3.1.100 \(\int \frac {(a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{7/2}} \, dx\) [100]

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

[Out]

-2/3*a^2*tan(f*x+e)/f/(c-c*sec(f*x+e))^(7/2)/(a+a*sec(f*x+e))^(1/2)-1/2*a^2*tan(f*x+e)/c/f/(c-c*sec(f*x+e))^(5
/2)/(a+a*sec(f*x+e))^(1/2)-a^2*tan(f*x+e)/c^2/f/(c-c*sec(f*x+e))^(3/2)/(a+a*sec(f*x+e))^(1/2)+a^2*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 = 0.25, antiderivative size = 196, normalized size of antiderivative = 1.00, 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 = {3993, 3992, 3996, 31} \begin {gather*} \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)}{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)}{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}} \end {gather*}

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 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 3993

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 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 {(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 ) \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 {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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 2.21, size = 199, normalized size = 1.02 \begin {gather*} \frac {a \left (-50+30 i f x-3 i f x \cos (3 (e+f x))+6 i \cos (2 (e+f x)) \left (4 i+3 f x+6 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 )+\cos (e+f x) \left (66-45 i f x+90 \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[(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 = 0.29, size = 289, normalized size = 1.47

method	result	size
default	\(-\frac {\left (-1+\cos \left (f x +e \right )\right ) \left (48 \left (\cos ^{3}\left (f x +e \right )\right ) \ln \left (-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right )-24 \left (\cos ^{3}\left (f x +e \right )\right ) \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )-35 \left (\cos ^{3}\left (f x +e \right )\right )-144 \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right )+72 \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )+9 \left (\cos ^{2}\left (f x +e \right )\right )+144 \cos \left (f x +e \right ) \ln \left (-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right )-72 \cos \left (f x +e \right ) \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )+27 \cos \left (f x +e \right )-48 \ln \left (-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right )+24 \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )-17\right ) \sqrt {\frac {a \left (\cos \left (f x +e \right )+1\right )}{\cos \left (f x +e \right )}}\, a}{24 f \left (\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}\right )^{\frac {7}{2}} \sin \left (f x +e \right ) \cos \left (f x +e \right )^{3}}\)	\(289\)
risch	\(\frac {a \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) x}{c^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}}-\frac {2 a \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) \left (f x +e \right )}{c^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}+\frac {2 i a \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left (12 \,{\mathrm e}^{5 i \left (f x +e \right )}-33 \,{\mathrm e}^{4 i \left (f x +e \right )}+50 \,{\mathrm e}^{3 i \left (f x +e \right )}-33 \,{\mathrm e}^{2 i \left (f x +e \right )}+12 \,{\mathrm e}^{i \left (f x +e \right )}\right )}{3 c^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{5} \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}-\frac {2 i a \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{c^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\)	\(448\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 3777 vs. \(2 (189) = 378\).
time = 3.33, size = 3777, normalized size = 19.27 \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))^(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...

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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))^(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(-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))**(3/2)/(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 = 1.72, size = 200, normalized size = 1.02 \begin {gather*} -\frac {\frac {24 \, \sqrt {-a c} a^{2} \log \left ({\left | a \right |} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}\right )}{c^{4} {\left | a \right |}} - \frac {24 \, \sqrt {-a c} a^{2} \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 {44 \, {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a\right )}^{3} \sqrt {-a c} a^{2} + 108 \, {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a\right )}^{2} \sqrt {-a c} a^{3} + 93 \, {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a\right )} \sqrt {-a c} a^{4} + 27 \, \sqrt {-a c} a^{5}}{a^{3} c^{4} {\left | a \right |} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6}}}{24 \, f} \end {gather*}

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]

-1/24*(24*sqrt(-a*c)*a^2*log(abs(a)*tan(1/2*f*x + 1/2*e)^2)/(c^4*abs(a)) - 24*sqrt(-a*c)*a^2*log(abs(-a*tan(1/
2*f*x + 1/2*e)^2 - a))/(c^4*abs(a)) - (44*(a*tan(1/2*f*x + 1/2*e)^2 - a)^3*sqrt(-a*c)*a^2 + 108*(a*tan(1/2*f*x
+ 1/2*e)^2 - a)^2*sqrt(-a*c)*a^3 + 93*(a*tan(1/2*f*x + 1/2*e)^2 - a)*sqrt(-a*c)*a^4 + 27*sqrt(-a*c)*a^5)/(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 {{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{3/2}}{{\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))^(3/2)/(c - c/cos(e + f*x))^(7/2),x)

[Out]

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

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