3.217 \(\int \frac{\cos (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx\)

Optimal. Leaf size=18 \[ \frac{\log \left (\sin ^2(c+d x)+1\right )}{2 d} \]

[Out]

Log[1 + Sin[c + d*x]^2]/(2*d)

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Rubi [A]  time = 0.0311779, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {4334, 260} \[ \frac{\log \left (\sin ^2(c+d x)+1\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[1 + Sin[c + d*x]^2]/(2*d)

Rule 4334

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, Dist[d/(b
*c), Subst[Int[SubstFor[1, Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d], x] /; FunctionOfQ[Sin[c*(a +
b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Cos] || EqQ[F, cos])

Rule 260

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

Rubi steps

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

Mathematica [A]  time = 0.107006, size = 20, normalized size = 1.11 \[ \frac{\log (3-\cos (2 (c+d x)))}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[3 - Cos[2*(c + d*x)]]/(2*d)

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Maple [A]  time = 0.042, size = 17, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{2}-2 \right ) }{2\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.67404, size = 22, normalized size = 1.22 \begin{align*} \frac{\log \left (\sin \left (d x + c\right )^{2} + 1\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(sin(d*x + c)^2 + 1)/d

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Fricas [A]  time = 0.492568, size = 43, normalized size = 2.39 \begin{align*} \frac{\log \left (-\cos \left (d x + c\right )^{2} + 2\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)/(csc(d*x+c)+sin(d*x+c)),x, algorithm="fricas")

[Out]

1/2*log(-cos(d*x + c)^2 + 2)/d

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

[Out]

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

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Giac [A]  time = 1.15153, size = 22, normalized size = 1.22 \begin{align*} \frac{\log \left (\sin \left (d x + c\right )^{2} + 1\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(sin(d*x + c)^2 + 1)/d