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(f*x+e))/f

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Rubi [A]  time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.00, 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*}

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Mathematica [B]  time = 0.02, 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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fricas [B]  time = 0.45, size = 38, normalized size = 2.24 \[ \frac {2 \, b f x - a \log \left (\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) + a \log \left (-\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right )}{2 \, f} \]

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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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

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]

Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)2/f*(a/2*ln(abs(tan((f*x+exp(1))/2)))+2*b/2*(f*x+exp(1))/2)

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maple [A]  time = 0.11, size = 32, normalized size = 1.88 \[ b x +\frac {a \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{f}+\frac {b e}{f} \]

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 = 0.64, size = 29, normalized size = 1.71 \[ \frac {{\left (f x + e\right )} b - a \log \left (\cot \left (f x + e\right ) + \csc \left (f x + e\right )\right )}{f} \]

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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mupad [B]  time = 6.80, size = 85, normalized size = 5.00 \[ \frac {a\,\ln \left (\frac {\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}\right )}{f}+\frac {2\,b\,\mathrm {atan}\left (\frac {b\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )+a\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}{a\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )-b\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}\right )}{f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 6.76, size = 51, normalized size = 3.00 \[ a \left (\begin {cases} \frac {x \cot {\relax (e )} \csc {\relax (e )}}{\cot {\relax (e )} + \csc {\relax (e )}} + \frac {x \csc ^{2}{\relax (e )}}{\cot {\relax (e )} + \csc {\relax (e )}} & \text {for}\: f = 0 \\- \frac {\log {\left (\cot {\left (e + f x \right )} + \csc {\left (e + f x \right )} \right )}}{f} & \text {otherwise} \end {cases}\right ) + b x \]

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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