3.1.93 \(\int \frac {\sqrt {a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{7/2}} \, dx\) [93]

Optimal. Leaf size=188 \[ -\frac {a \tan (e+f x)}{3 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{7/2}}-\frac {a \tan (e+f x)}{2 c f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2}}-\frac {a \tan (e+f x)}{c^2 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2}}+\frac {a \log (1-\cos (e+f x)) \tan (e+f x)}{c^3 f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \]

[Out]

-1/3*a*tan(f*x+e)/f/(c-c*sec(f*x+e))^(7/2)/(a+a*sec(f*x+e))^(1/2)-1/2*a*tan(f*x+e)/c/f/(c-c*sec(f*x+e))^(5/2)/
(a+a*sec(f*x+e))^(1/2)-a*tan(f*x+e)/c^2/f/(c-c*sec(f*x+e))^(3/2)/(a+a*sec(f*x+e))^(1/2)+a*ln(1-cos(f*x+e))*tan
(f*x+e)/c^3/f/(a+a*sec(f*x+e))^(1/2)/(c-c*sec(f*x+e))^(1/2)

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Rubi [A]
time = 0.24, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3992, 3996, 31} \begin {gather*} \frac {a \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 \tan (e+f x)}{c^2 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}-\frac {a \tan (e+f x)}{2 c f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}}-\frac {a \tan (e+f x)}{3 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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*} \int \frac {\sqrt {a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{7/2}} \, dx &=-\frac {a \tan (e+f x)}{3 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{7/2}}+\frac {\int \frac {\sqrt {a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{5/2}} \, dx}{c}\\ &=-\frac {a \tan (e+f x)}{3 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{7/2}}-\frac {a \tan (e+f x)}{2 c f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2}}+\frac {\int \frac {\sqrt {a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{3/2}} \, dx}{c^2}\\ &=-\frac {a \tan (e+f x)}{3 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{7/2}}-\frac {a \tan (e+f x)}{2 c f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2}}-\frac {a \tan (e+f x)}{c^2 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2}}+\frac {\int \frac {\sqrt {a+a \sec (e+f x)}}{\sqrt {c-c \sec (e+f x)}} \, dx}{c^3}\\ &=-\frac {a \tan (e+f x)}{3 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{7/2}}-\frac {a \tan (e+f x)}{2 c f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2}}-\frac {a \tan (e+f x)}{c^2 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2}}+\frac {(a \tan (e+f x)) \text {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 {a \tan (e+f x)}{3 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{7/2}}-\frac {a \tan (e+f x)}{2 c f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2}}-\frac {a \tan (e+f x)}{c^2 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2}}+\frac {a \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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 1.95, size = 198, normalized size = 1.05 \begin {gather*} \frac {\left (-40+30 i f x-3 i f x \cos (3 (e+f x))+18 i \cos (2 (e+f x)) \left (i+f x+2 i \log \left (1-e^{i (e+f x)}\right )\right )-60 \log \left (1-e^{i (e+f x)}\right )+6 \cos (3 (e+f x)) \log \left (1-e^{i (e+f x)}\right )+9 \cos (e+f x) \left (6-5 i f x+10 \log \left (1-e^{i (e+f x)}\right )\right )\right ) \sqrt {a (1+\sec (e+f x))} \tan \left (\frac {1}{2} (e+f x)\right )}{12 c^3 f (-1+\cos (e+f x))^3 \sqrt {c-c \sec (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-40 + (30*I)*f*x - (3*I)*f*x*Cos[3*(e + f*x)] + (18*I)*Cos[2*(e + f*x)]*(I + f*x + (2*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))] + 9*Cos[e + f*x]*(6 - (5*I)
*f*x + 10*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 = 0.27, size = 288, normalized size = 1.53

method	result	size
default	\(\frac {\left (-1+\cos \left (f x +e \right )\right ) \left (6 \left (\cos ^{3}\left (f x +e \right )\right ) \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )-12 \left (\cos ^{3}\left (f x +e \right )\right ) \ln \left (-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right )+7 \left (\cos ^{3}\left (f x +e \right )\right )-18 \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )+36 \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right )-3 \left (\cos ^{2}\left (f x +e \right )\right )+18 \cos \left (f x +e \right ) \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )-36 \cos \left (f x +e \right ) \ln \left (-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right )-6 \cos \left (f x +e \right )-6 \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )+12 \ln \left (-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right )+4\right ) \sqrt {\frac {a \left (\cos \left (f x +e \right )+1\right )}{\cos \left (f x +e \right )}}}{6 f \left (\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}\right )^{\frac {7}{2}} \sin \left (f x +e \right ) \cos \left (f x +e \right )^{3}}\)	\(288\)
risch	\(\frac {\sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) x}{c^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}}-\frac {2 \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) \left (f x +e \right )}{c^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}+\frac {2 i \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left (9 \,{\mathrm e}^{5 i \left (f x +e \right )}-27 \,{\mathrm e}^{4 i \left (f x +e \right )}+40 \,{\mathrm e}^{3 i \left (f x +e \right )}-27 \,{\mathrm e}^{2 i \left (f x +e \right )}+9 \,{\mathrm e}^{i \left (f x +e \right )}\right )}{3 c^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{5} \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}-\frac {2 i \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{c^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\)	\(444\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sec(f*x+e))^(1/2)/(c-c*sec(f*x+e))^(7/2),x,method=_RETURNVERBOSE)

[Out]

1/6/f*(-1+cos(f*x+e))*(6*cos(f*x+e)^3*ln(2/(cos(f*x+e)+1))-12*cos(f*x+e)^3*ln(-(-1+cos(f*x+e))/sin(f*x+e))+7*c
os(f*x+e)^3-18*cos(f*x+e)^2*ln(2/(cos(f*x+e)+1))+36*cos(f*x+e)^2*ln(-(-1+cos(f*x+e))/sin(f*x+e))-3*cos(f*x+e)^
2+18*cos(f*x+e)*ln(2/(cos(f*x+e)+1))-36*cos(f*x+e)*ln(-(-1+cos(f*x+e))/sin(f*x+e))-6*cos(f*x+e)-6*ln(2/(cos(f*
x+e)+1))+12*ln(-(-1+cos(f*x+e))/sin(f*x+e))+4)*(a*(cos(f*x+e)+1)/cos(f*x+e))^(1/2)/(c*(-1+cos(f*x+e))/cos(f*x+
e))^(7/2)/sin(f*x+e)/cos(f*x+e)^3

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 2681 vs. \(2 (181) = 362\).
time = 3.32, size = 2681, normalized size = 14.26 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(f*x+e))^(1/2)/(c-c*sec(f*x+e))^(7/2),x, algorithm="maxima")

[Out]

-1/3*(3*(f*x + e)*cos(6*f*x + 6*e)^2 + 108*(f*x + e)*cos(5*f*x + 5*e)^2 + 675*(f*x + e)*cos(4*f*x + 4*e)^2 + 1
200*(f*x + e)*cos(3*f*x + 3*e)^2 + 675*(f*x + e)*cos(2*f*x + 2*e)^2 + 108*(f*x + e)*cos(f*x + e)^2 + 3*(f*x +
e)*sin(6*f*x + 6*e)^2 + 108*(f*x + e)*sin(5*f*x + 5*e)^2 + 675*(f*x + e)*sin(4*f*x + 4*e)^2 + 1200*(f*x + e)*s
in(3*f*x + 3*e)^2 + 675*(f*x + e)*sin(2*f*x + 2*e)^2 + 108*(f*x + e)*sin(f*x + e)^2 + 3*f*x + 6*(2*(6*cos(5*f*
x + 5*e) - 15*cos(4*f*x + 4*e) + 20*cos(3*f*x + 3*e) - 15*cos(2*f*x + 2*e) + 6*cos(f*x + e) - 1)*cos(6*f*x + 6
*e) - cos(6*f*x + 6*e)^2 + 12*(15*cos(4*f*x + 4*e) - 20*cos(3*f*x + 3*e) + 15*cos(2*f*x + 2*e) - 6*cos(f*x + e
) + 1)*cos(5*f*x + 5*e) - 36*cos(5*f*x + 5*e)^2 + 30*(20*cos(3*f*x + 3*e) - 15*cos(2*f*x + 2*e) + 6*cos(f*x +
e) - 1)*cos(4*f*x + 4*e) - 225*cos(4*f*x + 4*e)^2 + 40*(15*cos(2*f*x + 2*e) - 6*cos(f*x + e) + 1)*cos(3*f*x +
3*e) - 400*cos(3*f*x + 3*e)^2 + 30*(6*cos(f*x + e) - 1)*cos(2*f*x + 2*e) - 225*cos(2*f*x + 2*e)^2 - 36*cos(f*x
+ e)^2 + 2*(6*sin(5*f*x + 5*e) - 15*sin(4*f*x + 4*e) + 20*sin(3*f*x + 3*e) - 15*sin(2*f*x + 2*e) + 6*sin(f*x
+ e))*sin(6*f*x + 6*e) - sin(6*f*x + 6*e)^2 + 12*(15*sin(4*f*x + 4*e) - 20*sin(3*f*x + 3*e) + 15*sin(2*f*x + 2
*e) - 6*sin(f*x + e))*sin(5*f*x + 5*e) - 36*sin(5*f*x + 5*e)^2 + 30*(20*sin(3*f*x + 3*e) - 15*sin(2*f*x + 2*e)
+ 6*sin(f*x + e))*sin(4*f*x + 4*e) - 225*sin(4*f*x + 4*e)^2 + 120*(5*sin(2*f*x + 2*e) - 2*sin(f*x + e))*sin(3
*f*x + 3*e) - 400*sin(3*f*x + 3*e)^2 - 225*sin(2*f*x + 2*e)^2 + 180*sin(2*f*x + 2*e)*sin(f*x + e) - 36*sin(f*x
+ e)^2 + 12*cos(f*x + e) - 1)*arctan2(sin(f*x + e), cos(f*x + e) - 1) + 2*(3*f*x - 18*(f*x + e)*cos(5*f*x + 5
*e) + 45*(f*x + e)*cos(4*f*x + 4*e) - 60*(f*x + e)*cos(3*f*x + 3*e) + 45*(f*x + e)*cos(2*f*x + 2*e) - 18*(f*x
+ e)*cos(f*x + e) + 3*e + 9*sin(5*f*x + 5*e) - 27*sin(4*f*x + 4*e) + 40*sin(3*f*x + 3*e) - 27*sin(2*f*x + 2*e)
+ 9*sin(f*x + e))*cos(6*f*x + 6*e) - 6*(6*f*x + 90*(f*x + e)*cos(4*f*x + 4*e) - 120*(f*x + e)*cos(3*f*x + 3*e
) + 90*(f*x + e)*cos(2*f*x + 2*e) - 36*(f*x + e)*cos(f*x + e) + 6*e - 9*sin(4*f*x + 4*e) + 20*sin(3*f*x + 3*e)
- 9*sin(2*f*x + 2*e))*cos(5*f*x + 5*e) + 6*(15*f*x - 300*(f*x + e)*cos(3*f*x + 3*e) + 225*(f*x + e)*cos(2*f*x
+ 2*e) - 90*(f*x + e)*cos(f*x + e) + 15*e + 20*sin(3*f*x + 3*e) - 9*sin(f*x + e))*cos(4*f*x + 4*e) - 120*(f*x
+ 15*(f*x + e)*cos(2*f*x + 2*e) - 6*(f*x + e)*cos(f*x + e) + e + sin(2*f*x + 2*e) - sin(f*x + e))*cos(3*f*x +
3*e) + 18*(5*f*x - 30*(f*x + e)*cos(f*x + e) + 5*e - 3*sin(f*x + e))*cos(2*f*x + 2*e) - 36*(f*x + e)*cos(f*x
+ e) - 2*(18*(f*x + e)*sin(5*f*x + 5*e) - 45*(f*x + e)*sin(4*f*x + 4*e) + 60*(f*x + e)*sin(3*f*x + 3*e) - 45*(
f*x + e)*sin(2*f*x + 2*e) + 18*(f*x + e)*sin(f*x + e) + 9*cos(5*f*x + 5*e) - 27*cos(4*f*x + 4*e) + 40*cos(3*f*
x + 3*e) - 27*cos(2*f*x + 2*e) + 9*cos(f*x + e))*sin(6*f*x + 6*e) - 6*(90*(f*x + e)*sin(4*f*x + 4*e) - 120*(f*
x + e)*sin(3*f*x + 3*e) + 90*(f*x + e)*sin(2*f*x + 2*e) - 36*(f*x + e)*sin(f*x + e) + 9*cos(4*f*x + 4*e) - 20*
cos(3*f*x + 3*e) + 9*cos(2*f*x + 2*e) - 3)*sin(5*f*x + 5*e) - 6*(300*(f*x + e)*sin(3*f*x + 3*e) - 225*(f*x + e
)*sin(2*f*x + 2*e) + 90*(f*x + e)*sin(f*x + e) + 20*cos(3*f*x + 3*e) - 9*cos(f*x + e) + 9)*sin(4*f*x + 4*e) -
40*(45*(f*x + e)*sin(2*f*x + 2*e) - 18*(f*x + e)*sin(f*x + e) - 3*cos(2*f*x + 2*e) + 3*cos(f*x + e) - 2)*sin(3
*f*x + 3*e) - 54*(10*(f*x + e)*sin(f*x + e) - cos(f*x + e) + 1)*sin(2*f*x + 2*e) + 3*e + 18*sin(f*x + e))*sqrt
(a)*sqrt(c)/((c^4*cos(6*f*x + 6*e)^2 + 36*c^4*cos(5*f*x + 5*e)^2 + 225*c^4*cos(4*f*x + 4*e)^2 + 400*c^4*cos(3*
f*x + 3*e)^2 + 225*c^4*cos(2*f*x + 2*e)^2 + 36*c^4*cos(f*x + e)^2 + c^4*sin(6*f*x + 6*e)^2 + 36*c^4*sin(5*f*x
+ 5*e)^2 + 225*c^4*sin(4*f*x + 4*e)^2 + 400*c^4*sin(3*f*x + 3*e)^2 + 225*c^4*sin(2*f*x + 2*e)^2 - 180*c^4*sin(
2*f*x + 2*e)*sin(f*x + e) + 36*c^4*sin(f*x + e)^2 - 12*c^4*cos(f*x + e) + c^4 - 2*(6*c^4*cos(5*f*x + 5*e) - 15
*c^4*cos(4*f*x + 4*e) + 20*c^4*cos(3*f*x + 3*e) - 15*c^4*cos(2*f*x + 2*e) + 6*c^4*cos(f*x + e) - c^4)*cos(6*f*
x + 6*e) - 12*(15*c^4*cos(4*f*x + 4*e) - 20*c^4*cos(3*f*x + 3*e) + 15*c^4*cos(2*f*x + 2*e) - 6*c^4*cos(f*x + e
) + c^4)*cos(5*f*x + 5*e) - 30*(20*c^4*cos(3*f*x + 3*e) - 15*c^4*cos(2*f*x + 2*e) + 6*c^4*cos(f*x + e) - c^4)*
cos(4*f*x + 4*e) - 40*(15*c^4*cos(2*f*x + 2*e) - 6*c^4*cos(f*x + e) + c^4)*cos(3*f*x + 3*e) - 30*(6*c^4*cos(f*
x + e) - c^4)*cos(2*f*x + 2*e) - 2*(6*c^4*sin(5*f*x + 5*e) - 15*c^4*sin(4*f*x + 4*e) + 20*c^4*sin(3*f*x + 3*e)
- 15*c^4*sin(2*f*x + 2*e) + 6*c^4*sin(f*x + e))*sin(6*f*x + 6*e) - 12*(15*c^4*sin(4*f*x + 4*e) - 20*c^4*sin(3
*f*x + 3*e) + 15*c^4*sin(2*f*x + 2*e) - 6*c^4*sin(f*x + e))*sin(5*f*x + 5*e) - 30*(20*c^4*sin(3*f*x + 3*e) - 1
5*c^4*sin(2*f*x + 2*e) + 6*c^4*sin(f*x + e))*sin(4*f*x + 4*e) - 120*(5*c^4*sin(2*f*x + 2*e) - 2*c^4*sin(f*x +
e))*sin(3*f*x + 3*e))*f)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(f*x+e))^(1/2)/(c-c*sec(f*x+e))^(7/2),x, algorithm="fricas")

[Out]

integral(sqrt(a*sec(f*x + e) + a)*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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(f*x+e))**(1/2)/(c-c*sec(f*x+e))**(7/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4370 deep

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Giac [A]
time = 1.93, size = 208, normalized size = 1.11 \begin {gather*} -\frac {\sqrt {2} {\left (\frac {24 \, \sqrt {2} \sqrt {-a c} a \log \left (2 \, {\left | a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} \right |}\right )}{c^{4} {\left | a \right |}} - \frac {24 \, \sqrt {2} \sqrt {-a c} a \log \left ({\left | -a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a \right |}\right )}{c^{4} {\left | a \right |}} - \frac {\sqrt {2} {\left (44 \, {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a\right )}^{3} \sqrt {-a c} a + 111 \, {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a\right )}^{2} \sqrt {-a c} a^{2} + 96 \, {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a\right )} \sqrt {-a c} a^{3} + 28 \, \sqrt {-a c} a^{4}\right )}}{a^{3} c^{4} {\left | a \right |} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6}}\right )}}{48 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(f*x+e))^(1/2)/(c-c*sec(f*x+e))^(7/2),x, algorithm="giac")

[Out]

-1/48*sqrt(2)*(24*sqrt(2)*sqrt(-a*c)*a*log(2*abs(a*tan(1/2*f*x + 1/2*e)^2))/(c^4*abs(a)) - 24*sqrt(2)*sqrt(-a*
c)*a*log(abs(-a*tan(1/2*f*x + 1/2*e)^2 - a))/(c^4*abs(a)) - sqrt(2)*(44*(a*tan(1/2*f*x + 1/2*e)^2 - a)^3*sqrt(
-a*c)*a + 111*(a*tan(1/2*f*x + 1/2*e)^2 - a)^2*sqrt(-a*c)*a^2 + 96*(a*tan(1/2*f*x + 1/2*e)^2 - a)*sqrt(-a*c)*a
^3 + 28*sqrt(-a*c)*a^4)/(a^3*c^4*abs(a)*tan(1/2*f*x + 1/2*e)^6))/f

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}}{{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a/cos(e + f*x))^(1/2)/(c - c/cos(e + f*x))^(7/2),x)

[Out]

int((a + a/cos(e + f*x))^(1/2)/(c - c/cos(e + f*x))^(7/2), x)

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