3.504 \(\int \cot ^5(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0705625, 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} \[ -\frac{a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^4(c+d x)}{4 d}+\frac{2 a \csc ^3(c+d x)}{3 d}+\frac{a \csc ^2(c+d x)}{d}-\frac{a \csc (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^5*Csc[c + d*x]*(a + a*Sin[c + d*x]),x]

[Out]

-((a*Csc[c + d*x])/d) + (a*Csc[c + d*x]^2)/d + (2*a*Csc[c + d*x]^3)/(3*d) - (a*Csc[c + d*x]^4)/(4*d) - (a*Csc[
c + d*x]^5)/(5*d) + (a*Log[Sin[c + d*x]])/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 \cot ^5(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^6 (a-x)^2 (a+x)^3}{x^6} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{a \operatorname{Subst}\left (\int \frac{(a-x)^2 (a+x)^3}{x^6} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a \operatorname{Subst}\left (\int \left (\frac{a^5}{x^6}+\frac{a^4}{x^5}-\frac{2 a^3}{x^4}-\frac{2 a^2}{x^3}+\frac{a}{x^2}+\frac{1}{x}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{a \csc (c+d x)}{d}+\frac{a \csc ^2(c+d x)}{d}+\frac{2 a \csc ^3(c+d x)}{3 d}-\frac{a \csc ^4(c+d x)}{4 d}-\frac{a \csc ^5(c+d x)}{5 d}+\frac{a \log (\sin (c+d x))}{d}\\ \end{align*}

Mathematica [A]  time = 0.164674, size = 92, normalized size = 1.07 \[ -\frac{a \csc ^5(c+d x)}{5 d}+\frac{2 a \csc ^3(c+d x)}{3 d}-\frac{a \csc (c+d x)}{d}+\frac{a \left (-\cot ^4(c+d x)+2 \cot ^2(c+d x)+4 \log (\tan (c+d x))+4 \log (\cos (c+d x))\right )}{4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^5*Csc[c + d*x]*(a + a*Sin[c + d*x]),x]

[Out]

-((a*Csc[c + d*x])/d) + (2*a*Csc[c + d*x]^3)/(3*d) - (a*Csc[c + d*x]^5)/(5*d) + (a*(2*Cot[c + d*x]^2 - Cot[c +
d*x]^4 + 4*Log[Cos[c + d*x]] + 4*Log[Tan[c + d*x]]))/(4*d)

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Maple [A]  time = 0.061, size = 160, normalized size = 1.9 \begin{align*} -{\frac{a \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+{\frac{a \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+{\frac{a\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{5\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}+{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{15\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{5\,d\sin \left ( dx+c \right ) }}-{\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5*csc(d*x+c)^6*(a+a*sin(d*x+c)),x)

[Out]

-1/4/d*a*cot(d*x+c)^4+1/2*a*cot(d*x+c)^2/d+a*ln(sin(d*x+c))/d-1/5/d*a/sin(d*x+c)^5*cos(d*x+c)^6+1/15/d*a/sin(d
*x+c)^3*cos(d*x+c)^6-1/5/d*a/sin(d*x+c)*cos(d*x+c)^6-8/15*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

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Maxima [A]  time = 1.07469, size = 97, normalized size = 1.13 \begin{align*} \frac{60 \, a \log \left (\sin \left (d x + c\right )\right ) - \frac{60 \, a \sin \left (d x + c\right )^{4} - 60 \, a \sin \left (d x + c\right )^{3} - 40 \, a \sin \left (d x + c\right )^{2} + 15 \, a \sin \left (d x + c\right ) + 12 \, a}{\sin \left (d x + c\right )^{5}}}{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)^6*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.1246, size = 332, normalized size = 3.86 \begin{align*} -\frac{60 \, a \cos \left (d x + c\right )^{4} - 80 \, a \cos \left (d x + c\right )^{2} - 60 \,{\left (a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{2} + a\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) + 15 \,{\left (4 \, a \cos \left (d x + c\right )^{2} - 3 \, a\right )} \sin \left (d x + c\right ) + 32 \, a}{60 \,{\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^6*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/60*(60*a*cos(d*x + c)^4 - 80*a*cos(d*x + c)^2 - 60*(a*cos(d*x + c)^4 - 2*a*cos(d*x + c)^2 + a)*log(1/2*sin(
d*x + c))*sin(d*x + c) + 15*(4*a*cos(d*x + c)^2 - 3*a)*sin(d*x + c) + 32*a)/((d*cos(d*x + c)^4 - 2*d*cos(d*x +
c)^2 + d)*sin(d*x + c))

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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)**6*(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.315, size = 113, normalized size = 1.31 \begin{align*} \frac{60 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - \frac{137 \, a \sin \left (d x + c\right )^{5} + 60 \, a \sin \left (d x + c\right )^{4} - 60 \, a \sin \left (d x + c\right )^{3} - 40 \, a \sin \left (d x + c\right )^{2} + 15 \, a \sin \left (d x + c\right ) + 12 \, a}{\sin \left (d x + c\right )^{5}}}{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)^6*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/60*(60*a*log(abs(sin(d*x + c))) - (137*a*sin(d*x + c)^5 + 60*a*sin(d*x + c)^4 - 60*a*sin(d*x + c)^3 - 40*a*s
in(d*x + c)^2 + 15*a*sin(d*x + c) + 12*a)/sin(d*x + c)^5)/d