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

Optimal. Leaf size=22 \[ -\frac {2}{a d \sqrt {a \sin (c+d x)+a}} \]

[Out]

-2/a/d/(a+a*sin(d*x+c))^(1/2)

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Rubi [A]  time = 0.03, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2667, 32} \[ -\frac {2}{a d \sqrt {a \sin (c+d x)+a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2/(a*d*Sqrt[a + a*Sin[c + d*x]])

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2667

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 22, normalized size = 1.00 \[ -\frac {2}{a d \sqrt {a \sin (c+d x)+a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2/(a*d*Sqrt[a + a*Sin[c + d*x]])

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fricas [A]  time = 0.57, size = 33, normalized size = 1.50 \[ -\frac {2 \, \sqrt {a \sin \left (d x + c\right ) + a}}{a^{2} d \sin \left (d x + c\right ) + a^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(a*sin(d*x + c) + a)/(a^2*d*sin(d*x + c) + a^2*d)

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giac [A]  time = 1.88, size = 20, normalized size = 0.91 \[ -\frac {2}{\sqrt {a \sin \left (d x + c\right ) + a} a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(sqrt(a*sin(d*x + c) + a)*a*d)

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maple [A]  time = 0.02, size = 21, normalized size = 0.95 \[ -\frac {2}{a d \sqrt {a +a \sin \left (d x +c \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/a/d/(a+a*sin(d*x+c))^(1/2)

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maxima [A]  time = 0.46, size = 20, normalized size = 0.91 \[ -\frac {2}{\sqrt {a \sin \left (d x + c\right ) + a} a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(sqrt(a*sin(d*x + c) + a)*a*d)

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mupad [B]  time = 4.88, size = 50, normalized size = 2.27 \[ -\frac {4\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (\sin \left (c+d\,x\right )+1\right )}{a^2\,d\,\left (2\,{\sin \left (c+d\,x\right )}^2+4\,\sin \left (c+d\,x\right )+2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(4*(a*(sin(c + d*x) + 1))^(1/2)*(sin(c + d*x) + 1))/(a^2*d*(4*sin(c + d*x) + 2*sin(c + d*x)^2 + 2))

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sympy [A]  time = 3.46, size = 56, normalized size = 2.55 \[ \begin {cases} \text {NaN} & \text {for}\: \left (c = \frac {3 \pi }{2} \vee c = - d x + \frac {3 \pi }{2}\right ) \wedge \left (c = - d x + \frac {3 \pi }{2} \vee d = 0\right ) \\\frac {x \cos {\relax (c )}}{\left (a \sin {\relax (c )} + a\right )^{\frac {3}{2}}} & \text {for}\: d = 0 \\- \frac {2}{a d \sqrt {a \sin {\left (c + d x \right )} + a}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((nan, (Eq(d, 0) | Eq(c, -d*x + 3*pi/2)) & (Eq(c, 3*pi/2) | Eq(c, -d*x + 3*pi/2))), (x*cos(c)/(a*sin(
c) + a)**(3/2), Eq(d, 0)), (-2/(a*d*sqrt(a*sin(c + d*x) + a)), True))

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