3.1.5 \(\int \frac {1}{4+3 \cos (x)+4 \sin (x)} \, dx\) [5]

Optimal. Leaf size=21 \[ -\frac {1}{3} \log \left (4+3 \cot \left (\frac {\pi }{4}+\frac {x}{2}\right )\right ) \]

[Out]

-1/3*ln(4+3*cot(1/4*Pi+1/2*x))

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Rubi [A]
time = 0.01, antiderivative size = 21, 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 = {3202, 31} \begin {gather*} -\frac {1}{3} \log \left (3 \cot \left (\frac {x}{2}+\frac {\pi }{4}\right )+4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4 + 3*Cos[x] + 4*Sin[x])^(-1),x]

[Out]

-1/3*Log[4 + 3*Cot[Pi/4 + x/2]]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 3202

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(-1), x_Symbol] :> Module[{f = Free
Factors[Cot[(d + e*x)/2 + Pi/4], x]}, Dist[-f/e, Subst[Int[1/(a + b*f*x), x], x, Cot[(d + e*x)/2 + Pi/4]/f], x
]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[a - c, 0] && NeQ[a - b, 0]

Rubi steps

\begin {align*} \int \frac {1}{4+3 \cos (x)+4 \sin (x)} \, dx &=-\text {Subst}\left (\int \frac {1}{4+3 x} \, dx,x,\cot \left (\frac {\pi }{4}+\frac {x}{2}\right )\right )\\ &=-\frac {1}{3} \log \left (4+3 \cot \left (\frac {\pi }{4}+\frac {x}{2}\right )\right )\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 39, normalized size = 1.86 \begin {gather*} \frac {1}{3} \log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )-\frac {1}{3} \log \left (7 \cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 + 3*Cos[x] + 4*Sin[x])^(-1),x]

[Out]

Log[Cos[x/2] + Sin[x/2]]/3 - Log[7*Cos[x/2] + Sin[x/2]]/3

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Mathics [A]
time = 1.88, size = 19, normalized size = 0.90 \begin {gather*} -\frac {\text {Log}\left [7+\text {Tan}\left [\frac {x}{2}\right ]\right ]}{3}+\frac {\text {Log}\left [1+\text {Tan}\left [\frac {x}{2}\right ]\right ]}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[1/(4 + 3*Cos[x] + 4*Sin[x]),x]')

[Out]

-Log[7 + Tan[x / 2]] / 3 + Log[1 + Tan[x / 2]] / 3

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Maple [A]
time = 0.08, size = 20, normalized size = 0.95

method	result	size
default	\(-\frac {\ln \left (\tan \left (\frac {x}{2}\right )+7\right )}{3}+\frac {\ln \left (1+\tan \left (\frac {x}{2}\right )\right )}{3}\)	\(20\)
norman	\(-\frac {\ln \left (\tan \left (\frac {x}{2}\right )+7\right )}{3}+\frac {\ln \left (1+\tan \left (\frac {x}{2}\right )\right )}{3}\)	\(20\)
risch	\(-\frac {\ln \left ({\mathrm e}^{i x}+\frac {24}{25}+\frac {7 i}{25}\right )}{3}+\frac {\ln \left ({\mathrm e}^{i x}+i\right )}{3}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(4+3*cos(x)+4*sin(x)),x,method=_RETURNVERBOSE)

[Out]

-1/3*ln(tan(1/2*x)+7)+1/3*ln(1+tan(1/2*x))

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Maxima [A]
time = 0.24, size = 29, normalized size = 1.38 \begin {gather*} -\frac {1}{3} \, \log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1} + 7\right ) + \frac {1}{3} \, \log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4+3*cos(x)+4*sin(x)),x, algorithm="maxima")

[Out]

-1/3*log(sin(x)/(cos(x) + 1) + 7) + 1/3*log(sin(x)/(cos(x) + 1) + 1)

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Fricas [A]
time = 0.32, size = 21, normalized size = 1.00 \begin {gather*} -\frac {1}{6} \, \log \left (24 \, \cos \left (x\right ) + 7 \, \sin \left (x\right ) + 25\right ) + \frac {1}{6} \, \log \left (\sin \left (x\right ) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4+3*cos(x)+4*sin(x)),x, algorithm="fricas")

[Out]

-1/6*log(24*cos(x) + 7*sin(x) + 25) + 1/6*log(sin(x) + 1)

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Sympy [A]
time = 0.13, size = 19, normalized size = 0.90 \begin {gather*} \frac {\log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )}}{3} - \frac {\log {\left (\tan {\left (\frac {x}{2} \right )} + 7 \right )}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4+3*cos(x)+4*sin(x)),x)

[Out]

log(tan(x/2) + 1)/3 - log(tan(x/2) + 7)/3

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Giac [A]
time = 0.00, size = 28, normalized size = 1.33 \begin {gather*} 2 \left (\frac {\ln \left |\tan \left (\frac {x}{2}\right )+1\right |}{6}-\frac {\ln \left |\tan \left (\frac {x}{2}\right )+7\right |}{6}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4+3*cos(x)+4*sin(x)),x)

[Out]

-1/3*log(abs(tan(1/2*x) + 7)) + 1/3*log(abs(tan(1/2*x) + 1))

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Mupad [B]
time = 0.41, size = 11, normalized size = 0.52 \begin {gather*} -\frac {2\,\mathrm {atanh}\left (\frac {\mathrm {tan}\left (\frac {x}{2}\right )}{3}+\frac {4}{3}\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3*cos(x) + 4*sin(x) + 4),x)

[Out]

-(2*atanh(tan(x/2)/3 + 4/3))/3

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