3.231 \(\int \frac{\cot (c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=82 \[ -\frac{\csc ^3(c+d x)}{3 a d}+\frac{\csc ^2(c+d x)}{2 a d}-\frac{\csc (c+d x)}{a d}-\frac{\log (\sin (c+d x))}{a d}+\frac{\log (\sin (c+d x)+1)}{a d} \]

[Out]

-(Csc[c + d*x]/(a*d)) + Csc[c + d*x]^2/(2*a*d) - Csc[c + d*x]^3/(3*a*d) - Log[Sin[c + d*x]]/(a*d) + Log[1 + Si
n[c + d*x]]/(a*d)

________________________________________________________________________________________

Rubi [A]  time = 0.082485, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2833, 12, 44} \[ -\frac{\csc ^3(c+d x)}{3 a d}+\frac{\csc ^2(c+d x)}{2 a d}-\frac{\csc (c+d x)}{a d}-\frac{\log (\sin (c+d x))}{a d}+\frac{\log (\sin (c+d x)+1)}{a d} \]

Antiderivative was successfully verified.

[In]

Int[(Cot[c + d*x]*Csc[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) - Csc[c + d*x]^3/(3*a*d) - Log[Sin[c + d*x]]/(a*d) + Log[1 + Si
n[c + d*x]]/(a*d)

Rule 2833

Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)
])^(n_.), x_Symbol] :> Dist[1/(b*f), Subst[Int[(a + x)^m*(c + (d*x)/b)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\cot (c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^4}{x^4 (a+x)} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{x^4 (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^3 \operatorname{Subst}\left (\int \left (\frac{1}{a x^4}-\frac{1}{a^2 x^3}+\frac{1}{a^3 x^2}-\frac{1}{a^4 x}+\frac{1}{a^4 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{\csc (c+d x)}{a d}+\frac{\csc ^2(c+d x)}{2 a d}-\frac{\csc ^3(c+d x)}{3 a d}-\frac{\log (\sin (c+d x))}{a d}+\frac{\log (1+\sin (c+d x))}{a d}\\ \end{align*}

Mathematica [A]  time = 0.0440093, size = 82, normalized size = 1. \[ -\frac{\csc ^3(c+d x)}{3 a d}+\frac{\csc ^2(c+d x)}{2 a d}-\frac{\csc (c+d x)}{a d}-\frac{\log (\sin (c+d x))}{a d}+\frac{\log (\sin (c+d x)+1)}{a d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cot[c + d*x]*Csc[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) - Csc[c + d*x]^3/(3*a*d) - Log[Sin[c + d*x]]/(a*d) + Log[1 + Si
n[c + d*x]]/(a*d)

________________________________________________________________________________________

Maple [A]  time = 0.043, size = 81, normalized size = 1. \begin{align*}{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{da}}-{\frac{1}{3\,da \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{1}{da\sin \left ( dx+c \right ) }}+{\frac{1}{2\,da \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{da}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)*csc(d*x+c)^4/(a+a*sin(d*x+c)),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.10643, size = 88, normalized size = 1.07 \begin{align*} \frac{\frac{6 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac{6 \, \log \left (\sin \left (d x + c\right )\right )}{a} - \frac{6 \, \sin \left (d x + c\right )^{2} - 3 \, \sin \left (d x + c\right ) + 2}{a \sin \left (d x + c\right )^{3}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*csc(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.47274, size = 281, normalized size = 3.43 \begin{align*} -\frac{6 \,{\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) - 6 \,{\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 6 \, \cos \left (d x + c\right )^{2} + 3 \, \sin \left (d x + c\right ) - 8}{6 \,{\left (a d \cos \left (d x + c\right )^{2} - a 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)*csc(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/6*(6*(cos(d*x + c)^2 - 1)*log(1/2*sin(d*x + c))*sin(d*x + c) - 6*(cos(d*x + c)^2 - 1)*log(sin(d*x + c) + 1)
*sin(d*x + c) + 6*cos(d*x + c)^2 + 3*sin(d*x + c) - 8)/((a*d*cos(d*x + c)^2 - a*d)*sin(d*x + c))

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\cos{\left (c + d x \right )} \csc ^{4}{\left (c + d x \right )}}{\sin{\left (c + d x \right )} + 1}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*csc(d*x+c)**4/(a+a*sin(d*x+c)),x)

[Out]

Integral(cos(c + d*x)*csc(c + d*x)**4/(sin(c + d*x) + 1), x)/a

________________________________________________________________________________________

Giac [A]  time = 1.19256, size = 90, normalized size = 1.1 \begin{align*} \frac{\frac{6 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac{6 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} - \frac{6 \, \sin \left (d x + c\right )^{2} - 3 \, \sin \left (d x + c\right ) + 2}{a \sin \left (d x + c\right )^{3}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*csc(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

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