3.52 \(\int \frac {1}{3+5 \csc (c+d x)} \, dx\)

Optimal. Leaf size=31 \[ -\frac {5 \tan ^{-1}\left (\frac {\cos (c+d x)}{\sin (c+d x)+3}\right )}{6 d}-\frac {x}{12} \]

[Out]

-1/12*x-5/6*arctan(cos(d*x+c)/(3+sin(d*x+c)))/d

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Rubi [A]  time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3783, 2657} \[ -\frac {5 \tan ^{-1}\left (\frac {\cos (c+d x)}{\sin (c+d x)+3}\right )}{6 d}-\frac {x}{12} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 5*Csc[c + d*x])^(-1),x]

[Out]

-x/12 - (5*ArcTan[Cos[c + d*x]/(3 + Sin[c + d*x])])/(6*d)

Rule 2657

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{q = Rt[a^2 - b^2, 2]}, Simp[x/q, x] + Simp
[(2*ArcTan[(b*Cos[c + d*x])/(a + q + b*Sin[c + d*x])])/(d*q), x]] /; FreeQ[{a, b, c, d}, x] && GtQ[a^2 - b^2,
0] && PosQ[a]

Rule 3783

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(-1), x_Symbol] :> Simp[x/a, x] - Dist[1/a, Int[1/(1 + (a*Sin[c + d
*x])/b), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]

Rubi steps

\begin {align*} \int \frac {1}{3+5 \csc (c+d x)} \, dx &=\frac {x}{3}-\frac {1}{3} \int \frac {1}{1+\frac {3}{5} \sin (c+d x)} \, dx\\ &=-\frac {x}{12}-\frac {5 \tan ^{-1}\left (\frac {\cos (c+d x)}{3+\sin (c+d x)}\right )}{6 d}\\ \end {align*}

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Mathematica [B]  time = 0.05, size = 66, normalized size = 2.13 \[ \frac {2 (c+d x)-5 \tan ^{-1}\left (\frac {2 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}\right )}{6 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*Csc[c + d*x])^(-1),x]

[Out]

(2*(c + d*x) - 5*ArcTan[(2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])])/(6*d
)

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fricas [A]  time = 0.51, size = 33, normalized size = 1.06 \[ \frac {4 \, d x - 5 \, \arctan \left (\frac {5 \, \sin \left (d x + c\right ) + 3}{4 \, \cos \left (d x + c\right )}\right )}{12 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3+5*csc(d*x+c)),x, algorithm="fricas")

[Out]

1/12*(4*d*x - 5*arctan(1/4*(5*sin(d*x + c) + 3)/cos(d*x + c)))/d

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giac [A]  time = 0.59, size = 49, normalized size = 1.58 \[ -\frac {d x + c + 10 \, \arctan \left (-\frac {3 \, \cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 3}{\cos \left (d x + c\right ) - 3 \, \sin \left (d x + c\right ) - 9}\right )}{12 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3+5*csc(d*x+c)),x, algorithm="giac")

[Out]

-1/12*(d*x + c + 10*arctan(-(3*cos(d*x + c) + sin(d*x + c) + 3)/(cos(d*x + c) - 3*sin(d*x + c) - 9)))/d

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maple [A]  time = 0.56, size = 36, normalized size = 1.16 \[ -\frac {5 \arctan \left (\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}+\frac {3}{4}\right )}{6 d}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3+5*csc(d*x+c)),x)

[Out]

-5/6/d*arctan(5/4*tan(1/2*d*x+1/2*c)+3/4)+2/3/d*arctan(tan(1/2*d*x+1/2*c))

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maxima [A]  time = 0.41, size = 49, normalized size = 1.58 \[ -\frac {5 \, \arctan \left (\frac {5 \, \sin \left (d x + c\right )}{4 \, {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {3}{4}\right ) - 4 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{6 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3+5*csc(d*x+c)),x, algorithm="maxima")

[Out]

-1/6*(5*arctan(5/4*sin(d*x + c)/(cos(d*x + c) + 1) + 3/4) - 4*arctan(sin(d*x + c)/(cos(d*x + c) + 1)))/d

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mupad [B]  time = 0.25, size = 39, normalized size = 1.26 \[ \frac {x}{3}-\frac {5\,\mathrm {atan}\left (\frac {7\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-15}{24\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+20}\right )}{6\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(5/sin(c + d*x) + 3),x)

[Out]

x/3 - (5*atan((7*tan(c/2 + (d*x)/2) - 15)/(24*tan(c/2 + (d*x)/2) + 20)))/(6*d)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{5 \csc {\left (c + d x \right )} + 3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3+5*csc(d*x+c)),x)

[Out]

Integral(1/(5*csc(c + d*x) + 3), x)

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