3.2.47 \(\int \frac {\sec (e+f x) \sqrt {c-c \sec (e+f x)}}{(a+a \sec (e+f x))^{5/2}} \, dx\) [147]

Optimal. Leaf size=43 \[ \frac {c \tan (e+f x)}{2 f (a+a \sec (e+f x))^{5/2} \sqrt {c-c \sec (e+f x)}} \]

[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)

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Rubi [A]
time = 0.09, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {4038} \begin {gather*} \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[(Sec[e + f*x]*Sqrt[c - c*Sec[e + f*x]])/(a + a*Sec[e + f*x])^(5/2),x]

[Out]

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

Rule 4038

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)]

Rubi steps

\begin {align*} \int \frac {\sec (e+f x) \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)}}\\ \end {align*}

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Mathematica [A]
time = 0.25, size = 71, normalized size = 1.65 \begin {gather*} \frac {(1+2 \cos (e+f x)) \csc \left (\frac {1}{2} (e+f x)\right ) \sec ^3\left (\frac {1}{2} (e+f x)\right ) \sqrt {c-c \sec (e+f x)}}{8 a^2 f \sqrt {a (1+\sec (e+f x))}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((1 + 2*Cos[e + f*x])*Csc[(e + f*x)/2]*Sec[(e + f*x)/2]^3*Sqrt[c - c*Sec[e + f*x]])/(8*a^2*f*Sqrt[a*(1 + Sec[e
+ f*x])])

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Maple [A]
time = 3.05, size = 75, normalized size = 1.74

method	result	size
default	\(-\frac {\sqrt {\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}}\, \left (-1+\cos \left (f x +e \right )\right )^{2} \left (\cos ^{3}\left (f x +e \right )\right ) \sqrt {\frac {a \left (\cos \left (f x +e \right )+1\right )}{\cos \left (f x +e \right )}}}{2 f \sin \left (f x +e \right )^{5} a^{3}}\)	\(75\)
risch	\(\frac {2 i \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left ({\mathrm e}^{3 i \left (f x +e \right )}+{\mathrm e}^{2 i \left (f x +e \right )}+{\mathrm e}^{i \left (f x +e \right )}\right )}{a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{3} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) f}\)	\(124\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.50, size = 62, normalized size = 1.44 \begin {gather*} -\frac {\sqrt {c} {\left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}^{2} {\left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}^{2}}{8 \, \sqrt {-a} a^{2} f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*sqrt(c)*(sin(f*x + e)/(cos(f*x + e) + 1) + 1)^2*(sin(f*x + e)/(cos(f*x + e) + 1) - 1)^2/(sqrt(-a)*a^2*f)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (40) = 80\).
time = 1.71, size = 113, normalized size = 2.63 \begin {gather*} \frac {{\left (2 \, \cos \left (f x + e\right )^{2} + \cos \left (f x + e\right )\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 )}}}{2 \, {\left (a^{3} f \cos \left (f x + e\right )^{2} + 2 \, a^{3} f \cos \left (f x + e\right ) + a^{3} f\right )} \sin \left (f x + e\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*cos(f*x + e)^2 + cos(f*x + e))*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt((c*cos(f*x + e) - c)/cos(f*
x + e))/((a^3*f*cos(f*x + e)^2 + 2*a^3*f*cos(f*x + e) + a^3*f)*sin(f*x + e))

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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 )} \sec {\left (e + f x \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(sec(f*x+e)*(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))*sec(e + f*x)/(a*(sec(e + f*x) + 1))**(5/2), x)

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Giac [A]
time = 1.54, size = 37, normalized size = 0.86 \begin {gather*} -\frac {{\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{2}}{8 \, \sqrt {-a c} a^{2} f {\left | c \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(c*tan(1/2*f*x + 1/2*e)^2 - c)^2/(sqrt(-a*c)*a^2*f*abs(c))

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Mupad [B]
time = 3.25, size = 120, normalized size = 2.79 \begin {gather*} -\frac {2\,\left (3\,\sin \left (e+f\,x\right )+3\,\sin \left (2\,e+2\,f\,x\right )+\sin \left (3\,e+3\,f\,x\right )\right )\,\sqrt {\frac {c\,\left (\cos \left (e+f\,x\right )-1\right )}{\cos \left (e+f\,x\right )}}}{a^2\,f\,\sqrt {\frac {a\,\left (\cos \left (e+f\,x\right )+1\right )}{\cos \left (e+f\,x\right )}}\,\left (4\,\cos \left (2\,e+2\,f\,x\right )-4\,\cos \left (e+f\,x\right )+4\,\cos \left (3\,e+3\,f\,x\right )+\cos \left (4\,e+4\,f\,x\right )-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*(3*sin(e + f*x) + 3*sin(2*e + 2*f*x) + sin(3*e + 3*f*x))*((c*(cos(e + f*x) - 1))/cos(e + f*x))^(1/2))/(a^2
*f*((a*(cos(e + f*x) + 1))/cos(e + f*x))^(1/2)*(4*cos(2*e + 2*f*x) - 4*cos(e + f*x) + 4*cos(3*e + 3*f*x) + cos
(4*e + 4*f*x) - 5))

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