3.119 \(\int \frac{\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{5/2}} \, dx\)

Optimal. Leaf size=42 \[ -\frac{\tan (e+f x) (a \sec (e+f x)+a)^{3/2}}{4 f (c-c \sec (e+f x))^{5/2}} \]

[Out]

-((a + a*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(4*f*(c - c*Sec[e + f*x])^(5/2))

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Rubi [A]  time = 0.149108, antiderivative size = 42, 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 = {3950} \[ -\frac{\tan (e+f x) (a \sec (e+f x)+a)^{3/2}}{4 f (c-c \sec (e+f x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(Sec[e + f*x]*(a + a*Sec[e + f*x])^(3/2))/(c - c*Sec[e + f*x])^(5/2),x]

[Out]

-((a + a*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(4*f*(c - c*Sec[e + f*x])^(5/2))

Rule 3950

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

Rubi steps

\begin{align*} \int \frac{\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{5/2}} \, dx &=-\frac{(a+a \sec (e+f x))^{3/2} \tan (e+f x)}{4 f (c-c \sec (e+f x))^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.492289, size = 63, normalized size = 1.5 \[ \frac{a \tan \left (\frac{1}{2} (e+f x)\right ) \sec (e+f x) \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)}}{c^3 f (\sec (e+f x)-1)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^(3/2))/(c - c*Sec[e + f*x])^(5/2),x]

[Out]

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

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Maple [A]  time = 0.267, size = 73, normalized size = 1.7 \begin{align*}{\frac{a \left ( \sin \left ( fx+e \right ) \right ) ^{3}}{4\,f \left ( -1+\cos \left ( fx+e \right ) \right ) \left ( \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{5}{2}}}} \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))^(5/2),x)

[Out]

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

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Maxima [B]  time = 1.82011, size = 720, normalized size = 17.14 \begin{align*} \frac{2 \,{\left (6 \, a \cos \left (3 \, f x + 3 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) + 6 \, a \cos \left (f x + e\right ) \sin \left (2 \, f x + 2 \, e\right ) - 6 \, a \cos \left (2 \, f x + 2 \, e\right ) \sin \left (f x + e\right ) -{\left (a \sin \left (3 \, f x + 3 \, e\right ) + a \sin \left (f x + e\right )\right )} \cos \left (4 \, f x + 4 \, e\right ) +{\left (a \cos \left (3 \, f x + 3 \, e\right ) + a \cos \left (f x + e\right )\right )} \sin \left (4 \, f x + 4 \, e\right ) -{\left (6 \, a \cos \left (2 \, f x + 2 \, e\right ) + a\right )} \sin \left (3 \, f x + 3 \, e\right ) - a \sin \left (f x + e\right )\right )} \sqrt{a} \sqrt{c}}{{\left (c^{3} \cos \left (4 \, f x + 4 \, e\right )^{2} + 16 \, c^{3} \cos \left (3 \, f x + 3 \, e\right )^{2} + 36 \, c^{3} \cos \left (2 \, f x + 2 \, e\right )^{2} + 16 \, c^{3} \cos \left (f x + e\right )^{2} + c^{3} \sin \left (4 \, f x + 4 \, e\right )^{2} + 16 \, c^{3} \sin \left (3 \, f x + 3 \, e\right )^{2} + 36 \, c^{3} \sin \left (2 \, f x + 2 \, e\right )^{2} - 48 \, c^{3} \sin \left (2 \, f x + 2 \, e\right ) \sin \left (f x + e\right ) + 16 \, c^{3} \sin \left (f x + e\right )^{2} - 8 \, c^{3} \cos \left (f x + e\right ) + c^{3} - 2 \,{\left (4 \, c^{3} \cos \left (3 \, f x + 3 \, e\right ) - 6 \, c^{3} \cos \left (2 \, f x + 2 \, e\right ) + 4 \, c^{3} \cos \left (f x + e\right ) - c^{3}\right )} \cos \left (4 \, f x + 4 \, e\right ) - 8 \,{\left (6 \, c^{3} \cos \left (2 \, f x + 2 \, e\right ) - 4 \, c^{3} \cos \left (f x + e\right ) + c^{3}\right )} \cos \left (3 \, f x + 3 \, e\right ) - 12 \,{\left (4 \, c^{3} \cos \left (f x + e\right ) - c^{3}\right )} \cos \left (2 \, f x + 2 \, e\right ) - 4 \,{\left (2 \, c^{3} \sin \left (3 \, f x + 3 \, e\right ) - 3 \, c^{3} \sin \left (2 \, f x + 2 \, e\right ) + 2 \, c^{3} \sin \left (f x + e\right )\right )} \sin \left (4 \, f x + 4 \, e\right ) - 16 \,{\left (3 \, c^{3} \sin \left (2 \, f x + 2 \, e\right ) - 2 \, c^{3} \sin \left (f x + e\right )\right )} \sin \left (3 \, f x + 3 \, e\right )\right )} f} \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))^(5/2),x, algorithm="maxima")

[Out]

2*(6*a*cos(3*f*x + 3*e)*sin(2*f*x + 2*e) + 6*a*cos(f*x + e)*sin(2*f*x + 2*e) - 6*a*cos(2*f*x + 2*e)*sin(f*x +
e) - (a*sin(3*f*x + 3*e) + a*sin(f*x + e))*cos(4*f*x + 4*e) + (a*cos(3*f*x + 3*e) + a*cos(f*x + e))*sin(4*f*x
+ 4*e) - (6*a*cos(2*f*x + 2*e) + a)*sin(3*f*x + 3*e) - a*sin(f*x + e))*sqrt(a)*sqrt(c)/((c^3*cos(4*f*x + 4*e)^
2 + 16*c^3*cos(3*f*x + 3*e)^2 + 36*c^3*cos(2*f*x + 2*e)^2 + 16*c^3*cos(f*x + e)^2 + c^3*sin(4*f*x + 4*e)^2 + 1
6*c^3*sin(3*f*x + 3*e)^2 + 36*c^3*sin(2*f*x + 2*e)^2 - 48*c^3*sin(2*f*x + 2*e)*sin(f*x + e) + 16*c^3*sin(f*x +
e)^2 - 8*c^3*cos(f*x + e) + c^3 - 2*(4*c^3*cos(3*f*x + 3*e) - 6*c^3*cos(2*f*x + 2*e) + 4*c^3*cos(f*x + e) - c
^3)*cos(4*f*x + 4*e) - 8*(6*c^3*cos(2*f*x + 2*e) - 4*c^3*cos(f*x + e) + c^3)*cos(3*f*x + 3*e) - 12*(4*c^3*cos(
f*x + e) - c^3)*cos(2*f*x + 2*e) - 4*(2*c^3*sin(3*f*x + 3*e) - 3*c^3*sin(2*f*x + 2*e) + 2*c^3*sin(f*x + e))*si
n(4*f*x + 4*e) - 16*(3*c^3*sin(2*f*x + 2*e) - 2*c^3*sin(f*x + e))*sin(3*f*x + 3*e))*f)

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Fricas [B]  time = 0.470585, size = 225, normalized size = 5.36 \begin{align*} \frac{a \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 )}} \cos \left (f x + e\right )^{2}}{{\left (c^{3} f \cos \left (f x + e\right )^{2} - 2 \, c^{3} f \cos \left (f x + e\right ) + c^{3} 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)*(a+a*sec(f*x+e))^(3/2)/(c-c*sec(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

a*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt((c*cos(f*x + e) - c)/cos(f*x + e))*cos(f*x + e)^2/((c^3*f*cos(f
*x + e)^2 - 2*c^3*f*cos(f*x + e) + c^3*f)*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))**(5/2),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))^(5/2),x, algorithm="giac")

[Out]

Timed out