3.1.12 \(\int \frac {\sin ^4(x)}{(a+a \sin (x))^2} \, dx\) [12]

Optimal. Leaf size=66 \[ \frac {7 x}{2 a^2}+\frac {16 \cos (x)}{3 a^2}-\frac {7 \cos (x) \sin (x)}{2 a^2}+\frac {8 \cos (x) \sin ^2(x)}{3 a^2 (1+\sin (x))}+\frac {\cos (x) \sin ^3(x)}{3 (a+a \sin (x))^2} \]

[Out]

7/2*x/a^2+16/3*cos(x)/a^2-7/2*cos(x)*sin(x)/a^2+8/3*cos(x)*sin(x)^2/a^2/(1+sin(x))+1/3*cos(x)*sin(x)^3/(a+a*si
n(x))^2

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Rubi [A]
time = 0.09, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2844, 3056, 2813} \begin {gather*} \frac {7 x}{2 a^2}+\frac {16 \cos (x)}{3 a^2}+\frac {8 \sin ^2(x) \cos (x)}{3 a^2 (\sin (x)+1)}-\frac {7 \sin (x) \cos (x)}{2 a^2}+\frac {\sin ^3(x) \cos (x)}{3 (a \sin (x)+a)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[x]^4/(a + a*Sin[x])^2,x]

[Out]

(7*x)/(2*a^2) + (16*Cos[x])/(3*a^2) - (7*Cos[x]*Sin[x])/(2*a^2) + (8*Cos[x]*Sin[x]^2)/(3*a^2*(1 + Sin[x])) + (
Cos[x]*Sin[x]^3)/(3*(a + a*Sin[x])^2)

Rule 2813

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

Rule 2844

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n - 1)/(a*f*(2*m + 1))), x] + Dist[1/
(a*b*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 2)*Simp[b*(c^2*(m + 1) + d^2*(n -
1)) + a*c*d*(m - n + 1) + d*(a*d*(m - n + 1) + b*c*(m + n))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e,
f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ
[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 0]))

Rule 3056

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x]
)^n/(a*f*(2*m + 1))), x] - Dist[1/(a*b*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n -
1)*Simp[A*(a*d*n - b*c*(m + 1)) - B*(a*c*m + b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x],
x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ
[m, -2^(-1)] && GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])

Rubi steps

\begin {align*} \int \frac {\sin ^4(x)}{(a+a \sin (x))^2} \, dx &=\frac {\cos (x) \sin ^3(x)}{3 (a+a \sin (x))^2}-\frac {\int \frac {\sin ^2(x) (3 a-5 a \sin (x))}{a+a \sin (x)} \, dx}{3 a^2}\\ &=\frac {8 \cos (x) \sin ^2(x)}{3 a^2 (1+\sin (x))}+\frac {\cos (x) \sin ^3(x)}{3 (a+a \sin (x))^2}-\frac {\int \sin (x) \left (16 a^2-21 a^2 \sin (x)\right ) \, dx}{3 a^4}\\ &=\frac {7 x}{2 a^2}+\frac {16 \cos (x)}{3 a^2}-\frac {7 \cos (x) \sin (x)}{2 a^2}+\frac {8 \cos (x) \sin ^2(x)}{3 a^2 (1+\sin (x))}+\frac {\cos (x) \sin ^3(x)}{3 (a+a \sin (x))^2}\\ \end {align*}

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Mathematica [A]
time = 0.18, size = 100, normalized size = 1.52 \begin {gather*} \frac {\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \left (21 (-7+12 x) \cos \left (\frac {x}{2}\right )+(239-84 x) \cos \left (\frac {3 x}{2}\right )+3 \left (-5 \cos \left (\frac {5 x}{2}\right )+\cos \left (\frac {7 x}{2}\right )+2 (-50+56 x+(27+28 x) \cos (x)+6 \cos (2 x)+\cos (3 x)) \sin \left (\frac {x}{2}\right )\right )\right )}{48 a^2 (1+\sin (x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]^4/(a + a*Sin[x])^2,x]

[Out]

((Cos[x/2] + Sin[x/2])*(21*(-7 + 12*x)*Cos[x/2] + (239 - 84*x)*Cos[(3*x)/2] + 3*(-5*Cos[(5*x)/2] + Cos[(7*x)/2
] + 2*(-50 + 56*x + (27 + 28*x)*Cos[x] + 6*Cos[2*x] + Cos[3*x])*Sin[x/2])))/(48*a^2*(1 + Sin[x])^2)

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Maple [A]
time = 0.15, size = 80, normalized size = 1.21

method	result	size
default	\(\frac {-\frac {4}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {2}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {6}{\tan \left (\frac {x}{2}\right )+1}+\frac {2 \left (\frac {\left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{2}+2 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-\frac {\tan \left (\frac {x}{2}\right )}{2}+2\right )}{\left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{2}}+7 \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{a^{2}}\)	\(80\)
risch	\(\frac {7 x}{2 a^{2}}+\frac {i {\mathrm e}^{2 i x}}{8 a^{2}}+\frac {{\mathrm e}^{i x}}{a^{2}}+\frac {{\mathrm e}^{-i x}}{a^{2}}-\frac {i {\mathrm e}^{-2 i x}}{8 a^{2}}+\frac {8 \,{\mathrm e}^{2 i x}-\frac {22}{3}+14 i {\mathrm e}^{i x}}{\left ({\mathrm e}^{i x}+i\right )^{3} a^{2}}\)	\(80\)
norman	\(\frac {\frac {7 \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{a}+\frac {25 \tan \left (\frac {x}{2}\right )}{a}+\frac {92 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{a}+\frac {7 x}{2 a}+\frac {32}{3 a}+\frac {21 x \tan \left (\frac {x}{2}\right )}{2 a}+\frac {49 x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {91 x \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {63 x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{a}+\frac {77 x \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{a}+\frac {77 x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{a}+\frac {63 x \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{a}+\frac {91 x \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {49 x \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {21 x \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {7 x \left (\tan ^{11}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {21 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{a}+\frac {108 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{a}+\frac {84 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{a}+\frac {161 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{3 a}+\frac {130 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{a}+\frac {314 \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{3 a}+\frac {140 \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{3 a}}{\left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{4} a \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}\)	\(273\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^4/(a+a*sin(x))^2,x,method=_RETURNVERBOSE)

[Out]

32/a^2*(-1/24/(tan(1/2*x)+1)^3+1/16/(tan(1/2*x)+1)^2+3/16/(tan(1/2*x)+1)+1/16*(1/2*tan(1/2*x)^3+2*tan(1/2*x)^2
-1/2*tan(1/2*x)+2)/(tan(1/2*x)^2+1)^2+7/32*arctan(tan(1/2*x)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 198 vs. \(2 (56) = 112\).
time = 0.73, size = 198, normalized size = 3.00 \begin {gather*} \frac {\frac {75 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {97 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {126 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {98 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {63 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {21 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + 32}{3 \, {\left (a^{2} + \frac {3 \, a^{2} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {5 \, a^{2} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {7 \, a^{2} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {7 \, a^{2} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {5 \, a^{2} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {3 \, a^{2} \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac {a^{2} \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}}\right )}} + \frac {7 \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^4/(a+a*sin(x))^2,x, algorithm="maxima")

[Out]

1/3*(75*sin(x)/(cos(x) + 1) + 97*sin(x)^2/(cos(x) + 1)^2 + 126*sin(x)^3/(cos(x) + 1)^3 + 98*sin(x)^4/(cos(x) +
1)^4 + 63*sin(x)^5/(cos(x) + 1)^5 + 21*sin(x)^6/(cos(x) + 1)^6 + 32)/(a^2 + 3*a^2*sin(x)/(cos(x) + 1) + 5*a^2
*sin(x)^2/(cos(x) + 1)^2 + 7*a^2*sin(x)^3/(cos(x) + 1)^3 + 7*a^2*sin(x)^4/(cos(x) + 1)^4 + 5*a^2*sin(x)^5/(cos
(x) + 1)^5 + 3*a^2*sin(x)^6/(cos(x) + 1)^6 + a^2*sin(x)^7/(cos(x) + 1)^7) + 7*arctan(sin(x)/(cos(x) + 1))/a^2

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Fricas [A]
time = 0.36, size = 105, normalized size = 1.59 \begin {gather*} -\frac {3 \, \cos \left (x\right )^{4} - {\left (21 \, x - 31\right )} \cos \left (x\right )^{2} - 6 \, \cos \left (x\right )^{3} + {\left (21 \, x + 38\right )} \cos \left (x\right ) + {\left (3 \, \cos \left (x\right )^{3} + {\left (21 \, x + 40\right )} \cos \left (x\right ) + 9 \, \cos \left (x\right )^{2} + 42 \, x + 2\right )} \sin \left (x\right ) + 42 \, x - 2}{6 \, {\left (a^{2} \cos \left (x\right )^{2} - a^{2} \cos \left (x\right ) - 2 \, a^{2} - {\left (a^{2} \cos \left (x\right ) + 2 \, a^{2}\right )} \sin \left (x\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^4/(a+a*sin(x))^2,x, algorithm="fricas")

[Out]

-1/6*(3*cos(x)^4 - (21*x - 31)*cos(x)^2 - 6*cos(x)^3 + (21*x + 38)*cos(x) + (3*cos(x)^3 + (21*x + 40)*cos(x) +
9*cos(x)^2 + 42*x + 2)*sin(x) + 42*x - 2)/(a^2*cos(x)^2 - a^2*cos(x) - 2*a^2 - (a^2*cos(x) + 2*a^2)*sin(x))

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1423 vs. \(2 (70) = 140\).
time = 5.05, size = 1423, normalized size = 21.56 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)**4/(a+a*sin(x))**2,x)

[Out]

21*x*tan(x/2)**7/(6*a**2*tan(x/2)**7 + 18*a**2*tan(x/2)**6 + 30*a**2*tan(x/2)**5 + 42*a**2*tan(x/2)**4 + 42*a*
*2*tan(x/2)**3 + 30*a**2*tan(x/2)**2 + 18*a**2*tan(x/2) + 6*a**2) + 63*x*tan(x/2)**6/(6*a**2*tan(x/2)**7 + 18*
a**2*tan(x/2)**6 + 30*a**2*tan(x/2)**5 + 42*a**2*tan(x/2)**4 + 42*a**2*tan(x/2)**3 + 30*a**2*tan(x/2)**2 + 18*
a**2*tan(x/2) + 6*a**2) + 105*x*tan(x/2)**5/(6*a**2*tan(x/2)**7 + 18*a**2*tan(x/2)**6 + 30*a**2*tan(x/2)**5 +
42*a**2*tan(x/2)**4 + 42*a**2*tan(x/2)**3 + 30*a**2*tan(x/2)**2 + 18*a**2*tan(x/2) + 6*a**2) + 147*x*tan(x/2)*
*4/(6*a**2*tan(x/2)**7 + 18*a**2*tan(x/2)**6 + 30*a**2*tan(x/2)**5 + 42*a**2*tan(x/2)**4 + 42*a**2*tan(x/2)**3
+ 30*a**2*tan(x/2)**2 + 18*a**2*tan(x/2) + 6*a**2) + 147*x*tan(x/2)**3/(6*a**2*tan(x/2)**7 + 18*a**2*tan(x/2)
**6 + 30*a**2*tan(x/2)**5 + 42*a**2*tan(x/2)**4 + 42*a**2*tan(x/2)**3 + 30*a**2*tan(x/2)**2 + 18*a**2*tan(x/2)
+ 6*a**2) + 105*x*tan(x/2)**2/(6*a**2*tan(x/2)**7 + 18*a**2*tan(x/2)**6 + 30*a**2*tan(x/2)**5 + 42*a**2*tan(x
/2)**4 + 42*a**2*tan(x/2)**3 + 30*a**2*tan(x/2)**2 + 18*a**2*tan(x/2) + 6*a**2) + 63*x*tan(x/2)/(6*a**2*tan(x/
2)**7 + 18*a**2*tan(x/2)**6 + 30*a**2*tan(x/2)**5 + 42*a**2*tan(x/2)**4 + 42*a**2*tan(x/2)**3 + 30*a**2*tan(x/
2)**2 + 18*a**2*tan(x/2) + 6*a**2) + 21*x/(6*a**2*tan(x/2)**7 + 18*a**2*tan(x/2)**6 + 30*a**2*tan(x/2)**5 + 42
*a**2*tan(x/2)**4 + 42*a**2*tan(x/2)**3 + 30*a**2*tan(x/2)**2 + 18*a**2*tan(x/2) + 6*a**2) + 42*tan(x/2)**6/(6
*a**2*tan(x/2)**7 + 18*a**2*tan(x/2)**6 + 30*a**2*tan(x/2)**5 + 42*a**2*tan(x/2)**4 + 42*a**2*tan(x/2)**3 + 30
*a**2*tan(x/2)**2 + 18*a**2*tan(x/2) + 6*a**2) + 126*tan(x/2)**5/(6*a**2*tan(x/2)**7 + 18*a**2*tan(x/2)**6 + 3
0*a**2*tan(x/2)**5 + 42*a**2*tan(x/2)**4 + 42*a**2*tan(x/2)**3 + 30*a**2*tan(x/2)**2 + 18*a**2*tan(x/2) + 6*a*
*2) + 196*tan(x/2)**4/(6*a**2*tan(x/2)**7 + 18*a**2*tan(x/2)**6 + 30*a**2*tan(x/2)**5 + 42*a**2*tan(x/2)**4 +
42*a**2*tan(x/2)**3 + 30*a**2*tan(x/2)**2 + 18*a**2*tan(x/2) + 6*a**2) + 252*tan(x/2)**3/(6*a**2*tan(x/2)**7 +
18*a**2*tan(x/2)**6 + 30*a**2*tan(x/2)**5 + 42*a**2*tan(x/2)**4 + 42*a**2*tan(x/2)**3 + 30*a**2*tan(x/2)**2 +
18*a**2*tan(x/2) + 6*a**2) + 194*tan(x/2)**2/(6*a**2*tan(x/2)**7 + 18*a**2*tan(x/2)**6 + 30*a**2*tan(x/2)**5
+ 42*a**2*tan(x/2)**4 + 42*a**2*tan(x/2)**3 + 30*a**2*tan(x/2)**2 + 18*a**2*tan(x/2) + 6*a**2) + 150*tan(x/2)/
(6*a**2*tan(x/2)**7 + 18*a**2*tan(x/2)**6 + 30*a**2*tan(x/2)**5 + 42*a**2*tan(x/2)**4 + 42*a**2*tan(x/2)**3 +
30*a**2*tan(x/2)**2 + 18*a**2*tan(x/2) + 6*a**2) + 64/(6*a**2*tan(x/2)**7 + 18*a**2*tan(x/2)**6 + 30*a**2*tan(
x/2)**5 + 42*a**2*tan(x/2)**4 + 42*a**2*tan(x/2)**3 + 30*a**2*tan(x/2)**2 + 18*a**2*tan(x/2) + 6*a**2)

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Giac [A]
time = 0.58, size = 72, normalized size = 1.09 \begin {gather*} \frac {7 \, x}{2 \, a^{2}} + \frac {\tan \left (\frac {1}{2} \, x\right )^{3} + 4 \, \tan \left (\frac {1}{2} \, x\right )^{2} - \tan \left (\frac {1}{2} \, x\right ) + 4}{{\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}^{2} a^{2}} + \frac {2 \, {\left (9 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 21 \, \tan \left (\frac {1}{2} \, x\right ) + 10\right )}}{3 \, a^{2} {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^4/(a+a*sin(x))^2,x, algorithm="giac")

[Out]

7/2*x/a^2 + (tan(1/2*x)^3 + 4*tan(1/2*x)^2 - tan(1/2*x) + 4)/((tan(1/2*x)^2 + 1)^2*a^2) + 2/3*(9*tan(1/2*x)^2
+ 21*tan(1/2*x) + 10)/(a^2*(tan(1/2*x) + 1)^3)

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Mupad [B]
time = 6.81, size = 77, normalized size = 1.17 \begin {gather*} \frac {7\,x}{2\,a^2}+\frac {7\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6+21\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5+\frac {98\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4}{3}+42\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+\frac {97\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{3}+25\,\mathrm {tan}\left (\frac {x}{2}\right )+\frac {32}{3}}{a^2\,{\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )}^2\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^4/(a + a*sin(x))^2,x)

[Out]

(7*x)/(2*a^2) + (25*tan(x/2) + (97*tan(x/2)^2)/3 + 42*tan(x/2)^3 + (98*tan(x/2)^4)/3 + 21*tan(x/2)^5 + 7*tan(x
/2)^6 + 32/3)/(a^2*(tan(x/2)^2 + 1)^2*(tan(x/2) + 1)^3)

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