\(\int \frac {\sin (a+b x)}{(d \cos (a+b x))^{3/2}} \, dx\) [190]

Optimal result

Integrand size = 19, antiderivative size = 20 \[ \int \frac {\sin (a+b x)}{(d \cos (a+b x))^{3/2}} \, dx=\frac {2}{b d \sqrt {d \cos (a+b x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2645, 30} \[ \int \frac {\sin (a+b x)}{(d \cos (a+b x))^{3/2}} \, dx=\frac {2}{b d \sqrt {d \cos (a+b x)}} \]

[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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2645

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])

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {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] (verified)

Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\sin (a+b x)}{(d \cos (a+b x))^{3/2}} \, dx=\frac {2}{b d \sqrt {d \cos (a+b x)}} \]

[In]

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

[Out]

2/(b*d*Sqrt[d*Cos[a + b*x]])

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95

method	result	size
derivativedivides	\(\frac {2}{b d \sqrt {d \cos \left (b x +a \right )}}\)	\(19\)
default	\(\frac {2}{b d \sqrt {d \cos \left (b x +a \right )}}\)	\(19\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.30 \[ \int \frac {\sin (a+b x)}{(d \cos (a+b x))^{3/2}} \, dx=\frac {2 \, \sqrt {d \cos \left (b x + a\right )}}{b d^{2} \cos \left (b x + a\right )} \]

[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))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (15) = 30\).

Time = 0.77 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.70 \[ \int \frac {\sin (a+b x)}{(d \cos (a+b x))^{3/2}} \, dx=\begin {cases} \frac {2 \cos {\left (a + b x \right )}}{b \left (d \cos {\left (a + b x \right )}\right )^{\frac {3}{2}}} & \text {for}\: b \neq 0 \\\frac {x \sin {\left (a \right )}}{\left (d \cos {\left (a \right )}\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {\sin (a+b x)}{(d \cos (a+b x))^{3/2}} \, dx=\frac {2}{\sqrt {d \cos \left (b x + a\right )} b d} \]

[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)

Giac [F]

\[ \int \frac {\sin (a+b x)}{(d \cos (a+b x))^{3/2}} \, dx=\int { \frac {\sin \left (b x + a\right )}{\left (d \cos \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.85 \[ \int \frac {\sin (a+b x)}{(d \cos (a+b x))^{3/2}} \, dx=\frac {4\,\cos \left (a+b\,x\right )\,\sqrt {d\,\cos \left (a+b\,x\right )}}{b\,d^2\,\left (\cos \left (2\,a+2\,b\,x\right )+1\right )} \]

[In]

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

[Out]

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