3.5.76 \(\int \frac {1}{(a+a \sin (e+f x))^3} \, dx\) [476]

Optimal. Leaf size=83 \[ -\frac {\cos (e+f x)}{5 f (a+a \sin (e+f x))^3}-\frac {2 \cos (e+f x)}{15 a f (a+a \sin (e+f x))^2}-\frac {2 \cos (e+f x)}{15 f \left (a^3+a^3 \sin (e+f x)\right )} \]

[Out]

-1/5*cos(f*x+e)/f/(a+a*sin(f*x+e))^3-2/15*cos(f*x+e)/a/f/(a+a*sin(f*x+e))^2-2/15*cos(f*x+e)/f/(a^3+a^3*sin(f*x
+e))

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Rubi [A]
time = 0.03, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2729, 2727} \begin {gather*} -\frac {2 \cos (e+f x)}{15 f \left (a^3 \sin (e+f x)+a^3\right )}-\frac {2 \cos (e+f x)}{15 a f (a \sin (e+f x)+a)^2}-\frac {\cos (e+f x)}{5 f (a \sin (e+f x)+a)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])^(-3),x]

[Out]

-1/5*Cos[e + f*x]/(f*(a + a*Sin[e + f*x])^3) - (2*Cos[e + f*x])/(15*a*f*(a + a*Sin[e + f*x])^2) - (2*Cos[e + f
*x])/(15*f*(a^3 + a^3*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]

Rule 2729

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

Rubi steps

\begin {align*} \int \frac {1}{(a+a \sin (e+f x))^3} \, dx &=-\frac {\cos (e+f x)}{5 f (a+a \sin (e+f x))^3}+\frac {2 \int \frac {1}{(a+a \sin (e+f x))^2} \, dx}{5 a}\\ &=-\frac {\cos (e+f x)}{5 f (a+a \sin (e+f x))^3}-\frac {2 \cos (e+f x)}{15 a f (a+a \sin (e+f x))^2}+\frac {2 \int \frac {1}{a+a \sin (e+f x)} \, dx}{15 a^2}\\ &=-\frac {\cos (e+f x)}{5 f (a+a \sin (e+f x))^3}-\frac {2 \cos (e+f x)}{15 a f (a+a \sin (e+f x))^2}-\frac {2 \cos (e+f x)}{15 f \left (a^3+a^3 \sin (e+f x)\right )}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 76, normalized size = 0.92 \begin {gather*} \frac {10-15 \cos (e+f x)-6 \cos (2 (e+f x))+\cos (3 (e+f x))+15 \sin (e+f x)-6 \sin (2 (e+f x))-\sin (3 (e+f x))}{30 a^3 f (1+\sin (e+f x))^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[e + f*x])^(-3),x]

[Out]

(10 - 15*Cos[e + f*x] - 6*Cos[2*(e + f*x)] + Cos[3*(e + f*x)] + 15*Sin[e + f*x] - 6*Sin[2*(e + f*x)] - Sin[3*(
e + f*x)])/(30*a^3*f*(1 + Sin[e + f*x])^3)

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Maple [A]
time = 0.25, size = 85, normalized size = 1.02

method	result	size
risch	\(\frac {-\frac {4}{15}+\frac {8 \,{\mathrm e}^{2 i \left (f x +e \right )}}{3}+\frac {4 i {\mathrm e}^{i \left (f x +e \right )}}{3}}{f \,a^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{5}}\)	\(48\)
derivativedivides	\(\frac {\frac {4}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}+\frac {4}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {8}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {16}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {2}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}}{f \,a^{3}}\)	\(85\)
default	\(\frac {\frac {4}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}+\frac {4}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {8}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {16}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {2}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}}{f \,a^{3}}\)	\(85\)
norman	\(\frac {-\frac {2 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {14}{15 a f}-\frac {4 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {8 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{3 a f}-\frac {16 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f}}{a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}\)	\(101\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+a*sin(f*x+e))^3,x,method=_RETURNVERBOSE)

[Out]

2/f/a^3*(2/(tan(1/2*f*x+1/2*e)+1)^4+2/(tan(1/2*f*x+1/2*e)+1)^2-4/5/(tan(1/2*f*x+1/2*e)+1)^5-8/3/(tan(1/2*f*x+1
/2*e)+1)^3-1/(tan(1/2*f*x+1/2*e)+1))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (83) = 166\).
time = 0.28, size = 221, normalized size = 2.66 \begin {gather*} -\frac {2 \, {\left (\frac {20 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {40 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {30 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + 7\right )}}{15 \, {\left (a^{3} + \frac {5 \, a^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )} f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sin(f*x+e))^3,x, algorithm="maxima")

[Out]

-2/15*(20*sin(f*x + e)/(cos(f*x + e) + 1) + 40*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 30*sin(f*x + e)^3/(cos(f*
x + e) + 1)^3 + 15*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 7)/((a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e) + 1) + 10
*a^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 10*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 5*a^3*sin(f*x + e)^4/(
cos(f*x + e) + 1)^4 + a^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5)*f)

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Fricas [A]
time = 0.35, size = 159, normalized size = 1.92 \begin {gather*} -\frac {2 \, \cos \left (f x + e\right )^{3} - 4 \, \cos \left (f x + e\right )^{2} - {\left (2 \, \cos \left (f x + e\right )^{2} + 6 \, \cos \left (f x + e\right ) - 3\right )} \sin \left (f x + e\right ) - 9 \, \cos \left (f x + e\right ) - 3}{15 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f + {\left (a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f\right )} \sin \left (f x + e\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sin(f*x+e))^3,x, algorithm="fricas")

[Out]

-1/15*(2*cos(f*x + e)^3 - 4*cos(f*x + e)^2 - (2*cos(f*x + e)^2 + 6*cos(f*x + e) - 3)*sin(f*x + e) - 9*cos(f*x
+ e) - 3)/(a^3*f*cos(f*x + e)^3 + 3*a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f + (a^3*f*cos(f*x + e
)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f)*sin(f*x + e))

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 558 vs. \(2 (73) = 146\).
time = 1.51, size = 558, normalized size = 6.72 \begin {gather*} \begin {cases} - \frac {30 \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{15 a^{3} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 15 a^{3} f} - \frac {60 \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{15 a^{3} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 15 a^{3} f} - \frac {80 \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{15 a^{3} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 15 a^{3} f} - \frac {40 \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{15 a^{3} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 15 a^{3} f} - \frac {14}{15 a^{3} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 15 a^{3} f} & \text {for}\: f \neq 0 \\\frac {x}{\left (a \sin {\left (e \right )} + a\right )^{3}} & \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))**3,x)

[Out]

Piecewise((-30*tan(e/2 + f*x/2)**4/(15*a**3*f*tan(e/2 + f*x/2)**5 + 75*a**3*f*tan(e/2 + f*x/2)**4 + 150*a**3*f
*tan(e/2 + f*x/2)**3 + 150*a**3*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) - 60*tan(e/2 +
f*x/2)**3/(15*a**3*f*tan(e/2 + f*x/2)**5 + 75*a**3*f*tan(e/2 + f*x/2)**4 + 150*a**3*f*tan(e/2 + f*x/2)**3 + 1
50*a**3*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) - 80*tan(e/2 + f*x/2)**2/(15*a**3*f*ta
n(e/2 + f*x/2)**5 + 75*a**3*f*tan(e/2 + f*x/2)**4 + 150*a**3*f*tan(e/2 + f*x/2)**3 + 150*a**3*f*tan(e/2 + f*x/
2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) - 40*tan(e/2 + f*x/2)/(15*a**3*f*tan(e/2 + f*x/2)**5 + 75*a**3
*f*tan(e/2 + f*x/2)**4 + 150*a**3*f*tan(e/2 + f*x/2)**3 + 150*a**3*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 +
f*x/2) + 15*a**3*f) - 14/(15*a**3*f*tan(e/2 + f*x/2)**5 + 75*a**3*f*tan(e/2 + f*x/2)**4 + 150*a**3*f*tan(e/2
+ f*x/2)**3 + 150*a**3*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f), Ne(f, 0)), (x/(a*sin(e
) + a)**3, True))

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Giac [A]
time = 0.48, size = 78, normalized size = 0.94 \begin {gather*} -\frac {2 \, {\left (15 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 30 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 40 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 20 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 7\right )}}{15 \, a^{3} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sin(f*x+e))^3,x, algorithm="giac")

[Out]

-2/15*(15*tan(1/2*f*x + 1/2*e)^4 + 30*tan(1/2*f*x + 1/2*e)^3 + 40*tan(1/2*f*x + 1/2*e)^2 + 20*tan(1/2*f*x + 1/
2*e) + 7)/(a^3*f*(tan(1/2*f*x + 1/2*e) + 1)^5)

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Mupad [B]
time = 6.96, size = 133, normalized size = 1.60 \begin {gather*} -\frac {2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (7\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+20\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )+40\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+30\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+15\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\right )}{15\,a^3\,f\,{\left (\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + a*sin(e + f*x))^3,x)

[Out]

-(2*cos(e/2 + (f*x)/2)*(7*cos(e/2 + (f*x)/2)^4 + 15*sin(e/2 + (f*x)/2)^4 + 30*cos(e/2 + (f*x)/2)*sin(e/2 + (f*
x)/2)^3 + 20*cos(e/2 + (f*x)/2)^3*sin(e/2 + (f*x)/2) + 40*cos(e/2 + (f*x)/2)^2*sin(e/2 + (f*x)/2)^2))/(15*a^3*
f*(cos(e/2 + (f*x)/2) + sin(e/2 + (f*x)/2))^5)

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