3.2.27 \(\int \frac {\sqrt {c-c \sec (e+f x)}}{(a+a \sec (e+f x))^{5/2}} \, dx\) [127]

Optimal. Leaf size=140 \[ -\frac {c \tan (e+f x)}{2 f (a+a \sec (e+f x))^{5/2} \sqrt {c-c \sec (e+f x)}}-\frac {c \tan (e+f x)}{a f (a+a \sec (e+f x))^{3/2} \sqrt {c-c \sec (e+f x)}}+\frac {c \log (1+\cos (e+f x)) \tan (e+f x)}{a^2 f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.18, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 4, 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 {c \tan (e+f x) \log (\cos (e+f x)+1)}{a^2 f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}-\frac {c \tan (e+f x)}{a f (a \sec (e+f x)+a)^{3/2} \sqrt {c-c \sec (e+f x)}}-\frac {c \tan (e+f x)}{2 f (a \sec (e+f x)+a)^{5/2} \sqrt {c-c \sec (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[c - c*Sec[e + f*x]]/(a + a*Sec[e + f*x])^(5/2),x]

[Out]

-1/2*(c*Tan[e + f*x])/(f*(a + a*Sec[e + f*x])^(5/2)*Sqrt[c - c*Sec[e + f*x]]) - (c*Tan[e + f*x])/(a*f*(a + a*S
ec[e + f*x])^(3/2)*Sqrt[c - c*Sec[e + f*x]]) + (c*Log[1 + Cos[e + f*x]]*Tan[e + f*x])/(a^2*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 {c-c \sec (e+f x)}}{(a+a \sec (e+f x))^{5/2}} \, dx &=-\frac {c \tan (e+f x)}{2 f (a+a \sec (e+f x))^{5/2} \sqrt {c-c \sec (e+f x)}}+\frac {\int \frac {\sqrt {c-c \sec (e+f x)}}{(a+a \sec (e+f x))^{3/2}} \, dx}{a}\\ &=-\frac {c \tan (e+f x)}{2 f (a+a \sec (e+f x))^{5/2} \sqrt {c-c \sec (e+f x)}}-\frac {c \tan (e+f x)}{a f (a+a \sec (e+f x))^{3/2} \sqrt {c-c \sec (e+f x)}}+\frac {\int \frac {\sqrt {c-c \sec (e+f x)}}{\sqrt {a+a \sec (e+f x)}} \, dx}{a^2}\\ &=-\frac {c \tan (e+f x)}{2 f (a+a \sec (e+f x))^{5/2} \sqrt {c-c \sec (e+f x)}}-\frac {c \tan (e+f x)}{a f (a+a \sec (e+f x))^{3/2} \sqrt {c-c \sec (e+f x)}}+\frac {(c \tan (e+f x)) \text {Subst}\left (\int \frac {1}{a+a x} \, dx,x,\cos (e+f x)\right )}{a f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}\\ &=-\frac {c \tan (e+f x)}{2 f (a+a \sec (e+f x))^{5/2} \sqrt {c-c \sec (e+f x)}}-\frac {c \tan (e+f x)}{a f (a+a \sec (e+f x))^{3/2} \sqrt {c-c \sec (e+f x)}}+\frac {c \log (1+\cos (e+f x)) \tan (e+f x)}{a^2 f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains complex when optimal does not.
time = 0.69, size = 151, normalized size = 1.08 \begin {gather*} \frac {i \cot \left (\frac {1}{2} (e+f x)\right ) \left (3 i+3 f x+\cos (2 (e+f x)) \left (f x+2 i \log \left (1+e^{i (e+f x)}\right )\right )+4 \cos (e+f x) \left (i+f x+2 i \log \left (1+e^{i (e+f x)}\right )\right )+6 i \log \left (1+e^{i (e+f x)}\right )\right ) \sqrt {c-c \sec (e+f x)}}{2 a^2 f (1+\cos (e+f x))^2 \sqrt {a (1+\sec (e+f x))}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c - c*Sec[e + f*x]]/(a + a*Sec[e + f*x])^(5/2),x]

[Out]

((I/2)*Cot[(e + f*x)/2]*(3*I + 3*f*x + Cos[2*(e + f*x)]*(f*x + (2*I)*Log[1 + E^(I*(e + f*x))]) + 4*Cos[e + f*x
]*(I + f*x + (2*I)*Log[1 + E^(I*(e + f*x))]) + (6*I)*Log[1 + E^(I*(e + f*x))])*Sqrt[c - c*Sec[e + f*x]])/(a^2*
f*(1 + Cos[e + f*x])^2*Sqrt[a*(1 + Sec[e + f*x])])

________________________________________________________________________________________

Maple [A]
time = 0.31, size = 152, normalized size = 1.09

method	result	size
default	\(\frac {\left (-1+\cos \left (f x +e \right )\right )^{2} \left (8 \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )+7 \left (\cos ^{2}\left (f x +e \right )\right )+16 \cos \left (f x +e \right ) \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )-2 \cos \left (f x +e \right )+8 \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )-5\right ) \sqrt {\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}}\, \sqrt {\frac {a \left (\cos \left (f x +e \right )+1\right )}{\cos \left (f x +e \right )}}\, \cos \left (f x +e \right )}{8 f \sin \left (f x +e \right )^{5} a^{3}}\)	\(152\)
risch	\(\frac {\left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, x}{a^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}-\frac {2 \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left (f x +e \right )}{a^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) f}-\frac {2 i \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left (2 \,{\mathrm e}^{3 i \left (f x +e \right )}+3 \,{\mathrm e}^{2 i \left (f x +e \right )}+2 \,{\mathrm e}^{i \left (f x +e \right )}\right )}{a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{3} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) f}-\frac {2 i \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{a^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) f}\)	\(422\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c-c*sec(f*x+e))^(1/2)/(a+a*sec(f*x+e))^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/8/f*(-1+cos(f*x+e))^2*(8*cos(f*x+e)^2*ln(2/(cos(f*x+e)+1))+7*cos(f*x+e)^2+16*cos(f*x+e)*ln(2/(cos(f*x+e)+1))
-2*cos(f*x+e)+8*ln(2/(cos(f*x+e)+1))-5)*(c*(-1+cos(f*x+e))/cos(f*x+e))^(1/2)*(a*(cos(f*x+e)+1)/cos(f*x+e))^(1/
2)*cos(f*x+e)/sin(f*x+e)^5/a^3

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1280 vs. \(2 (136) = 272\).
time = 0.73, size = 1280, normalized size = 9.14 \begin {gather*} -\frac {{\left ({\left (f x + e\right )} \cos \left (4 \, f x + 4 \, e\right )^{2} + 16 \, {\left (f x + e\right )} \cos \left (3 \, f x + 3 \, e\right )^{2} + 36 \, {\left (f x + e\right )} \cos \left (2 \, f x + 2 \, e\right )^{2} + 16 \, {\left (f x + e\right )} \cos \left (f x + e\right )^{2} + {\left (f x + e\right )} \sin \left (4 \, f x + 4 \, e\right )^{2} + 16 \, {\left (f x + e\right )} \sin \left (3 \, f x + 3 \, e\right )^{2} + 36 \, {\left (f x + e\right )} \sin \left (2 \, f x + 2 \, e\right )^{2} + 16 \, {\left (f x + e\right )} \sin \left (f x + e\right )^{2} + f x - 2 \, {\left (2 \, {\left (4 \, \cos \left (3 \, f x + 3 \, e\right ) + 6 \, \cos \left (2 \, f x + 2 \, e\right ) + 4 \, \cos \left (f x + e\right ) + 1\right )} \cos \left (4 \, f x + 4 \, e\right ) + \cos \left (4 \, f x + 4 \, e\right )^{2} + 8 \, {\left (6 \, \cos \left (2 \, f x + 2 \, e\right ) + 4 \, \cos \left (f x + e\right ) + 1\right )} \cos \left (3 \, f x + 3 \, e\right ) + 16 \, \cos \left (3 \, f x + 3 \, e\right )^{2} + 12 \, {\left (4 \, \cos \left (f x + e\right ) + 1\right )} \cos \left (2 \, f x + 2 \, e\right ) + 36 \, \cos \left (2 \, f x + 2 \, e\right )^{2} + 16 \, \cos \left (f x + e\right )^{2} + 4 \, {\left (2 \, \sin \left (3 \, f x + 3 \, e\right ) + 3 \, \sin \left (2 \, f x + 2 \, e\right ) + 2 \, \sin \left (f x + e\right )\right )} \sin \left (4 \, f x + 4 \, e\right ) + \sin \left (4 \, f x + 4 \, e\right )^{2} + 16 \, {\left (3 \, \sin \left (2 \, f x + 2 \, e\right ) + 2 \, \sin \left (f x + e\right )\right )} \sin \left (3 \, f x + 3 \, e\right ) + 16 \, \sin \left (3 \, f x + 3 \, e\right )^{2} + 36 \, \sin \left (2 \, f x + 2 \, e\right )^{2} + 48 \, \sin \left (2 \, f x + 2 \, e\right ) \sin \left (f x + e\right ) + 16 \, \sin \left (f x + e\right )^{2} + 8 \, \cos \left (f x + e\right ) + 1\right )} \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right ) + 1\right ) + 2 \, {\left (f x + 4 \, {\left (f x + e\right )} \cos \left (3 \, f x + 3 \, e\right ) + 6 \, {\left (f x + e\right )} \cos \left (2 \, f x + 2 \, e\right ) + 4 \, {\left (f x + e\right )} \cos \left (f x + e\right ) + e - 2 \, \sin \left (3 \, f x + 3 \, e\right ) - 3 \, \sin \left (2 \, f x + 2 \, e\right ) - 2 \, \sin \left (f x + e\right )\right )} \cos \left (4 \, f x + 4 \, e\right ) + 8 \, {\left (f x + 6 \, {\left (f x + e\right )} \cos \left (2 \, f x + 2 \, e\right ) + 4 \, {\left (f x + e\right )} \cos \left (f x + e\right ) + e\right )} \cos \left (3 \, f x + 3 \, e\right ) + 12 \, {\left (f x + 4 \, {\left (f x + e\right )} \cos \left (f x + e\right ) + e\right )} \cos \left (2 \, f x + 2 \, e\right ) + 8 \, {\left (f x + e\right )} \cos \left (f x + e\right ) + 2 \, {\left (4 \, {\left (f x + e\right )} \sin \left (3 \, f x + 3 \, e\right ) + 6 \, {\left (f x + e\right )} \sin \left (2 \, f x + 2 \, e\right ) + 4 \, {\left (f x + e\right )} \sin \left (f x + e\right ) + 2 \, \cos \left (3 \, f x + 3 \, e\right ) + 3 \, \cos \left (2 \, f x + 2 \, e\right ) + 2 \, \cos \left (f x + e\right )\right )} \sin \left (4 \, f x + 4 \, e\right ) + 4 \, {\left (12 \, {\left (f x + e\right )} \sin \left (2 \, f x + 2 \, e\right ) + 8 \, {\left (f x + e\right )} \sin \left (f x + e\right ) - 1\right )} \sin \left (3 \, f x + 3 \, e\right ) + 6 \, {\left (8 \, {\left (f x + e\right )} \sin \left (f x + e\right ) - 1\right )} \sin \left (2 \, f x + 2 \, e\right ) + e - 4 \, \sin \left (f x + e\right )\right )} \sqrt {a} \sqrt {c}}{{\left (a^{3} \cos \left (4 \, f x + 4 \, e\right )^{2} + 16 \, a^{3} \cos \left (3 \, f x + 3 \, e\right )^{2} + 36 \, a^{3} \cos \left (2 \, f x + 2 \, e\right )^{2} + 16 \, a^{3} \cos \left (f x + e\right )^{2} + a^{3} \sin \left (4 \, f x + 4 \, e\right )^{2} + 16 \, a^{3} \sin \left (3 \, f x + 3 \, e\right )^{2} + 36 \, a^{3} \sin \left (2 \, f x + 2 \, e\right )^{2} + 48 \, a^{3} \sin \left (2 \, f x + 2 \, e\right ) \sin \left (f x + e\right ) + 16 \, a^{3} \sin \left (f x + e\right )^{2} + 8 \, a^{3} \cos \left (f x + e\right ) + a^{3} + 2 \, {\left (4 \, a^{3} \cos \left (3 \, f x + 3 \, e\right ) + 6 \, a^{3} \cos \left (2 \, f x + 2 \, e\right ) + 4 \, a^{3} \cos \left (f x + e\right ) + a^{3}\right )} \cos \left (4 \, f x + 4 \, e\right ) + 8 \, {\left (6 \, a^{3} \cos \left (2 \, f x + 2 \, e\right ) + 4 \, a^{3} \cos \left (f x + e\right ) + a^{3}\right )} \cos \left (3 \, f x + 3 \, e\right ) + 12 \, {\left (4 \, a^{3} \cos \left (f x + e\right ) + a^{3}\right )} \cos \left (2 \, f x + 2 \, e\right ) + 4 \, {\left (2 \, a^{3} \sin \left (3 \, f x + 3 \, e\right ) + 3 \, a^{3} \sin \left (2 \, f x + 2 \, e\right ) + 2 \, a^{3} \sin \left (f x + e\right )\right )} \sin \left (4 \, f x + 4 \, e\right ) + 16 \, {\left (3 \, a^{3} \sin \left (2 \, f x + 2 \, e\right ) + 2 \, a^{3} \sin \left (f x + e\right )\right )} \sin \left (3 \, f x + 3 \, e\right )\right )} f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c*sec(f*x+e))^(1/2)/(a+a*sec(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

-((f*x + e)*cos(4*f*x + 4*e)^2 + 16*(f*x + e)*cos(3*f*x + 3*e)^2 + 36*(f*x + e)*cos(2*f*x + 2*e)^2 + 16*(f*x +
e)*cos(f*x + e)^2 + (f*x + e)*sin(4*f*x + 4*e)^2 + 16*(f*x + e)*sin(3*f*x + 3*e)^2 + 36*(f*x + e)*sin(2*f*x +
2*e)^2 + 16*(f*x + e)*sin(f*x + e)^2 + f*x - 2*(2*(4*cos(3*f*x + 3*e) + 6*cos(2*f*x + 2*e) + 4*cos(f*x + e) +
1)*cos(4*f*x + 4*e) + cos(4*f*x + 4*e)^2 + 8*(6*cos(2*f*x + 2*e) + 4*cos(f*x + e) + 1)*cos(3*f*x + 3*e) + 16*
cos(3*f*x + 3*e)^2 + 12*(4*cos(f*x + e) + 1)*cos(2*f*x + 2*e) + 36*cos(2*f*x + 2*e)^2 + 16*cos(f*x + e)^2 + 4*
(2*sin(3*f*x + 3*e) + 3*sin(2*f*x + 2*e) + 2*sin(f*x + e))*sin(4*f*x + 4*e) + sin(4*f*x + 4*e)^2 + 16*(3*sin(2
*f*x + 2*e) + 2*sin(f*x + e))*sin(3*f*x + 3*e) + 16*sin(3*f*x + 3*e)^2 + 36*sin(2*f*x + 2*e)^2 + 48*sin(2*f*x
+ 2*e)*sin(f*x + e) + 16*sin(f*x + e)^2 + 8*cos(f*x + e) + 1)*arctan2(sin(f*x + e), cos(f*x + e) + 1) + 2*(f*x
+ 4*(f*x + e)*cos(3*f*x + 3*e) + 6*(f*x + e)*cos(2*f*x + 2*e) + 4*(f*x + e)*cos(f*x + e) + e - 2*sin(3*f*x +
3*e) - 3*sin(2*f*x + 2*e) - 2*sin(f*x + e))*cos(4*f*x + 4*e) + 8*(f*x + 6*(f*x + e)*cos(2*f*x + 2*e) + 4*(f*x
+ e)*cos(f*x + e) + e)*cos(3*f*x + 3*e) + 12*(f*x + 4*(f*x + e)*cos(f*x + e) + e)*cos(2*f*x + 2*e) + 8*(f*x +
e)*cos(f*x + e) + 2*(4*(f*x + e)*sin(3*f*x + 3*e) + 6*(f*x + e)*sin(2*f*x + 2*e) + 4*(f*x + e)*sin(f*x + e) +
2*cos(3*f*x + 3*e) + 3*cos(2*f*x + 2*e) + 2*cos(f*x + e))*sin(4*f*x + 4*e) + 4*(12*(f*x + e)*sin(2*f*x + 2*e)
+ 8*(f*x + e)*sin(f*x + e) - 1)*sin(3*f*x + 3*e) + 6*(8*(f*x + e)*sin(f*x + e) - 1)*sin(2*f*x + 2*e) + e - 4*s
in(f*x + e))*sqrt(a)*sqrt(c)/((a^3*cos(4*f*x + 4*e)^2 + 16*a^3*cos(3*f*x + 3*e)^2 + 36*a^3*cos(2*f*x + 2*e)^2
+ 16*a^3*cos(f*x + e)^2 + a^3*sin(4*f*x + 4*e)^2 + 16*a^3*sin(3*f*x + 3*e)^2 + 36*a^3*sin(2*f*x + 2*e)^2 + 48*
a^3*sin(2*f*x + 2*e)*sin(f*x + e) + 16*a^3*sin(f*x + e)^2 + 8*a^3*cos(f*x + e) + a^3 + 2*(4*a^3*cos(3*f*x + 3*
e) + 6*a^3*cos(2*f*x + 2*e) + 4*a^3*cos(f*x + e) + a^3)*cos(4*f*x + 4*e) + 8*(6*a^3*cos(2*f*x + 2*e) + 4*a^3*c
os(f*x + e) + a^3)*cos(3*f*x + 3*e) + 12*(4*a^3*cos(f*x + e) + a^3)*cos(2*f*x + 2*e) + 4*(2*a^3*sin(3*f*x + 3*
e) + 3*a^3*sin(2*f*x + 2*e) + 2*a^3*sin(f*x + e))*sin(4*f*x + 4*e) + 16*(3*a^3*sin(2*f*x + 2*e) + 2*a^3*sin(f*
x + e))*sin(3*f*x + 3*e))*f)

________________________________________________________________________________________

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((c-c*sec(f*x+e))^(1/2)/(a+a*sec(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- c \left (\sec {\left (e + f x \right )} - 1\right )}}{\left (a \left (\sec {\left (e + f x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c*sec(f*x+e))**(1/2)/(a+a*sec(f*x+e))**(5/2),x)

[Out]

Integral(sqrt(-c*(sec(e + f*x) - 1))/(a*(sec(e + f*x) + 1))**(5/2), x)

________________________________________________________________________________________

Giac [A]
time = 1.58, size = 124, normalized size = 0.89 \begin {gather*} -\frac {\sqrt {2} {\left (\frac {8 \, \sqrt {2} \sqrt {-a c} c \log \left ({\left | c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + c \right |}\right )}{a^{3} {\left | c \right |}} + \frac {\sqrt {2} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{2} \sqrt {-a c} a^{3} c {\left | c \right |} - 4 \, \sqrt {2} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )} \sqrt {-a c} a^{3} c^{2} {\left | c \right |}}{a^{6} c^{4}}\right )}}{16 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c*sec(f*x+e))^(1/2)/(a+a*sec(f*x+e))^(5/2),x, algorithm="giac")

[Out]

-1/16*sqrt(2)*(8*sqrt(2)*sqrt(-a*c)*c*log(abs(c*tan(1/2*f*x + 1/2*e)^2 + c))/(a^3*abs(c)) + (sqrt(2)*(c*tan(1/
2*f*x + 1/2*e)^2 - c)^2*sqrt(-a*c)*a^3*c*abs(c) - 4*sqrt(2)*(c*tan(1/2*f*x + 1/2*e)^2 - c)*sqrt(-a*c)*a^3*c^2*
abs(c))/(a^6*c^4))/f

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {c-\frac {c}{\cos \left (e+f\,x\right )}}}{{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c - c/cos(e + f*x))^(1/2)/(a + a/cos(e + f*x))^(5/2),x)

[Out]

int((c - c/cos(e + f*x))^(1/2)/(a + a/cos(e + f*x))^(5/2), x)

________________________________________________________________________________________