3.76 \(\int \frac{\cos ^7(c+d x)}{(a+a \sin (c+d x))^3} \, dx\)

Optimal. Leaf size=23 \[ -\frac{(a-a \sin (c+d x))^4}{4 a^7 d} \]

[Out]

-(a - a*Sin[c + d*x])^4/(4*a^7*d)

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Rubi [A]  time = 0.0428404, antiderivative size = 23, normalized size of antiderivative = 1., 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} \[ -\frac{(a-a \sin (c+d x))^4}{4 a^7 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^7/(a + a*Sin[c + d*x])^3,x]

[Out]

-(a - a*Sin[c + d*x])^4/(4*a^7*d)

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])

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]

Rubi steps

\begin{align*} \int \frac{\cos ^7(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac{\operatorname{Subst}\left (\int (a-x)^3 \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=-\frac{(a-a \sin (c+d x))^4}{4 a^7 d}\\ \end{align*}

Mathematica [A]  time = 0.158463, size = 44, normalized size = 1.91 \[ -\frac{\sin (c+d x) \left (\sin ^3(c+d x)-4 \sin ^2(c+d x)+6 \sin (c+d x)-4\right )}{4 a^3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^7/(a + a*Sin[c + d*x])^3,x]

[Out]

-(Sin[c + d*x]*(-4 + 6*Sin[c + d*x] - 4*Sin[c + d*x]^2 + Sin[c + d*x]^3))/(4*a^3*d)

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Maple [A]  time = 0.069, size = 19, normalized size = 0.8 \begin{align*} -{\frac{ \left ( \sin \left ( dx+c \right ) -1 \right ) ^{4}}{4\,d{a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^7/(a+a*sin(d*x+c))^3,x)

[Out]

-1/4/d/a^3*(sin(d*x+c)-1)^4

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Maxima [B]  time = 0.970327, size = 61, normalized size = 2.65 \begin{align*} -\frac{\sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{3} + 6 \, \sin \left (d x + c\right )^{2} - 4 \, \sin \left (d x + c\right )}{4 \, a^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7/(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/4*(sin(d*x + c)^4 - 4*sin(d*x + c)^3 + 6*sin(d*x + c)^2 - 4*sin(d*x + c))/(a^3*d)

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Fricas [B]  time = 1.9082, size = 119, normalized size = 5.17 \begin{align*} -\frac{\cos \left (d x + c\right )^{4} - 8 \, \cos \left (d x + c\right )^{2} + 4 \,{\left (\cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right )}{4 \, a^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7/(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/4*(cos(d*x + c)^4 - 8*cos(d*x + c)^2 + 4*(cos(d*x + c)^2 - 2)*sin(d*x + c))/(a^3*d)

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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(cos(d*x+c)**7/(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

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Giac [B]  time = 1.17269, size = 61, normalized size = 2.65 \begin{align*} -\frac{\sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{3} + 6 \, \sin \left (d x + c\right )^{2} - 4 \, \sin \left (d x + c\right )}{4 \, a^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7/(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

-1/4*(sin(d*x + c)^4 - 4*sin(d*x + c)^3 + 6*sin(d*x + c)^2 - 4*sin(d*x + c))/(a^3*d)