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]

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

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Rubi [A]  time = 0.04, antiderivative size = 30, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.10, size = 30, normalized size = 1.00 \[ -\frac {2 b}{3 f \sin ^{\frac {3}{2}}(e+f x) (b \sec (e+f x))^{3/2}} \]

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))

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fricas [A]  time = 0.89, size = 48, normalized size = 1.60 \[ \frac {2 \, \sqrt {\frac {b}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )^{2} \sqrt {\sin \left (f x + e\right )}}{3 \, {\left (b f \cos \left (f x + e\right )^{2} - b f\right )}} \]

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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \sec \left (f x + e\right )} \sin \left (f x + e\right )^{\frac {5}{2}}}\,{d x} \]

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)

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maple [B]  time = 0.14, size = 70, normalized size = 2.33 \[ -\frac {8 \cos \left (f x +e \right ) \left (-1+\cos \left (f x +e \right )\right )^{2}}{3 f \sin \left (f x +e \right )^{\frac {3}{2}} \left (\sin ^{2}\left (f x +e \right )+\cos ^{2}\left (f x +e \right )-2 \cos \left (f x +e \right )+1\right )^{2} \sqrt {\frac {b}{\cos \left (f x +e \right )}}} \]

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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \sec \left (f x + e\right )} \sin \left (f x + e\right )^{\frac {5}{2}}}\,{d x} \]

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)

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mupad [B]  time = 1.25, size = 57, normalized size = 1.90 \[ \frac {\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}\,\left (\sin \left (e+f\,x\right )+\sin \left (3\,e+3\,f\,x\right )\right )}{3\,b\,f\,\sqrt {\sin \left (e+f\,x\right )}\,\left (\cos \left (2\,e+2\,f\,x\right )-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

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