3.100 \(\int \frac{(a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{7/2}} \, dx\)

Optimal. Leaf size=196 \[ -\frac{a^2 \tan (e+f x)}{c^2 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}+\frac{a^2 \tan (e+f x) \log (1-\cos (e+f x))}{c^3 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a^2 \tan (e+f x)}{2 c f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}}-\frac{2 a^2 \tan (e+f x)}{3 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}} \]

[Out]

(-2*a^2*Tan[e + f*x])/(3*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(7/2)) - (a^2*Tan[e + f*x])/(2*c*f*Sq
rt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(5/2)) - (a^2*Tan[e + f*x])/(c^2*f*Sqrt[a + a*Sec[e + f*x]]*(c - c
*Sec[e + f*x])^(3/2)) + (a^2*Log[1 - Cos[e + f*x]]*Tan[e + f*x])/(c^3*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Se
c[e + f*x]])

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Rubi [A]  time = 0.382437, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {3908, 3907, 3911, 31} \[ -\frac{a^2 \tan (e+f x)}{c^2 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}+\frac{a^2 \tan (e+f x) \log (1-\cos (e+f x))}{c^3 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a^2 \tan (e+f x)}{2 c f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}}-\frac{2 a^2 \tan (e+f x)}{3 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sec[e + f*x])^(3/2)/(c - c*Sec[e + f*x])^(7/2),x]

[Out]

(-2*a^2*Tan[e + f*x])/(3*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(7/2)) - (a^2*Tan[e + f*x])/(2*c*f*Sq
rt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(5/2)) - (a^2*Tan[e + f*x])/(c^2*f*Sqrt[a + a*Sec[e + f*x]]*(c - c
*Sec[e + f*x])^(3/2)) + (a^2*Log[1 - Cos[e + f*x]]*Tan[e + f*x])/(c^3*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Se
c[e + f*x]])

Rule 3908

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(3/2)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.), x_Symbol] :> Si
mp[(-4*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/c, Int[Sqr
t[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 3907

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 3911

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{(a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{7/2}} \, dx &=-\frac{2 a^2 \tan (e+f x)}{3 f \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^{7/2}}+\frac{a \int \frac{\sqrt{a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{5/2}} \, dx}{c}\\ &=-\frac{2 a^2 \tan (e+f x)}{3 f \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^{7/2}}-\frac{a^2 \tan (e+f x)}{2 c f \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2}}+\frac{a \int \frac{\sqrt{a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{3/2}} \, dx}{c^2}\\ &=-\frac{2 a^2 \tan (e+f x)}{3 f \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^{7/2}}-\frac{a^2 \tan (e+f x)}{2 c f \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2}}-\frac{a^2 \tan (e+f x)}{c^2 f \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2}}+\frac{a \int \frac{\sqrt{a+a \sec (e+f x)}}{\sqrt{c-c \sec (e+f x)}} \, dx}{c^3}\\ &=-\frac{2 a^2 \tan (e+f x)}{3 f \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^{7/2}}-\frac{a^2 \tan (e+f x)}{2 c f \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2}}-\frac{a^2 \tan (e+f x)}{c^2 f \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2}}+\frac{\left (a^2 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{-c+c x} \, dx,x,\cos (e+f x)\right )}{c^2 f \sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)}}\\ &=-\frac{2 a^2 \tan (e+f x)}{3 f \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^{7/2}}-\frac{a^2 \tan (e+f x)}{2 c f \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2}}-\frac{a^2 \tan (e+f x)}{c^2 f \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2}}+\frac{a^2 \log (1-\cos (e+f x)) \tan (e+f x)}{c^3 f \sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)}}\\ \end{align*}

Mathematica [C]  time = 2.0955, size = 199, normalized size = 1.02 \[ \frac{a \tan \left (\frac{1}{2} (e+f x)\right ) \sqrt{a (\sec (e+f x)+1)} \left (-60 \log \left (1-e^{i (e+f x)}\right )-3 i f x \cos (3 (e+f x))+6 i \left (6 i \log \left (1-e^{i (e+f x)}\right )+3 f x+4 i\right ) \cos (2 (e+f x))+6 \log \left (1-e^{i (e+f x)}\right ) \cos (3 (e+f x))+\left (90 \log \left (1-e^{i (e+f x)}\right )-45 i f x+66\right ) \cos (e+f x)+30 i f x-50\right )}{12 c^3 f (\cos (e+f x)-1)^3 \sqrt{c-c \sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sec[e + f*x])^(3/2)/(c - c*Sec[e + f*x])^(7/2),x]

[Out]

(a*(-50 + (30*I)*f*x - (3*I)*f*x*Cos[3*(e + f*x)] + (6*I)*Cos[2*(e + f*x)]*(4*I + 3*f*x + (6*I)*Log[1 - E^(I*(
e + f*x))]) - 60*Log[1 - E^(I*(e + f*x))] + 6*Cos[3*(e + f*x)]*Log[1 - E^(I*(e + f*x))] + Cos[e + f*x]*(66 - (
45*I)*f*x + 90*Log[1 - E^(I*(e + f*x))]))*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(12*c^3*f*(-1 + Cos[e +
f*x])^3*Sqrt[c - c*Sec[e + f*x]])

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Maple [A]  time = 0.275, size = 289, normalized size = 1.5 \begin{align*} -{\frac{a \left ( -1+\cos \left ( fx+e \right ) \right ) }{24\,f\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{3}} \left ( 48\, \left ( \cos \left ( fx+e \right ) \right ) ^{3}\ln \left ( -{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) -24\, \left ( \cos \left ( fx+e \right ) \right ) ^{3}\ln \left ( 2\, \left ( 1+\cos \left ( fx+e \right ) \right ) ^{-1} \right ) -144\,\ln \left ( -{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}-35\, \left ( \cos \left ( fx+e \right ) \right ) ^{3}+72\,\ln \left ( 2\, \left ( 1+\cos \left ( fx+e \right ) \right ) ^{-1} \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}+144\,\cos \left ( fx+e \right ) \ln \left ( -{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) +9\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}-72\,\cos \left ( fx+e \right ) \ln \left ( 2\, \left ( 1+\cos \left ( fx+e \right ) \right ) ^{-1} \right ) -48\,\ln \left ( -{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) +27\,\cos \left ( fx+e \right ) +24\,\ln \left ( 2\, \left ( 1+\cos \left ( fx+e \right ) \right ) ^{-1} \right ) -17 \right ) \sqrt{{\frac{a \left ( 1+\cos \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) }}} \left ({\frac{c \left ( -1+\cos \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) }} \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sec(f*x+e))^(3/2)/(c-c*sec(f*x+e))^(7/2),x)

[Out]

-1/24/f*a*(-1+cos(f*x+e))*(48*cos(f*x+e)^3*ln(-(-1+cos(f*x+e))/sin(f*x+e))-24*cos(f*x+e)^3*ln(2/(1+cos(f*x+e))
)-144*ln(-(-1+cos(f*x+e))/sin(f*x+e))*cos(f*x+e)^2-35*cos(f*x+e)^3+72*ln(2/(1+cos(f*x+e)))*cos(f*x+e)^2+144*co
s(f*x+e)*ln(-(-1+cos(f*x+e))/sin(f*x+e))+9*cos(f*x+e)^2-72*cos(f*x+e)*ln(2/(1+cos(f*x+e)))-48*ln(-(-1+cos(f*x+
e))/sin(f*x+e))+27*cos(f*x+e)+24*ln(2/(1+cos(f*x+e)))-17)*(1/cos(f*x+e)*a*(1+cos(f*x+e)))^(1/2)/sin(f*x+e)/cos
(f*x+e)^3/(c*(-1+cos(f*x+e))/cos(f*x+e))^(7/2)

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Maxima [B]  time = 11.3007, size = 4698, normalized size = 23.97 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(f*x+e))^(3/2)/(c-c*sec(f*x+e))^(7/2),x, algorithm="maxima")

[Out]

-1/3*(3*(f*x + e)*a*cos(6*f*x + 6*e)^2 + 675*(f*x + e)*a*cos(4*f*x + 4*e)^2 + 675*(f*x + e)*a*cos(2*f*x + 2*e)
^2 + 108*(f*x + e)*a*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 1200*(f*x + e)*a*cos(3/2*arctan2
(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 108*(f*x + e)*a*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))
^2 + 3*(f*x + e)*a*sin(6*f*x + 6*e)^2 + 675*(f*x + e)*a*sin(4*f*x + 4*e)^2 + 675*(f*x + e)*a*sin(2*f*x + 2*e)^
2 + 108*(f*x + e)*a*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 1200*(f*x + e)*a*sin(3/2*arctan2(
sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 108*(f*x + e)*a*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^
2 + 90*(f*x + e)*a*cos(2*f*x + 2*e) + 3*(f*x + e)*a - 6*(a*cos(6*f*x + 6*e)^2 + 225*a*cos(4*f*x + 4*e)^2 + 225
*a*cos(2*f*x + 2*e)^2 + 36*a*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 400*a*cos(3/2*arctan2(si
n(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 36*a*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + a*sin(6*f
*x + 6*e)^2 + 225*a*sin(4*f*x + 4*e)^2 + 450*a*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + 225*a*sin(2*f*x + 2*e)^2 +
36*a*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 400*a*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*
x + 2*e)))^2 + 36*a*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 2*(15*a*cos(4*f*x + 4*e) + 15*a*c
os(2*f*x + 2*e) + a)*cos(6*f*x + 6*e) + 30*(15*a*cos(2*f*x + 2*e) + a)*cos(4*f*x + 4*e) + 30*a*cos(2*f*x + 2*e
) - 12*(a*cos(6*f*x + 6*e) + 15*a*cos(4*f*x + 4*e) + 15*a*cos(2*f*x + 2*e) - 20*a*cos(3/2*arctan2(sin(2*f*x +
2*e), cos(2*f*x + 2*e))) - 6*a*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + a)*cos(5/2*arctan2(sin(2
*f*x + 2*e), cos(2*f*x + 2*e))) - 40*(a*cos(6*f*x + 6*e) + 15*a*cos(4*f*x + 4*e) + 15*a*cos(2*f*x + 2*e) - 6*a
*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + a)*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))
) - 12*(a*cos(6*f*x + 6*e) + 15*a*cos(4*f*x + 4*e) + 15*a*cos(2*f*x + 2*e) + a)*cos(1/2*arctan2(sin(2*f*x + 2*
e), cos(2*f*x + 2*e))) + 30*(a*sin(4*f*x + 4*e) + a*sin(2*f*x + 2*e))*sin(6*f*x + 6*e) - 12*(a*sin(6*f*x + 6*e
) + 15*a*sin(4*f*x + 4*e) + 15*a*sin(2*f*x + 2*e) - 20*a*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))
- 6*a*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)
)) - 40*(a*sin(6*f*x + 6*e) + 15*a*sin(4*f*x + 4*e) + 15*a*sin(2*f*x + 2*e) - 6*a*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))) - 12*(a*sin(6*f*x + 6*e) + 15*a
*sin(4*f*x + 4*e) + 15*a*sin(2*f*x + 2*e))*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + a)*arctan2(s
in(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)
+ 6*(15*(f*x + e)*a*cos(4*f*x + 4*e) + 15*(f*x + e)*a*cos(2*f*x + 2*e) + (f*x + e)*a - 11*a*sin(4*f*x + 4*e)
- 11*a*sin(2*f*x + 2*e))*cos(6*f*x + 6*e) + 90*(15*(f*x + e)*a*cos(2*f*x + 2*e) + (f*x + e)*a)*cos(4*f*x + 4*e
) - 12*(3*(f*x + e)*a*cos(6*f*x + 6*e) + 45*(f*x + e)*a*cos(4*f*x + 4*e) + 45*(f*x + e)*a*cos(2*f*x + 2*e) - 6
0*(f*x + e)*a*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 18*(f*x + e)*a*cos(1/2*arctan2(sin(2*f*x
+ 2*e), cos(2*f*x + 2*e))) + 3*(f*x + e)*a + 2*a*sin(6*f*x + 6*e) - 3*a*sin(4*f*x + 4*e) - 3*a*sin(2*f*x + 2*e
) + 10*a*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))) - 20*(6*(f*x + e)*a*cos(6*f*x + 6*e) + 90*(f*x + e)*a*cos(4*f*x + 4*e) + 90*(f*x + e)*a*cos(2*f*x + 2*e)
- 36*(f*x + e)*a*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 6*(f*x + e)*a + 5*a*sin(6*f*x + 6*e)
+ 9*a*sin(4*f*x + 4*e) + 9*a*sin(2*f*x + 2*e) - 6*a*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))) - 12*(3*(f*x + e)*a*cos(6*f*x + 6*e) + 45*(f*x + e)*a*cos(4*f
*x + 4*e) + 45*(f*x + e)*a*cos(2*f*x + 2*e) + 3*(f*x + e)*a + 2*a*sin(6*f*x + 6*e) - 3*a*sin(4*f*x + 4*e) - 3*
a*sin(2*f*x + 2*e))*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 6*(15*(f*x + e)*a*sin(4*f*x + 4*e)
+ 15*(f*x + e)*a*sin(2*f*x + 2*e) + 11*a*cos(4*f*x + 4*e) + 11*a*cos(2*f*x + 2*e))*sin(6*f*x + 6*e) + 6*(225*(
f*x + e)*a*sin(2*f*x + 2*e) - 11*a)*sin(4*f*x + 4*e) - 66*a*sin(2*f*x + 2*e) - 12*(3*(f*x + e)*a*sin(6*f*x + 6
*e) + 45*(f*x + e)*a*sin(4*f*x + 4*e) + 45*(f*x + e)*a*sin(2*f*x + 2*e) - 60*(f*x + e)*a*sin(3/2*arctan2(sin(2
*f*x + 2*e), cos(2*f*x + 2*e))) - 18*(f*x + e)*a*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 2*a*co
s(6*f*x + 6*e) + 3*a*cos(4*f*x + 4*e) + 3*a*cos(2*f*x + 2*e) - 10*a*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*
x + 2*e))) - 2*a)*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 20*(6*(f*x + e)*a*sin(6*f*x + 6*e) +
90*(f*x + e)*a*sin(4*f*x + 4*e) + 90*(f*x + e)*a*sin(2*f*x + 2*e) - 36*(f*x + e)*a*sin(1/2*arctan2(sin(2*f*x +
2*e), cos(2*f*x + 2*e))) - 5*a*cos(6*f*x + 6*e) - 9*a*cos(4*f*x + 4*e) - 9*a*cos(2*f*x + 2*e) + 6*a*cos(1/2*a
rctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 5*a)*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 12*(
3*(f*x + e)*a*sin(6*f*x + 6*e) + 45*(f*x + e)*a*sin(4*f*x + 4*e) + 45*(f*x + e)*a*sin(2*f*x + 2*e) - 2*a*cos(6
*f*x + 6*e) + 3*a*cos(4*f*x + 4*e) + 3*a*cos(2*f*x + 2*e) - 2*a)*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x +
2*e))))*sqrt(a)*sqrt(c)/((c^4*cos(6*f*x + 6*e)^2 + 225*c^4*cos(4*f*x + 4*e)^2 + 225*c^4*cos(2*f*x + 2*e)^2 +
36*c^4*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 400*c^4*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(
2*f*x + 2*e)))^2 + 36*c^4*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + c^4*sin(6*f*x + 6*e)^2 + 22
5*c^4*sin(4*f*x + 4*e)^2 + 450*c^4*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + 225*c^4*sin(2*f*x + 2*e)^2 + 36*c^4*sin
(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 400*c^4*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*
e)))^2 + 36*c^4*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 30*c^4*cos(2*f*x + 2*e) + c^4 + 2*(15
*c^4*cos(4*f*x + 4*e) + 15*c^4*cos(2*f*x + 2*e) + c^4)*cos(6*f*x + 6*e) + 30*(15*c^4*cos(2*f*x + 2*e) + c^4)*c
os(4*f*x + 4*e) - 12*(c^4*cos(6*f*x + 6*e) + 15*c^4*cos(4*f*x + 4*e) + 15*c^4*cos(2*f*x + 2*e) - 20*c^4*cos(3/
2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 6*c^4*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) +
c^4)*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 40*(c^4*cos(6*f*x + 6*e) + 15*c^4*cos(4*f*x + 4*e)
+ 15*c^4*cos(2*f*x + 2*e) - 6*c^4*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + c^4)*cos(3/2*arctan2
(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 12*(c^4*cos(6*f*x + 6*e) + 15*c^4*cos(4*f*x + 4*e) + 15*c^4*cos(2*f*x
+ 2*e) + c^4)*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 30*(c^4*sin(4*f*x + 4*e) + c^4*sin(2*f*x
+ 2*e))*sin(6*f*x + 6*e) - 12*(c^4*sin(6*f*x + 6*e) + 15*c^4*sin(4*f*x + 4*e) + 15*c^4*sin(2*f*x + 2*e) - 20*c
^4*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 6*c^4*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))) - 40*(c^4*sin(6*f*x + 6*e) + 15*c^4*sin(4*f*x + 4
*e) + 15*c^4*sin(2*f*x + 2*e) - 6*c^4*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*sin(3/2*arctan2(si
n(2*f*x + 2*e), cos(2*f*x + 2*e))) - 12*(c^4*sin(6*f*x + 6*e) + 15*c^4*sin(4*f*x + 4*e) + 15*c^4*sin(2*f*x + 2
*e))*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*f)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (a \sec \left (f x + e\right ) + a\right )}^{\frac{3}{2}} \sqrt{-c \sec \left (f x + e\right ) + c}}{c^{4} \sec \left (f x + e\right )^{4} - 4 \, c^{4} \sec \left (f x + e\right )^{3} + 6 \, c^{4} \sec \left (f x + e\right )^{2} - 4 \, c^{4} \sec \left (f x + e\right ) + c^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(f*x+e))^(3/2)/(c-c*sec(f*x+e))^(7/2),x, algorithm="fricas")

[Out]

integral((a*sec(f*x + e) + a)^(3/2)*sqrt(-c*sec(f*x + e) + c)/(c^4*sec(f*x + e)^4 - 4*c^4*sec(f*x + e)^3 + 6*c
^4*sec(f*x + e)^2 - 4*c^4*sec(f*x + e) + c^4), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(f*x+e))**(3/2)/(c-c*sec(f*x+e))**(7/2),x)

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(f*x+e))^(3/2)/(c-c*sec(f*x+e))^(7/2),x, algorithm="giac")

[Out]

Timed out