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

Optimal. Leaf size=46 \[ -\frac{\csc (c+d x)}{a d}-\frac{\log (\sin (c+d x))}{a d}+\frac{\log (\sin (c+d x)+1)}{a d} \]

[Out]

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

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Rubi [A]  time = 0.0617029, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2833, 12, 44} \[ -\frac{\csc (c+d x)}{a d}-\frac{\log (\sin (c+d x))}{a d}+\frac{\log (\sin (c+d x)+1)}{a d} \]

Antiderivative was successfully verified.

[In]

Int[(Cot[c + d*x]*Csc[c + d*x])/(a + a*Sin[c + d*x]),x]

[Out]

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

Mathematica [A]  time = 0.032843, size = 46, normalized size = 1. \[ -\frac{\csc (c+d x)}{a d}-\frac{\log (\sin (c+d x))}{a d}+\frac{\log (\sin (c+d x)+1)}{a d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cot[c + d*x]*Csc[c + d*x])/(a + a*Sin[c + d*x]),x]

[Out]

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

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Maple [A]  time = 0.037, size = 49, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{da}}-{\frac{1}{da\sin \left ( dx+c \right ) }}-{\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)^2/(a+a*sin(d*x+c)),x)

[Out]

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

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Maxima [A]  time = 1.03582, size = 58, normalized size = 1.26 \begin{align*} \frac{\frac{\log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac{\log \left (\sin \left (d x + c\right )\right )}{a} - \frac{1}{a \sin \left (d x + c\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*csc(d*x+c)^2/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.5621, size = 134, normalized size = 2.91 \begin{align*} -\frac{\log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) - \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 1}{a d \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)^2/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

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

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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 ^{2}{\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)**2/(a+a*sin(d*x+c)),x)

[Out]

Integral(cos(c + d*x)*csc(c + d*x)**2/(sin(c + d*x) + 1), x)/a

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Giac [A]  time = 1.22712, size = 61, normalized size = 1.33 \begin{align*} \frac{\frac{\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac{\log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} - \frac{1}{a \sin \left (d x + c\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*csc(d*x+c)^2/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

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