3.28 \(\int \cot ^3(c+d x) (a+a \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=98 \[ -\frac{a^3 \sin ^3(c+d x)}{3 d}-\frac{3 a^3 \sin ^2(c+d x)}{2 d}-\frac{2 a^3 \sin (c+d x)}{d}-\frac{a^3 \csc ^2(c+d x)}{2 d}-\frac{3 a^3 \csc (c+d x)}{d}+\frac{2 a^3 \log (\sin (c+d x))}{d} \]

[Out]

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

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Rubi [A]  time = 0.0648456, antiderivative size = 98, 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 = {2707, 75} \[ -\frac{a^3 \sin ^3(c+d x)}{3 d}-\frac{3 a^3 \sin ^2(c+d x)}{2 d}-\frac{2 a^3 \sin (c+d x)}{d}-\frac{a^3 \csc ^2(c+d x)}{2 d}-\frac{3 a^3 \csc (c+d x)}{d}+\frac{2 a^3 \log (\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]

[Out]

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

Rule 2707

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p_.), x_Symbol] :> Dist[1/f, Subst[I
nt[(x^p*(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]
&& EqQ[a^2 - b^2, 0] && IntegerQ[(p + 1)/2]

Rule 75

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])

Rubi steps

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

Mathematica [A]  time = 0.201659, size = 67, normalized size = 0.68 \[ -\frac{a^3 \left (2 \sin ^3(c+d x)+9 \sin ^2(c+d x)+12 \sin (c+d x)+3 \csc ^2(c+d x)+18 \csc (c+d x)-12 \log (\sin (c+d x))+30\right )}{6 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]

[Out]

-(a^3*(30 + 18*Csc[c + d*x] + 3*Csc[c + d*x]^2 - 12*Log[Sin[c + d*x]] + 12*Sin[c + d*x] + 9*Sin[c + d*x]^2 + 2
*Sin[c + d*x]^3))/(6*d)

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Maple [A]  time = 0.053, size = 109, normalized size = 1.1 \begin{align*} -{\frac{8\,{a}^{3}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3\,d}}-{\frac{16\,{a}^{3}\sin \left ( dx+c \right ) }{3\,d}}+{\frac{3\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+2\,{\frac{{a}^{3}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-3\,{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{d\sin \left ( dx+c \right ) }}-{\frac{{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^3*(a+a*sin(d*x+c))^3,x)

[Out]

-8/3/d*a^3*sin(d*x+c)*cos(d*x+c)^2-16/3*a^3*sin(d*x+c)/d+3/2/d*a^3*cos(d*x+c)^2+2*a^3*ln(sin(d*x+c))/d-3/d*a^3
/sin(d*x+c)*cos(d*x+c)^4-1/2/d*a^3*cot(d*x+c)^2

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Maxima [A]  time = 1.05985, size = 108, normalized size = 1.1 \begin{align*} -\frac{2 \, a^{3} \sin \left (d x + c\right )^{3} + 9 \, a^{3} \sin \left (d x + c\right )^{2} - 12 \, a^{3} \log \left (\sin \left (d x + c\right )\right ) + 12 \, a^{3} \sin \left (d x + c\right ) + \frac{3 \,{\left (6 \, a^{3} \sin \left (d x + c\right ) + a^{3}\right )}}{\sin \left (d x + c\right )^{2}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(2*a^3*sin(d*x + c)^3 + 9*a^3*sin(d*x + c)^2 - 12*a^3*log(sin(d*x + c)) + 12*a^3*sin(d*x + c) + 3*(6*a^3*
sin(d*x + c) + a^3)/sin(d*x + c)^2)/d

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Fricas [A]  time = 1.48336, size = 284, normalized size = 2.9 \begin{align*} \frac{18 \, a^{3} \cos \left (d x + c\right )^{4} - 27 \, a^{3} \cos \left (d x + c\right )^{2} + 15 \, a^{3} + 24 \,{\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) + 4 \,{\left (a^{3} \cos \left (d x + c\right )^{4} - 8 \, a^{3} \cos \left (d x + c\right )^{2} + 16 \, a^{3}\right )} \sin \left (d x + c\right )}{12 \,{\left (d \cos \left (d x + c\right )^{2} - d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(18*a^3*cos(d*x + c)^4 - 27*a^3*cos(d*x + c)^2 + 15*a^3 + 24*(a^3*cos(d*x + c)^2 - a^3)*log(1/2*sin(d*x +
c)) + 4*(a^3*cos(d*x + c)^4 - 8*a^3*cos(d*x + c)^2 + 16*a^3)*sin(d*x + c))/(d*cos(d*x + c)^2 - d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a^{3} \left (\int 3 \sin{\left (c + d x \right )} \cot ^{3}{\left (c + d x \right )}\, dx + \int 3 \sin ^{2}{\left (c + d x \right )} \cot ^{3}{\left (c + d x \right )}\, dx + \int \sin ^{3}{\left (c + d x \right )} \cot ^{3}{\left (c + d x \right )}\, dx + \int \cot ^{3}{\left (c + d x \right )}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**3*(a+a*sin(d*x+c))**3,x)

[Out]

a**3*(Integral(3*sin(c + d*x)*cot(c + d*x)**3, x) + Integral(3*sin(c + d*x)**2*cot(c + d*x)**3, x) + Integral(
sin(c + d*x)**3*cot(c + d*x)**3, x) + Integral(cot(c + d*x)**3, x))

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Giac [A]  time = 1.37156, size = 127, normalized size = 1.3 \begin{align*} -\frac{2 \, a^{3} \sin \left (d x + c\right )^{3} + 9 \, a^{3} \sin \left (d x + c\right )^{2} - 12 \, a^{3} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 12 \, a^{3} \sin \left (d x + c\right ) + \frac{3 \,{\left (6 \, a^{3} \sin \left (d x + c\right )^{2} + 6 \, a^{3} \sin \left (d x + c\right ) + a^{3}\right )}}{\sin \left (d x + c\right )^{2}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(2*a^3*sin(d*x + c)^3 + 9*a^3*sin(d*x + c)^2 - 12*a^3*log(abs(sin(d*x + c))) + 12*a^3*sin(d*x + c) + 3*(6
*a^3*sin(d*x + c)^2 + 6*a^3*sin(d*x + c) + a^3)/sin(d*x + c)^2)/d