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

Optimal. Leaf size=20 \[ -\frac{2 d}{3 b (d \sec (a+b x))^{3/2}} \]

[Out]

(-2*d)/(3*b*(d*Sec[a + b*x])^(3/2))

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Rubi [A]  time = 0.0326505, antiderivative size = 20, 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} \[ -\frac{2 d}{3 b (d \sec (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d)/(3*b*(d*Sec[a + b*x])^(3/2))

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 \frac{\sin (a+b x)}{\sqrt{d \sec (a+b x)}} \, dx &=\frac{d \operatorname{Subst}\left (\int \frac{1}{x^{5/2}} \, dx,x,d \sec (a+b x)\right )}{b}\\ &=-\frac{2 d}{3 b (d \sec (a+b x))^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0529927, size = 20, normalized size = 1. \[ -\frac{2 d}{3 b (d \sec (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d)/(3*b*(d*Sec[a + b*x])^(3/2))

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Maple [A]  time = 0.025, size = 17, normalized size = 0.9 \begin{align*} -{\frac{2\,d}{3\,b} \left ( d\sec \left ( bx+a \right ) \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*d/b/(d*sec(b*x+a))^(3/2)

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Maxima [A]  time = 1.18467, size = 31, normalized size = 1.55 \begin{align*} -\frac{2 \, \cos \left (b x + a\right )}{3 \, b \sqrt{\frac{d}{\cos \left (b x + a\right )}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*cos(b*x + a)/(b*sqrt(d/cos(b*x + a)))

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Fricas [A]  time = 1.70856, size = 65, normalized size = 3.25 \begin{align*} -\frac{2 \, \sqrt{\frac{d}{\cos \left (b x + a\right )}} \cos \left (b x + a\right )^{2}}{3 \, b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*sqrt(d/cos(b*x + a))*cos(b*x + a)^2/(b*d)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 3.62161, size = 47, normalized size = 2.35 \begin{align*} -\frac{2 \, \sqrt{d \cos \left (b x + a\right )}{\left | b \right |} \cos \left (b x + a\right ) \mathrm{sgn}\left (b\right ) \mathrm{sgn}\left (\cos \left (b x + a\right )\right )}{3 \, b^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*sqrt(d*cos(b*x + a))*abs(b)*cos(b*x + a)*sgn(b)*sgn(cos(b*x + a))/(b^2*d)