3.63 \(\int \frac{\cos ^7(c+d x)}{(a+a \sin (c+d x))^2} \, dx\)

Optimal. Leaf size=47 \[ \frac{(a-a \sin (c+d x))^5}{5 a^7 d}-\frac{(a-a \sin (c+d x))^4}{2 a^6 d} \]

[Out]

-(a - a*Sin[c + d*x])^4/(2*a^6*d) + (a - a*Sin[c + d*x])^5/(5*a^7*d)

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Rubi [A]  time = 0.0520617, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2667, 43} \[ \frac{(a-a \sin (c+d x))^5}{5 a^7 d}-\frac{(a-a \sin (c+d x))^4}{2 a^6 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a - a*Sin[c + d*x])^4/(2*a^6*d) + (a - a*Sin[c + d*x])^5/(5*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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\cos ^7(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac{\operatorname{Subst}\left (\int (a-x)^3 (a+x) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (2 a (a-x)^3-(a-x)^4\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=-\frac{(a-a \sin (c+d x))^4}{2 a^6 d}+\frac{(a-a \sin (c+d x))^5}{5 a^7 d}\\ \end{align*}

Mathematica [A]  time = 0.159816, size = 46, normalized size = 0.98 \[ -\frac{\sin (c+d x) \left (2 \sin ^4(c+d x)-5 \sin ^3(c+d x)+10 \sin (c+d x)-10\right )}{10 a^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sin[c + d*x]*(-10 + 10*Sin[c + d*x] - 5*Sin[c + d*x]^3 + 2*Sin[c + d*x]^4))/(10*a^2*d)

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Maple [A]  time = 0.068, size = 45, normalized size = 1. \begin{align*}{\frac{1}{d{a}^{2}} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{5}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{2}}- \left ( \sin \left ( dx+c \right ) \right ) ^{2}+\sin \left ( dx+c \right ) \right ) } \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))^2,x)

[Out]

1/d/a^2*(-1/5*sin(d*x+c)^5+1/2*sin(d*x+c)^4-sin(d*x+c)^2+sin(d*x+c))

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Maxima [A]  time = 0.937682, size = 63, normalized size = 1.34 \begin{align*} -\frac{2 \, \sin \left (d x + c\right )^{5} - 5 \, \sin \left (d x + c\right )^{4} + 10 \, \sin \left (d x + c\right )^{2} - 10 \, \sin \left (d x + c\right )}{10 \, a^{2} 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))^2,x, algorithm="maxima")

[Out]

-1/10*(2*sin(d*x + c)^5 - 5*sin(d*x + c)^4 + 10*sin(d*x + c)^2 - 10*sin(d*x + c))/(a^2*d)

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Fricas [A]  time = 1.93265, size = 122, normalized size = 2.6 \begin{align*} \frac{5 \, \cos \left (d x + c\right )^{4} - 2 \,{\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} - 4\right )} \sin \left (d x + c\right )}{10 \, a^{2} 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))^2,x, algorithm="fricas")

[Out]

1/10*(5*cos(d*x + c)^4 - 2*(cos(d*x + c)^4 - 2*cos(d*x + c)^2 - 4)*sin(d*x + c))/(a^2*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))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.14583, size = 63, normalized size = 1.34 \begin{align*} -\frac{2 \, \sin \left (d x + c\right )^{5} - 5 \, \sin \left (d x + c\right )^{4} + 10 \, \sin \left (d x + c\right )^{2} - 10 \, \sin \left (d x + c\right )}{10 \, a^{2} 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))^2,x, algorithm="giac")

[Out]

-1/10*(2*sin(d*x + c)^5 - 5*sin(d*x + c)^4 + 10*sin(d*x + c)^2 - 10*sin(d*x + c))/(a^2*d)