3.192 \(\int \frac{\sin (a+b x)}{(d \cos (a+b x))^{7/2}} \, dx\)

Optimal. Leaf size=22 \[ \frac{2}{5 b d (d \cos (a+b x))^{5/2}} \]

[Out]

2/(5*b*d*(d*Cos[a + b*x])^(5/2))

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Rubi [A]  time = 0.0263088, 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} \[ \frac{2}{5 b d (d \cos (a+b x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*x]/(d*Cos[a + b*x])^(7/2),x]

[Out]

2/(5*b*d*(d*Cos[a + b*x])^(5/2))

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))^{7/2}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x^{7/2}} \, dx,x,d \cos (a+b x)\right )}{b d}\\ &=\frac{2}{5 b d (d \cos (a+b x))^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0390279, size = 22, normalized size = 1. \[ \frac{2}{5 b d (d \cos (a+b x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*x]/(d*Cos[a + b*x])^(7/2),x]

[Out]

2/(5*b*d*(d*Cos[a + b*x])^(5/2))

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Maple [A]  time = 0.004, size = 19, normalized size = 0.9 \begin{align*}{\frac{2}{5\,bd} \left ( d\cos \left ( bx+a \right ) \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(b*x+a)/(d*cos(b*x+a))^(7/2),x)

[Out]

2/5/b/d/(d*cos(b*x+a))^(5/2)

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Maxima [A]  time = 0.961368, size = 24, normalized size = 1.09 \begin{align*} \frac{2}{5 \, \left (d \cos \left (b x + a\right )\right )^{\frac{5}{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))^(7/2),x, algorithm="maxima")

[Out]

2/5/((d*cos(b*x + a))^(5/2)*b*d)

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Fricas [A]  time = 1.76498, size = 66, normalized size = 3. \begin{align*} \frac{2 \, \sqrt{d \cos \left (b x + a\right )}}{5 \, b d^{4} \cos \left (b x + a\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)/(d*cos(b*x+a))^(7/2),x, algorithm="fricas")

[Out]

2/5*sqrt(d*cos(b*x + a))/(b*d^4*cos(b*x + a)^3)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)/(d*cos(b*x+a))**(7/2),x)

[Out]

Timed out

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Giac [A]  time = 1.15505, size = 35, normalized size = 1.59 \begin{align*} \frac{2}{5 \, \sqrt{d \cos \left (b x + a\right )} 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))^(7/2),x, algorithm="giac")

[Out]

2/5/(sqrt(d*cos(b*x + a))*b*d^3*cos(b*x + a)^2)