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

Optimal. Leaf size=54 \[ -\frac{a \sin (c+d x)}{d}-\frac{a \csc ^2(c+d x)}{2 d}-\frac{a \csc (c+d x)}{d}-\frac{a \log (\sin (c+d x))}{d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0341207, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2707, 75} \[ -\frac{a \sin (c+d x)}{d}-\frac{a \csc ^2(c+d x)}{2 d}-\frac{a \csc (c+d x)}{d}-\frac{a \log (\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.1128, size = 60, normalized size = 1.11 \[ -\frac{a \sin (c+d x)}{d}-\frac{a \csc (c+d x)}{d}-\frac{a \left (\cot ^2(c+d x)+2 \log (\tan (c+d x))+2 \log (\cos (c+d x))\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.056, size = 83, normalized size = 1.5 \begin{align*} -{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{d\sin \left ( dx+c \right ) }}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) a}{d}}-2\,{\frac{a\sin \left ( dx+c \right ) }{d}}-{\frac{a \left ( \cot \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)^3*csc(d*x+c)^3*(a+a*sin(d*x+c)),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.00127, size = 61, normalized size = 1.13 \begin{align*} -\frac{2 \, a \log \left (\sin \left (d x + c\right )\right ) + 2 \, a \sin \left (d x + c\right ) + \frac{2 \, a \sin \left (d x + c\right ) + a}{\sin \left (d x + c\right )^{2}}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.45479, size = 167, normalized size = 3.09 \begin{align*} -\frac{2 \,{\left (a \cos \left (d x + c\right )^{2} - a\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) + 2 \,{\left (a \cos \left (d x + c\right )^{2} - 2 \, a\right )} \sin \left (d x + c\right ) - a}{2 \,{\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(cos(d*x+c)^3*csc(d*x+c)^3*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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)**3*csc(d*x+c)**3*(a+a*sin(d*x+c)),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.3406, size = 62, normalized size = 1.15 \begin{align*} -\frac{2 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 2 \, a \sin \left (d x + c\right ) + \frac{2 \, a \sin \left (d x + c\right ) + a}{\sin \left (d x + c\right )^{2}}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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