3.3.14 \(\int \frac {\sin ^5(c+d x)}{(a-b \sin ^4(c+d x))^2} \, dx\) [214]

Optimal. Leaf size=217 \[ \frac {\left (\sqrt {a}-2 \sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 \sqrt {a} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} b^{5/4} d}+\frac {\left (\sqrt {a}+2 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 \sqrt {a} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} b^{5/4} d}-\frac {\cos (c+d x) \left (a+b-b \cos ^2(c+d x)\right )}{4 (a-b) b d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )} \]

[Out]

-1/4*cos(d*x+c)*(a+b-b*cos(d*x+c)^2)/(a-b)/b/d/(a-b+2*b*cos(d*x+c)^2-b*cos(d*x+c)^4)+1/8*arctan(b^(1/4)*cos(d*
x+c)/(a^(1/2)-b^(1/2))^(1/2))*(a^(1/2)-2*b^(1/2))/b^(5/4)/d/a^(1/2)/(a^(1/2)-b^(1/2))^(3/2)+1/8*arctanh(b^(1/4
)*cos(d*x+c)/(a^(1/2)+b^(1/2))^(1/2))*(a^(1/2)+2*b^(1/2))/b^(5/4)/d/a^(1/2)/(a^(1/2)+b^(1/2))^(3/2)

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Rubi [A]
time = 0.19, antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3294, 1219, 1180, 211, 214} \begin {gather*} \frac {\left (\sqrt {a}-2 \sqrt {b}\right ) \text {ArcTan}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 \sqrt {a} b^{5/4} d \left (\sqrt {a}-\sqrt {b}\right )^{3/2}}+\frac {\left (\sqrt {a}+2 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 \sqrt {a} b^{5/4} d \left (\sqrt {a}+\sqrt {b}\right )^{3/2}}-\frac {\cos (c+d x) \left (a-b \cos ^2(c+d x)+b\right )}{4 b d (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]^5/(a - b*Sin[c + d*x]^4)^2,x]

[Out]

((Sqrt[a] - 2*Sqrt[b])*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(8*Sqrt[a]*(Sqrt[a] - Sqrt[b])^
(3/2)*b^(5/4)*d) + ((Sqrt[a] + 2*Sqrt[b])*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(8*Sqrt[a]*
(Sqrt[a] + Sqrt[b])^(3/2)*b^(5/4)*d) - (Cos[c + d*x]*(a + b - b*Cos[c + d*x]^2))/(4*(a - b)*b*d*(a - b + 2*b*C
os[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 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 ^5(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx &=-\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{\left (a-b+2 b x^2-b x^4\right )^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {\cos (c+d x) \left (a+b-b \cos ^2(c+d x)\right )}{4 (a-b) 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 {2 a (a-3 b)+2 a b x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cos (c+d x)\right )}{8 a (a-b) b d}\\ &=-\frac {\cos (c+d x) \left (a+b-b \cos ^2(c+d x)\right )}{4 (a-b) b d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}-\frac {\left (\sqrt {a}-2 \sqrt {b}\right ) \text {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{8 \sqrt {a} \left (\sqrt {a}-\sqrt {b}\right ) \sqrt {b} d}+\frac {\left (a+\sqrt {a} \sqrt {b}-2 b\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{8 \sqrt {a} (a-b) \sqrt {b} d}\\ &=\frac {\left (\sqrt {a}-2 \sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 \sqrt {a} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} b^{5/4} d}+\frac {\left (\sqrt {a}+2 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 \sqrt {a} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} b^{5/4} d}-\frac {\cos (c+d x) \left (a+b-b \cos ^2(c+d x)\right )}{4 (a-b) 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 = 0.44, size = 469, normalized size = 2.16 \begin {gather*} -\frac {\frac {32 \cos (c+d x) (2 a+b-b \cos (2 (c+d x)))}{8 a-3 b+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))}+i \text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {-2 b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )-8 a \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+22 b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+4 i a \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-11 i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2+8 a \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-22 b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-4 i a \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4+11 i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4+2 b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^6-i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^6}{-b \text {$\#$1}-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-3 b \text {$\#$1}^5+b \text {$\#$1}^7}\&\right ]}{32 (a-b) b d} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Sin[c + d*x]^5/(a - b*Sin[c + d*x]^4)^2,x]

[Out]

-1/32*((32*Cos[c + d*x]*(2*a + b - b*Cos[2*(c + d*x)]))/(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)]
) + I*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-2*b*ArcTan[Sin[c + d*x]/(Cos[c + d
*x] - #1)] + I*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - 8*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + 22*b*
ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (4*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (11*I)*b*Log[
1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + 8*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - 22*b*ArcTan[Sin[c + d
*x]/(Cos[c + d*x] - #1)]*#1^4 - (4*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + (11*I)*b*Log[1 - 2*Cos[c + d*
x]*#1 + #1^2]*#1^4 + 2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 - 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) & ])/((a - b)*b*d)

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Maple [A]
time = 0.88, size = 188, normalized size = 0.87

method	result	size
derivativedivides	\(\frac {-\frac {-\frac {\cos ^{3}\left (d x +c \right )}{4 \left (a -b \right )}+\frac {\left (a +b \right ) \cos \left (d x +c \right )}{4 b \left (a -b \right )}}{a -b +2 b \left (\cos ^{2}\left (d x +c \right )\right )-b \left (\cos ^{4}\left (d x +c \right )\right )}-\frac {\frac {\left (\sqrt {a b}+2 b -a \right ) \arctan \left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}-b \right ) b}}-\frac {\left (\sqrt {a b}-2 b +a \right ) \arctanh \left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+b \right ) b}}}{4 \left (a -b \right )}}{d}\)	\(188\)
default	\(\frac {-\frac {-\frac {\cos ^{3}\left (d x +c \right )}{4 \left (a -b \right )}+\frac {\left (a +b \right ) \cos \left (d x +c \right )}{4 b \left (a -b \right )}}{a -b +2 b \left (\cos ^{2}\left (d x +c \right )\right )-b \left (\cos ^{4}\left (d x +c \right )\right )}-\frac {\frac {\left (\sqrt {a b}+2 b -a \right ) \arctan \left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}-b \right ) b}}-\frac {\left (\sqrt {a b}-2 b +a \right ) \arctanh \left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+b \right ) b}}}{4 \left (a -b \right )}}{d}\)	\(188\)
risch	\(-\frac {b \,{\mathrm e}^{7 i \left (d x +c \right )}-4 a \,{\mathrm e}^{5 i \left (d x +c \right )}-b \,{\mathrm e}^{5 i \left (d x +c \right )}-4 a \,{\mathrm e}^{3 i \left (d x +c \right )}-b \,{\mathrm e}^{3 i \left (d x +c \right )}+b \,{\mathrm e}^{i \left (d x +c \right )}}{2 b \left (a -b \right ) d \left (b \,{\mathrm e}^{8 i \left (d x +c \right )}-4 b \,{\mathrm e}^{6 i \left (d x +c \right )}-16 a \,{\mathrm e}^{4 i \left (d x +c \right )}+6 b \,{\mathrm e}^{4 i \left (d x +c \right )}-4 b \,{\mathrm e}^{2 i \left (d x +c \right )}+b \right )}-\frac {i \left (\munderset {\textit {\_R} =\RootOf \left (\left (a^{5} b^{5} d^{4}-3 a^{4} b^{6} d^{4}+3 a^{3} b^{7} d^{4}-a^{2} b^{8} d^{4}\right ) \textit {\_Z}^{4}+\left (8 a^{3} b^{3} d^{2}-8 a^{2} b^{4} d^{2}-32 a \,b^{5} d^{2}\right ) \textit {\_Z}^{2}-16 a^{2}+128 a b -256 b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\left (-\frac {i a^{4} b^{5} d^{3}}{2 a^{3}-18 a^{2} b +56 a \,b^{2}-64 b^{3}}+\frac {3 i a^{3} b^{6} d^{3}}{2 a^{3}-18 a^{2} b +56 a \,b^{2}-64 b^{3}}-\frac {3 i a^{2} b^{7} d^{3}}{2 a^{3}-18 a^{2} b +56 a \,b^{2}-64 b^{3}}+\frac {i a \,b^{8} d^{3}}{2 a^{3}-18 a^{2} b +56 a \,b^{2}-64 b^{3}}\right ) \textit {\_R}^{3}+\left (-\frac {2 i a^{4} b d}{2 a^{3}-18 a^{2} b +56 a \,b^{2}-64 b^{3}}+\frac {16 i a^{3} b^{2} d}{2 a^{3}-18 a^{2} b +56 a \,b^{2}-64 b^{3}}-\frac {50 i a^{2} b^{3} d}{2 a^{3}-18 a^{2} b +56 a \,b^{2}-64 b^{3}}+\frac {52 i a \,b^{4} d}{2 a^{3}-18 a^{2} b +56 a \,b^{2}-64 b^{3}}+\frac {16 i b^{5} d}{2 a^{3}-18 a^{2} b +56 a \,b^{2}-64 b^{3}}\right ) \textit {\_R} \right ) {\mathrm e}^{i \left (d x +c \right )}+1\right )\right )}{32}\)	\(604\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)^5/(a-b*sin(d*x+c)^4)^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(-(-1/4/(a-b)*cos(d*x+c)^3+1/4*(a+b)/b/(a-b)*cos(d*x+c))/(a-b+2*b*cos(d*x+c)^2-b*cos(d*x+c)^4)-1/4/(a-b)*(
1/2*((a*b)^(1/2)+2*b-a)/(a*b)^(1/2)/(((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*b)^(1/2)-2*b+a)/(a*b)^(1/2)/(((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)^5/(a-b*sin(d*x+c)^4)^2,x, algorithm="maxima")

[Out]

1/2*(4*b^2*cos(2*d*x + 2*c)*cos(d*x + c) + 4*b^2*sin(2*d*x + 2*c)*sin(d*x + c) - b^2*cos(d*x + c) - 4*(4*a*b +
b^2)*sin(3*d*x + 3*c)*sin(2*d*x + 2*c) - (b^2*cos(7*d*x + 7*c) + b^2*cos(d*x + c) - (4*a*b + b^2)*cos(5*d*x +
5*c) - (4*a*b + b^2)*cos(3*d*x + 3*c))*cos(8*d*x + 8*c) + (4*b^2*cos(6*d*x + 6*c) + 4*b^2*cos(2*d*x + 2*c) -
b^2 + 2*(8*a*b - 3*b^2)*cos(4*d*x + 4*c))*cos(7*d*x + 7*c) + 4*(b^2*cos(d*x + c) - (4*a*b + b^2)*cos(5*d*x + 5
*c) - (4*a*b + b^2)*cos(3*d*x + 3*c))*cos(6*d*x + 6*c) + (4*a*b + b^2 - 2*(32*a^2 - 4*a*b - 3*b^2)*cos(4*d*x +
4*c) - 4*(4*a*b + b^2)*cos(2*d*x + 2*c))*cos(5*d*x + 5*c) - 2*((32*a^2 - 4*a*b - 3*b^2)*cos(3*d*x + 3*c) - (8
*a*b - 3*b^2)*cos(d*x + c))*cos(4*d*x + 4*c) + (4*a*b + b^2 - 4*(4*a*b + b^2)*cos(2*d*x + 2*c))*cos(3*d*x + 3*
c) + 2*((a*b^3 - b^4)*d*cos(8*d*x + 8*c)^2 + 16*(a*b^3 - b^4)*d*cos(6*d*x + 6*c)^2 + 4*(64*a^3*b - 112*a^2*b^2
+ 57*a*b^3 - 9*b^4)*d*cos(4*d*x + 4*c)^2 + 16*(a*b^3 - b^4)*d*cos(2*d*x + 2*c)^2 + (a*b^3 - b^4)*d*sin(8*d*x
+ 8*c)^2 + 16*(a*b^3 - b^4)*d*sin(6*d*x + 6*c)^2 + 4*(64*a^3*b - 112*a^2*b^2 + 57*a*b^3 - 9*b^4)*d*sin(4*d*x +
4*c)^2 + 16*(8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*(a*b^3 - b^4)*d*sin(2*d*x
+ 2*c)^2 - 8*(a*b^3 - b^4)*d*cos(2*d*x + 2*c) + (a*b^3 - b^4)*d - 2*(4*(a*b^3 - b^4)*d*cos(6*d*x + 6*c) + 2*(
8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*cos(4*d*x + 4*c) + 4*(a*b^3 - b^4)*d*cos(2*d*x + 2*c) - (a*b^3 - b^4)*d)*cos(8
*d*x + 8*c) + 8*(2*(8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*cos(4*d*x + 4*c) + 4*(a*b^3 - b^4)*d*cos(2*d*x + 2*c) - (a
*b^3 - b^4)*d)*cos(6*d*x + 6*c) + 4*(4*(8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*cos(2*d*x + 2*c) - (8*a^2*b^2 - 11*a*b
^3 + 3*b^4)*d)*cos(4*d*x + 4*c) - 4*(2*(a*b^3 - b^4)*d*sin(6*d*x + 6*c) + (8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*sin
(4*d*x + 4*c) + 2*(a*b^3 - b^4)*d*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 16*((8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*si
n(4*d*x + 4*c) + 2*(a*b^3 - b^4)*d*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*integrate(-1/2*(4*b^2*cos(d*x + c)*sin(
2*d*x + 2*c) - 4*b^2*cos(2*d*x + 2*c)*sin(d*x + c) + 4*(4*a*b - 11*b^2)*cos(3*d*x + 3*c)*sin(2*d*x + 2*c) + b^
2*sin(d*x + c) - (b^2*sin(7*d*x + 7*c) - b^2*sin(d*x + c) + (4*a*b - 11*b^2)*sin(5*d*x + 5*c) - (4*a*b - 11*b^
2)*sin(3*d*x + 3*c))*cos(8*d*x + 8*c) - 2*(2*b^2*sin(6*d*x + 6*c) + 2*b^2*sin(2*d*x + 2*c) + (8*a*b - 3*b^2)*s
in(4*d*x + 4*c))*cos(7*d*x + 7*c) - 4*(b^2*sin(d*x + c) - (4*a*b - 11*b^2)*sin(5*d*x + 5*c) + (4*a*b - 11*b^2)
*sin(3*d*x + 3*c))*cos(6*d*x + 6*c) - 2*((32*a^2 - 100*a*b + 33*b^2)*sin(4*d*x + 4*c) + 2*(4*a*b - 11*b^2)*sin
(2*d*x + 2*c))*cos(5*d*x + 5*c) - 2*((32*a^2 - 100*a*b + 33*b^2)*sin(3*d*x + 3*c) + (8*a*b - 3*b^2)*sin(d*x +
c))*cos(4*d*x + 4*c) + (b^2*cos(7*d*x + 7*c) - b^2*cos(d*x + c) + (4*a*b - 11*b^2)*cos(5*d*x + 5*c) - (4*a*b -
11*b^2)*cos(3*d*x + 3*c))*sin(8*d*x + 8*c) + (4*b^2*cos(6*d*x + 6*c) + 4*b^2*cos(2*d*x + 2*c) - b^2 + 2*(8*a*
b - 3*b^2)*cos(4*d*x + 4*c))*sin(7*d*x + 7*c) + 4*(b^2*cos(d*x + c) - (4*a*b - 11*b^2)*cos(5*d*x + 5*c) + (4*a
*b - 11*b^2)*cos(3*d*x + 3*c))*sin(6*d*x + 6*c) - (4*a*b - 11*b^2 - 2*(32*a^2 - 100*a*b + 33*b^2)*cos(4*d*x +
4*c) - 4*(4*a*b - 11*b^2)*cos(2*d*x + 2*c))*sin(5*d*x + 5*c) + 2*((32*a^2 - 100*a*b + 33*b^2)*cos(3*d*x + 3*c)
+ (8*a*b - 3*b^2)*cos(d*x + c))*sin(4*d*x + 4*c) + (4*a*b - 11*b^2 - 4*(4*a*b - 11*b^2)*cos(2*d*x + 2*c))*sin
(3*d*x + 3*c))/(a*b^3 - b^4 + (a*b^3 - b^4)*cos(8*d*x + 8*c)^2 + 16*(a*b^3 - b^4)*cos(6*d*x + 6*c)^2 + 4*(64*a
^3*b - 112*a^2*b^2 + 57*a*b^3 - 9*b^4)*cos(4*d*x + 4*c)^2 + 16*(a*b^3 - b^4)*cos(2*d*x + 2*c)^2 + (a*b^3 - b^4
)*sin(8*d*x + 8*c)^2 + 16*(a*b^3 - b^4)*sin(6*d*x + 6*c)^2 + 4*(64*a^3*b - 112*a^2*b^2 + 57*a*b^3 - 9*b^4)*sin
(4*d*x + 4*c)^2 + 16*(8*a^2*b^2 - 11*a*b^3 + 3*b^4)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*(a*b^3 - b^4)*sin(2
*d*x + 2*c)^2 + 2*(a*b^3 - b^4 - 4*(a*b^3 - b^4)*cos(6*d*x + 6*c) - 2*(8*a^2*b^2 - 11*a*b^3 + 3*b^4)*cos(4*d*x
+ 4*c) - 4*(a*b^3 - b^4)*cos(2*d*x + 2*c))*cos(8*d*x + 8*c) - 8*(a*b^3 - b^4 - 2*(8*a^2*b^2 - 11*a*b^3 + 3*b^
4)*cos(4*d*x + 4*c) - 4*(a*b^3 - b^4)*cos(2*d*x + 2*c))*cos(6*d*x + 6*c) - 4*(8*a^2*b^2 - 11*a*b^3 + 3*b^4 - 4
*(8*a^2*b^2 - 11*a*b^3 + 3*b^4)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - 8*(a*b^3 - b^4)*cos(2*d*x + 2*c) - 4*(2*(
a*b^3 - b^4)*sin(6*d*x + 6*c) + (8*a^2*b^2 - 11*a*b^3 + 3*b^4)*sin(4*d*x + 4*c) + 2*(a*b^3 - b^4)*sin(2*d*x +
2*c))*sin(8*d*x + 8*c) + 16*((8*a^2*b^2 - 11*a*b^3 + 3*b^4)*sin(4*d*x + 4*c) + 2*(a*b^3 - b^4)*sin(2*d*x + 2*c
))*sin(6*d*x + 6*c)), x) - (b^2*sin(7*d*x + 7*c) + b^2*sin(d*x + c) - (4*a*b + b^2)*sin(5*d*x + 5*c) - (4*a*b
+ b^2)*sin(3*d*x + 3*c))*sin(8*d*x + 8*c) + 2*(2*b^2*sin(6*d*x + 6*c) + 2*b^2*sin(2*d*x + 2*c) + (8*a*b - 3*b^
2)*sin(4*d*x + 4*c))*sin(7*d*x + 7*c) + 4*(b^2*sin(d*x + c) - (4*a*b + b^2)*sin(5*d*x + 5*c) - (4*a*b + b^2)*s
in(3*d*x + 3*c))*sin(6*d*x + 6*c) - 2*((32*a^2 - 4*a*b - 3*b^2)*sin(4*d*x + 4*c) + 2*(4*a*b + b^2)*sin(2*d*x +
2*c))*sin(5*d*x + 5*c) - 2*((32*a^2 - 4*a*b - 3*b^2)*sin(3*d*x + 3*c) - (8*a*b - 3*b^2)*sin(d*x + c))*sin(4*d
*x + 4*c))/((a*b^3 - b^4)*d*cos(8*d*x + 8*c)^2 ...

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2507 vs. \(2 (169) = 338\).
time = 0.66, size = 2507, normalized size = 11.55 \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)^5/(a-b*sin(d*x+c)^4)^2,x, algorithm="fricas")

[Out]

-1/16*(4*b*cos(d*x + c)^3 - ((a*b^2 - b^3)*d*cos(d*x + c)^4 - 2*(a*b^2 - b^3)*d*cos(d*x + c)^2 - (a^2*b - 2*a*
b^2 + b^3)*d)*sqrt(((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d^2*sqrt((a^4 - 10*a^3*b + 41*a^2*b^2 - 80*a*b^3
+ 64*b^4)/((a^7*b^5 - 6*a^6*b^6 + 15*a^5*b^7 - 20*a^4*b^8 + 15*a^3*b^9 - 6*a^2*b^10 + a*b^11)*d^4)) + a^2 - a
*b - 4*b^2)/((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d^2))*log((a^3 - 9*a^2*b + 28*a*b^2 - 32*b^3)*cos(d*x +
c) - (2*(a^4*b^5 - 3*a^3*b^6 + 3*a^2*b^7 - a*b^8)*d^3*sqrt((a^4 - 10*a^3*b + 41*a^2*b^2 - 80*a*b^3 + 64*b^4)/
((a^7*b^5 - 6*a^6*b^6 + 15*a^5*b^7 - 20*a^4*b^8 + 15*a^3*b^9 - 6*a^2*b^10 + a*b^11)*d^4)) - (a^4*b - 8*a^3*b^2
+ 23*a^2*b^3 - 24*a*b^4)*d)*sqrt(((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d^2*sqrt((a^4 - 10*a^3*b + 41*a^2
*b^2 - 80*a*b^3 + 64*b^4)/((a^7*b^5 - 6*a^6*b^6 + 15*a^5*b^7 - 20*a^4*b^8 + 15*a^3*b^9 - 6*a^2*b^10 + a*b^11)*
d^4)) + a^2 - a*b - 4*b^2)/((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d^2))) + ((a*b^2 - b^3)*d*cos(d*x + c)^4
- 2*(a*b^2 - b^3)*d*cos(d*x + c)^2 - (a^2*b - 2*a*b^2 + b^3)*d)*sqrt(-((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b
^5)*d^2*sqrt((a^4 - 10*a^3*b + 41*a^2*b^2 - 80*a*b^3 + 64*b^4)/((a^7*b^5 - 6*a^6*b^6 + 15*a^5*b^7 - 20*a^4*b^8
+ 15*a^3*b^9 - 6*a^2*b^10 + a*b^11)*d^4)) - a^2 + a*b + 4*b^2)/((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d^2
))*log((a^3 - 9*a^2*b + 28*a*b^2 - 32*b^3)*cos(d*x + c) - (2*(a^4*b^5 - 3*a^3*b^6 + 3*a^2*b^7 - a*b^8)*d^3*sqr
t((a^4 - 10*a^3*b + 41*a^2*b^2 - 80*a*b^3 + 64*b^4)/((a^7*b^5 - 6*a^6*b^6 + 15*a^5*b^7 - 20*a^4*b^8 + 15*a^3*b
^9 - 6*a^2*b^10 + a*b^11)*d^4)) + (a^4*b - 8*a^3*b^2 + 23*a^2*b^3 - 24*a*b^4)*d)*sqrt(-((a^4*b^2 - 3*a^3*b^3 +
3*a^2*b^4 - a*b^5)*d^2*sqrt((a^4 - 10*a^3*b + 41*a^2*b^2 - 80*a*b^3 + 64*b^4)/((a^7*b^5 - 6*a^6*b^6 + 15*a^5*
b^7 - 20*a^4*b^8 + 15*a^3*b^9 - 6*a^2*b^10 + a*b^11)*d^4)) - a^2 + a*b + 4*b^2)/((a^4*b^2 - 3*a^3*b^3 + 3*a^2*
b^4 - a*b^5)*d^2))) + ((a*b^2 - b^3)*d*cos(d*x + c)^4 - 2*(a*b^2 - b^3)*d*cos(d*x + c)^2 - (a^2*b - 2*a*b^2 +
b^3)*d)*sqrt(((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d^2*sqrt((a^4 - 10*a^3*b + 41*a^2*b^2 - 80*a*b^3 + 64*
b^4)/((a^7*b^5 - 6*a^6*b^6 + 15*a^5*b^7 - 20*a^4*b^8 + 15*a^3*b^9 - 6*a^2*b^10 + a*b^11)*d^4)) + a^2 - a*b - 4
*b^2)/((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d^2))*log(-(a^3 - 9*a^2*b + 28*a*b^2 - 32*b^3)*cos(d*x + c) -
(2*(a^4*b^5 - 3*a^3*b^6 + 3*a^2*b^7 - a*b^8)*d^3*sqrt((a^4 - 10*a^3*b + 41*a^2*b^2 - 80*a*b^3 + 64*b^4)/((a^7
*b^5 - 6*a^6*b^6 + 15*a^5*b^7 - 20*a^4*b^8 + 15*a^3*b^9 - 6*a^2*b^10 + a*b^11)*d^4)) - (a^4*b - 8*a^3*b^2 + 23
*a^2*b^3 - 24*a*b^4)*d)*sqrt(((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d^2*sqrt((a^4 - 10*a^3*b + 41*a^2*b^2
- 80*a*b^3 + 64*b^4)/((a^7*b^5 - 6*a^6*b^6 + 15*a^5*b^7 - 20*a^4*b^8 + 15*a^3*b^9 - 6*a^2*b^10 + a*b^11)*d^4))
+ a^2 - a*b - 4*b^2)/((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d^2))) - ((a*b^2 - b^3)*d*cos(d*x + c)^4 - 2*
(a*b^2 - b^3)*d*cos(d*x + c)^2 - (a^2*b - 2*a*b^2 + b^3)*d)*sqrt(-((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d
^2*sqrt((a^4 - 10*a^3*b + 41*a^2*b^2 - 80*a*b^3 + 64*b^4)/((a^7*b^5 - 6*a^6*b^6 + 15*a^5*b^7 - 20*a^4*b^8 + 15
*a^3*b^9 - 6*a^2*b^10 + a*b^11)*d^4)) - a^2 + a*b + 4*b^2)/((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d^2))*lo
g(-(a^3 - 9*a^2*b + 28*a*b^2 - 32*b^3)*cos(d*x + c) - (2*(a^4*b^5 - 3*a^3*b^6 + 3*a^2*b^7 - a*b^8)*d^3*sqrt((a
^4 - 10*a^3*b + 41*a^2*b^2 - 80*a*b^3 + 64*b^4)/((a^7*b^5 - 6*a^6*b^6 + 15*a^5*b^7 - 20*a^4*b^8 + 15*a^3*b^9 -
6*a^2*b^10 + a*b^11)*d^4)) + (a^4*b - 8*a^3*b^2 + 23*a^2*b^3 - 24*a*b^4)*d)*sqrt(-((a^4*b^2 - 3*a^3*b^3 + 3*a
^2*b^4 - a*b^5)*d^2*sqrt((a^4 - 10*a^3*b + 41*a^2*b^2 - 80*a*b^3 + 64*b^4)/((a^7*b^5 - 6*a^6*b^6 + 15*a^5*b^7
- 20*a^4*b^8 + 15*a^3*b^9 - 6*a^2*b^10 + a*b^11)*d^4)) - a^2 + a*b + 4*b^2)/((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4
- a*b^5)*d^2))) - 4*(a + b)*cos(d*x + c))/((a*b^2 - b^3)*d*cos(d*x + c)^4 - 2*(a*b^2 - b^3)*d*cos(d*x + c)^2 -
(a^2*b - 2*a*b^2 + b^3)*d)

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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(d*x+c)**5/(a-b*sin(d*x+c)**4)**2,x)

[Out]

Timed out

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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)^5/(a-b*sin(d*x+c)^4)^2,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]=[-
57,-84]Warning, need

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Mupad [B]
time = 16.63, size = 2500, normalized size = 11.52 \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)^5/(a - b*sin(c + d*x)^4)^2,x)

[Out]

(cos(c + d*x)^3/(4*(a - b)) - (cos(c + d*x)*(a + b))/(4*b*(a - b)))/(d*(a - b + 2*b*cos(c + d*x)^2 - b*cos(c +
d*x)^4)) - (atan(((((768*a*b^4 - 1024*a^2*b^3 + 256*a^3*b^2)/(64*(a^2 - 2*a*b + b^2)) - (cos(c + d*x)*(-(a^2*
(a^3*b^5)^(1/2) + 8*b^2*(a^3*b^5)^(1/2) - 4*a*b^5 - a^2*b^4 + a^3*b^3 - 5*a*b*(a^3*b^5)^(1/2))/(256*(a^2*b^8 -
3*a^3*b^7 + 3*a^4*b^6 - a^5*b^5)))^(1/2)*(256*a*b^6 - 512*a^2*b^5 + 256*a^3*b^4))/(4*(a^2 - 2*a*b + b^2)))*(-
(a^2*(a^3*b^5)^(1/2) + 8*b^2*(a^3*b^5)^(1/2) - 4*a*b^5 - a^2*b^4 + a^3*b^3 - 5*a*b*(a^3*b^5)^(1/2))/(256*(a^2*
b^8 - 3*a^3*b^7 + 3*a^4*b^6 - a^5*b^5)))^(1/2) + (cos(c + d*x)*(a^2*b - 3*a*b^2 + 4*b^3))/(4*(a^2 - 2*a*b + b^
2)))*(-(a^2*(a^3*b^5)^(1/2) + 8*b^2*(a^3*b^5)^(1/2) - 4*a*b^5 - a^2*b^4 + a^3*b^3 - 5*a*b*(a^3*b^5)^(1/2))/(25
6*(a^2*b^8 - 3*a^3*b^7 + 3*a^4*b^6 - a^5*b^5)))^(1/2)*1i - (((768*a*b^4 - 1024*a^2*b^3 + 256*a^3*b^2)/(64*(a^2
- 2*a*b + b^2)) + (cos(c + d*x)*(-(a^2*(a^3*b^5)^(1/2) + 8*b^2*(a^3*b^5)^(1/2) - 4*a*b^5 - a^2*b^4 + a^3*b^3
- 5*a*b*(a^3*b^5)^(1/2))/(256*(a^2*b^8 - 3*a^3*b^7 + 3*a^4*b^6 - a^5*b^5)))^(1/2)*(256*a*b^6 - 512*a^2*b^5 + 2
56*a^3*b^4))/(4*(a^2 - 2*a*b + b^2)))*(-(a^2*(a^3*b^5)^(1/2) + 8*b^2*(a^3*b^5)^(1/2) - 4*a*b^5 - a^2*b^4 + a^3
*b^3 - 5*a*b*(a^3*b^5)^(1/2))/(256*(a^2*b^8 - 3*a^3*b^7 + 3*a^4*b^6 - a^5*b^5)))^(1/2) - (cos(c + d*x)*(a^2*b
- 3*a*b^2 + 4*b^3))/(4*(a^2 - 2*a*b + b^2)))*(-(a^2*(a^3*b^5)^(1/2) + 8*b^2*(a^3*b^5)^(1/2) - 4*a*b^5 - a^2*b^
4 + a^3*b^3 - 5*a*b*(a^3*b^5)^(1/2))/(256*(a^2*b^8 - 3*a^3*b^7 + 3*a^4*b^6 - a^5*b^5)))^(1/2)*1i)/((((768*a*b^
4 - 1024*a^2*b^3 + 256*a^3*b^2)/(64*(a^2 - 2*a*b + b^2)) - (cos(c + d*x)*(-(a^2*(a^3*b^5)^(1/2) + 8*b^2*(a^3*b
^5)^(1/2) - 4*a*b^5 - a^2*b^4 + a^3*b^3 - 5*a*b*(a^3*b^5)^(1/2))/(256*(a^2*b^8 - 3*a^3*b^7 + 3*a^4*b^6 - a^5*b
^5)))^(1/2)*(256*a*b^6 - 512*a^2*b^5 + 256*a^3*b^4))/(4*(a^2 - 2*a*b + b^2)))*(-(a^2*(a^3*b^5)^(1/2) + 8*b^2*(
a^3*b^5)^(1/2) - 4*a*b^5 - a^2*b^4 + a^3*b^3 - 5*a*b*(a^3*b^5)^(1/2))/(256*(a^2*b^8 - 3*a^3*b^7 + 3*a^4*b^6 -
a^5*b^5)))^(1/2) + (cos(c + d*x)*(a^2*b - 3*a*b^2 + 4*b^3))/(4*(a^2 - 2*a*b + b^2)))*(-(a^2*(a^3*b^5)^(1/2) +
8*b^2*(a^3*b^5)^(1/2) - 4*a*b^5 - a^2*b^4 + a^3*b^3 - 5*a*b*(a^3*b^5)^(1/2))/(256*(a^2*b^8 - 3*a^3*b^7 + 3*a^4
*b^6 - a^5*b^5)))^(1/2) - (a - 4*b)/(32*(a^2 - 2*a*b + b^2)) + (((768*a*b^4 - 1024*a^2*b^3 + 256*a^3*b^2)/(64*
(a^2 - 2*a*b + b^2)) + (cos(c + d*x)*(-(a^2*(a^3*b^5)^(1/2) + 8*b^2*(a^3*b^5)^(1/2) - 4*a*b^5 - a^2*b^4 + a^3*
b^3 - 5*a*b*(a^3*b^5)^(1/2))/(256*(a^2*b^8 - 3*a^3*b^7 + 3*a^4*b^6 - a^5*b^5)))^(1/2)*(256*a*b^6 - 512*a^2*b^5
+ 256*a^3*b^4))/(4*(a^2 - 2*a*b + b^2)))*(-(a^2*(a^3*b^5)^(1/2) + 8*b^2*(a^3*b^5)^(1/2) - 4*a*b^5 - a^2*b^4 +
a^3*b^3 - 5*a*b*(a^3*b^5)^(1/2))/(256*(a^2*b^8 - 3*a^3*b^7 + 3*a^4*b^6 - a^5*b^5)))^(1/2) - (cos(c + d*x)*(a^
2*b - 3*a*b^2 + 4*b^3))/(4*(a^2 - 2*a*b + b^2)))*(-(a^2*(a^3*b^5)^(1/2) + 8*b^2*(a^3*b^5)^(1/2) - 4*a*b^5 - a^
2*b^4 + a^3*b^3 - 5*a*b*(a^3*b^5)^(1/2))/(256*(a^2*b^8 - 3*a^3*b^7 + 3*a^4*b^6 - a^5*b^5)))^(1/2)))*(-(a^2*(a^
3*b^5)^(1/2) + 8*b^2*(a^3*b^5)^(1/2) - 4*a*b^5 - a^2*b^4 + a^3*b^3 - 5*a*b*(a^3*b^5)^(1/2))/(256*(a^2*b^8 - 3*
a^3*b^7 + 3*a^4*b^6 - a^5*b^5)))^(1/2)*2i)/d - (atan(((((768*a*b^4 - 1024*a^2*b^3 + 256*a^3*b^2)/(64*(a^2 - 2*
a*b + b^2)) - (cos(c + d*x)*((a^2*(a^3*b^5)^(1/2) + 8*b^2*(a^3*b^5)^(1/2) + 4*a*b^5 + a^2*b^4 - a^3*b^3 - 5*a*
b*(a^3*b^5)^(1/2))/(256*(a^2*b^8 - 3*a^3*b^7 + 3*a^4*b^6 - a^5*b^5)))^(1/2)*(256*a*b^6 - 512*a^2*b^5 + 256*a^3
*b^4))/(4*(a^2 - 2*a*b + b^2)))*((a^2*(a^3*b^5)^(1/2) + 8*b^2*(a^3*b^5)^(1/2) + 4*a*b^5 + a^2*b^4 - a^3*b^3 -
5*a*b*(a^3*b^5)^(1/2))/(256*(a^2*b^8 - 3*a^3*b^7 + 3*a^4*b^6 - a^5*b^5)))^(1/2) + (cos(c + d*x)*(a^2*b - 3*a*b
^2 + 4*b^3))/(4*(a^2 - 2*a*b + b^2)))*((a^2*(a^3*b^5)^(1/2) + 8*b^2*(a^3*b^5)^(1/2) + 4*a*b^5 + a^2*b^4 - a^3*
b^3 - 5*a*b*(a^3*b^5)^(1/2))/(256*(a^2*b^8 - 3*a^3*b^7 + 3*a^4*b^6 - a^5*b^5)))^(1/2)*1i - (((768*a*b^4 - 1024
*a^2*b^3 + 256*a^3*b^2)/(64*(a^2 - 2*a*b + b^2)) + (cos(c + d*x)*((a^2*(a^3*b^5)^(1/2) + 8*b^2*(a^3*b^5)^(1/2)
+ 4*a*b^5 + a^2*b^4 - a^3*b^3 - 5*a*b*(a^3*b^5)^(1/2))/(256*(a^2*b^8 - 3*a^3*b^7 + 3*a^4*b^6 - a^5*b^5)))^(1/
2)*(256*a*b^6 - 512*a^2*b^5 + 256*a^3*b^4))/(4*(a^2 - 2*a*b + b^2)))*((a^2*(a^3*b^5)^(1/2) + 8*b^2*(a^3*b^5)^(
1/2) + 4*a*b^5 + a^2*b^4 - a^3*b^3 - 5*a*b*(a^3*b^5)^(1/2))/(256*(a^2*b^8 - 3*a^3*b^7 + 3*a^4*b^6 - a^5*b^5)))
^(1/2) - (cos(c + d*x)*(a^2*b - 3*a*b^2 + 4*b^3))/(4*(a^2 - 2*a*b + b^2)))*((a^2*(a^3*b^5)^(1/2) + 8*b^2*(a^3*
b^5)^(1/2) + 4*a*b^5 + a^2*b^4 - a^3*b^3 - 5*a*b*(a^3*b^5)^(1/2))/(256*(a^2*b^8 - 3*a^3*b^7 + 3*a^4*b^6 - a^5*
b^5)))^(1/2)*1i)/((((768*a*b^4 - 1024*a^2*b^3 + 256*a^3*b^2)/(64*(a^2 - 2*a*b + b^2)) - (cos(c + d*x)*((a^2*(a
^3*b^5)^(1/2) + 8*b^2*(a^3*b^5)^(1/2) + 4*a*b^5 + a^2*b^4 - a^3*b^3 - 5*a*b*(a^3*b^5)^(1/2))/(256*(a^2*b^8 - 3
*a^3*b^7 + 3*a^4*b^6 - a^5*b^5)))^(1/2)*(256*a*b^6 - 512*a^2*b^5 + 256*a^3*b^4))/(4*(a^2 - 2*a*b + b^2)))*((a^
2*(a^3*b^5)^(1/2) + 8*b^2*(a^3*b^5)^(1/2) + 4*a*b^5 + a^2*b^4 - a^3*b^3 - 5*a*b*(a^3*b^5)^(1/2))/(256*(a^2*b^8
- 3*a^3*b^7 + 3*a^4*b^6 - a^5*b^5)))^(1/2) + (...

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