3.226 \(\int \sqrt{d \sec (a+b x)} \sin (a+b x) \, dx\)

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

[Out]

(-2*d)/(b*Sqrt[d*Sec[a + b*x]])

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Rubi [A]  time = 0.032692, 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} \[ -\frac{2 d}{b \sqrt{d \sec (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[d*Sec[a + b*x]]*Sin[a + b*x],x]

[Out]

(-2*d)/(b*Sqrt[d*Sec[a + b*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{d \sec (a+b x)} \sin (a+b x) \, dx &=\frac{d \operatorname{Subst}\left (\int \frac{1}{x^{3/2}} \, dx,x,d \sec (a+b x)\right )}{b}\\ &=-\frac{2 d}{b \sqrt{d \sec (a+b x)}}\\ \end{align*}

Mathematica [A]  time = 0.0376551, size = 18, normalized size = 1. \[ -\frac{2 d}{b \sqrt{d \sec (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[d*Sec[a + b*x]]*Sin[a + b*x],x]

[Out]

(-2*d)/(b*Sqrt[d*Sec[a + b*x]])

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Maple [A]  time = 0.025, size = 17, normalized size = 0.9 \begin{align*} -2\,{\frac{d}{b\sqrt{d\sec \left ( bx+a \right ) }}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*sec(b*x+a))^(1/2)*sin(b*x+a),x)

[Out]

-2*d/b/(d*sec(b*x+a))^(1/2)

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Maxima [A]  time = 1.05814, size = 31, normalized size = 1.72 \begin{align*} -\frac{2 \, \sqrt{\frac{d}{\cos \left (b x + a\right )}} \cos \left (b x + a\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(d/cos(b*x + a))*cos(b*x + a)/b

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Fricas [A]  time = 1.6582, size = 54, normalized size = 3. \begin{align*} -\frac{2 \, \sqrt{\frac{d}{\cos \left (b x + a\right )}} \cos \left (b x + a\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(d/cos(b*x + a))*cos(b*x + a)/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sec(b*x+a))**(1/2)*sin(b*x+a),x)

[Out]

Integral(sqrt(d*sec(a + b*x))*sin(a + b*x), x)

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Giac [A]  time = 1.32468, size = 30, normalized size = 1.67 \begin{align*} -\frac{2 \, \sqrt{d \cos \left (b x + a\right )} \mathrm{sgn}\left (\cos \left (b x + a\right )\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(d*cos(b*x + a))*sgn(cos(b*x + a))/b