3.220 \(\int \frac{\sec (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx\)

Optimal. Leaf size=16 \[ \frac{\tanh ^{-1}\left (\sin ^2(c+d x)\right )}{2 d} \]

[Out]

ArcTanh[Sin[c + d*x]^2]/(2*d)

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Rubi [A]  time = 0.0317371, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {275, 206} \[ \frac{\tanh ^{-1}\left (\sin ^2(c+d x)\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTanh[Sin[c + d*x]^2]/(2*d)

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\sec (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x}{1-x^4} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sin ^2(c+d x)\right )}{2 d}\\ &=\frac{\tanh ^{-1}\left (\sin ^2(c+d x)\right )}{2 d}\\ \end{align*}

Mathematica [A]  time = 0.0475111, size = 30, normalized size = 1.88 \[ \frac{\log \left (2-\cos ^2(c+d x)\right )-2 \log (\cos (c+d x))}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Log[Cos[c + d*x]] + Log[2 - Cos[c + d*x]^2])/(4*d)

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Maple [A]  time = 0.05, size = 19, normalized size = 1.2 \begin{align*}{\frac{\ln \left ( 2\, \left ( \sec \left ( dx+c \right ) \right ) ^{2}-1 \right ) }{4\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/d*ln(2*sec(d*x+c)^2-1)

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Maxima [B]  time = 1.57702, size = 53, normalized size = 3.31 \begin{align*} \frac{\log \left (\sin \left (d x + c\right )^{2} + 1\right ) - \log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(log(sin(d*x + c)^2 + 1) - log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1))/d

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Fricas [B]  time = 0.496821, size = 77, normalized size = 4.81 \begin{align*} \frac{\log \left (-\cos \left (d x + c\right )^{2} + 2\right ) - 2 \, \log \left (-\cos \left (d x + c\right )\right )}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(log(-cos(d*x + c)^2 + 2) - 2*log(-cos(d*x + c)))/d

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

[Out]

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

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Giac [B]  time = 1.23052, size = 107, normalized size = 6.69 \begin{align*} -\frac{2 \, \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) - \log \left ({\left | -\frac{6 \,{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1 \right |}\right )}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)) - log(abs(-6*(cos(d*x + c) - 1)/(cos(d*x + c) + 1
) + (cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 1)))/d