3.5.57 \(\int \frac {1}{a+a \sin (e+f x)} \, dx\) [457]

Optimal. Leaf size=23 \[ -\frac {\cos (e+f x)}{f (a+a \sin (e+f x))} \]

[Out]

-cos(f*x+e)/f/(a+a*sin(f*x+e))

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Rubi [A]
time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2727} \begin {gather*} -\frac {\cos (e+f x)}{f (a \sin (e+f x)+a)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])^(-1),x]

[Out]

-(Cos[e + f*x]/(f*(a + a*Sin[e + f*x])))

Rule 2727

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

Rubi steps

\begin {align*} \int \frac {1}{a+a \sin (e+f x)} \, dx &=-\frac {\cos (e+f x)}{f (a+a \sin (e+f x))}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(48\) vs. \(2(23)=46\).
time = 0.03, size = 48, normalized size = 2.09 \begin {gather*} \frac {2 \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )}{f (a+a \sin (e+f x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[e + f*x])^(-1),x]

[Out]

(2*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))/(f*(a + a*Sin[e + f*x]))

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Maple [A]
time = 0.18, size = 22, normalized size = 0.96

method	result	size
derivativedivides	\(-\frac {2}{f a \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}\)	\(22\)
default	\(-\frac {2}{f a \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}\)	\(22\)
risch	\(-\frac {2}{f a \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}\)	\(23\)
norman	\(\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a f \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+a*sin(f*x+e)),x,method=_RETURNVERBOSE)

[Out]

-2/f/a/(tan(1/2*f*x+1/2*e)+1)

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Maxima [A]
time = 0.29, size = 29, normalized size = 1.26 \begin {gather*} -\frac {2}{{\left (a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )} f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sin(f*x+e)),x, algorithm="maxima")

[Out]

-2/((a + a*sin(f*x + e)/(cos(f*x + e) + 1))*f)

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Fricas [A]
time = 0.35, size = 46, normalized size = 2.00 \begin {gather*} -\frac {\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1}{a f \cos \left (f x + e\right ) + a f \sin \left (f x + e\right ) + a f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sin(f*x+e)),x, algorithm="fricas")

[Out]

-(cos(f*x + e) - sin(f*x + e) + 1)/(a*f*cos(f*x + e) + a*f*sin(f*x + e) + a*f)

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Sympy [A]
time = 0.39, size = 27, normalized size = 1.17 \begin {gather*} \begin {cases} - \frac {2}{a f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f} & \text {for}\: f \neq 0 \\\frac {x}{a \sin {\left (e \right )} + a} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sin(f*x+e)),x)

[Out]

Piecewise((-2/(a*f*tan(e/2 + f*x/2) + a*f), Ne(f, 0)), (x/(a*sin(e) + a), True))

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Giac [A]
time = 0.42, size = 22, normalized size = 0.96 \begin {gather*} -\frac {2}{a f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sin(f*x+e)),x, algorithm="giac")

[Out]

-2/(a*f*(tan(1/2*f*x + 1/2*e) + 1))

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Mupad [B]
time = 6.97, size = 21, normalized size = 0.91 \begin {gather*} -\frac {2}{a\,f\,\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + a*sin(e + f*x)),x)

[Out]

-2/(a*f*(tan(e/2 + (f*x)/2) + 1))

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