3.468 \(\int \frac{1}{\sqrt{b \sec (e+f x)} \sin ^{\frac{5}{2}}(e+f x)} \, dx\)

Optimal. Leaf size=30 \[ -\frac{2 b}{3 f \sin ^{\frac{3}{2}}(e+f x) (b \sec (e+f x))^{3/2}} \]

[Out]

(-2*b)/(3*f*(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.0378593, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {2578} \[ -\frac{2 b}{3 f \sin ^{\frac{3}{2}}(e+f x) (b \sec (e+f x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[b*Sec[e + f*x]]*Sin[e + f*x]^(5/2)),x]

[Out]

(-2*b)/(3*f*(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^(3/2))

Rule 2578

Int[((b_.)*sec[(e_.) + (f_.)*(x_)])^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Simp[(b*(a*Sin[e
+ f*x])^(m + 1)*(b*Sec[e + f*x])^(n - 1))/(a*f*(m + 1)), x] /; FreeQ[{a, b, e, f, m, n}, x] && EqQ[m - n + 2,
0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{b \sec (e+f x)} \sin ^{\frac{5}{2}}(e+f x)} \, dx &=-\frac{2 b}{3 f (b \sec (e+f x))^{3/2} \sin ^{\frac{3}{2}}(e+f x)}\\ \end{align*}

Mathematica [A]  time = 0.102346, size = 30, normalized size = 1. \[ -\frac{2 b}{3 f \sin ^{\frac{3}{2}}(e+f x) (b \sec (e+f x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[b*Sec[e + f*x]]*Sin[e + f*x]^(5/2)),x]

[Out]

(-2*b)/(3*f*(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^(3/2))

________________________________________________________________________________________

Maple [B]  time = 0.096, size = 70, normalized size = 2.3 \begin{align*} -{\frac{8\,\cos \left ( fx+e \right ) \left ( -1+\cos \left ( fx+e \right ) \right ) ^{2}}{3\,f \left ( \left ( \sin \left ( fx+e \right ) \right ) ^{2}+ \left ( \cos \left ( fx+e \right ) \right ) ^{2}-2\,\cos \left ( fx+e \right ) +1 \right ) ^{2}} \left ( \sin \left ( fx+e \right ) \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{\frac{b}{\cos \left ( fx+e \right ) }}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/sin(f*x+e)^(5/2)/(b*sec(f*x+e))^(1/2),x)

[Out]

-8/3/f*cos(f*x+e)*(-1+cos(f*x+e))^2/sin(f*x+e)^(3/2)/(sin(f*x+e)^2+cos(f*x+e)^2-2*cos(f*x+e)+1)^2/(b/cos(f*x+e
))^(1/2)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \sec \left (f x + e\right )} \sin \left (f x + e\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sin(f*x+e)^(5/2)/(b*sec(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(b*sec(f*x + e))*sin(f*x + e)^(5/2)), x)

________________________________________________________________________________________

Fricas [B]  time = 2.32199, size = 117, normalized size = 3.9 \begin{align*} \frac{2 \, \sqrt{\frac{b}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )^{2} \sqrt{\sin \left (f x + e\right )}}{3 \,{\left (b f \cos \left (f x + e\right )^{2} - b f\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sin(f*x+e)^(5/2)/(b*sec(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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(1/sin(f*x+e)**(5/2)/(b*sec(f*x+e))**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \sec \left (f x + e\right )} \sin \left (f x + e\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sin(f*x+e)^(5/2)/(b*sec(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(b*sec(f*x + e))*sin(f*x + e)^(5/2)), x)