3.78 \(\int \frac{\cos ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx\)

Optimal. Leaf size=50 \[ \frac{\sin ^2(c+d x)}{2 a^3 d}-\frac{3 \sin (c+d x)}{a^3 d}+\frac{4 \log (\sin (c+d x)+1)}{a^3 d} \]

[Out]

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

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Rubi [A]  time = 0.0499715, antiderivative size = 50, 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} \[ \frac{\sin ^2(c+d x)}{2 a^3 d}-\frac{3 \sin (c+d x)}{a^3 d}+\frac{4 \log (\sin (c+d x)+1)}{a^3 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^5/(a + a*Sin[c + d*x])^3,x]

[Out]

(4*Log[1 + Sin[c + d*x]])/(a^3*d) - (3*Sin[c + d*x])/(a^3*d) + Sin[c + d*x]^2/(2*a^3*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 ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^2}{a+x} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-3 a+x+\frac{4 a^2}{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{4 \log (1+\sin (c+d x))}{a^3 d}-\frac{3 \sin (c+d x)}{a^3 d}+\frac{\sin ^2(c+d x)}{2 a^3 d}\\ \end{align*}

Mathematica [A]  time = 0.0500167, size = 38, normalized size = 0.76 \[ \frac{\sin ^2(c+d x)-6 \sin (c+d x)+8 \log (\sin (c+d x)+1)}{2 a^3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^5/(a + a*Sin[c + d*x])^3,x]

[Out]

(8*Log[1 + Sin[c + d*x]] - 6*Sin[c + d*x] + Sin[c + d*x]^2)/(2*a^3*d)

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Maple [A]  time = 0.069, size = 49, normalized size = 1. \begin{align*} 4\,{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{{a}^{3}d}}-3\,{\frac{\sin \left ( dx+c \right ) }{{a}^{3}d}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2\,{a}^{3}d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5/(a+a*sin(d*x+c))^3,x)

[Out]

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

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Maxima [A]  time = 0.942104, size = 55, normalized size = 1.1 \begin{align*} \frac{\frac{\sin \left (d x + c\right )^{2} - 6 \, \sin \left (d x + c\right )}{a^{3}} + \frac{8 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3}}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5/(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/2*((sin(d*x + c)^2 - 6*sin(d*x + c))/a^3 + 8*log(sin(d*x + c) + 1)/a^3)/d

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Fricas [A]  time = 2.01086, size = 100, normalized size = 2. \begin{align*} -\frac{\cos \left (d x + c\right )^{2} - 8 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 6 \, \sin \left (d x + c\right )}{2 \, a^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5/(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/2*(cos(d*x + c)^2 - 8*log(sin(d*x + c) + 1) + 6*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)**5/(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

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Giac [B]  time = 1.16831, size = 155, normalized size = 3.1 \begin{align*} -\frac{2 \,{\left (\frac{2 \, \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}{a^{3}} - \frac{4 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{3}} - \frac{3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 7 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{2} a^{3}}\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5/(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

-2*(2*log(tan(1/2*d*x + 1/2*c)^2 + 1)/a^3 - 4*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^3 - (3*tan(1/2*d*x + 1/2*c)
^4 - 3*tan(1/2*d*x + 1/2*c)^3 + 7*tan(1/2*d*x + 1/2*c)^2 - 3*tan(1/2*d*x + 1/2*c) + 3)/((tan(1/2*d*x + 1/2*c)^
2 + 1)^2*a^3))/d