3.218 \(\int \frac{\tan (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx\)

Optimal. Leaf size=29 \[ \frac{\tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{\tan ^{-1}(\sin (c+d x))}{2 d} \]

[Out]

-ArcTan[Sin[c + d*x]]/(2*d) + ArcTanh[Sin[c + d*x]]/(2*d)

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Rubi [A]  time = 0.0375861, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {298, 203, 206} \[ \frac{\tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{\tan ^{-1}(\sin (c+d x))}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-ArcTan[Sin[c + d*x]]/(2*d) + ArcTanh[Sin[c + d*x]]/(2*d)

Rule 298

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
&  !GtQ[a/b, 0]

Rule 203

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

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

Mathematica [A]  time = 0.0352153, size = 24, normalized size = 0.83 \[ \frac{\tanh ^{-1}(\sin (c+d x))-\tan ^{-1}(\sin (c+d x))}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-ArcTan[Sin[c + d*x]] + ArcTanh[Sin[c + d*x]])/(2*d)

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Maple [A]  time = 0.102, size = 42, normalized size = 1.5 \begin{align*} -{\frac{\arctan \left ( \sin \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{4\,d}}-{\frac{\ln \left ( \sin \left ( dx+c \right ) -1 \right ) }{4\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*arctan(sin(d*x+c))/d+1/4/d*ln(1+sin(d*x+c))-1/4/d*ln(sin(d*x+c)-1)

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Maxima [A]  time = 1.61021, size = 47, normalized size = 1.62 \begin{align*} -\frac{2 \, \arctan \left (\sin \left (d x + c\right )\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(tan(d*x+c)/(csc(d*x+c)+sin(d*x+c)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.519186, size = 111, normalized size = 3.83 \begin{align*} -\frac{2 \, \arctan \left (\sin \left (d x + c\right )\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(tan(d*x+c)/(csc(d*x+c)+sin(d*x+c)),x, algorithm="fricas")

[Out]

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

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

[Out]

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

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Giac [A]  time = 1.19726, size = 50, normalized size = 1.72 \begin{align*} -\frac{2 \, \arctan \left (\sin \left (d x + c\right )\right ) - \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) + \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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