3.230 \(\int \frac{\cot (c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=63 \[ -\frac{\csc ^2(c+d x)}{2 a d}+\frac{\csc (c+d x)}{a d}+\frac{\log (\sin (c+d x))}{a d}-\frac{\log (\sin (c+d x)+1)}{a d} \]

[Out]

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

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Rubi [A]  time = 0.0786564, antiderivative size = 63, 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} \[ -\frac{\csc ^2(c+d x)}{2 a d}+\frac{\csc (c+d x)}{a d}+\frac{\log (\sin (c+d x))}{a d}-\frac{\log (\sin (c+d x)+1)}{a d} \]

Antiderivative was successfully verified.

[In]

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

Mathematica [A]  time = 0.0356259, size = 63, normalized size = 1. \[ -\frac{\csc ^2(c+d x)}{2 a d}+\frac{\csc (c+d x)}{a d}+\frac{\log (\sin (c+d x))}{a d}-\frac{\log (\sin (c+d x)+1)}{a d} \]

Antiderivative was successfully verified.

[In]

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

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Maple [A]  time = 0.04, size = 64, normalized size = 1. \begin{align*} -{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{da}}-{\frac{1}{2\,da \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{da}}+{\frac{1}{da\sin \left ( dx+c \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.09842, size = 74, normalized size = 1.17 \begin{align*} -\frac{\frac{2 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac{2 \, \log \left (\sin \left (d x + c\right )\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)*csc(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.58783, size = 198, normalized size = 3.14 \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} - 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 2 \, \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)*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 - 1)*log(sin(d*x + c) + 1) - 2*sin(d*x +
c) + 1)/(a*d*cos(d*x + c)^2 - a*d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\cos{\left (c + d x \right )} \csc ^{3}{\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)**3/(a+a*sin(d*x+c)),x)

[Out]

Integral(cos(c + d*x)*csc(c + d*x)**3/(sin(c + d*x) + 1), x)/a

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Giac [A]  time = 1.30154, size = 77, normalized size = 1.22 \begin{align*} -\frac{\frac{2 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac{2 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\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)*csc(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

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