3.374 \(\int \sqrt{b \sec (e+f x)} \sin (e+f x) \, dx\)

Optimal. Leaf size=18 \[ -\frac{2 b}{f \sqrt{b \sec (e+f x)}} \]

[Out]

(-2*b)/(f*Sqrt[b*Sec[e + f*x]])

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Rubi [A]  time = 0.0296487, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2622, 30} \[ -\frac{2 b}{f \sqrt{b \sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[b*Sec[e + f*x]]*Sin[e + f*x],x]

[Out]

(-2*b)/(f*Sqrt[b*Sec[e + f*x]])

Rule 2622

Int[csc[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[1/(f*a^n), Subst[Int
[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n
+ 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0336027, size = 18, normalized size = 1. \[ -\frac{2 b}{f \sqrt{b \sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b*Sec[e + f*x]]*Sin[e + f*x],x]

[Out]

(-2*b)/(f*Sqrt[b*Sec[e + f*x]])

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Maple [A]  time = 0.019, size = 17, normalized size = 0.9 \begin{align*} -2\,{\frac{b}{f\sqrt{b\sec \left ( fx+e \right ) }}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*b/f/(b*sec(f*x+e))^(1/2)

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Maxima [A]  time = 1.01509, size = 31, normalized size = 1.72 \begin{align*} -\frac{2 \, \sqrt{\frac{b}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.11954, size = 54, normalized size = 3. \begin{align*} -\frac{2 \, \sqrt{\frac{b}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \sec{\left (e + f x \right )}} \sin{\left (e + f x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(b*sec(e + f*x))*sin(e + f*x), x)

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Giac [A]  time = 1.15806, size = 32, normalized size = 1.78 \begin{align*} -\frac{2 \, \sqrt{b \cos \left (f x + e\right )} \mathrm{sgn}\left (\cos \left (f x + e\right )\right )}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(b*cos(f*x + e))*sgn(cos(f*x + e))/f