3.499 \(\int \cos ^4(c+d x) \cot (c+d x) (a+a \sin (c+d x)) \, dx\)

Optimal. Leaf size=86 \[ \frac{a \sin ^5(c+d x)}{5 d}+\frac{a \sin ^4(c+d x)}{4 d}-\frac{2 a \sin ^3(c+d x)}{3 d}-\frac{a \sin ^2(c+d x)}{d}+\frac{a \sin (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d} \]

[Out]

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

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Rubi [A]  time = 0.0666494, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2836, 12, 88} \[ \frac{a \sin ^5(c+d x)}{5 d}+\frac{a \sin ^4(c+d x)}{4 d}-\frac{2 a \sin ^3(c+d x)}{3 d}-\frac{a \sin ^2(c+d x)}{d}+\frac{a \sin (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^4*Cot[c + d*x]*(a + a*Sin[c + d*x]),x]

[Out]

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

Rule 2836

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)
*(x_)])^(n_.), x_Symbol] :> Dist[1/(b^p*f), Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d*x)/b
)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2,
0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

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

Mathematica [A]  time = 0.0357863, size = 86, normalized size = 1. \[ \frac{a \sin ^5(c+d x)}{5 d}+\frac{a \sin ^4(c+d x)}{4 d}-\frac{2 a \sin ^3(c+d x)}{3 d}-\frac{a \sin ^2(c+d x)}{d}+\frac{a \sin (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^4*Cot[c + d*x]*(a + a*Sin[c + d*x]),x]

[Out]

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

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Maple [A]  time = 0.054, size = 94, normalized size = 1.1 \begin{align*}{\frac{8\,a\sin \left ( dx+c \right ) }{15\,d}}+{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}a}{5\,d}}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) a}{15\,d}}+{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+{\frac{a\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/15*a*sin(d*x+c)/d+1/5/d*cos(d*x+c)^4*sin(d*x+c)*a+4/15/d*cos(d*x+c)^2*sin(d*x+c)*a+1/4*a*cos(d*x+c)^4/d+1/2*
a*cos(d*x+c)^2/d+a*ln(sin(d*x+c))/d

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Maxima [A]  time = 1.0433, size = 93, normalized size = 1.08 \begin{align*} \frac{12 \, a \sin \left (d x + c\right )^{5} + 15 \, a \sin \left (d x + c\right )^{4} - 40 \, a \sin \left (d x + c\right )^{3} - 60 \, a \sin \left (d x + c\right )^{2} + 60 \, a \log \left (\sin \left (d x + c\right )\right ) + 60 \, a \sin \left (d x + c\right )}{60 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(12*a*sin(d*x + c)^5 + 15*a*sin(d*x + c)^4 - 40*a*sin(d*x + c)^3 - 60*a*sin(d*x + c)^2 + 60*a*log(sin(d*x
+ c)) + 60*a*sin(d*x + c))/d

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Fricas [A]  time = 1.18617, size = 197, normalized size = 2.29 \begin{align*} \frac{15 \, a \cos \left (d x + c\right )^{4} + 30 \, a \cos \left (d x + c\right )^{2} + 60 \, a \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) + 4 \,{\left (3 \, a \cos \left (d x + c\right )^{4} + 4 \, a \cos \left (d x + c\right )^{2} + 8 \, a\right )} \sin \left (d x + c\right )}{60 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(15*a*cos(d*x + c)^4 + 30*a*cos(d*x + c)^2 + 60*a*log(1/2*sin(d*x + c)) + 4*(3*a*cos(d*x + c)^4 + 4*a*cos
(d*x + c)^2 + 8*a)*sin(d*x + c))/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*csc(d*x+c)*(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.21495, size = 95, normalized size = 1.1 \begin{align*} \frac{12 \, a \sin \left (d x + c\right )^{5} + 15 \, a \sin \left (d x + c\right )^{4} - 40 \, a \sin \left (d x + c\right )^{3} - 60 \, a \sin \left (d x + c\right )^{2} + 60 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 60 \, a \sin \left (d x + c\right )}{60 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(12*a*sin(d*x + c)^5 + 15*a*sin(d*x + c)^4 - 40*a*sin(d*x + c)^3 - 60*a*sin(d*x + c)^2 + 60*a*log(abs(sin
(d*x + c))) + 60*a*sin(d*x + c))/d