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

Optimal. Leaf size=43 \[ \frac{a \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{2 f \sqrt{a \sec (e+f x)+a}} \]

[Out]

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

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Rubi [A]  time = 0.132618, antiderivative size = 43, normalized size of antiderivative = 1., 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 = {3953} \[ \frac{a \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{2 f \sqrt{a \sec (e+f x)+a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3953

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

Mathematica [A]  time = 0.301323, size = 73, normalized size = 1.7 \[ \frac{c (2 \cos (e+f x)-1) \csc \left (\frac{1}{2} (e+f x)\right ) \sec \left (\frac{1}{2} (e+f x)\right ) \sec (e+f x) \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)}}{4 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(-1 + 2*Cos[e + f*x])*Csc[(e + f*x)/2]*Sec[(e + f*x)/2]*Sec[e + f*x]*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*
Sec[e + f*x]])/(4*f)

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Maple [A]  time = 0.313, size = 72, normalized size = 1.7 \begin{align*} -{\frac{\sin \left ( fx+e \right ) \left ( 3\,\cos \left ( fx+e \right ) -1 \right ) }{2\,f \left ( -1+\cos \left ( fx+e \right ) \right ) ^{2}}\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{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(f*x+e)*(c-c*sec(f*x+e))^(3/2)*(a+a*sec(f*x+e))^(1/2),x)

[Out]

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

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Maxima [B]  time = 1.86547, size = 402, normalized size = 9.35 \begin{align*} \frac{2 \,{\left (2 \, c \cos \left (3 \, f x + 3 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) - 2 \, c \cos \left (2 \, f x + 2 \, e\right ) \sin \left (f x + e\right ) -{\left (c \sin \left (3 \, f x + 3 \, e\right ) - c \sin \left (2 \, f x + 2 \, e\right ) + c \sin \left (f x + e\right )\right )} \cos \left (4 \, f x + 4 \, e\right ) +{\left (c \cos \left (3 \, f x + 3 \, e\right ) - c \cos \left (2 \, f x + 2 \, e\right ) + c \cos \left (f x + e\right )\right )} \sin \left (4 \, f x + 4 \, e\right ) -{\left (2 \, c \cos \left (2 \, f x + 2 \, e\right ) + c\right )} \sin \left (3 \, f x + 3 \, e\right ) +{\left (2 \, c \cos \left (f x + e\right ) + c\right )} \sin \left (2 \, f x + 2 \, e\right ) - c \sin \left (f x + e\right )\right )} \sqrt{a} \sqrt{c}}{{\left (2 \,{\left (2 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )} \cos \left (4 \, f x + 4 \, e\right ) + \cos \left (4 \, f x + 4 \, e\right )^{2} + 4 \, \cos \left (2 \, f x + 2 \, e\right )^{2} + \sin \left (4 \, f x + 4 \, e\right )^{2} + 4 \, \sin \left (4 \, f x + 4 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) + 4 \, \sin \left (2 \, f x + 2 \, e\right )^{2} + 4 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(c-c*sec(f*x+e))^(3/2)*(a+a*sec(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 0.4648, size = 186, normalized size = 4.33 \begin{align*} \frac{{\left (2 \, c \cos \left (f x + e\right ) - c\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 \, f \cos \left (f x + e\right ) \sin \left (f x + e\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(c-c*sec(f*x+e))^(3/2)*(a+a*sec(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

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

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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(sec(f*x+e)*(c-c*sec(f*x+e))**(3/2)*(a+a*sec(f*x+e))**(1/2),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(sec(f*x+e)*(c-c*sec(f*x+e))^(3/2)*(a+a*sec(f*x+e))^(1/2),x, algorithm="giac")

[Out]

Timed out