3.190 \(\int \frac{\sin (a+b x)}{(d \cos (a+b x))^{3/2}} \, dx\)

Optimal. Leaf size=20 \[ \frac{2}{b d \sqrt{d \cos (a+b x)}} \]

[Out]

2/(b*d*Sqrt[d*Cos[a + b*x]])

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Rubi [A]  time = 0.0258174, 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 = {2565, 30} \[ \frac{2}{b d \sqrt{d \cos (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*x]/(d*Cos[a + b*x])^(3/2),x]

[Out]

2/(b*d*Sqrt[d*Cos[a + b*x]])

Rule 2565

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[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)}{(d \cos (a+b x))^{3/2}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x^{3/2}} \, dx,x,d \cos (a+b x)\right )}{b d}\\ &=\frac{2}{b d \sqrt{d \cos (a+b x)}}\\ \end{align*}

Mathematica [A]  time = 0.0235167, size = 20, normalized size = 1. \[ \frac{2}{b d \sqrt{d \cos (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*x]/(d*Cos[a + b*x])^(3/2),x]

[Out]

2/(b*d*Sqrt[d*Cos[a + b*x]])

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Maple [A]  time = 0.004, size = 19, normalized size = 1. \begin{align*} 2\,{\frac{1}{bd\sqrt{d\cos \left ( bx+a \right ) }}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(b*x+a)/(d*cos(b*x+a))^(3/2),x)

[Out]

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

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Maxima [A]  time = 0.978707, size = 24, normalized size = 1.2 \begin{align*} \frac{2}{\sqrt{d \cos \left (b x + a\right )} b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)/(d*cos(b*x+a))^(3/2),x, algorithm="maxima")

[Out]

2/(sqrt(d*cos(b*x + a))*b*d)

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Fricas [A]  time = 1.8419, size = 61, normalized size = 3.05 \begin{align*} \frac{2 \, \sqrt{d \cos \left (b x + a\right )}}{b d^{2} \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*cos(b*x+a))^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 6.07928, size = 31, normalized size = 1.55 \begin{align*} \begin{cases} \frac{2}{b d^{\frac{3}{2}} \sqrt{\cos{\left (a + b x \right )}}} & \text{for}\: b \neq 0 \\\frac{x \sin{\left (a \right )}}{\left (d \cos{\left (a \right )}\right )^{\frac{3}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)/(d*cos(b*x+a))**(3/2),x)

[Out]

Piecewise((2/(b*d**(3/2)*sqrt(cos(a + b*x))), Ne(b, 0)), (x*sin(a)/(d*cos(a))**(3/2), True))

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Giac [A]  time = 1.17022, size = 24, normalized size = 1.2 \begin{align*} \frac{2}{\sqrt{d \cos \left (b x + a\right )} b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)/(d*cos(b*x+a))^(3/2),x, algorithm="giac")

[Out]

2/(sqrt(d*cos(b*x + a))*b*d)