3.17 \(\int \cot ^3(c+d x) (a+a \sin (c+d x))^2 \, dx\)

Optimal. Leaf size=30 \[ -\frac{\csc ^2(c+d x) (a \sin (c+d x)+a)^4}{2 a^2 d} \]

[Out]

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

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Rubi [A]  time = 0.0392531, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2707, 74} \[ -\frac{\csc ^2(c+d x) (a \sin (c+d x)+a)^4}{2 a^2 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]

[Out]

-(Csc[c + d*x]^2*(a + a*Sin[c + d*x])^4)/(2*a^2*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 74

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

Rubi steps

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

Mathematica [A]  time = 0.0412311, size = 28, normalized size = 0.93 \[ -\frac{a^2 (\sin (c+d x)+1)^4 \csc ^2(c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]

[Out]

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

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Maple [B]  time = 0.049, size = 94, normalized size = 3.1 \begin{align*}{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-2\,{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{d\sin \left ( dx+c \right ) }}-2\,{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) }{d}}-4\,{\frac{{a}^{2}\sin \left ( dx+c \right ) }{d}}-{\frac{{a}^{2} \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))^2,x)

[Out]

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

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Maxima [A]  time = 1.11304, size = 72, normalized size = 2.4 \begin{align*} -\frac{a^{2} \sin \left (d x + c\right )^{2} + 4 \, a^{2} \sin \left (d x + c\right ) + \frac{4 \, a^{2} \sin \left (d x + c\right ) + a^{2}}{\sin \left (d x + c\right )^{2}}}{2 \, 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))^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.44029, size = 173, normalized size = 5.77 \begin{align*} \frac{2 \, a^{2} \cos \left (d x + c\right )^{4} - 3 \, a^{2} \cos \left (d x + c\right )^{2} + 3 \, a^{2} - 8 \,{\left (a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2}\right )} \sin \left (d x + c\right )}{4 \,{\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))^2,x, algorithm="fricas")

[Out]

1/4*(2*a^2*cos(d*x + c)^4 - 3*a^2*cos(d*x + c)^2 + 3*a^2 - 8*(a^2*cos(d*x + c)^2 - 2*a^2)*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^{2} \left (\int 2 \sin{\left (c + d x \right )} \cot ^{3}{\left (c + d x \right )}\, dx + \int \sin ^{2}{\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))**2,x)

[Out]

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

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Giac [A]  time = 1.37063, size = 63, normalized size = 2.1 \begin{align*} -\frac{a^{2}{\left (\frac{1}{\sin \left (d x + c\right )} + \sin \left (d x + c\right )\right )}^{2} + 4 \, a^{2}{\left (\frac{1}{\sin \left (d x + c\right )} + \sin \left (d x + c\right )\right )}}{2 \, 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))^2,x, algorithm="giac")

[Out]

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