3.153 \(\int \csc (e+f x) (a+b \sin (e+f x)) \, dx\)

Optimal. Leaf size=17 \[ b x-\frac{a \tanh ^{-1}(\cos (e+f x))}{f} \]

[Out]

b*x - (a*ArcTanh[Cos[e + f*x]])/f

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Rubi [A]  time = 0.0215194, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2735, 3770} \[ b x-\frac{a \tanh ^{-1}(\cos (e+f x))}{f} \]

Antiderivative was successfully verified.

[In]

Int[Csc[e + f*x]*(a + b*Sin[e + f*x]),x]

[Out]

b*x - (a*ArcTanh[Cos[e + f*x]])/f

Rule 2735

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*x)/d
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

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

Mathematica [B]  time = 0.0204587, size = 43, normalized size = 2.53 \[ \frac{a \log \left (\sin \left (\frac{e}{2}+\frac{f x}{2}\right )\right )}{f}-\frac{a \log \left (\cos \left (\frac{e}{2}+\frac{f x}{2}\right )\right )}{f}+b x \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[e + f*x]*(a + b*Sin[e + f*x]),x]

[Out]

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

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Maple [A]  time = 0.027, size = 32, normalized size = 1.9 \begin{align*} bx+{\frac{a\ln \left ( \csc \left ( fx+e \right ) -\cot \left ( fx+e \right ) \right ) }{f}}+{\frac{be}{f}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(f*x+e)*(a+b*sin(f*x+e)),x)

[Out]

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

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Maxima [A]  time = 1.74925, size = 39, normalized size = 2.29 \begin{align*} \frac{{\left (f x + e\right )} b - a \log \left (\cot \left (f x + e\right ) + \csc \left (f x + e\right )\right )}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)*(a+b*sin(f*x+e)),x, algorithm="maxima")

[Out]

((f*x + e)*b - a*log(cot(f*x + e) + csc(f*x + e)))/f

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Fricas [B]  time = 1.75978, size = 111, normalized size = 6.53 \begin{align*} \frac{2 \, b f x - a \log \left (\frac{1}{2} \, \cos \left (f x + e\right ) + \frac{1}{2}\right ) + a \log \left (-\frac{1}{2} \, \cos \left (f x + e\right ) + \frac{1}{2}\right )}{2 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)*(a+b*sin(f*x+e)),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 7.32409, size = 51, normalized size = 3. \begin{align*} a \left (\begin{cases} \frac{x \cot{\left (e \right )} \csc{\left (e \right )}}{\cot{\left (e \right )} + \csc{\left (e \right )}} + \frac{x \csc ^{2}{\left (e \right )}}{\cot{\left (e \right )} + \csc{\left (e \right )}} & \text{for}\: f = 0 \\- \frac{\log{\left (\cot{\left (e + f x \right )} + \csc{\left (e + f x \right )} \right )}}{f} & \text{otherwise} \end{cases}\right ) + b x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)*(a+b*sin(f*x+e)),x)

[Out]

a*Piecewise((x*cot(e)*csc(e)/(cot(e) + csc(e)) + x*csc(e)**2/(cot(e) + csc(e)), Eq(f, 0)), (-log(cot(e + f*x)
+ csc(e + f*x))/f, True)) + b*x

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Giac [A]  time = 2.15185, size = 36, normalized size = 2.12 \begin{align*} \frac{{\left (f x + e\right )} b + a \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) \right |}\right )}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)*(a+b*sin(f*x+e)),x, algorithm="giac")

[Out]

((f*x + e)*b + a*log(abs(tan(1/2*f*x + 1/2*e))))/f