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

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

[Out]

-arctanh(cos(b*x+a))/b

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Rubi [A]  time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, 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*}

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Mathematica [B]  time = 0.01, 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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fricas [B]  time = 0.46, size = 30, normalized size = 2.50 \[ -\frac {\log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) - \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right )}{2 \, b} \]

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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giac [B]  time = 1.16, size = 51, normalized size = 4.25 \[ \frac {\log \left ({\left | -\frac {\cos \left (b x + a\right )}{b} + \frac {1}{{\left | b \right |}} \right |}\right )}{2 \, {\left | b \right |}} - \frac {\log \left ({\left | -\frac {\cos \left (b x + a\right )}{b} - \frac {1}{{\left | b \right |}} \right |}\right )}{2 \, {\left | b \right |}} \]

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)

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maple [A]  time = 0.02, size = 21, normalized size = 1.75 \[ \frac {\ln \left (-\cot \left (b x +a \right )+\csc \left (b x +a \right )\right )}{b} \]

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 = 0.42, size = 26, normalized size = 2.17 \[ -\frac {\log \left (\cos \left (b x + a\right ) + 1\right ) - \log \left (\cos \left (b x + a\right ) - 1\right )}{2 \, b} \]

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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mupad [B]  time = 0.12, size = 12, normalized size = 1.00 \[ -\frac {\mathrm {atanh}\left (\cos \left (a+b\,x\right )\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/sin(a + b*x),x)

[Out]

-atanh(cos(a + b*x))/b

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sympy [A]  time = 0.53, size = 17, normalized size = 1.42 \[ \begin {cases} \frac {\log {\left (\tan {\left (\frac {a}{2} + \frac {b x}{2} \right )} \right )}}{b} & \text {for}\: b \neq 0 \\\frac {x}{\sin {\relax (a )}} & \text {otherwise} \end {cases} \]

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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