3.1.93 \(\int \frac {\sin ^3(a+b x)}{\sin ^{\frac {5}{2}}(2 a+2 b x)} \, dx\) [93]

Optimal. Leaf size=28 \[ \frac {\sin ^3(a+b x)}{3 b \sin ^{\frac {3}{2}}(2 a+2 b x)} \]

[Out]

1/3*sin(b*x+a)^3/b/sin(2*b*x+2*a)^(3/2)

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Rubi [A]
time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {4377} \begin {gather*} \frac {\sin ^3(a+b x)}{3 b \sin ^{\frac {3}{2}}(2 a+2 b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*x]^3/Sin[2*a + 2*b*x]^(5/2),x]

[Out]

Sin[a + b*x]^3/(3*b*Sin[2*a + 2*b*x]^(3/2))

Rule 4377

Int[((e_.)*sin[(a_.) + (b_.)*(x_)])^(m_.)*((g_.)*sin[(c_.) + (d_.)*(x_)])^(p_), x_Symbol] :> Simp[(e*Sin[a + b
*x])^m*((g*Sin[c + d*x])^(p + 1)/(b*g*m)), x] /; FreeQ[{a, b, c, d, e, g, m, p}, x] && EqQ[b*c - a*d, 0] && Eq
Q[d/b, 2] &&  !IntegerQ[p] && EqQ[m + 2*p + 2, 0]

Rubi steps

\begin {align*} \int \frac {\sin ^3(a+b x)}{\sin ^{\frac {5}{2}}(2 a+2 b x)} \, dx &=\frac {\sin ^3(a+b x)}{3 b \sin ^{\frac {3}{2}}(2 a+2 b x)}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 27, normalized size = 0.96 \begin {gather*} \frac {\sin ^3(a+b x)}{3 b \sin ^{\frac {3}{2}}(2 (a+b x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*x]^3/Sin[2*a + 2*b*x]^(5/2),x]

[Out]

Sin[a + b*x]^3/(3*b*Sin[2*(a + b*x)]^(3/2))

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 170.08, size = 727, normalized size = 25.96

method	result	size
default	\(-\frac {\sqrt {-\frac {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )}{\tan ^{2}\left (\frac {a}{2}+\frac {x b}{2}\right )-1}}\, \left (\tan ^{2}\left (\frac {a}{2}+\frac {x b}{2}\right )-1\right ) \left (6 \sqrt {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )+1}\, \sqrt {-2 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )+2}\, \sqrt {-\tan \left (\frac {a}{2}+\frac {x b}{2}\right )}\, \EllipticE \left (\sqrt {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )+1}, \frac {\sqrt {2}}{2}\right ) \left (\tan ^{6}\left (\frac {a}{2}+\frac {x b}{2}\right )\right )-3 \sqrt {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )+1}\, \sqrt {-2 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )+2}\, \sqrt {-\tan \left (\frac {a}{2}+\frac {x b}{2}\right )}\, \EllipticF \left (\sqrt {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )+1}, \frac {\sqrt {2}}{2}\right ) \left (\tan ^{6}\left (\frac {a}{2}+\frac {x b}{2}\right )\right )+18 \sqrt {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )+1}\, \sqrt {-2 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )+2}\, \sqrt {-\tan \left (\frac {a}{2}+\frac {x b}{2}\right )}\, \EllipticE \left (\sqrt {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )+1}, \frac {\sqrt {2}}{2}\right ) \left (\tan ^{4}\left (\frac {a}{2}+\frac {x b}{2}\right )\right )-9 \sqrt {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )+1}\, \sqrt {-2 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )+2}\, \sqrt {-\tan \left (\frac {a}{2}+\frac {x b}{2}\right )}\, \EllipticF \left (\sqrt {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )+1}, \frac {\sqrt {2}}{2}\right ) \left (\tan ^{4}\left (\frac {a}{2}+\frac {x b}{2}\right )\right )+6 \left (\tan ^{8}\left (\frac {a}{2}+\frac {x b}{2}\right )\right )+18 \sqrt {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )+1}\, \sqrt {-2 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )+2}\, \sqrt {-\tan \left (\frac {a}{2}+\frac {x b}{2}\right )}\, \EllipticE \left (\sqrt {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )+1}, \frac {\sqrt {2}}{2}\right ) \left (\tan ^{2}\left (\frac {a}{2}+\frac {x b}{2}\right )\right )-9 \sqrt {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )+1}\, \sqrt {-2 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )+2}\, \sqrt {-\tan \left (\frac {a}{2}+\frac {x b}{2}\right )}\, \EllipticF \left (\sqrt {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )+1}, \frac {\sqrt {2}}{2}\right ) \left (\tan ^{2}\left (\frac {a}{2}+\frac {x b}{2}\right )\right )-2 \left (\tan ^{6}\left (\frac {a}{2}+\frac {x b}{2}\right )\right )+6 \sqrt {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )+1}\, \sqrt {-2 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )+2}\, \sqrt {-\tan \left (\frac {a}{2}+\frac {x b}{2}\right )}\, \EllipticE \left (\sqrt {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )+1}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )+1}\, \sqrt {-2 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )+2}\, \sqrt {-\tan \left (\frac {a}{2}+\frac {x b}{2}\right )}\, \EllipticF \left (\sqrt {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )+1}, \frac {\sqrt {2}}{2}\right )+10 \left (\tan ^{4}\left (\frac {a}{2}+\frac {x b}{2}\right )\right )-14 \left (\tan ^{2}\left (\frac {a}{2}+\frac {x b}{2}\right )\right )\right )}{48 \sqrt {\tan \left (\frac {a}{2}+\frac {x b}{2}\right ) \left (\tan ^{2}\left (\frac {a}{2}+\frac {x b}{2}\right )-1\right )}\, \left (1+\tan ^{2}\left (\frac {a}{2}+\frac {x b}{2}\right )\right )^{3} \sqrt {\tan ^{3}\left (\frac {a}{2}+\frac {x b}{2}\right )-\tan \left (\frac {a}{2}+\frac {x b}{2}\right )}\, b}\)	\(727\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(b*x+a)^3/sin(2*b*x+2*a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/48*(-tan(1/2*a+1/2*x*b)/(tan(1/2*a+1/2*x*b)^2-1))^(1/2)*(tan(1/2*a+1/2*x*b)^2-1)*(6*(tan(1/2*a+1/2*x*b)+1)^
(1/2)*(-2*tan(1/2*a+1/2*x*b)+2)^(1/2)*(-tan(1/2*a+1/2*x*b))^(1/2)*EllipticE((tan(1/2*a+1/2*x*b)+1)^(1/2),1/2*2
^(1/2))*tan(1/2*a+1/2*x*b)^6-3*(tan(1/2*a+1/2*x*b)+1)^(1/2)*(-2*tan(1/2*a+1/2*x*b)+2)^(1/2)*(-tan(1/2*a+1/2*x*
b))^(1/2)*EllipticF((tan(1/2*a+1/2*x*b)+1)^(1/2),1/2*2^(1/2))*tan(1/2*a+1/2*x*b)^6+18*(tan(1/2*a+1/2*x*b)+1)^(
1/2)*(-2*tan(1/2*a+1/2*x*b)+2)^(1/2)*(-tan(1/2*a+1/2*x*b))^(1/2)*EllipticE((tan(1/2*a+1/2*x*b)+1)^(1/2),1/2*2^
(1/2))*tan(1/2*a+1/2*x*b)^4-9*(tan(1/2*a+1/2*x*b)+1)^(1/2)*(-2*tan(1/2*a+1/2*x*b)+2)^(1/2)*(-tan(1/2*a+1/2*x*b
))^(1/2)*EllipticF((tan(1/2*a+1/2*x*b)+1)^(1/2),1/2*2^(1/2))*tan(1/2*a+1/2*x*b)^4+6*tan(1/2*a+1/2*x*b)^8+18*(t
an(1/2*a+1/2*x*b)+1)^(1/2)*(-2*tan(1/2*a+1/2*x*b)+2)^(1/2)*(-tan(1/2*a+1/2*x*b))^(1/2)*EllipticE((tan(1/2*a+1/
2*x*b)+1)^(1/2),1/2*2^(1/2))*tan(1/2*a+1/2*x*b)^2-9*(tan(1/2*a+1/2*x*b)+1)^(1/2)*(-2*tan(1/2*a+1/2*x*b)+2)^(1/
2)*(-tan(1/2*a+1/2*x*b))^(1/2)*EllipticF((tan(1/2*a+1/2*x*b)+1)^(1/2),1/2*2^(1/2))*tan(1/2*a+1/2*x*b)^2-2*tan(
1/2*a+1/2*x*b)^6+6*(tan(1/2*a+1/2*x*b)+1)^(1/2)*(-2*tan(1/2*a+1/2*x*b)+2)^(1/2)*(-tan(1/2*a+1/2*x*b))^(1/2)*El
lipticE((tan(1/2*a+1/2*x*b)+1)^(1/2),1/2*2^(1/2))-3*(tan(1/2*a+1/2*x*b)+1)^(1/2)*(-2*tan(1/2*a+1/2*x*b)+2)^(1/
2)*(-tan(1/2*a+1/2*x*b))^(1/2)*EllipticF((tan(1/2*a+1/2*x*b)+1)^(1/2),1/2*2^(1/2))+10*tan(1/2*a+1/2*x*b)^4-14*
tan(1/2*a+1/2*x*b)^2)/(tan(1/2*a+1/2*x*b)*(tan(1/2*a+1/2*x*b)^2-1))^(1/2)/(1+tan(1/2*a+1/2*x*b)^2)^3/(tan(1/2*
a+1/2*x*b)^3-tan(1/2*a+1/2*x*b))^(1/2)/b

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)^3/sin(2*b*x+2*a)^(5/2),x, algorithm="maxima")

[Out]

integrate(sin(b*x + a)^3/sin(2*b*x + 2*a)^(5/2), x)

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Fricas [A]
time = 4.32, size = 48, normalized size = 1.71 \begin {gather*} -\frac {\cos \left (b x + a\right )^{2} - \sqrt {2} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} \sin \left (b x + a\right )}{12 \, b \cos \left (b x + a\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)^3/sin(2*b*x+2*a)^(5/2),x, algorithm="fricas")

[Out]

-1/12*(cos(b*x + a)^2 - sqrt(2)*sqrt(cos(b*x + a)*sin(b*x + a))*sin(b*x + a))/(b*cos(b*x + a)^2)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)**3/sin(2*b*x+2*a)**(5/2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 15648 vs. \(2 (24) = 48\).
time = 88.46, size = 15648, normalized size = 558.86 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)^3/sin(2*b*x+2*a)^(5/2),x, algorithm="giac")

[Out]

-1/6*sqrt(2)*sqrt(-tan(1/2*b*x)^4*tan(1/2*a)^3 - tan(1/2*b*x)^3*tan(1/2*a)^4 + tan(1/2*b*x)^4*tan(1/2*a) + 6*t
an(1/2*b*x)^3*tan(1/2*a)^2 + 6*tan(1/2*b*x)^2*tan(1/2*a)^3 + tan(1/2*b*x)*tan(1/2*a)^4 - tan(1/2*b*x)^3 - 6*ta
n(1/2*b*x)^2*tan(1/2*a) - 6*tan(1/2*b*x)*tan(1/2*a)^2 - tan(1/2*a)^3 + tan(1/2*b*x) + tan(1/2*a))*((((2*(4*(2*
(sqrt(2)*tan(1/2*a)^54 + 17*sqrt(2)*tan(1/2*a)^52 + 132*sqrt(2)*tan(1/2*a)^50 + 612*sqrt(2)*tan(1/2*a)^48 + 18
42*sqrt(2)*tan(1/2*a)^46 + 3570*sqrt(2)*tan(1/2*a)^44 + 3668*sqrt(2)*tan(1/2*a)^42 - 1292*sqrt(2)*tan(1/2*a)^4
0 - 11457*sqrt(2)*tan(1/2*a)^38 - 19057*sqrt(2)*tan(1/2*a)^36 - 12920*sqrt(2)*tan(1/2*a)^34 + 7752*sqrt(2)*tan
(1/2*a)^32 + 27132*sqrt(2)*tan(1/2*a)^30 + 27132*sqrt(2)*tan(1/2*a)^28 + 7752*sqrt(2)*tan(1/2*a)^26 - 12920*sq
rt(2)*tan(1/2*a)^24 - 19057*sqrt(2)*tan(1/2*a)^22 - 11457*sqrt(2)*tan(1/2*a)^20 - 1292*sqrt(2)*tan(1/2*a)^18 +
3668*sqrt(2)*tan(1/2*a)^16 + 3570*sqrt(2)*tan(1/2*a)^14 + 1842*sqrt(2)*tan(1/2*a)^12 + 612*sqrt(2)*tan(1/2*a)
^10 + 132*sqrt(2)*tan(1/2*a)^8 + 17*sqrt(2)*tan(1/2*a)^6 + sqrt(2)*tan(1/2*a)^4)*tan(1/2*b*x)/(tan(1/2*a)^51 +
23*tan(1/2*a)^49 + 252*tan(1/2*a)^47 + 1748*tan(1/2*a)^45 + 8602*tan(1/2*a)^43 + 31878*tan(1/2*a)^41 + 92092*
tan(1/2*a)^39 + 211508*tan(1/2*a)^37 + 389367*tan(1/2*a)^35 + 572033*tan(1/2*a)^33 + 653752*tan(1/2*a)^31 + 53
4888*tan(1/2*a)^29 + 208012*tan(1/2*a)^27 - 208012*tan(1/2*a)^25 - 534888*tan(1/2*a)^23 - 653752*tan(1/2*a)^21
- 572033*tan(1/2*a)^19 - 389367*tan(1/2*a)^17 - 211508*tan(1/2*a)^15 - 92092*tan(1/2*a)^13 - 31878*tan(1/2*a)
^11 - 8602*tan(1/2*a)^9 - 1748*tan(1/2*a)^7 - 252*tan(1/2*a)^5 - 23*tan(1/2*a)^3 - tan(1/2*a)) + 3*(sqrt(2)*ta
n(1/2*a)^55 + 12*sqrt(2)*tan(1/2*a)^53 + 43*sqrt(2)*tan(1/2*a)^51 - 120*sqrt(2)*tan(1/2*a)^49 - 1818*sqrt(2)*t
an(1/2*a)^47 - 8688*sqrt(2)*tan(1/2*a)^45 - 24598*sqrt(2)*tan(1/2*a)^43 - 44328*sqrt(2)*tan(1/2*a)^41 - 44365*
sqrt(2)*tan(1/2*a)^39 + 4028*sqrt(2)*tan(1/2*a)^37 + 93993*sqrt(2)*tan(1/2*a)^35 + 160208*sqrt(2)*tan(1/2*a)^3
3 + 127908*sqrt(2)*tan(1/2*a)^31 - 127908*sqrt(2)*tan(1/2*a)^27 - 160208*sqrt(2)*tan(1/2*a)^25 - 93993*sqrt(2)
*tan(1/2*a)^23 - 4028*sqrt(2)*tan(1/2*a)^21 + 44365*sqrt(2)*tan(1/2*a)^19 + 44328*sqrt(2)*tan(1/2*a)^17 + 2459
8*sqrt(2)*tan(1/2*a)^15 + 8688*sqrt(2)*tan(1/2*a)^13 + 1818*sqrt(2)*tan(1/2*a)^11 + 120*sqrt(2)*tan(1/2*a)^9 -
43*sqrt(2)*tan(1/2*a)^7 - 12*sqrt(2)*tan(1/2*a)^5 - sqrt(2)*tan(1/2*a)^3)/(tan(1/2*a)^51 + 23*tan(1/2*a)^49 +
252*tan(1/2*a)^47 + 1748*tan(1/2*a)^45 + 8602*tan(1/2*a)^43 + 31878*tan(1/2*a)^41 + 92092*tan(1/2*a)^39 + 211
508*tan(1/2*a)^37 + 389367*tan(1/2*a)^35 + 572033*tan(1/2*a)^33 + 653752*tan(1/2*a)^31 + 534888*tan(1/2*a)^29
+ 208012*tan(1/2*a)^27 - 208012*tan(1/2*a)^25 - 534888*tan(1/2*a)^23 - 653752*tan(1/2*a)^21 - 572033*tan(1/2*a
)^19 - 389367*tan(1/2*a)^17 - 211508*tan(1/2*a)^15 - 92092*tan(1/2*a)^13 - 31878*tan(1/2*a)^11 - 8602*tan(1/2*
a)^9 - 1748*tan(1/2*a)^7 - 252*tan(1/2*a)^5 - 23*tan(1/2*a)^3 - tan(1/2*a)))*tan(1/2*b*x) + 3*(sqrt(2)*tan(1/2
*a)^56 - 17*sqrt(2)*tan(1/2*a)^54 - 429*sqrt(2)*tan(1/2*a)^52 - 3555*sqrt(2)*tan(1/2*a)^50 - 16130*sqrt(2)*tan
(1/2*a)^48 - 43550*sqrt(2)*tan(1/2*a)^46 - 59310*sqrt(2)*tan(1/2*a)^44 + 32910*sqrt(2)*tan(1/2*a)^42 + 348955*
sqrt(2)*tan(1/2*a)^40 + 818805*sqrt(2)*tan(1/2*a)^38 + 1026665*sqrt(2)*tan(1/2*a)^36 + 479655*sqrt(2)*tan(1/2*
a)^34 - 755820*sqrt(2)*tan(1/2*a)^32 - 1828180*sqrt(2)*tan(1/2*a)^30 - 1828180*sqrt(2)*tan(1/2*a)^28 - 755820*
sqrt(2)*tan(1/2*a)^26 + 479655*sqrt(2)*tan(1/2*a)^24 + 1026665*sqrt(2)*tan(1/2*a)^22 + 818805*sqrt(2)*tan(1/2*
a)^20 + 348955*sqrt(2)*tan(1/2*a)^18 + 32910*sqrt(2)*tan(1/2*a)^16 - 59310*sqrt(2)*tan(1/2*a)^14 - 43550*sqrt(
2)*tan(1/2*a)^12 - 16130*sqrt(2)*tan(1/2*a)^10 - 3555*sqrt(2)*tan(1/2*a)^8 - 429*sqrt(2)*tan(1/2*a)^6 - 17*sqr
t(2)*tan(1/2*a)^4 + sqrt(2)*tan(1/2*a)^2)/(tan(1/2*a)^51 + 23*tan(1/2*a)^49 + 252*tan(1/2*a)^47 + 1748*tan(1/2
*a)^45 + 8602*tan(1/2*a)^43 + 31878*tan(1/2*a)^41 + 92092*tan(1/2*a)^39 + 211508*tan(1/2*a)^37 + 389367*tan(1/
2*a)^35 + 572033*tan(1/2*a)^33 + 653752*tan(1/2*a)^31 + 534888*tan(1/2*a)^29 + 208012*tan(1/2*a)^27 - 208012*t
an(1/2*a)^25 - 534888*tan(1/2*a)^23 - 653752*tan(1/2*a)^21 - 572033*tan(1/2*a)^19 - 389367*tan(1/2*a)^17 - 211
508*tan(1/2*a)^15 - 92092*tan(1/2*a)^13 - 31878*tan(1/2*a)^11 - 8602*tan(1/2*a)^9 - 1748*tan(1/2*a)^7 - 252*ta
n(1/2*a)^5 - 23*tan(1/2*a)^3 - tan(1/2*a)))*tan(1/2*b*x) - (sqrt(2)*tan(1/2*a)^57 + 98*sqrt(2)*tan(1/2*a)^55 +
1092*sqrt(2)*tan(1/2*a)^53 + 3814*sqrt(2)*tan(1/2*a)^51 - 10975*sqrt(2)*tan(1/2*a)^49 - 165060*sqrt(2)*tan(1/
2*a)^47 - 803600*sqrt(2)*tan(1/2*a)^45 - 2367580*sqrt(2)*tan(1/2*a)^43 - 4642995*sqrt(2)*tan(1/2*a)^41 - 58894
50*sqrt(2)*tan(1/2*a)^39 - 3619500*sqrt(2)*tan(1/2*a)^37 + 2314770*sqrt(2)*tan(1/2*a)^35 + 7653485*sqrt(2)*tan
(1/2*a)^33 + 6963880*sqrt(2)*tan(1/2*a)^31 - 6963880*sqrt(2)*tan(1/2*a)^27 - 7653485*sqrt(2)*tan(1/2*a)^25 - 2
314770*sqrt(2)*tan(1/2*a)^23 + 3619500*sqrt(2)*tan(1/2*a)^21 + 5889450*sqrt(2)*tan(1/2*a)^19 + 4642995*sqrt(2)
*tan(1/2*a)^17 + 2367580*sqrt(2)*tan(1/2*a)^15 ...

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Mupad [B]
time = 2.19, size = 85, normalized size = 3.04 \begin {gather*} -\frac {\sqrt {\sin \left (2\,a+2\,b\,x\right )}\,\left (2\,\sin \left (a+b\,x\right )+3\,\sin \left (3\,a+3\,b\,x\right )+\sin \left (5\,a+5\,b\,x\right )\right )}{6\,b\,\left (30\,{\sin \left (a+b\,x\right )}^2+12\,{\sin \left (2\,a+2\,b\,x\right )}^2+2\,{\sin \left (3\,a+3\,b\,x\right )}^2-32\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*x)^3/sin(2*a + 2*b*x)^(5/2),x)

[Out]

-(sin(2*a + 2*b*x)^(1/2)*(2*sin(a + b*x) + 3*sin(3*a + 3*b*x) + sin(5*a + 5*b*x)))/(6*b*(12*sin(2*a + 2*b*x)^2
+ 2*sin(3*a + 3*b*x)^2 + 30*sin(a + b*x)^2 - 32))

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