3.1283 \(\int \frac{\cot (c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx\)

Optimal. Leaf size=50 \[ -\frac{b \log (\sin (c+d x))}{a^2 d}+\frac{b \log (a+b \sin (c+d x))}{a^2 d}-\frac{\csc (c+d x)}{a d} \]

[Out]

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

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Rubi [A]  time = 0.0724541, antiderivative size = 50, 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} \[ -\frac{b \log (\sin (c+d x))}{a^2 d}+\frac{b \log (a+b \sin (c+d x))}{a^2 d}-\frac{\csc (c+d x)}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0415074, size = 50, normalized size = 1. \[ -\frac{b \log (\sin (c+d x))}{a^2 d}+\frac{b \log (a+b \sin (c+d x))}{a^2 d}-\frac{\csc (c+d x)}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.037, size = 35, normalized size = 0.7 \begin{align*} -{\frac{\csc \left ( dx+c \right ) }{da}}+{\frac{b\ln \left ( a\csc \left ( dx+c \right ) +b \right ) }{d{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)*csc(d*x+c)^2/(a+b*sin(d*x+c)),x)

[Out]

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

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Maxima [A]  time = 0.971688, size = 63, normalized size = 1.26 \begin{align*} \frac{\frac{b \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{2}} - \frac{b \log \left (\sin \left (d x + c\right )\right )}{a^{2}} - \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+b*sin(d*x+c)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.37955, size = 143, normalized size = 2.86 \begin{align*} \frac{b \log \left (b \sin \left (d x + c\right ) + a\right ) \sin \left (d x + c\right ) - b \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) - a}{a^{2} 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+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos{\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}}{a + b \sin{\left (c + d x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*csc(d*x+c)**2/(a+b*sin(d*x+c)),x)

[Out]

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

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Giac [A]  time = 1.2312, size = 66, normalized size = 1.32 \begin{align*} \frac{\frac{b \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{2}} - \frac{b \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{2}} - \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+b*sin(d*x+c)),x, algorithm="giac")

[Out]

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