3.1.82 \(\int \frac {\cos (c+d x)}{(a+a \sin (c+d x))^3} \, dx\) [82]

Optimal. Leaf size=22 \[ -\frac {1}{2 a d (a+a \sin (c+d x))^2} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.00, 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 = {2746, 32} \begin {gather*} -\frac {1}{2 a d (a \sin (c+d x)+a)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*1/(a*d*(a + a*Sin[c + d*x])^2)

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 2746

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

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Mathematica [A]
time = 0.04, size = 33, normalized size = 1.50 \begin {gather*} -\frac {1}{2 a^3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*1/(a^3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)

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Maple [A]
time = 0.11, size = 21, normalized size = 0.95

method	result	size
derivativedivides	\(-\frac {1}{2 a d \left (a +a \sin \left (d x +c \right )\right )^{2}}\)	\(21\)
default	\(-\frac {1}{2 a d \left (a +a \sin \left (d x +c \right )\right )^{2}}\)	\(21\)
risch	\(\frac {2 \,{\mathrm e}^{2 i \left (d x +c \right )}}{d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{4}}\)	\(32\)
norman	\(\frac {\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}+\frac {2 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {4 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {6 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {6 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}\)	\(146\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 20, normalized size = 0.91 \begin {gather*} -\frac {1}{2 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{2} a d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/((a*sin(d*x + c) + a)^2*a*d)

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Fricas [A]
time = 0.34, size = 36, normalized size = 1.64 \begin {gather*} \frac {1}{2 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \sin \left (d x + c\right ) - 2 \, a^{3} d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/(a^3*d*cos(d*x + c)^2 - 2*a^3*d*sin(d*x + c) - 2*a^3*d)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (19) = 38\).
time = 0.58, size = 51, normalized size = 2.32 \begin {gather*} \begin {cases} - \frac {1}{2 a^{3} d \sin ^{2}{\left (c + d x \right )} + 4 a^{3} d \sin {\left (c + d x \right )} + 2 a^{3} d} & \text {for}\: d \neq 0 \\\frac {x \cos {\left (c \right )}}{\left (a \sin {\left (c \right )} + a\right )^{3}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-1/(2*a**3*d*sin(c + d*x)**2 + 4*a**3*d*sin(c + d*x) + 2*a**3*d), Ne(d, 0)), (x*cos(c)/(a*sin(c) +
a)**3, True))

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Giac [A]
time = 6.34, size = 20, normalized size = 0.91 \begin {gather*} -\frac {1}{2 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{2} a d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/((a*sin(d*x + c) + a)^2*a*d)

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Mupad [B]
time = 4.46, size = 18, normalized size = 0.82 \begin {gather*} -\frac {1}{2\,a^3\,d\,{\left (\sin \left (c+d\,x\right )+1\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(2*a^3*d*(sin(c + d*x) + 1)^2)

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