3.191 \(\int \frac{\sin (a+b x)}{(d \cos (a+b x))^{5/2}} \, dx\)

Optimal. Leaf size=22 \[ \frac{2}{3 b d (d \cos (a+b x))^{3/2}} \]

[Out]

2/(3*b*d*(d*Cos[a + b*x])^(3/2))

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Rubi [A]  time = 0.0268243, antiderivative size = 22, 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} \[ \frac{2}{3 b d (d \cos (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*x]/(d*Cos[a + b*x])^(5/2),x]

[Out]

2/(3*b*d*(d*Cos[a + b*x])^(3/2))

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

Mathematica [A]  time = 0.0276373, size = 22, normalized size = 1. \[ \frac{2}{3 b d (d \cos (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*x]/(d*Cos[a + b*x])^(5/2),x]

[Out]

2/(3*b*d*(d*Cos[a + b*x])^(3/2))

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Maple [A]  time = 0.006, size = 19, normalized size = 0.9 \begin{align*}{\frac{2}{3\,bd} \left ( d\cos \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*cos(b*x+a))^(5/2),x)

[Out]

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

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Maxima [A]  time = 0.976698, size = 24, normalized size = 1.09 \begin{align*} \frac{2}{3 \, \left (d \cos \left (b x + a\right )\right )^{\frac{3}{2}} 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))^(5/2),x, algorithm="maxima")

[Out]

2/3/((d*cos(b*x + a))^(3/2)*b*d)

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Fricas [A]  time = 1.82662, size = 66, normalized size = 3. \begin{align*} \frac{2 \, \sqrt{d \cos \left (b x + a\right )}}{3 \, b d^{3} \cos \left (b x + a\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)/(d*cos(b*x+a))^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 66.8572, size = 32, normalized size = 1.45 \begin{align*} \begin{cases} \frac{2}{3 b d^{\frac{5}{2}} \cos ^{\frac{3}{2}}{\left (a + b x \right )}} & \text{for}\: b \neq 0 \\\frac{x \sin{\left (a \right )}}{\left (d \cos{\left (a \right )}\right )^{\frac{5}{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))**(5/2),x)

[Out]

Piecewise((2/(3*b*d**(5/2)*cos(a + b*x)**(3/2)), Ne(b, 0)), (x*sin(a)/(d*cos(a))**(5/2), True))

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Giac [A]  time = 1.16396, size = 35, normalized size = 1.59 \begin{align*} \frac{2}{3 \, \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))^(5/2),x, algorithm="giac")

[Out]

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