3.2.53 \(\int \csc (e+f x) (a+b \sin (e+f x)) \, dx\) [153]

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 = {2814, 3855} \begin {gather*} b x-\frac {a \tanh ^{-1}(\cos (e+f x))}{f} \end {gather*}

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 2814

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 3855

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] Leaf count is larger than twice the leaf count of optimal. \(43\) vs. \(2(17)=34\).
time = 0.01, size = 43, normalized size = 2.53 \begin {gather*} b x-\frac {a \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{f}+\frac {a \log \left (\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{f} \end {gather*}

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 = 0.18, size = 31, normalized size = 1.82

method	result	size
derivativedivides	\(\frac {a \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )+b \left (f x +e \right )}{f}\)	\(31\)
default	\(\frac {a \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )+b \left (f x +e \right )}{f}\)	\(31\)
risch	\(b x +\frac {a \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{f}-\frac {a \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{f}\)	\(40\)
norman	\(\frac {b x +b x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}+\frac {a \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 32, normalized size = 1.88 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (18) = 36\).
time = 0.35, size = 40, normalized size = 2.35 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (14) = 28\).
time = 2.81, size = 51, normalized size = 3.00 \begin {gather*} 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 {gather*}

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 = 0.46, size = 27, normalized size = 1.59 \begin {gather*} \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 {gather*}

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

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Mupad [B]
time = 6.80, size = 85, normalized size = 5.00 \begin {gather*} \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} \end {gather*}

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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