3.6.23 \(\int \frac {1}{3+5 \sec (c+d x)} \, dx\) [523]

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

[Out]

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

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Rubi [A]
time = 0.02, 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 = {3868, 2736} \begin {gather*} \frac {5 \text {ArcTan}\left (\frac {\sin (c+d x)}{\cos (c+d x)+3}\right )}{6 d}-\frac {x}{12} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + 5*Sec[c + d*x])^(-1),x]

[Out]

-1/12*x + (5*ArcTan[Sin[c + d*x]/(3 + Cos[c + d*x])])/(6*d)

Rule 2736

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

Rule 3868

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

Rubi steps

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

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Mathematica [A]
time = 0.06, size = 30, normalized size = 0.97 \begin {gather*} \frac {2 (c+d x)+5 \text {ArcTan}\left (2 \cot \left (\frac {1}{2} (c+d x)\right )\right )}{6 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(c + d*x) + 5*ArcTan[2*Cot[(c + d*x)/2]])/(6*d)

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Maple [A]
time = 0.06, size = 32, normalized size = 1.03

method	result	size
derivativedivides	\(\frac {\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {5 \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}\right )}{6}}{d}\)	\(32\)
default	\(\frac {\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {5 \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}\right )}{6}}{d}\)	\(32\)
risch	\(\frac {x}{3}-\frac {5 i \ln \left ({\mathrm e}^{i \left (d x +c \right )}+3\right )}{12 d}+\frac {5 i \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {1}{3}\right )}{12 d}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3+5*sec(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.46, size = 47, normalized size = 1.52 \begin {gather*} \frac {4 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right ) - 5 \, \arctan \left (\frac {\sin \left (d x + c\right )}{2 \, {\left (\cos \left (d x + c\right ) + 1\right )}}\right )}{6 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(4*arctan(sin(d*x + c)/(cos(d*x + c) + 1)) - 5*arctan(1/2*sin(d*x + c)/(cos(d*x + c) + 1)))/d

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Fricas [A]
time = 2.79, size = 33, normalized size = 1.06 \begin {gather*} \frac {4 \, d x + 5 \, \arctan \left (\frac {5 \, \cos \left (d x + c\right ) + 3}{4 \, \sin \left (d x + c\right )}\right )}{12 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{5 \sec {\left (c + d x \right )} + 3}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(5*sec(c + d*x) + 3), x)

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Giac [A]
time = 0.42, size = 30, normalized size = 0.97 \begin {gather*} -\frac {d x + c - 10 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 3}\right )}{12 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.82, size = 21, normalized size = 0.68 \begin {gather*} \frac {x}{3}-\frac {5\,\mathrm {atan}\left (\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}\right )}{6\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(5/cos(c + d*x) + 3),x)

[Out]

x/3 - (5*atan(tan(c/2 + (d*x)/2)/2))/(6*d)

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