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

Optimal. Leaf size=89 \[ -\frac{c^2 \tan (e+f x) (a \sec (e+f x)+a)^{3/2}}{3 f \sqrt{c-c \sec (e+f x)}}-\frac{c \tan (e+f x) (a \sec (e+f x)+a)^{3/2} \sqrt{c-c \sec (e+f x)}}{3 f} \]

[Out]

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

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Rubi [A]  time = 0.274389, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {3955, 3953} \[ -\frac{c^2 \tan (e+f x) (a \sec (e+f x)+a)^{3/2}}{3 f \sqrt{c-c \sec (e+f x)}}-\frac{c \tan (e+f x) (a \sec (e+f x)+a)^{3/2} \sqrt{c-c \sec (e+f x)}}{3 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3955

Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)
)^(n_), x_Symbol] :> -Simp[(d*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^(n - 1))/(f*(m + n)), x
] + Dist[(c*(2*n - 1))/(m + n), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^(n - 1), x], x] /
; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[n - 1/2, 0] &&  !LtQ[m, -2
^(-1)] &&  !(IGtQ[m - 1/2, 0] && LtQ[m, n])

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) (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{3/2} \, dx &=-\frac{c (a+a \sec (e+f x))^{3/2} \sqrt{c-c \sec (e+f x)} \tan (e+f x)}{3 f}+\frac{1}{3} (2 c) \int \sec (e+f x) (a+a \sec (e+f x))^{3/2} \sqrt{c-c \sec (e+f x)} \, dx\\ &=-\frac{c^2 (a+a \sec (e+f x))^{3/2} \tan (e+f x)}{3 f \sqrt{c-c \sec (e+f x)}}-\frac{c (a+a \sec (e+f x))^{3/2} \sqrt{c-c \sec (e+f x)} \tan (e+f x)}{3 f}\\ \end{align*}

Mathematica [A]  time = 0.430983, size = 78, normalized size = 0.88 \[ \frac{a c (3 \cos (2 (e+f x))+1) \csc \left (\frac{1}{2} (e+f x)\right ) \sec \left (\frac{1}{2} (e+f x)\right ) \sec ^2(e+f x) \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)}}{12 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*c*(1 + 3*Cos[2*(e + f*x)])*Csc[(e + f*x)/2]*Sec[(e + f*x)/2]*Sec[e + f*x]^2*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt
[c - c*Sec[e + f*x]])/(12*f)

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Maple [A]  time = 0.27, size = 83, normalized size = 0.9 \begin{align*}{\frac{a \left ( \sin \left ( fx+e \right ) \right ) ^{3} \left ( 2\,\cos \left ( fx+e \right ) -1 \right ) }{3\,f \left ( -1+\cos \left ( fx+e \right ) \right ) ^{3}\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}}}\sqrt{{\frac{a \left ( 1+\cos \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.84834, size = 743, normalized size = 8.35 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(6*a*c*cos(3*f*x + 3*e)*sin(2*f*x + 2*e) + 9*a*c*cos(f*x + e)*sin(2*f*x + 2*e) - 9*a*c*cos(2*f*x + 2*e)*si
n(f*x + e) - 3*a*c*sin(f*x + e) - (3*a*c*sin(5*f*x + 5*e) + 2*a*c*sin(3*f*x + 3*e) + 3*a*c*sin(f*x + e))*cos(6
*f*x + 6*e) + 9*(a*c*sin(4*f*x + 4*e) + a*c*sin(2*f*x + 2*e))*cos(5*f*x + 5*e) - 3*(2*a*c*sin(3*f*x + 3*e) + 3
*a*c*sin(f*x + e))*cos(4*f*x + 4*e) + (3*a*c*cos(5*f*x + 5*e) + 2*a*c*cos(3*f*x + 3*e) + 3*a*c*cos(f*x + e))*s
in(6*f*x + 6*e) - 3*(3*a*c*cos(4*f*x + 4*e) + 3*a*c*cos(2*f*x + 2*e) + a*c)*sin(5*f*x + 5*e) + 3*(2*a*c*cos(3*
f*x + 3*e) + 3*a*c*cos(f*x + e))*sin(4*f*x + 4*e) - 2*(3*a*c*cos(2*f*x + 2*e) + a*c)*sin(3*f*x + 3*e))*sqrt(a)
*sqrt(c)/((2*(3*cos(4*f*x + 4*e) + 3*cos(2*f*x + 2*e) + 1)*cos(6*f*x + 6*e) + cos(6*f*x + 6*e)^2 + 6*(3*cos(2*
f*x + 2*e) + 1)*cos(4*f*x + 4*e) + 9*cos(4*f*x + 4*e)^2 + 9*cos(2*f*x + 2*e)^2 + 6*(sin(4*f*x + 4*e) + sin(2*f
*x + 2*e))*sin(6*f*x + 6*e) + sin(6*f*x + 6*e)^2 + 9*sin(4*f*x + 4*e)^2 + 18*sin(4*f*x + 4*e)*sin(2*f*x + 2*e)
+ 9*sin(2*f*x + 2*e)^2 + 6*cos(2*f*x + 2*e) + 1)*f)

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Fricas [A]  time = 0.471238, size = 197, normalized size = 2.21 \begin{align*} \frac{{\left (3 \, a c \cos \left (f x + e\right )^{2} - a 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 )}}}{3 \, f \cos \left (f x + e\right )^{2} \sin \left (f x + e\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out