\(\int \frac {1}{(a+a \sin (x))^3} \, dx\) [27]

Optimal result

Integrand size = 8, antiderivative size = 50 \[ \int \frac {1}{(a+a \sin (x))^3} \, dx=-\frac {\cos (x)}{5 (a+a \sin (x))^3}-\frac {2 \cos (x)}{15 a (a+a \sin (x))^2}-\frac {2 \cos (x)}{15 \left (a^3+a^3 \sin (x)\right )} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2729, 2727} \[ \int \frac {1}{(a+a \sin (x))^3} \, dx=-\frac {2 \cos (x)}{15 \left (a^3 \sin (x)+a^3\right )}-\frac {2 \cos (x)}{15 a (a \sin (x)+a)^2}-\frac {\cos (x)}{5 (a \sin (x)+a)^3} \]

[In]

Int[(a + a*Sin[x])^(-3),x]

[Out]

-1/5*Cos[x]/(a + a*Sin[x])^3 - (2*Cos[x])/(15*a*(a + a*Sin[x])^2) - (2*Cos[x])/(15*(a^3 + a^3*Sin[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*} \text {integral}& = -\frac {\cos (x)}{5 (a+a \sin (x))^3}+\frac {2 \int \frac {1}{(a+a \sin (x))^2} \, dx}{5 a} \\ & = -\frac {\cos (x)}{5 (a+a \sin (x))^3}-\frac {2 \cos (x)}{15 a (a+a \sin (x))^2}+\frac {2 \int \frac {1}{a+a \sin (x)} \, dx}{15 a^2} \\ & = -\frac {\cos (x)}{5 (a+a \sin (x))^3}-\frac {2 \cos (x)}{15 a (a+a \sin (x))^2}-\frac {2 \cos (x)}{15 \left (a^3+a^3 \sin (x)\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(a+a \sin (x))^3} \, dx=-\frac {5 \cos \left (\frac {3 x}{2}\right )-10 \sin \left (\frac {x}{2}\right )+\sin \left (\frac {5 x}{2}\right )}{15 a^3 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5} \]

[In]

Integrate[(a + a*Sin[x])^(-3),x]

[Out]

-1/15*(5*Cos[(3*x)/2] - 10*Sin[x/2] + Sin[(5*x)/2])/(a^3*(Cos[x/2] + Sin[x/2])^5)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.33 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.66

method	result	size
risch	\(\frac {-\frac {4}{15}+\frac {8 \,{\mathrm e}^{2 i x}}{3}+\frac {4 i {\mathrm e}^{i x}}{3}}{\left ({\mathrm e}^{i x}+i\right )^{5} a^{3}}\)	\(33\)
parallelrisch	\(\frac {12 \tan \left (x \right ) \left (\sec ^{4}\left (x \right )\right )-12 \left (\sec ^{5}\left (x \right )\right )+\tan \left (x \right ) \left (\sec ^{2}\left (x \right )\right )+5 \left (\sec ^{3}\left (x \right )\right )+2 \tan \left (x \right )-7}{15 a^{3}}\)	\(39\)
default	\(\frac {-\frac {8}{5 \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}-\frac {16}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {4}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {4}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {2}{\tan \left (\frac {x}{2}\right )+1}}{a^{3}}\)	\(57\)
norman	\(\frac {-\frac {2 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{a}-\frac {14}{15 a}-\frac {4 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{a}-\frac {8 \tan \left (\frac {x}{2}\right )}{3 a}-\frac {16 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{3 a}}{a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}\)	\(61\)

[In]

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

[Out]

4/15*(-1+10*exp(2*I*x)+5*I*exp(I*x))/(exp(I*x)+I)^5/a^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (44) = 88\).

Time = 0.26 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.84 \[ \int \frac {1}{(a+a \sin (x))^3} \, dx=-\frac {2 \, \cos \left (x\right )^{3} - 4 \, \cos \left (x\right )^{2} - {\left (2 \, \cos \left (x\right )^{2} + 6 \, \cos \left (x\right ) - 3\right )} \sin \left (x\right ) - 9 \, \cos \left (x\right ) - 3}{15 \, {\left (a^{3} \cos \left (x\right )^{3} + 3 \, a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3} + {\left (a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3}\right )} \sin \left (x\right )\right )}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 348 vs. \(2 (49) = 98\).

Time = 0.60 (sec) , antiderivative size = 348, normalized size of antiderivative = 6.96 \[ \int \frac {1}{(a+a \sin (x))^3} \, dx=- \frac {30 \tan ^{4}{\left (\frac {x}{2} \right )}}{15 a^{3} \tan ^{5}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{3}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan {\left (\frac {x}{2} \right )} + 15 a^{3}} - \frac {60 \tan ^{3}{\left (\frac {x}{2} \right )}}{15 a^{3} \tan ^{5}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{3}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan {\left (\frac {x}{2} \right )} + 15 a^{3}} - \frac {80 \tan ^{2}{\left (\frac {x}{2} \right )}}{15 a^{3} \tan ^{5}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{3}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan {\left (\frac {x}{2} \right )} + 15 a^{3}} - \frac {40 \tan {\left (\frac {x}{2} \right )}}{15 a^{3} \tan ^{5}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{3}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan {\left (\frac {x}{2} \right )} + 15 a^{3}} - \frac {14}{15 a^{3} \tan ^{5}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{3}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan {\left (\frac {x}{2} \right )} + 15 a^{3}} \]

[In]

integrate(1/(a+a*sin(x))**3,x)

[Out]

-30*tan(x/2)**4/(15*a**3*tan(x/2)**5 + 75*a**3*tan(x/2)**4 + 150*a**3*tan(x/2)**3 + 150*a**3*tan(x/2)**2 + 75*
a**3*tan(x/2) + 15*a**3) - 60*tan(x/2)**3/(15*a**3*tan(x/2)**5 + 75*a**3*tan(x/2)**4 + 150*a**3*tan(x/2)**3 +
150*a**3*tan(x/2)**2 + 75*a**3*tan(x/2) + 15*a**3) - 80*tan(x/2)**2/(15*a**3*tan(x/2)**5 + 75*a**3*tan(x/2)**4
+ 150*a**3*tan(x/2)**3 + 150*a**3*tan(x/2)**2 + 75*a**3*tan(x/2) + 15*a**3) - 40*tan(x/2)/(15*a**3*tan(x/2)**
5 + 75*a**3*tan(x/2)**4 + 150*a**3*tan(x/2)**3 + 150*a**3*tan(x/2)**2 + 75*a**3*tan(x/2) + 15*a**3) - 14/(15*a
**3*tan(x/2)**5 + 75*a**3*tan(x/2)**4 + 150*a**3*tan(x/2)**3 + 150*a**3*tan(x/2)**2 + 75*a**3*tan(x/2) + 15*a*
*3)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (44) = 88\).

Time = 0.20 (sec) , antiderivative size = 128, normalized size of antiderivative = 2.56 \[ \int \frac {1}{(a+a \sin (x))^3} \, dx=-\frac {2 \, {\left (\frac {20 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {40 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {30 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + 7\right )}}{15 \, {\left (a^{3} + \frac {5 \, a^{3} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {10 \, a^{3} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}}\right )}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(a+a \sin (x))^3} \, dx=-\frac {2 \, {\left (15 \, \tan \left (\frac {1}{2} \, x\right )^{4} + 30 \, \tan \left (\frac {1}{2} \, x\right )^{3} + 40 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 20 \, \tan \left (\frac {1}{2} \, x\right ) + 7\right )}}{15 \, a^{3} {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{5}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.95 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(a+a \sin (x))^3} \, dx=-\frac {2\,\left (15\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+30\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+40\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+20\,\mathrm {tan}\left (\frac {x}{2}\right )+7\right )}{15\,a^3\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^5} \]

[In]

int(1/(a + a*sin(x))^3,x)

[Out]

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