∫ sin7 (c+dx)___ (a−b sin4(c+dx))3 dx [225]

3.3.25 $\int \frac {\sin ^7(c+d x)}{(a-b \sin ^4(c+d x))^3} \, dx$ [225]

Optimal. Leaf size=290 \[ \frac {3 \left (\sqrt {a}-2 \sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 \sqrt {a} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} b^{7/4} d}-\frac {3 \left (\sqrt {a}+2 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 \sqrt {a} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} b^{7/4} d}-\frac {a \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 (a-b) b d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )^2}+\frac {\cos (c+d x) \left (5 a-17 b-3 (a-3 b) \cos ^2(c+d x)\right )}{32 (a-b)^2 b d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )} \]

[Out]

-1/8*a*cos(d*x+c)*(2-cos(d*x+c)^2)/(a-b)/b/d/(a-b+2*b*cos(d*x+c)^2-b*cos(d*x+c)^4)^2+1/32*cos(d*x+c)*(5*a-17*b

-3*(a-3*b)*cos(d*x+c)^2)/(a-b)^2/b/d/(a-b+2*b*cos(d*x+c)^2-b*cos(d*x+c)^4)+3/64*arctan(b^(1/4)*cos(d*x+c)/(a^(

1/2)-b^(1/2))^(1/2))*(a^(1/2)-2*b^(1/2))/b^(7/4)/d/a^(1/2)/(a^(1/2)-b^(1/2))^(5/2)-3/64*arctanh(b^(1/4)*cos(d*

x+c)/(a^(1/2)+b^(1/2))^(1/2))*(a^(1/2)+2*b^(1/2))/b^(7/4)/d/a^(1/2)/(a^(1/2)+b^(1/2))^(5/2)

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Rubi [A]
time = 0.30, antiderivative size = 290, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, $\frac {\text {number of rules}}{\text {integrand size}}$ = 0.250, Rules used = {3294, 1219, 1192, 1180, 211, 214} \begin {gather*} \frac {3 \left (\sqrt {a}-2 \sqrt {b}\right ) \text {ArcTan}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 \sqrt {a} b^{7/4} d \left (\sqrt {a}-\sqrt {b}\right )^{5/2}}-\frac {3 \left (\sqrt {a}+2 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 \sqrt {a} b^{7/4} d \left (\sqrt {a}+\sqrt {b}\right )^{5/2}}+\frac {\cos (c+d x) \left (-3 (a-3 b) \cos ^2(c+d x)+5 a-17 b\right )}{32 b d (a-b)^2 \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}-\frac {a \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 b d (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[Sin[c + d*x]^7/(a - b*Sin[c + d*x]^4)^3,x]

[Out]

(3*(Sqrt[a] - 2*Sqrt[b])*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(64*Sqrt[a]*(Sqrt[a] - Sqrt[b

])^(5/2)*b^(7/4)*d) - (3*(Sqrt[a] + 2*Sqrt[b])*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(64*Sq

rt[a]*(Sqrt[a] + Sqrt[b])^(5/2)*b^(7/4)*d) - (a*Cos[c + d*x]*(2 - Cos[c + d*x]^2))/(8*(a - b)*b*d*(a - b + 2*b

*Cos[c + d*x]^2 - b*Cos[c + d*x]^4)^2) + (Cos[c + d*x]*(5*a - 17*b - 3*(a - 3*b)*Cos[c + d*x]^2))/(32*(a - b)^

2*b*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4))

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]

&& PosQ[a/b]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},

x] && NegQ[a/b]

Rule 1180

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di

st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +

q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^

2 - 4*a*c]

Rule 1192

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[x*(a*b*e - d*(b^2 - 2*a

*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Dist[1/(2*a*(p + 1)

*(b^2 - 4*a*c)), Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a +

b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e

^2, 0] && LtQ[p, -1] && IntegerQ[2*p]

Rule 1219

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{f = Coeff[Polynom

ialRemainder[(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x

^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4)^(p + 1)*((a*b*g - f*(b^2 - 2*a*c) - c*(b*f - 2*a*g)*x^2)/(2

*a*(p + 1)*(b^2 - 4*a*c))), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToS

um[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[(d + e*x^2)^q, a + b*x^2 + c*x^4, x] + b^2*f*(2*p + 3) - 2*a*c

*f*(4*p + 5) - a*b*g + c*(4*p + 7)*(b*f - 2*a*g)*x^2, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*

a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[q, 1] && LtQ[p, -1]

Rule 3294

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = Free

Factors[Cos[e + f*x], x]}, Dist[-ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4

)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps

\begin {align*} \int \frac {\sin ^7(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx &=-\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^3}{\left (a-b+2 b x^2-b x^4\right )^3} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 (a-b) b d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )^2}+\frac {\text {Subst}\left (\int \frac {4 a (a-4 b)-2 a (3 a-8 b) x^2}{\left (a-b+2 b x^2-b x^4\right )^2} \, dx,x,\cos (c+d x)\right )}{16 a (a-b) b d}\\ &=-\frac {a \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 (a-b) b d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )^2}+\frac {\cos (c+d x) \left (5 a-17 b-3 (a-3 b) \cos ^2(c+d x)\right )}{32 (a-b)^2 b d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {-12 a^2 (a-5 b) b+12 a^2 (a-3 b) b x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cos (c+d x)\right )}{128 a^2 (a-b)^2 b^2 d}\\ &=-\frac {a \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 (a-b) b d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )^2}+\frac {\cos (c+d x) \left (5 a-17 b-3 (a-3 b) \cos ^2(c+d x)\right )}{32 (a-b)^2 b d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}-\frac {\left (3 \left (\sqrt {a}+2 \sqrt {b}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{64 \sqrt {a} \left (\sqrt {a}+\sqrt {b}\right )^2 b d}-\frac {\left (3 \left (a^{3/2}-3 \sqrt {a} b-2 b^{3/2}\right )\right ) \text {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{64 \sqrt {a} (a-b)^2 b d}\\ &=\frac {3 \left (\sqrt {a}-2 \sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 \sqrt {a} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} b^{7/4} d}-\frac {3 \left (\sqrt {a}+2 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 \sqrt {a} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} b^{7/4} d}-\frac {a \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 (a-b) b d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )^2}+\frac {\cos (c+d x) \left (5 a-17 b-3 (a-3 b) \cos ^2(c+d x)\right )}{32 (a-b)^2 b d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.77, size = 630, normalized size = 2.17 \begin {gather*} \frac {-\frac {32 \cos (c+d x) (-7 a+25 b+3 (a-3 b) \cos (2 (c+d x)))}{8 a-3 b+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))}+\frac {512 a (a-b) (-5 \cos (c+d x)+\cos (3 (c+d x)))}{(-8 a+3 b-4 b \cos (2 (c+d x))+b \cos (4 (c+d x)))^2}-3 i \text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {2 a \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-6 b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-i a \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+3 i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )-6 a \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+34 b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+3 i a \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-17 i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2+6 a \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-34 b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-3 i a \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4+17 i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4-2 a \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^6+6 b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^6+i a \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^6-3 i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^6}{-b \text {$\#$1}-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-3 b \text {$\#$1}^5+b \text {$\#$1}^7}\&\right ]}{256 (a-b)^2 b d} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Sin[c + d*x]^7/(a - b*Sin[c + d*x]^4)^3,x]

[Out]

((-32*Cos[c + d*x]*(-7*a + 25*b + 3*(a - 3*b)*Cos[2*(c + d*x)]))/(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(

c + d*x)]) + (512*a*(a - b)*(-5*Cos[c + d*x] + Cos[3*(c + d*x)]))/(-8*a + 3*b - 4*b*Cos[2*(c + d*x)] + b*Cos[4

*(c + d*x)])^2 - (3*I)*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (2*a*ArcTan[Sin[c +

d*x]/(Cos[c + d*x] - #1)] - 6*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - I*a*Log[1 - 2*Cos[c + d*x]*#1 + #1

^2] + (3*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - 6*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + 34*b*Arc

Tan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (3*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (17*I)*b*Log[1 -

2*Cos[c + d*x]*#1 + #1^2]*#1^2 + 6*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - 34*b*ArcTan[Sin[c + d*x]

/(Cos[c + d*x] - #1)]*#1^4 - (3*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + (17*I)*b*Log[1 - 2*Cos[c + d*x]*

#1 + #1^2]*#1^4 - 2*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 + 6*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] -

#1)]*#1^6 + I*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6 - (3*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6)/(-(b*

#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(256*(a - b)^2*b*d)

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Maple [A]
time = 1.50, size = 303, normalized size = 1.04 Too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)^7/(a-b*sin(d*x+c)^4)^3,x,method=_RETURNVERBOSE)

[Out]

1/d*((3/32*(a-3*b)/(a^2-2*a*b+b^2)*cos(d*x+c)^7-1/32*(11*a-35*b)/(a^2-2*a*b+b^2)*cos(d*x+c)^5+1/32*(a^2+18*a*b

-43*b^2)/b/(a^2-2*a*b+b^2)*cos(d*x+c)^3-1/32*(3*a+17*b)/(a-b)/b*cos(d*x+c))/(a-b+2*b*cos(d*x+c)^2-b*cos(d*x+c)

^4)^2+3/32/(a^2-2*a*b+b^2)*(1/2*(a*(a*b)^(1/2)-3*(a*b)^(1/2)*b-2*b^2)/(a*b)^(1/2)/b/(((a*b)^(1/2)-b)*b)^(1/2)*

arctan(b*cos(d*x+c)/(((a*b)^(1/2)-b)*b)^(1/2))-1/2*(a*(a*b)^(1/2)-3*(a*b)^(1/2)*b+2*b^2)/(a*b)^(1/2)/b/(((a*b)

^(1/2)+b)*b)^(1/2)*arctanh(b*cos(d*x+c)/(((a*b)^(1/2)+b)*b)^(1/2))))

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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(d*x+c)^7/(a-b*sin(d*x+c)^4)^3,x, algorithm="maxima")

[Out]

-1/16*(24*(a*b^3 - 3*b^4)*cos(2*d*x + 2*c)*cos(d*x + c) - 8*(23*a*b^3 - 77*b^4)*sin(3*d*x + 3*c)*sin(2*d*x + 2

*c) + 24*(a*b^3 - 3*b^4)*sin(2*d*x + 2*c)*sin(d*x + c) - (3*(a*b^3 - 3*b^4)*cos(15*d*x + 15*c) - (23*a*b^3 - 7

7*b^4)*cos(13*d*x + 13*c) + (16*a^2*b^2 + 131*a*b^3 - 177*b^4)*cos(11*d*x + 11*c) - (144*a^2*b^2 + 367*a*b^3 -

109*b^4)*cos(9*d*x + 9*c) - (144*a^2*b^2 + 367*a*b^3 - 109*b^4)*cos(7*d*x + 7*c) + (16*a^2*b^2 + 131*a*b^3 -

177*b^4)*cos(5*d*x + 5*c) - (23*a*b^3 - 77*b^4)*cos(3*d*x + 3*c) + 3*(a*b^3 - 3*b^4)*cos(d*x + c))*cos(16*d*x

+ 16*c) - 3*(a*b^3 - 3*b^4 - 8*(a*b^3 - 3*b^4)*cos(14*d*x + 14*c) - 4*(8*a^2*b^2 - 31*a*b^3 + 21*b^4)*cos(12*d

*x + 12*c) + 8*(16*a^2*b^2 - 55*a*b^3 + 21*b^4)*cos(10*d*x + 10*c) + 2*(128*a^3*b - 480*a^2*b^2 + 323*a*b^3 -

105*b^4)*cos(8*d*x + 8*c) + 8*(16*a^2*b^2 - 55*a*b^3 + 21*b^4)*cos(6*d*x + 6*c) - 4*(8*a^2*b^2 - 31*a*b^3 + 21

*b^4)*cos(4*d*x + 4*c) - 8*(a*b^3 - 3*b^4)*cos(2*d*x + 2*c))*cos(15*d*x + 15*c) - 8*((23*a*b^3 - 77*b^4)*cos(1

3*d*x + 13*c) - (16*a^2*b^2 + 131*a*b^3 - 177*b^4)*cos(11*d*x + 11*c) + (144*a^2*b^2 + 367*a*b^3 - 109*b^4)*co

s(9*d*x + 9*c) + (144*a^2*b^2 + 367*a*b^3 - 109*b^4)*cos(7*d*x + 7*c) - (16*a^2*b^2 + 131*a*b^3 - 177*b^4)*cos

(5*d*x + 5*c) + (23*a*b^3 - 77*b^4)*cos(3*d*x + 3*c) - 3*(a*b^3 - 3*b^4)*cos(d*x + c))*cos(14*d*x + 14*c) + (2

3*a*b^3 - 77*b^4 - 4*(184*a^2*b^2 - 777*a*b^3 + 539*b^4)*cos(12*d*x + 12*c) + 8*(368*a^2*b^2 - 1393*a*b^3 + 53

9*b^4)*cos(10*d*x + 10*c) + 2*(2944*a^3*b - 12064*a^2*b^2 + 8197*a*b^3 - 2695*b^4)*cos(8*d*x + 8*c) + 8*(368*a

^2*b^2 - 1393*a*b^3 + 539*b^4)*cos(6*d*x + 6*c) - 4*(184*a^2*b^2 - 777*a*b^3 + 539*b^4)*cos(4*d*x + 4*c) - 8*(

23*a*b^3 - 77*b^4)*cos(2*d*x + 2*c))*cos(13*d*x + 13*c) + 4*((128*a^3*b + 936*a^2*b^2 - 2333*a*b^3 + 1239*b^4)

*cos(11*d*x + 11*c) - (1152*a^3*b + 1928*a^2*b^2 - 3441*a*b^3 + 763*b^4)*cos(9*d*x + 9*c) - (1152*a^3*b + 1928

*a^2*b^2 - 3441*a*b^3 + 763*b^4)*cos(7*d*x + 7*c) + (128*a^3*b + 936*a^2*b^2 - 2333*a*b^3 + 1239*b^4)*cos(5*d*

x + 5*c) - (184*a^2*b^2 - 777*a*b^3 + 539*b^4)*cos(3*d*x + 3*c) + 3*(8*a^2*b^2 - 31*a*b^3 + 21*b^4)*cos(d*x +

c))*cos(12*d*x + 12*c) - (16*a^2*b^2 + 131*a*b^3 - 177*b^4 + 8*(256*a^3*b + 1984*a^2*b^2 - 3749*a*b^3 + 1239*b

^4)*cos(10*d*x + 10*c) + 2*(2048*a^4 + 15232*a^3*b - 34672*a^2*b^2 + 21577*a*b^3 - 6195*b^4)*cos(8*d*x + 8*c)

+ 8*(256*a^3*b + 1984*a^2*b^2 - 3749*a*b^3 + 1239*b^4)*cos(6*d*x + 6*c) - 4*(128*a^3*b + 936*a^2*b^2 - 2333*a*

b^3 + 1239*b^4)*cos(4*d*x + 4*c) - 8*(16*a^2*b^2 + 131*a*b^3 - 177*b^4)*cos(2*d*x + 2*c))*cos(11*d*x + 11*c) +

8*((2304*a^3*b + 4864*a^2*b^2 - 4313*a*b^3 + 763*b^4)*cos(9*d*x + 9*c) + (2304*a^3*b + 4864*a^2*b^2 - 4313*a*

b^3 + 763*b^4)*cos(7*d*x + 7*c) - (256*a^3*b + 1984*a^2*b^2 - 3749*a*b^3 + 1239*b^4)*cos(5*d*x + 5*c) + (368*a

^2*b^2 - 1393*a*b^3 + 539*b^4)*cos(3*d*x + 3*c) - 3*(16*a^2*b^2 - 55*a*b^3 + 21*b^4)*cos(d*x + c))*cos(10*d*x

+ 10*c) + (144*a^2*b^2 + 367*a*b^3 - 109*b^4 + 2*(18432*a^4 + 33152*a^3*b - 44144*a^2*b^2 + 23309*a*b^3 - 3815

*b^4)*cos(8*d*x + 8*c) + 8*(2304*a^3*b + 4864*a^2*b^2 - 4313*a*b^3 + 763*b^4)*cos(6*d*x + 6*c) - 4*(1152*a^3*b

+ 1928*a^2*b^2 - 3441*a*b^3 + 763*b^4)*cos(4*d*x + 4*c) - 8*(144*a^2*b^2 + 367*a*b^3 - 109*b^4)*cos(2*d*x + 2

*c))*cos(9*d*x + 9*c) + 2*((18432*a^4 + 33152*a^3*b - 44144*a^2*b^2 + 23309*a*b^3 - 3815*b^4)*cos(7*d*x + 7*c)

- (2048*a^4 + 15232*a^3*b - 34672*a^2*b^2 + 21577*a*b^3 - 6195*b^4)*cos(5*d*x + 5*c) + (2944*a^3*b - 12064*a^

2*b^2 + 8197*a*b^3 - 2695*b^4)*cos(3*d*x + 3*c) - 3*(128*a^3*b - 480*a^2*b^2 + 323*a*b^3 - 105*b^4)*cos(d*x +

c))*cos(8*d*x + 8*c) + (144*a^2*b^2 + 367*a*b^3 - 109*b^4 + 8*(2304*a^3*b + 4864*a^2*b^2 - 4313*a*b^3 + 763*b^

4)*cos(6*d*x + 6*c) - 4*(1152*a^3*b + 1928*a^2*b^2 - 3441*a*b^3 + 763*b^4)*cos(4*d*x + 4*c) - 8*(144*a^2*b^2 +

367*a*b^3 - 109*b^4)*cos(2*d*x + 2*c))*cos(7*d*x + 7*c) - 8*((256*a^3*b + 1984*a^2*b^2 - 3749*a*b^3 + 1239*b^

4)*cos(5*d*x + 5*c) - (368*a^2*b^2 - 1393*a*b^3 + 539*b^4)*cos(3*d*x + 3*c) + 3*(16*a^2*b^2 - 55*a*b^3 + 21*b^

4)*cos(d*x + c))*cos(6*d*x + 6*c) - (16*a^2*b^2 + 131*a*b^3 - 177*b^4 - 4*(128*a^3*b + 936*a^2*b^2 - 2333*a*b^

3 + 1239*b^4)*cos(4*d*x + 4*c) - 8*(16*a^2*b^2 + 131*a*b^3 - 177*b^4)*cos(2*d*x + 2*c))*cos(5*d*x + 5*c) - 4*(

(184*a^2*b^2 - 777*a*b^3 + 539*b^4)*cos(3*d*x + 3*c) - 3*(8*a^2*b^2 - 31*a*b^3 + 21*b^4)*cos(d*x + c))*cos(4*d

*x + 4*c) + (23*a*b^3 - 77*b^4 - 8*(23*a*b^3 - 77*b^4)*cos(2*d*x + 2*c))*cos(3*d*x + 3*c) - 3*(a*b^3 - 3*b^4)*

cos(d*x + c) + 16*((a^2*b^5 - 2*a*b^6 + b^7)*d*cos(16*d*x + 16*c)^2 + 64*(a^2*b^5 - 2*a*b^6 + b^7)*d*cos(14*d*

x + 14*c)^2 + 16*(64*a^4*b^3 - 240*a^3*b^4 + 337*a^2*b^5 - 210*a*b^6 + 49*b^7)*d*cos(12*d*x + 12*c)^2 + 64*(25

6*a^4*b^3 - 736*a^3*b^4 + 753*a^2*b^5 - 322*a*b^6 + 49*b^7)*d*cos(10*d*x + 10*c)^2 + 4*(16384*a^6*b - 57344*a^

5*b^2 + 83712*a^4*b^3 - 67648*a^3*b^4 + 32841*a^2*b^5 - 9170*a*b^6 + 1225*b^7)*d*cos(8*d*x + 8*c)^2 + 64*(256*

a^4*b^3 - 736*a^3*b^4 + 753*a^2*b^5 - 322*a*b^6 + 49*b^7)*d*cos(6*d*x + 6*c)^2 + 16*(64*a^4*b^3 - 240*a^3*b^4

+ 337*a^2*b^5 - 210*a*b^6 + 49*b^7)*d*cos(4*d*x...

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4185 vs. $2 (234) = 468$.
time = 0.88, size = 4185, normalized size = 14.43 \begin {gather*} \text {Too large to display} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^7/(a-b*sin(d*x+c)^4)^3,x, algorithm="fricas")

[Out]

1/128*(12*(a*b - 3*b^2)*cos(d*x + c)^7 - 4*(11*a*b - 35*b^2)*cos(d*x + c)^5 + 4*(a^2 + 18*a*b - 43*b^2)*cos(d*

x + c)^3 + 3*((a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^8 - 4*(a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^6 - 2*(a

^3*b^2 - 5*a^2*b^3 + 7*a*b^4 - 3*b^5)*d*cos(d*x + c)^4 + 4*(a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*d*cos(d*x + c

)^2 + (a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*d)*sqrt(-((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6

+ 5*a^2*b^7 - a*b^8)*d^2*sqrt((a^6 - 12*a^5*b + 46*a^4*b^2 - 28*a^3*b^3 - 167*a^2*b^4 + 160*a*b^5 + 256*b^6)/

((a^11*b^7 - 10*a^10*b^8 + 45*a^9*b^9 - 120*a^8*b^10 + 210*a^7*b^11 - 252*a^6*b^12 + 210*a^5*b^13 - 120*a^4*b^

14 + 45*a^3*b^15 - 10*a^2*b^16 + a*b^17)*d^4)) + a^3 - 10*a^2*b + 21*a*b^2 + 4*b^3)/((a^6*b^3 - 5*a^5*b^4 + 10

*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2))*log(27*(a^4 - 10*a^3*b + 29*a^2*b^2 - 4*a*b^3 - 64*b^4)*cos(d

*x + c) + 27*((a^8*b^5 - 8*a^7*b^6 + 23*a^6*b^7 - 30*a^5*b^8 + 15*a^4*b^9 + 4*a^3*b^10 - 7*a^2*b^11 + 2*a*b^12

)*d^3*sqrt((a^6 - 12*a^5*b + 46*a^4*b^2 - 28*a^3*b^3 - 167*a^2*b^4 + 160*a*b^5 + 256*b^6)/((a^11*b^7 - 10*a^10

*b^8 + 45*a^9*b^9 - 120*a^8*b^10 + 210*a^7*b^11 - 252*a^6*b^12 + 210*a^5*b^13 - 120*a^4*b^14 + 45*a^3*b^15 - 1

0*a^2*b^16 + a*b^17)*d^4)) - (a^5*b^2 - 11*a^4*b^3 + 35*a^3*b^4 - 9*a^2*b^5 - 80*a*b^6)*d)*sqrt(-((a^6*b^3 - 5

*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2*sqrt((a^6 - 12*a^5*b + 46*a^4*b^2 - 28*a^3*b^3 - 1

67*a^2*b^4 + 160*a*b^5 + 256*b^6)/((a^11*b^7 - 10*a^10*b^8 + 45*a^9*b^9 - 120*a^8*b^10 + 210*a^7*b^11 - 252*a^

6*b^12 + 210*a^5*b^13 - 120*a^4*b^14 + 45*a^3*b^15 - 10*a^2*b^16 + a*b^17)*d^4)) + a^3 - 10*a^2*b + 21*a*b^2 +

4*b^3)/((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2))) - 3*((a^2*b^3 - 2*a*b^4 +

b^5)*d*cos(d*x + c)^8 - 4*(a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^6 - 2*(a^3*b^2 - 5*a^2*b^3 + 7*a*b^4 - 3*b^

5)*d*cos(d*x + c)^4 + 4*(a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*d*cos(d*x + c)^2 + (a^4*b - 4*a^3*b^2 + 6*a^2*b^

3 - 4*a*b^4 + b^5)*d)*sqrt(((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2*sqrt((a^6

- 12*a^5*b + 46*a^4*b^2 - 28*a^3*b^3 - 167*a^2*b^4 + 160*a*b^5 + 256*b^6)/((a^11*b^7 - 10*a^10*b^8 + 45*a^9*b^

9 - 120*a^8*b^10 + 210*a^7*b^11 - 252*a^6*b^12 + 210*a^5*b^13 - 120*a^4*b^14 + 45*a^3*b^15 - 10*a^2*b^16 + a*b

^17)*d^4)) - a^3 + 10*a^2*b - 21*a*b^2 - 4*b^3)/((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 -

a*b^8)*d^2))*log(27*(a^4 - 10*a^3*b + 29*a^2*b^2 - 4*a*b^3 - 64*b^4)*cos(d*x + c) + 27*((a^8*b^5 - 8*a^7*b^6 +

23*a^6*b^7 - 30*a^5*b^8 + 15*a^4*b^9 + 4*a^3*b^10 - 7*a^2*b^11 + 2*a*b^12)*d^3*sqrt((a^6 - 12*a^5*b + 46*a^4*

b^2 - 28*a^3*b^3 - 167*a^2*b^4 + 160*a*b^5 + 256*b^6)/((a^11*b^7 - 10*a^10*b^8 + 45*a^9*b^9 - 120*a^8*b^10 + 2

10*a^7*b^11 - 252*a^6*b^12 + 210*a^5*b^13 - 120*a^4*b^14 + 45*a^3*b^15 - 10*a^2*b^16 + a*b^17)*d^4)) + (a^5*b^

2 - 11*a^4*b^3 + 35*a^3*b^4 - 9*a^2*b^5 - 80*a*b^6)*d)*sqrt(((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 +

5*a^2*b^7 - a*b^8)*d^2*sqrt((a^6 - 12*a^5*b + 46*a^4*b^2 - 28*a^3*b^3 - 167*a^2*b^4 + 160*a*b^5 + 256*b^6)/((a

^11*b^7 - 10*a^10*b^8 + 45*a^9*b^9 - 120*a^8*b^10 + 210*a^7*b^11 - 252*a^6*b^12 + 210*a^5*b^13 - 120*a^4*b^14

+ 45*a^3*b^15 - 10*a^2*b^16 + a*b^17)*d^4)) - a^3 + 10*a^2*b - 21*a*b^2 - 4*b^3)/((a^6*b^3 - 5*a^5*b^4 + 10*a^

4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2))) - 3*((a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^8 - 4*(a^2*b^3 -

2*a*b^4 + b^5)*d*cos(d*x + c)^6 - 2*(a^3*b^2 - 5*a^2*b^3 + 7*a*b^4 - 3*b^5)*d*cos(d*x + c)^4 + 4*(a^3*b^2 - 3*

a^2*b^3 + 3*a*b^4 - b^5)*d*cos(d*x + c)^2 + (a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*d)*sqrt(-((a^6*b^3

- 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2*sqrt((a^6 - 12*a^5*b + 46*a^4*b^2 - 28*a^3*b^3

- 167*a^2*b^4 + 160*a*b^5 + 256*b^6)/((a^11*b^7 - 10*a^10*b^8 + 45*a^9*b^9 - 120*a^8*b^10 + 210*a^7*b^11 - 25

2*a^6*b^12 + 210*a^5*b^13 - 120*a^4*b^14 + 45*a^3*b^15 - 10*a^2*b^16 + a*b^17)*d^4)) + a^3 - 10*a^2*b + 21*a*b

^2 + 4*b^3)/((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2))*log(-27*(a^4 - 10*a^3*b

+ 29*a^2*b^2 - 4*a*b^3 - 64*b^4)*cos(d*x + c) + 27*((a^8*b^5 - 8*a^7*b^6 + 23*a^6*b^7 - 30*a^5*b^8 + 15*a^4*b

^9 + 4*a^3*b^10 - 7*a^2*b^11 + 2*a*b^12)*d^3*sqrt((a^6 - 12*a^5*b + 46*a^4*b^2 - 28*a^3*b^3 - 167*a^2*b^4 + 16

0*a*b^5 + 256*b^6)/((a^11*b^7 - 10*a^10*b^8 + 45*a^9*b^9 - 120*a^8*b^10 + 210*a^7*b^11 - 252*a^6*b^12 + 210*a^

5*b^13 - 120*a^4*b^14 + 45*a^3*b^15 - 10*a^2*b^16 + a*b^17)*d^4)) - (a^5*b^2 - 11*a^4*b^3 + 35*a^3*b^4 - 9*a^2

*b^5 - 80*a*b^6)*d)*sqrt(-((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2*sqrt((a^6 -

12*a^5*b + 46*a^4*b^2 - 28*a^3*b^3 - 167*a^2*b^4 + 160*a*b^5 + 256*b^6)/((a^11*b^7 - 10*a^10*b^8 + 45*a^9*b^9

- 120*a^8*b^10 + 210*a^7*b^11 - 252*a^6*b^12 + 210*a^5*b^13 - 120*a^4*b^14 + 45*a^3*b^15 - 10*a^2*b^16 + a*b^

17)*d^4)) + a^3 - 10*a^2*b + 21*a*b^2 + 4*b^3)/((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a

*b^8)*d^2))) + 3*((a^2*b^3 - 2*a*b^4 + b^5)*d*c...

________________________________________________________________________________________

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(d*x+c)**7/(a-b*sin(d*x+c)**4)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^7/(a-b*sin(d*x+c)^4)^3,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Warning, need to choose a branch for the

root of a polynomial with parameters. This might be wrong.The choice was done assuming [sageVARa,sageVARb]=[6

1,-66]Warning, need t

________________________________________________________________________________________

Mupad [B]
time = 19.62, size = 2500, normalized size = 8.62 \begin {gather*} \text {Too large to display} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(sin(c + d*x)^7/(a - b*sin(c + d*x)^4)^3,x)

[Out]

((3*cos(c + d*x)^7*(a - 3*b))/(32*(a^2 - 2*a*b + b^2)) - (cos(c + d*x)^5*(11*a - 35*b))/(32*(a^2 - 2*a*b + b^2

)) + (cos(c + d*x)^3*(18*a*b + a^2 - 43*b^2))/(32*b*(a - b)^2) - (cos(c + d*x)*(3*a + 17*b))/(32*b*(a - b)))/(

d*(a^2 - 2*a*b + b^2 + cos(c + d*x)^2*(4*a*b - 4*b^2) - cos(c + d*x)^4*(2*a*b - 6*b^2) - 4*b^2*cos(c + d*x)^6

+ b^2*cos(c + d*x)^8)) + (atan(((((3*(81920*a*b^7 - 180224*a^2*b^6 + 114688*a^3*b^5 - 16384*a^4*b^4))/(32768*(

b^6 - 4*a*b^5 + 6*a^2*b^4 - 4*a^3*b^3 + a^4*b^2)) - (cos(c + d*x)*((9*(a^3*(a^3*b^7)^(1/2) + 16*b^3*(a^3*b^7)^

(1/2) + 4*a*b^7 + 21*a^2*b^6 - 10*a^3*b^5 + a^4*b^4 + 5*a*b^2*(a^3*b^7)^(1/2) - 6*a^2*b*(a^3*b^7)^(1/2)))/(163

84*(a^2*b^12 - 5*a^3*b^11 + 10*a^4*b^10 - 10*a^5*b^9 + 5*a^6*b^8 - a^7*b^7)))^(1/2)*(16384*a*b^8 - 65536*a^2*b

^7 + 98304*a^3*b^6 - 65536*a^4*b^5 + 16384*a^5*b^4))/(256*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)))*((9*(a

^3*(a^3*b^7)^(1/2) + 16*b^3*(a^3*b^7)^(1/2) + 4*a*b^7 + 21*a^2*b^6 - 10*a^3*b^5 + a^4*b^4 + 5*a*b^2*(a^3*b^7)^

(1/2) - 6*a^2*b*(a^3*b^7)^(1/2)))/(16384*(a^2*b^12 - 5*a^3*b^11 + 10*a^4*b^10 - 10*a^5*b^9 + 5*a^6*b^8 - a^7*b

^7)))^(1/2) + (cos(c + d*x)*(81*a*b^2 - 54*a^2*b + 9*a^3 + 36*b^3))/(256*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^

2*b^2)))*((9*(a^3*(a^3*b^7)^(1/2) + 16*b^3*(a^3*b^7)^(1/2) + 4*a*b^7 + 21*a^2*b^6 - 10*a^3*b^5 + a^4*b^4 + 5*a

*b^2*(a^3*b^7)^(1/2) - 6*a^2*b*(a^3*b^7)^(1/2)))/(16384*(a^2*b^12 - 5*a^3*b^11 + 10*a^4*b^10 - 10*a^5*b^9 + 5*

a^6*b^8 - a^7*b^7)))^(1/2)*1i - (((3*(81920*a*b^7 - 180224*a^2*b^6 + 114688*a^3*b^5 - 16384*a^4*b^4))/(32768*(

b^6 - 4*a*b^5 + 6*a^2*b^4 - 4*a^3*b^3 + a^4*b^2)) + (cos(c + d*x)*((9*(a^3*(a^3*b^7)^(1/2) + 16*b^3*(a^3*b^7)^

(1/2) + 4*a*b^7 + 21*a^2*b^6 - 10*a^3*b^5 + a^4*b^4 + 5*a*b^2*(a^3*b^7)^(1/2) - 6*a^2*b*(a^3*b^7)^(1/2)))/(163

84*(a^2*b^12 - 5*a^3*b^11 + 10*a^4*b^10 - 10*a^5*b^9 + 5*a^6*b^8 - a^7*b^7)))^(1/2)*(16384*a*b^8 - 65536*a^2*b

^7 + 98304*a^3*b^6 - 65536*a^4*b^5 + 16384*a^5*b^4))/(256*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)))*((9*(a

^3*(a^3*b^7)^(1/2) + 16*b^3*(a^3*b^7)^(1/2) + 4*a*b^7 + 21*a^2*b^6 - 10*a^3*b^5 + a^4*b^4 + 5*a*b^2*(a^3*b^7)^

(1/2) - 6*a^2*b*(a^3*b^7)^(1/2)))/(16384*(a^2*b^12 - 5*a^3*b^11 + 10*a^4*b^10 - 10*a^5*b^9 + 5*a^6*b^8 - a^7*b

^7)))^(1/2) - (cos(c + d*x)*(81*a*b^2 - 54*a^2*b + 9*a^3 + 36*b^3))/(256*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^

2*b^2)))*((9*(a^3*(a^3*b^7)^(1/2) + 16*b^3*(a^3*b^7)^(1/2) + 4*a*b^7 + 21*a^2*b^6 - 10*a^3*b^5 + a^4*b^4 + 5*a

*b^2*(a^3*b^7)^(1/2) - 6*a^2*b*(a^3*b^7)^(1/2)))/(16384*(a^2*b^12 - 5*a^3*b^11 + 10*a^4*b^10 - 10*a^5*b^9 + 5*

a^6*b^8 - a^7*b^7)))^(1/2)*1i)/((((3*(81920*a*b^7 - 180224*a^2*b^6 + 114688*a^3*b^5 - 16384*a^4*b^4))/(32768*(