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

Optimal. Leaf size=38 \[ \frac{\log (\sin (c+d x))}{a d}-\frac{\log \left (a+b \sin ^2(c+d x)\right )}{2 a d} \]

[Out]

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

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Rubi [A]  time = 0.0431496, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3194, 36, 29, 31} \[ \frac{\log (\sin (c+d x))}{a d}-\frac{\log \left (a+b \sin ^2(c+d x)\right )}{2 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3194

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = Free
Factors[Sin[e + f*x]^2, x]}, Dist[ff^((m + 1)/2)/(2*f), Subst[Int[(x^((m - 1)/2)*(a + b*ff*x)^p)/(1 - ff*x)^((
m + 1)/2), x], x, Sin[e + f*x]^2/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0215795, size = 38, normalized size = 1. \[ \frac{\log (\sin (c+d x))}{a d}-\frac{\log \left (a+b \sin ^2(c+d x)\right )}{2 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.044, size = 37, normalized size = 1. \begin{align*}{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{da}}-{\frac{\ln \left ( a+ \left ( \sin \left ( dx+c \right ) \right ) ^{2}b \right ) }{2\,da}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01851, size = 50, normalized size = 1.32 \begin{align*} -\frac{\frac{\log \left (b \sin \left (d x + c\right )^{2} + a\right )}{a} - \frac{\log \left (\sin \left (d x + c\right )^{2}\right )}{a}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.82779, size = 96, normalized size = 2.53 \begin{align*} -\frac{\log \left (-b \cos \left (d x + c\right )^{2} + a + b\right ) - 2 \, \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right )}{2 \, a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)/(a+b*sin(d*x+c)^2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.1071, size = 51, normalized size = 1.34 \begin{align*} \frac{\frac{\log \left (\sin \left (d x + c\right )^{2}\right )}{a} - \frac{\log \left ({\left | b \sin \left (d x + c\right )^{2} + a \right |}\right )}{a}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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