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

Optimal. Leaf size=89 \[ \frac{2 c \tan (e+f x) \sqrt{c-c \sec (e+f x)}}{3 f (a \sec (e+f x)+a)^2}-\frac{4 c^2 \tan (e+f x)}{3 f \left (a^2 \sec (e+f x)+a^2\right ) \sqrt{c-c \sec (e+f x)}} \]

[Out]

(-4*c^2*Tan[e + f*x])/(3*f*(a^2 + a^2*Sec[e + f*x])*Sqrt[c - c*Sec[e + f*x]]) + (2*c*Sqrt[c - c*Sec[e + f*x]]*
Tan[e + f*x])/(3*f*(a + a*Sec[e + f*x])^2)

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Rubi [A]  time = 0.219272, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {3954, 3953} \[ \frac{2 c \tan (e+f x) \sqrt{c-c \sec (e+f x)}}{3 f (a \sec (e+f x)+a)^2}-\frac{4 c^2 \tan (e+f x)}{3 f \left (a^2 \sec (e+f x)+a^2\right ) \sqrt{c-c \sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*c^2*Tan[e + f*x])/(3*f*(a^2 + a^2*Sec[e + f*x])*Sqrt[c - c*Sec[e + f*x]]) + (2*c*Sqrt[c - c*Sec[e + f*x]]*
Tan[e + f*x])/(3*f*(a + a*Sec[e + f*x])^2)

Rule 3954

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

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 \frac{\sec (e+f x) (c-c \sec (e+f x))^{3/2}}{(a+a \sec (e+f x))^2} \, dx &=\frac{2 c \sqrt{c-c \sec (e+f x)} \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}-\frac{(2 c) \int \frac{\sec (e+f x) \sqrt{c-c \sec (e+f x)}}{a+a \sec (e+f x)} \, dx}{3 a}\\ &=-\frac{4 c^2 \tan (e+f x)}{3 f \left (a^2+a^2 \sec (e+f x)\right ) \sqrt{c-c \sec (e+f x)}}+\frac{2 c \sqrt{c-c \sec (e+f x)} \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}\\ \end{align*}

Mathematica [A]  time = 0.248986, size = 60, normalized size = 0.67 \[ \frac{2 c \cos (e+f x) (\cos (e+f x)+3) \cot \left (\frac{1}{2} (e+f x)\right ) \sqrt{c-c \sec (e+f x)}}{3 a^2 f (\cos (e+f x)+1)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.207, size = 53, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 2\,\cos \left ( fx+e \right ) +6 \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}}{3\,f{a}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{3}} \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))^2,x)

[Out]

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

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Maxima [A]  time = 1.53611, size = 149, normalized size = 1.67 \begin{align*} -\frac{2 \, \sqrt{2} c^{\frac{3}{2}} - \frac{3 \, \sqrt{2} c^{\frac{3}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac{\sqrt{2} c^{\frac{3}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}}}{3 \, a^{2} f{\left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}^{\frac{3}{2}}{\left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}^{\frac{3}{2}}} \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))^2,x, algorithm="maxima")

[Out]

-1/3*(2*sqrt(2)*c^(3/2) - 3*sqrt(2)*c^(3/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + sqrt(2)*c^(3/2)*sin(f*x + e)
^6/(cos(f*x + e) + 1)^6)/(a^2*f*(sin(f*x + e)/(cos(f*x + e) + 1) + 1)^(3/2)*(sin(f*x + e)/(cos(f*x + e) + 1) -
1)^(3/2))

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Fricas [A]  time = 0.464593, size = 171, normalized size = 1.92 \begin{align*} \frac{2 \,{\left (c \cos \left (f x + e\right )^{2} + 3 \, c \cos \left (f x + e\right )\right )} \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{3 \,{\left (a^{2} f \cos \left (f x + e\right ) + a^{2} f\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))^2,x, algorithm="fricas")

[Out]

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

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

[Out]

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

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Giac [A]  time = 2.22855, size = 84, normalized size = 0.94 \begin{align*} \frac{\sqrt{2}{\left ({\left (c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c\right )}^{\frac{3}{2}} c + 3 \, \sqrt{c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c} c^{2}\right )}}{3 \, a^{2} c 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))^2,x, algorithm="giac")

[Out]

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