3.111 \(\int \csc (a+b x) \, dx\)

Optimal. Leaf size=12 \[ -\frac{\tanh ^{-1}(\cos (a+b x))}{b} \]

[Out]

-(ArcTanh[Cos[a + b*x]]/b)

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Rubi [A]  time = 0.0035878, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3770} \[ -\frac{\tanh ^{-1}(\cos (a+b x))}{b} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x],x]

[Out]

-(ArcTanh[Cos[a + b*x]]/b)

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \csc (a+b x) \, dx &=-\frac{\tanh ^{-1}(\cos (a+b x))}{b}\\ \end{align*}

Mathematica [B]  time = 0.0168323, size = 38, normalized size = 3.17 \[ \frac{\log \left (\sin \left (\frac{a}{2}+\frac{b x}{2}\right )\right )}{b}-\frac{\log \left (\cos \left (\frac{a}{2}+\frac{b x}{2}\right )\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x],x]

[Out]

-(Log[Cos[a/2 + (b*x)/2]]/b) + Log[Sin[a/2 + (b*x)/2]]/b

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Maple [A]  time = 0.006, size = 21, normalized size = 1.8 \begin{align*}{\frac{\ln \left ( \csc \left ( bx+a \right ) -\cot \left ( bx+a \right ) \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/sin(b*x+a),x)

[Out]

1/b*ln(csc(b*x+a)-cot(b*x+a))

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Maxima [B]  time = 0.964039, size = 35, normalized size = 2.92 \begin{align*} -\frac{\log \left (\cos \left (b x + a\right ) + 1\right ) - \log \left (\cos \left (b x + a\right ) - 1\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sin(b*x+a),x, algorithm="maxima")

[Out]

-1/2*(log(cos(b*x + a) + 1) - log(cos(b*x + a) - 1))/b

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Fricas [B]  time = 1.89001, size = 93, normalized size = 7.75 \begin{align*} -\frac{\log \left (\frac{1}{2} \, \cos \left (b x + a\right ) + \frac{1}{2}\right ) - \log \left (-\frac{1}{2} \, \cos \left (b x + a\right ) + \frac{1}{2}\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sin(b*x+a),x, algorithm="fricas")

[Out]

-1/2*(log(1/2*cos(b*x + a) + 1/2) - log(-1/2*cos(b*x + a) + 1/2))/b

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Sympy [A]  time = 0.538475, size = 17, normalized size = 1.42 \begin{align*} \begin{cases} \frac{\log{\left (\tan{\left (\frac{a}{2} + \frac{b x}{2} \right )} \right )}}{b} & \text{for}\: b \neq 0 \\\frac{x}{\sin{\left (a \right )}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sin(b*x+a),x)

[Out]

Piecewise((log(tan(a/2 + b*x/2))/b, Ne(b, 0)), (x/sin(a), True))

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Giac [B]  time = 1.08453, size = 69, normalized size = 5.75 \begin{align*} \frac{\log \left ({\left | -\frac{\cos \left (b x + a\right )}{b} + \frac{1}{{\left | b \right |}} \right |}\right )}{2 \,{\left | b \right |}} - \frac{\log \left ({\left | -\frac{\cos \left (b x + a\right )}{b} - \frac{1}{{\left | b \right |}} \right |}\right )}{2 \,{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sin(b*x+a),x, algorithm="giac")

[Out]

1/2*log(abs(-cos(b*x + a)/b + 1/abs(b)))/abs(b) - 1/2*log(abs(-cos(b*x + a)/b - 1/abs(b)))/abs(b)