 6*a^2*c^3*cos(2*f*x + 2*e
) + a^2*c^3)*sin(7*f*x + 7*e) + 10*(156*a^2*c^3*cos(5*f*x + 5*e) + 75*a^2*c^3*cos(4*f*x + 4*e) + 70*a^2*c^3*co
s(3*f*x + 3*e) + 30*a^2*c^3*cos(f*x + e) + 5*a^2*c^3)*sin(6*f*x + 6*e) - 78*(15*a^2*c^3*cos(4*f*x + 4*e) + 6*a
^2*c^3*cos(2*f*x + 2*e) + a^2*c^3)*sin(5*f*x + 5*e) + 75*(7*a^2*c^3*cos(3*f*x + 3*e) - 3*a^2*c^3*cos(2*f*x + 2
*e) + 3*a^2*c^3*cos(f*x + e))*sin(4*f*x + 4*e) - 35*(6*a^2*c^3*cos(2*f*x + 2*e) + a^2*c^3)*sin(3*f*x + 3*e) +
15*(6*a^2*c^3*cos(f*x + e) + a^2*c^3)*sin(2*f*x + 2*e))*sqrt(a)*sqrt(c)/((2*(6*cos(10*f*x + 10*e) + 15*cos(8*f
*x + 8*e) + 20*cos(6*f*x + 6*e) + 15*cos(4*f*x + 4*e) + 6*cos(2*f*x + 2*e) + 1)*cos(12*f*x + 12*e) + cos(12*f*
x + 12*e)^2 + 12*(15*cos(8*f*x + 8*e) + 20*cos(6*f*x + 6*e) + 15*cos(4*f*x + 4*e) + 6*cos(2*f*x + 2*e) + 1)*co
s(10*f*x + 10*e) + 36*cos(10*f*x + 10*e)^2 + 30*(20*cos(6*f*x + 6*e) + 15*cos(4*f*x + 4*e) + 6*cos(2*f*x + 2*e
) + 1)*cos(8*f*x + 8*e) + 225*cos(8*f*x + 8*e)^2 + 40*(15*cos(4*f*x + 4*e) + 6*cos(2*f*x + 2*e) + 1)*cos(6*f*x
+ 6*e) + 400*cos(6*f*x + 6*e)^2 + 30*(6*cos(2*f*x + 2*e) + 1)*cos(4*f*x + 4*e) + 225*cos(4*f*x + 4*e)^2 + 36*
cos(2*f*x + 2*e)^2 + 2*(6*sin(10*f*x + 10*e) + 15*sin(8*f*x + 8*e) + 20*sin(6*f*x + 6*e) + 15*sin(4*f*x + 4*e)
+ 6*sin(2*f*x + 2*e))*sin(12*f*x + 12*e) + sin(12*f*x + 12*e)^2 + 12*(15*sin(8*f*x + 8*e) + 20*sin(6*f*x + 6*
e) + 15*sin(4*f*x + 4*e) + 6*sin(2*f*x + 2*e))*sin(10*f*x + 10*e) + 36*sin(10*f*x + 10*e)^2 + 30*(20*sin(6*f*x
+ 6*e) + 15*sin(4*f*x + 4*e) + 6*sin(2*f*x + 2*e))*sin(8*f*x + 8*e) + 225*sin(8*f*x + 8*e)^2 + 120*(5*sin(4*f
*x + 4*e) + 2*sin(2*f*x + 2*e))*sin(6*f*x + 6*e) + 400*sin(6*f*x + 6*e)^2 + 225*sin(4*f*x + 4*e)^2 + 180*sin(4
*f*x + 4*e)*sin(2*f*x + 2*e) + 36*sin(2*f*x + 2*e)^2 + 12*cos(2*f*x + 2*e) + 1)*f)
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mupad [B] time = 6.23, size = 307, normalized size = 2.29 \[ \frac {\sqrt {c-\frac {c}{\cos \left (e+f\,x\right )}}\,\left (-\frac {a^2\,c^3\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,20{}\mathrm {i}}{3\,f}+\frac {a^2\,c^3\,\cos \left (e+f\,x\right )\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,104{}\mathrm {i}}{5\,f}+\frac {a^2\,c^3\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\cos \left (3\,e+3\,f\,x\right )\,\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,28{}\mathrm {i}}{3\,f}-\frac {a^2\,c^3\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\cos \left (4\,e+4\,f\,x\right )\,\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,4{}\mathrm {i}}{f}+\frac {a^2\,c^3\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\cos \left (5\,e+5\,f\,x\right )\,\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,4{}\mathrm {i}}{f}\right )}{{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sin \left (2\,e+2\,f\,x\right )\,10{}\mathrm {i}+{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sin \left (4\,e+4\,f\,x\right )\,8{}\mathrm {i}+{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sin \left (6\,e+6\,f\,x\right )\,2{}\mathrm {i}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((a + a/cos(e + f*x))^(5/2)*(c - c/cos(e + f*x))^(7/2))/cos(e + f*x),x)
[Out]
((c - c/cos(e + f*x))^(1/2)*((a^2*c^3*cos(e + f*x)*exp(e*6i + f*x*6i)*(a + a/cos(e + f*x))^(1/2)*104i)/(5*f) -
(a^2*c^3*exp(e*6i + f*x*6i)*(a + a/cos(e + f*x))^(1/2)*20i)/(3*f) + (a^2*c^3*exp(e*6i + f*x*6i)*cos(3*e + 3*f
*x)*(a + a/cos(e + f*x))^(1/2)*28i)/(3*f) - (a^2*c^3*exp(e*6i + f*x*6i)*cos(4*e + 4*f*x)*(a + a/cos(e + f*x))^
(1/2)*4i)/f + (a^2*c^3*exp(e*6i + f*x*6i)*cos(5*e + 5*f*x)*(a + a/cos(e + f*x))^(1/2)*4i)/f))/(exp(e*6i + f*x*
6i)*sin(2*e + 2*f*x)*10i + exp(e*6i + f*x*6i)*sin(4*e + 4*f*x)*8i + exp(e*6i + f*x*6i)*sin(6*e + 6*f*x)*2i)
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(f*x+e)*(a+a*sec(f*x+e))**(5/2)*(c-c*sec(f*x+e))**(7/2),x)
[Out]
Timed out
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