3.538 \(\int \frac{\cos ^2(c+d x) \cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=60 \[ \frac{\sin (c+d x)}{a d}-\frac{\csc ^2(c+d x)}{2 a d}+\frac{\csc (c+d x)}{a d}-\frac{\log (\sin (c+d x))}{a d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.105576, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {2836, 12, 75} \[ \frac{\sin (c+d x)}{a d}-\frac{\csc ^2(c+d x)}{2 a d}+\frac{\csc (c+d x)}{a d}-\frac{\log (\sin (c+d x))}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0862675, size = 45, normalized size = 0.75 \[ -\frac{-2 \sin (c+d x)+\csc ^2(c+d x)-2 \csc (c+d x)+2 \log (\sin (c+d x))+3}{2 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.113, size = 61, normalized size = 1. \begin{align*}{\frac{\sin \left ( dx+c \right ) }{da}}+{\frac{1}{da\sin \left ( dx+c \right ) }}-{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{da}}-{\frac{1}{2\,da \left ( \sin \left ( dx+c \right ) \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5*csc(d*x+c)^3/(a+a*sin(d*x+c)),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.00717, size = 70, normalized size = 1.17 \begin{align*} -\frac{\frac{2 \, \log \left (\sin \left (d x + c\right )\right )}{a} - \frac{2 \, \sin \left (d x + c\right )}{a} - \frac{2 \, \sin \left (d x + c\right ) - 1}{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)^5*csc(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.13329, size = 165, normalized size = 2.75 \begin{align*} -\frac{2 \,{\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) - 2 \,{\left (\cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right ) - 1}{2 \,{\left (a d \cos \left (d x + c\right )^{2} - a d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/2*(2*(cos(d*x + c)^2 - 1)*log(1/2*sin(d*x + c)) - 2*(cos(d*x + c)^2 - 2)*sin(d*x + c) - 1)/(a*d*cos(d*x + c
)^2 - a*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)**5*csc(d*x+c)**3/(a+a*sin(d*x+c)),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.2756, size = 85, normalized size = 1.42 \begin{align*} -\frac{\frac{2 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} - \frac{2 \, \sin \left (d x + c\right )}{a} - \frac{3 \, \sin \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) - 1}{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)^5*csc(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

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