3.4.93 \(\int \frac {\cos ^7(c+d x)}{(a+b \sin ^3(c+d x))^2} \, dx\) [393]

Optimal. Leaf size=288 \[ -\frac {2 \left (2 a^2+3 a^{4/3} b^{2/3}+b^2\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} b^{7/3} d}+\frac {2 \left (2 a^2-3 a^{4/3} b^{2/3}+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} b^{7/3} d}-\frac {\left (2 a^2-3 a^{4/3} b^{2/3}+b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{9 a^{5/3} b^{7/3} d}-\frac {\sin (c+d x)}{b^2 d}-\frac {\sin (c+d x) \left (a^2-b^2+3 a b \sin (c+d x)+3 b^2 \sin ^2(c+d x)\right )}{3 a b^2 d \left (a+b \sin ^3(c+d x)\right )} \]

[Out]

2/9*(2*a^2-3*a^(4/3)*b^(2/3)+b^2)*ln(a^(1/3)+b^(1/3)*sin(d*x+c))/a^(5/3)/b^(7/3)/d-1/9*(2*a^2-3*a^(4/3)*b^(2/3
)+b^2)*ln(a^(2/3)-a^(1/3)*b^(1/3)*sin(d*x+c)+b^(2/3)*sin(d*x+c)^2)/a^(5/3)/b^(7/3)/d-sin(d*x+c)/b^2/d-1/3*sin(
d*x+c)*(a^2-b^2+3*a*b*sin(d*x+c)+3*b^2*sin(d*x+c)^2)/a/b^2/d/(a+b*sin(d*x+c)^3)-2/9*(2*a^2+3*a^(4/3)*b^(2/3)+b
^2)*arctan(1/3*(a^(1/3)-2*b^(1/3)*sin(d*x+c))/a^(1/3)*3^(1/2))/a^(5/3)/b^(7/3)/d*3^(1/2)

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Rubi [A]
time = 0.22, antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {3302, 1872, 1901, 1874, 31, 648, 631, 210, 642} \begin {gather*} -\frac {\sin (c+d x) \left (a^2+3 a b \sin (c+d x)+3 b^2 \sin ^2(c+d x)-b^2\right )}{3 a b^2 d \left (a+b \sin ^3(c+d x)\right )}-\frac {2 \left (3 a^{4/3} b^{2/3}+2 a^2+b^2\right ) \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} b^{7/3} d}-\frac {\left (-3 a^{4/3} b^{2/3}+2 a^2+b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{9 a^{5/3} b^{7/3} d}+\frac {2 \left (-3 a^{4/3} b^{2/3}+2 a^2+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} b^{7/3} d}-\frac {\sin (c+d x)}{b^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^7/(a + b*Sin[c + d*x]^3)^2,x]

[Out]

(-2*(2*a^2 + 3*a^(4/3)*b^(2/3) + b^2)*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]
*a^(5/3)*b^(7/3)*d) + (2*(2*a^2 - 3*a^(4/3)*b^(2/3) + b^2)*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(9*a^(5/3)*b^(
7/3)*d) - ((2*a^2 - 3*a^(4/3)*b^(2/3) + b^2)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]
^2])/(9*a^(5/3)*b^(7/3)*d) - Sin[c + d*x]/(b^2*d) - (Sin[c + d*x]*(a^2 - b^2 + 3*a*b*Sin[c + d*x] + 3*b^2*Sin[
c + d*x]^2))/(3*a*b^2*d*(a + b*Sin[c + d*x]^3))

Rule 31

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

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1872

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient
[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x
]}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[(a + b*x^n)^(p + 1)*ExpandToSum[a*n*(p + 1)*Q + n*(p +
1)*R + D[x*R, x], x], x], x] + Simp[(-x)*R*((a + b*x^n)^(p + 1)/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1))), x]] /
; GeQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 1874

Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numerator[Rt[a/b, 3]], s = Denominator[R
t[a/b, 3]]}, Dist[(-r)*((B*r - A*s)/(3*a*s)), Int[1/(r + s*x), x], x] + Dist[r/(3*a*s), Int[(r*(B*r + 2*A*s) +
s*(B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] && NeQ[a*B^3 - b*A^3, 0] && PosQ[
a/b]

Rule 1901

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^n), x], x] /; FreeQ[{a, b}, x
] && PolyQ[Pq, x] && IntegerQ[n]

Rule 3302

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With
[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*(c*ff*x)^n)^p, x]
, x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 1)/2] && (EqQ[n, 4] || GtQ[m, 0
] || IGtQ[p, 0] || IntegersQ[m, p])

Rubi steps

\begin {align*} \int \frac {\cos ^7(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^3}{\left (a+b x^3\right )^2} \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac {\sin (c+d x) \left (a^2-b^2+3 a b \sin (c+d x)+3 b^2 \sin ^2(c+d x)\right )}{3 a b^2 d \left (a+b \sin ^3(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {-a^2-2 b^2-6 a b x+3 a b x^3}{a+b x^3} \, dx,x,\sin (c+d x)\right )}{3 a b^2 d}\\ &=-\frac {\sin (c+d x) \left (a^2-b^2+3 a b \sin (c+d x)+3 b^2 \sin ^2(c+d x)\right )}{3 a b^2 d \left (a+b \sin ^3(c+d x)\right )}-\frac {\text {Subst}\left (\int \left (3 a-\frac {2 \left (2 a^2+b^2+3 a b x\right )}{a+b x^3}\right ) \, dx,x,\sin (c+d x)\right )}{3 a b^2 d}\\ &=-\frac {\sin (c+d x)}{b^2 d}-\frac {\sin (c+d x) \left (a^2-b^2+3 a b \sin (c+d x)+3 b^2 \sin ^2(c+d x)\right )}{3 a b^2 d \left (a+b \sin ^3(c+d x)\right )}+\frac {2 \text {Subst}\left (\int \frac {2 a^2+b^2+3 a b x}{a+b x^3} \, dx,x,\sin (c+d x)\right )}{3 a b^2 d}\\ &=-\frac {\sin (c+d x)}{b^2 d}-\frac {\sin (c+d x) \left (a^2-b^2+3 a b \sin (c+d x)+3 b^2 \sin ^2(c+d x)\right )}{3 a b^2 d \left (a+b \sin ^3(c+d x)\right )}+\frac {2 \text {Subst}\left (\int \frac {\sqrt [3]{a} \left (3 a^{4/3} b+2 \sqrt [3]{b} \left (2 a^2+b^2\right )\right )+\sqrt [3]{b} \left (3 a^{4/3} b-\sqrt [3]{b} \left (2 a^2+b^2\right )\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sin (c+d x)\right )}{9 a^{5/3} b^{7/3} d}+\frac {\left (2 \left (2 a^2-3 a^{4/3} b^{2/3}+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,\sin (c+d x)\right )}{9 a^{5/3} b^2 d}\\ &=\frac {2 \left (2 a^2-3 a^{4/3} b^{2/3}+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} b^{7/3} d}-\frac {\sin (c+d x)}{b^2 d}-\frac {\sin (c+d x) \left (a^2-b^2+3 a b \sin (c+d x)+3 b^2 \sin ^2(c+d x)\right )}{3 a b^2 d \left (a+b \sin ^3(c+d x)\right )}-\frac {\left (2 a^2-3 a^{4/3} b^{2/3}+b^2\right ) \text {Subst}\left (\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sin (c+d x)\right )}{9 a^{5/3} b^{7/3} d}+\frac {\left (2 a^2+3 a^{4/3} b^{2/3}+b^2\right ) \text {Subst}\left (\int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sin (c+d x)\right )}{3 a^{4/3} b^2 d}\\ &=\frac {2 \left (2 a^2-3 a^{4/3} b^{2/3}+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} b^{7/3} d}-\frac {\left (2 a^2-3 a^{4/3} b^{2/3}+b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{9 a^{5/3} b^{7/3} d}-\frac {\sin (c+d x)}{b^2 d}-\frac {\sin (c+d x) \left (a^2-b^2+3 a b \sin (c+d x)+3 b^2 \sin ^2(c+d x)\right )}{3 a b^2 d \left (a+b \sin ^3(c+d x)\right )}+\frac {\left (2 \left (2 a^2+3 a^{4/3} b^{2/3}+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} \sin (c+d x)}{\sqrt [3]{a}}\right )}{3 a^{5/3} b^{7/3} d}\\ &=-\frac {2 \left (2 a^2+3 a^{4/3} b^{2/3}+b^2\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sin (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{3 \sqrt {3} a^{5/3} b^{7/3} d}+\frac {2 \left (2 a^2-3 a^{4/3} b^{2/3}+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} b^{7/3} d}-\frac {\left (2 a^2-3 a^{4/3} b^{2/3}+b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{9 a^{5/3} b^{7/3} d}-\frac {\sin (c+d x)}{b^2 d}-\frac {\sin (c+d x) \left (a^2-b^2+3 a b \sin (c+d x)+3 b^2 \sin ^2(c+d x)\right )}{3 a b^2 d \left (a+b \sin ^3(c+d x)\right )}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 2.44, size = 402, normalized size = 1.40 \begin {gather*} \frac {\frac {6 \sqrt [3]{-1} \left (2 \sqrt [3]{-1} a^{2/3}+3 b^{2/3}\right ) \log \left (-(-1)^{2/3} \sqrt [3]{a}-\sqrt [3]{b} \sin (c+d x)\right )}{\sqrt [3]{a} b^{7/3}}+\frac {6 \left (2 a^{2/3}-3 b^{2/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{\sqrt [3]{a} b^{7/3}}-\frac {6 \sqrt [3]{-1} \left (2 a^{2/3}+3 \sqrt [3]{-1} b^{2/3}\right ) \log \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} \sin (c+d x)\right )}{\sqrt [3]{a} b^{7/3}}+\frac {2 \left (a^2-b^2\right ) \left (2 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )-2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )+\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )\right )}{a^{5/3} b^{7/3}}-\frac {18 \sin (c+d x)}{b^2}-\frac {27 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};-\frac {b \sin ^3(c+d x)}{a}\right ) \sin ^2(c+d x)}{a b}+\frac {18}{b \left (a+b \sin ^3(c+d x)\right )}+\frac {6 \left (1-\frac {a^2}{b^2}\right ) \sin (c+d x)}{a \left (a+b \sin ^3(c+d x)\right )}}{18 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^7/(a + b*Sin[c + d*x]^3)^2,x]

[Out]

((6*(-1)^(1/3)*(2*(-1)^(1/3)*a^(2/3) + 3*b^(2/3))*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sin[c + d*x]])/(a^(1/3)*
b^(7/3)) + (6*(2*a^(2/3) - 3*b^(2/3))*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(a^(1/3)*b^(7/3)) - (6*(-1)^(1/3)*(
2*a^(2/3) + 3*(-1)^(1/3)*b^(2/3))*Log[a^(1/3) + (-1)^(2/3)*b^(1/3)*Sin[c + d*x]])/(a^(1/3)*b^(7/3)) + (2*(a^2
- b^2)*(2*Sqrt[3]*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))] - 2*Log[a^(1/3) + b^(1/3)*Sin[c
+ d*x]] + Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2]))/(a^(5/3)*b^(7/3)) - (18*Sin[
c + d*x])/b^2 - (27*Hypergeometric2F1[2/3, 2, 5/3, -((b*Sin[c + d*x]^3)/a)]*Sin[c + d*x]^2)/(a*b) + 18/(b*(a +
b*Sin[c + d*x]^3)) + (6*(1 - a^2/b^2)*Sin[c + d*x])/(a*(a + b*Sin[c + d*x]^3)))/(18*d)

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Maple [A]
time = 1.30, size = 310, normalized size = 1.08

method	result	size
derivativedivides	\(\frac {-\frac {\sin \left (d x +c \right )}{b^{2}}+\frac {\frac {-\left (\sin ^{2}\left (d x +c \right )\right ) b -\frac {\left (a^{2}-b^{2}\right ) \sin \left (d x +c \right )}{3 a}+b}{a +b \left (\sin ^{3}\left (d x +c \right )\right )}+\frac {\frac {2 \left (2 a^{2}+b^{2}\right ) \left (\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3}+2 a b \left (-\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a}}{b^{2}}}{d}\)	\(310\)
default	\(\frac {-\frac {\sin \left (d x +c \right )}{b^{2}}+\frac {\frac {-\left (\sin ^{2}\left (d x +c \right )\right ) b -\frac {\left (a^{2}-b^{2}\right ) \sin \left (d x +c \right )}{3 a}+b}{a +b \left (\sin ^{3}\left (d x +c \right )\right )}+\frac {\frac {2 \left (2 a^{2}+b^{2}\right ) \left (\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3}+2 a b \left (-\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a}}{b^{2}}}{d}\)	\(310\)
risch	\(\frac {i {\mathrm e}^{i \left (d x +c \right )}}{2 b^{2} d}-\frac {i {\mathrm e}^{-i \left (d x +c \right )}}{2 b^{2} d}-\frac {2 i \left (3 b \,{\mathrm e}^{5 i \left (d x +c \right )} a +6 a b \,{\mathrm e}^{3 i \left (d x +c \right )}+2 i a^{2} {\mathrm e}^{4 i \left (d x +c \right )}-2 i b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+3 b \,{\mathrm e}^{i \left (d x +c \right )} a -2 i a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+2 i b^{2} {\mathrm e}^{2 i \left (d x +c \right )}\right )}{3 b^{2} d a \left (b \,{\mathrm e}^{6 i \left (d x +c \right )}-3 b \,{\mathrm e}^{4 i \left (d x +c \right )}+3 b \,{\mathrm e}^{2 i \left (d x +c \right )}-8 i a \,{\mathrm e}^{3 i \left (d x +c \right )}-b \right )}+\left (\munderset {\textit {\_R} =\RootOf \left (729 a^{5} b^{7} d^{3} \textit {\_Z}^{3}+\left (648 a^{5} b^{3} d +324 a^{3} b^{5} d \right ) \textit {\_Z} -64 a^{6}+120 a^{4} b^{2}-48 a^{2} b^{4}-8 b^{6}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\frac {243 i a^{5} b^{5} d^{2} \textit {\_R}^{2}}{16 a^{6}+78 a^{4} b^{2}+12 a^{2} b^{4}+2 b^{6}}+\left (\frac {72 i a^{6} b^{2} d}{16 a^{6}+78 a^{4} b^{2}+12 a^{2} b^{4}+2 b^{6}}+\frac {72 i a^{4} b^{4} d}{16 a^{6}+78 a^{4} b^{2}+12 a^{2} b^{4}+2 b^{6}}+\frac {18 i a^{2} b^{6} d}{16 a^{6}+78 a^{4} b^{2}+12 a^{2} b^{4}+2 b^{6}}\right ) \textit {\_R} +\frac {144 i a^{5} b}{16 a^{6}+78 a^{4} b^{2}+12 a^{2} b^{4}+2 b^{6}}+\frac {72 i a^{3} b^{3}}{16 a^{6}+78 a^{4} b^{2}+12 a^{2} b^{4}+2 b^{6}}\right ) {\mathrm e}^{i \left (d x +c \right )}-\frac {16 a^{6}}{16 a^{6}+78 a^{4} b^{2}+12 a^{2} b^{4}+2 b^{6}}-\frac {78 a^{4} b^{2}}{16 a^{6}+78 a^{4} b^{2}+12 a^{2} b^{4}+2 b^{6}}-\frac {12 a^{2} b^{4}}{16 a^{6}+78 a^{4} b^{2}+12 a^{2} b^{4}+2 b^{6}}-\frac {2 b^{6}}{16 a^{6}+78 a^{4} b^{2}+12 a^{2} b^{4}+2 b^{6}}\right )\right )\)	\(674\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^7/(a+b*sin(d*x+c)^3)^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/b^2*sin(d*x+c)+1/b^2*((-sin(d*x+c)^2*b-1/3*(a^2-b^2)/a*sin(d*x+c)+b)/(a+b*sin(d*x+c)^3)+2/3/a*((2*a^2+
b^2)*(1/3/b/(1/b*a)^(2/3)*ln(sin(d*x+c)+(1/b*a)^(1/3))-1/6/b/(1/b*a)^(2/3)*ln(sin(d*x+c)^2-(1/b*a)^(1/3)*sin(d
*x+c)+(1/b*a)^(2/3))+1/3/b/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*sin(d*x+c)-1)))+3*a*b*(-1
/3/b/(1/b*a)^(1/3)*ln(sin(d*x+c)+(1/b*a)^(1/3))+1/6/b/(1/b*a)^(1/3)*ln(sin(d*x+c)^2-(1/b*a)^(1/3)*sin(d*x+c)+(
1/b*a)^(2/3))+1/3*3^(1/2)/b/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*sin(d*x+c)-1))))))

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Maxima [A]
time = 0.57, size = 263, normalized size = 0.91 \begin {gather*} -\frac {\frac {3 \, {\left (3 \, a b \sin \left (d x + c\right )^{2} - 3 \, a b + {\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )\right )}}{a b^{3} \sin \left (d x + c\right )^{3} + a^{2} b^{2}} + \frac {9 \, \sin \left (d x + c\right )}{b^{2}} - \frac {2 \, \sqrt {3} {\left (3 \, a b \left (\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, a^{2} + b^{2}\right )} \arctan \left (-\frac {\sqrt {3} {\left (\left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, \sin \left (d x + c\right )\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (3 \, a b \left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a^{2} - b^{2}\right )} \log \left (\sin \left (d x + c\right )^{2} - \left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x + c\right ) + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{a b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {2 \, {\left (3 \, a b \left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a^{2} - b^{2}\right )} \log \left (\left (\frac {a}{b}\right )^{\frac {1}{3}} + \sin \left (d x + c\right )\right )}{a b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}}}{9 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7/(a+b*sin(d*x+c)^3)^2,x, algorithm="maxima")

[Out]

-1/9*(3*(3*a*b*sin(d*x + c)^2 - 3*a*b + (a^2 - b^2)*sin(d*x + c))/(a*b^3*sin(d*x + c)^3 + a^2*b^2) + 9*sin(d*x
+ c)/b^2 - 2*sqrt(3)*(3*a*b*(a/b)^(1/3) + 2*a^2 + b^2)*arctan(-1/3*sqrt(3)*((a/b)^(1/3) - 2*sin(d*x + c))/(a/
b)^(1/3))/(a*b^3*(a/b)^(2/3)) - (3*a*b*(a/b)^(1/3) - 2*a^2 - b^2)*log(sin(d*x + c)^2 - (a/b)^(1/3)*sin(d*x + c
) + (a/b)^(2/3))/(a*b^3*(a/b)^(2/3)) + 2*(3*a*b*(a/b)^(1/3) - 2*a^2 - b^2)*log((a/b)^(1/3) + sin(d*x + c))/(a*
b^3*(a/b)^(2/3)))/d

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Fricas [C] Result contains complex when optimal does not.
time = 95.60, size = 6415, normalized size = 22.27 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7/(a+b*sin(d*x+c)^3)^2,x, algorithm="fricas")

[Out]

1/36*(12*(7*a^2*b - b^3)*cos(d*x + c)^4 + 84*a^2*b - 12*b^3 - 6*sqrt(1/3)*(a*b^4*d*cos(d*x + c)^6 - 3*a*b^4*d*
cos(d*x + c)^4 + 3*a*b^4*d*cos(d*x + c)^2 - (a^3*b^2 + a*b^4)*d + 2*(a^2*b^3*d*cos(d*x + c)^2 - a^2*b^3*d)*sin
(d*x + c))*sqrt(((4^(1/3)*(I*sqrt(3) + 1)*((8*a^6 + 39*a^4*b^2 + 6*a^2*b^4 + b^6)/(a^5*b^7*d^3) + (8*a^6 - 15*
a^4*b^2 + 6*a^2*b^4 + b^6)/(a^5*b^7*d^3))^(1/3) - 3*4^(2/3)*(2*a^2 + b^2)*(-I*sqrt(3) + 1)/(a^2*b^4*d^2*((8*a^
6 + 39*a^4*b^2 + 6*a^2*b^4 + b^6)/(a^5*b^7*d^3) + (8*a^6 - 15*a^4*b^2 + 6*a^2*b^4 + b^6)/(a^5*b^7*d^3))^(1/3))
)^2*a^2*b^4*d^2 + 384*a^2 + 192*b^2)/(a^2*b^4*d^2))*arctan(1/64*sqrt(1/3)*((8*a^11*b^7 + 39*a^9*b^9 + 6*a^7*b^
11 + a^5*b^13)*(4^(1/3)*(I*sqrt(3) + 1)*((8*a^6 + 39*a^4*b^2 + 6*a^2*b^4 + b^6)/(a^5*b^7*d^3) + (8*a^6 - 15*a^
4*b^2 + 6*a^2*b^4 + b^6)/(a^5*b^7*d^3))^(1/3) - 3*4^(2/3)*(2*a^2 + b^2)*(-I*sqrt(3) + 1)/(a^2*b^4*d^2*((8*a^6
+ 39*a^4*b^2 + 6*a^2*b^4 + b^6)/(a^5*b^7*d^3) + (8*a^6 - 15*a^4*b^2 + 6*a^2*b^4 + b^6)/(a^5*b^7*d^3))^(1/3)))^
2*d^3 + 8*(16*a^11*b^5 + 86*a^9*b^7 + 51*a^7*b^9 + 8*a^5*b^11 + a^3*b^13)*(4^(1/3)*(I*sqrt(3) + 1)*((8*a^6 + 3
9*a^4*b^2 + 6*a^2*b^4 + b^6)/(a^5*b^7*d^3) + (8*a^6 - 15*a^4*b^2 + 6*a^2*b^4 + b^6)/(a^5*b^7*d^3))^(1/3) - 3*4
^(2/3)*(2*a^2 + b^2)*(-I*sqrt(3) + 1)/(a^2*b^4*d^2*((8*a^6 + 39*a^4*b^2 + 6*a^2*b^4 + b^6)/(a^5*b^7*d^3) + (8*
a^6 - 15*a^4*b^2 + 6*a^2*b^4 + b^6)/(a^5*b^7*d^3))^(1/3)))*d^2*sin(d*x + c) - 288*(8*a^10*b^4 + 39*a^8*b^6 + 6
*a^6*b^8 + a^4*b^10)*d*sin(d*x + c) + 48*(16*a^11*b^3 + 86*a^9*b^5 + 51*a^7*b^7 + 8*a^5*b^9 + a^3*b^11)*d + 2*
sqrt(4096*a^12 + 28032*a^10*b^2 + 43920*a^8*b^4 + 14176*a^6*b^6 + 2784*a^4*b^8 + 288*a^2*b^10 + 16*b^12 - (3*(
8*a^11*b^5 + 39*a^9*b^7 + 6*a^7*b^9 + a^5*b^11)*d^2*sin(d*x + c) - (16*a^12*b^4 + 86*a^10*b^6 + 51*a^8*b^8 + 8
*a^6*b^10 + a^4*b^12)*d^2)*(4^(1/3)*(I*sqrt(3) + 1)*((8*a^6 + 39*a^4*b^2 + 6*a^2*b^4 + b^6)/(a^5*b^7*d^3) + (8
*a^6 - 15*a^4*b^2 + 6*a^2*b^4 + b^6)/(a^5*b^7*d^3))^(1/3) - 3*4^(2/3)*(2*a^2 + b^2)*(-I*sqrt(3) + 1)/(a^2*b^4*
d^2*((8*a^6 + 39*a^4*b^2 + 6*a^2*b^4 + b^6)/(a^5*b^7*d^3) + (8*a^6 - 15*a^4*b^2 + 6*a^2*b^4 + b^6)/(a^5*b^7*d^
3))^(1/3)))^2 - 16*(64*a^12 + 624*a^10*b^2 + 1617*a^8*b^4 + 484*a^6*b^6 + 114*a^4*b^8 + 12*a^2*b^10 + b^12)*co
s(d*x + c)^2 + 4*((32*a^12*b^2 + 188*a^10*b^4 + 188*a^8*b^6 + 67*a^6*b^8 + 10*a^4*b^10 + a^2*b^12)*d*sin(d*x +
c) + 9*(8*a^11*b^3 + 39*a^9*b^5 + 6*a^7*b^7 + a^5*b^9)*d)*(4^(1/3)*(I*sqrt(3) + 1)*((8*a^6 + 39*a^4*b^2 + 6*a
^2*b^4 + b^6)/(a^5*b^7*d^3) + (8*a^6 - 15*a^4*b^2 + 6*a^2*b^4 + b^6)/(a^5*b^7*d^3))^(1/3) - 3*4^(2/3)*(2*a^2 +
b^2)*(-I*sqrt(3) + 1)/(a^2*b^4*d^2*((8*a^6 + 39*a^4*b^2 + 6*a^2*b^4 + b^6)/(a^5*b^7*d^3) + (8*a^6 - 15*a^4*b^
2 + 6*a^2*b^4 + b^6)/(a^5*b^7*d^3))^(1/3))) - 288*(16*a^11*b + 86*a^9*b^3 + 51*a^7*b^5 + 8*a^5*b^7 + a^3*b^9)*
sin(d*x + c))*(36*a^4*b^4*d - (2*a^5*b^5 + a^3*b^7)*(4^(1/3)*(I*sqrt(3) + 1)*((8*a^6 + 39*a^4*b^2 + 6*a^2*b^4
+ b^6)/(a^5*b^7*d^3) + (8*a^6 - 15*a^4*b^2 + 6*a^2*b^4 + b^6)/(a^5*b^7*d^3))^(1/3) - 3*4^(2/3)*(2*a^2 + b^2)*(
-I*sqrt(3) + 1)/(a^2*b^4*d^2*((8*a^6 + 39*a^4*b^2 + 6*a^2*b^4 + b^6)/(a^5*b^7*d^3) + (8*a^6 - 15*a^4*b^2 + 6*a
^2*b^4 + b^6)/(a^5*b^7*d^3))^(1/3)))*d^2))*sqrt(((4^(1/3)*(I*sqrt(3) + 1)*((8*a^6 + 39*a^4*b^2 + 6*a^2*b^4 + b
^6)/(a^5*b^7*d^3) + (8*a^6 - 15*a^4*b^2 + 6*a^2*b^4 + b^6)/(a^5*b^7*d^3))^(1/3) - 3*4^(2/3)*(2*a^2 + b^2)*(-I*
sqrt(3) + 1)/(a^2*b^4*d^2*((8*a^6 + 39*a^4*b^2 + 6*a^2*b^4 + b^6)/(a^5*b^7*d^3) + (8*a^6 - 15*a^4*b^2 + 6*a^2*
b^4 + b^6)/(a^5*b^7*d^3))^(1/3)))^2*a^2*b^4*d^2 + 384*a^2 + 192*b^2)/(a^2*b^4*d^2))/(64*a^12 + 624*a^10*b^2 +
1617*a^8*b^4 + 484*a^6*b^6 + 114*a^4*b^8 + 12*a^2*b^10 + b^12)) + 6*sqrt(1/3)*(a*b^4*d*cos(d*x + c)^6 - 3*a*b^
4*d*cos(d*x + c)^4 + 3*a*b^4*d*cos(d*x + c)^2 - (a^3*b^2 + a*b^4)*d + 2*(a^2*b^3*d*cos(d*x + c)^2 - a^2*b^3*d)
*sin(d*x + c))*sqrt(((4^(1/3)*(I*sqrt(3) + 1)*((8*a^6 + 39*a^4*b^2 + 6*a^2*b^4 + b^6)/(a^5*b^7*d^3) + (8*a^6 -
15*a^4*b^2 + 6*a^2*b^4 + b^6)/(a^5*b^7*d^3))^(1/3) - 3*4^(2/3)*(2*a^2 + b^2)*(-I*sqrt(3) + 1)/(a^2*b^4*d^2*((
8*a^6 + 39*a^4*b^2 + 6*a^2*b^4 + b^6)/(a^5*b^7*d^3) + (8*a^6 - 15*a^4*b^2 + 6*a^2*b^4 + b^6)/(a^5*b^7*d^3))^(1
/3)))^2*a^2*b^4*d^2 + 384*a^2 + 192*b^2)/(a^2*b^4*d^2))*arctan(-1/64*sqrt(1/3)*((8*a^11*b^7 + 39*a^9*b^9 + 6*a
^7*b^11 + a^5*b^13)*(4^(1/3)*(I*sqrt(3) + 1)*((8*a^6 + 39*a^4*b^2 + 6*a^2*b^4 + b^6)/(a^5*b^7*d^3) + (8*a^6 -
15*a^4*b^2 + 6*a^2*b^4 + b^6)/(a^5*b^7*d^3))^(1/3) - 3*4^(2/3)*(2*a^2 + b^2)*(-I*sqrt(3) + 1)/(a^2*b^4*d^2*((8
*a^6 + 39*a^4*b^2 + 6*a^2*b^4 + b^6)/(a^5*b^7*d^3) + (8*a^6 - 15*a^4*b^2 + 6*a^2*b^4 + b^6)/(a^5*b^7*d^3))^(1/
3)))^2*d^3 + 8*(16*a^11*b^5 + 86*a^9*b^7 + 51*a^7*b^9 + 8*a^5*b^11 + a^3*b^13)*(4^(1/3)*(I*sqrt(3) + 1)*((8*a^
6 + 39*a^4*b^2 + 6*a^2*b^4 + b^6)/(a^5*b^7*d^3) + (8*a^6 - 15*a^4*b^2 + 6*a^2*b^4 + b^6)/(a^5*b^7*d^3))^(1/3)
- 3*4^(2/3)*(2*a^2 + b^2)*(-I*sqrt(3) + 1)/(a^2*b^4*d^2*((8*a^6 + 39*a^4*b^2 + 6*a^2*b^4 + b^6)/(a^5*b^7*d^3)
+ (8*a^6 - 15*a^4*b^2 + 6*a^2*b^4 + b^6)/(a^5*b^7*d^3))^(1/3)))*d^2*sin(d*x + c) - 288*(8*a^10*b^4 + 39*a^8*b^
6 + 6*a^6*b^8 + a^4*b^10)*d*sin(d*x + c) + 48*(...

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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(cos(d*x+c)**7/(a+b*sin(d*x+c)**3)**2,x)

[Out]

Timed out

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Giac [A]
time = 0.48, size = 277, normalized size = 0.96 \begin {gather*} -\frac {\frac {9 \, \sin \left (d x + c\right )}{b^{2}} + \frac {2 \, {\left (3 \, a b \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, a^{2} + b^{2}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | -\left (-\frac {a}{b}\right )^{\frac {1}{3}} + \sin \left (d x + c\right ) \right |}\right )}{a^{2} b^{2}} + \frac {2 \, \sqrt {3} {\left (3 \, \left (-a b^{2}\right )^{\frac {2}{3}} a - \left (-a b^{2}\right )^{\frac {1}{3}} {\left (2 \, a^{2} + b^{2}\right )}\right )} \arctan \left (\frac {\sqrt {3} {\left (\left (-\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, \sin \left (d x + c\right )\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a^{2} b^{3}} + \frac {3 \, {\left (3 \, a b \sin \left (d x + c\right )^{2} + a^{2} \sin \left (d x + c\right ) - b^{2} \sin \left (d x + c\right ) - 3 \, a b\right )}}{{\left (b \sin \left (d x + c\right )^{3} + a\right )} a b^{2}} - \frac {{\left (3 \, \left (-a b^{2}\right )^{\frac {2}{3}} a + \left (-a b^{2}\right )^{\frac {1}{3}} {\left (2 \, a^{2} + b^{2}\right )}\right )} \log \left (\sin \left (d x + c\right )^{2} + \left (-\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x + c\right ) + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{a^{2} b^{3}}}{9 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7/(a+b*sin(d*x+c)^3)^2,x, algorithm="giac")

[Out]

-1/9*(9*sin(d*x + c)/b^2 + 2*(3*a*b*(-a/b)^(1/3) + 2*a^2 + b^2)*(-a/b)^(1/3)*log(abs(-(-a/b)^(1/3) + sin(d*x +
c)))/(a^2*b^2) + 2*sqrt(3)*(3*(-a*b^2)^(2/3)*a - (-a*b^2)^(1/3)*(2*a^2 + b^2))*arctan(1/3*sqrt(3)*((-a/b)^(1/
3) + 2*sin(d*x + c))/(-a/b)^(1/3))/(a^2*b^3) + 3*(3*a*b*sin(d*x + c)^2 + a^2*sin(d*x + c) - b^2*sin(d*x + c) -
3*a*b)/((b*sin(d*x + c)^3 + a)*a*b^2) - (3*(-a*b^2)^(2/3)*a + (-a*b^2)^(1/3)*(2*a^2 + b^2))*log(sin(d*x + c)^
2 + (-a/b)^(1/3)*sin(d*x + c) + (-a/b)^(2/3))/(a^2*b^3))/d

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Mupad [B]
time = 0.42, size = 384, normalized size = 1.33 \begin {gather*} \frac {\sum _{k=1}^3\ln \left (\frac {8\,a^2+4\,b^2+{\mathrm {root}\left (729\,a^5\,b^7\,d^3+648\,a^5\,b^3\,d+324\,a^3\,b^5\,d+120\,a^4\,b^2-48\,a^2\,b^4-8\,b^6-64\,a^6,d,k\right )}^2\,a^2\,b^4\,27+12\,a\,b\,\sin \left (c+d\,x\right )+\mathrm {root}\left (729\,a^5\,b^7\,d^3+648\,a^5\,b^3\,d+324\,a^3\,b^5\,d+120\,a^4\,b^2-48\,a^2\,b^4-8\,b^6-64\,a^6,d,k\right )\,b^4\,\sin \left (c+d\,x\right )\,6+\mathrm {root}\left (729\,a^5\,b^7\,d^3+648\,a^5\,b^3\,d+324\,a^3\,b^5\,d+120\,a^4\,b^2-48\,a^2\,b^4-8\,b^6-64\,a^6,d,k\right )\,a^2\,b^2\,\sin \left (c+d\,x\right )\,12}{a\,b^2\,3}\right )\,\mathrm {root}\left (729\,a^5\,b^7\,d^3+648\,a^5\,b^3\,d+324\,a^3\,b^5\,d+120\,a^4\,b^2-48\,a^2\,b^4-8\,b^6-64\,a^6,d,k\right )}{d}-\frac {\sin \left (c+d\,x\right )}{b^2\,d}-\frac {b\,{\sin \left (c+d\,x\right )}^2-b+\frac {\sin \left (c+d\,x\right )\,\left (a^2-b^2\right )}{3\,a}}{d\,\left (b^3\,{\sin \left (c+d\,x\right )}^3+a\,b^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^7/(a + b*sin(c + d*x)^3)^2,x)

[Out]

symsum(log((8*a^2 + 4*b^2 + 27*root(729*a^5*b^7*d^3 + 648*a^5*b^3*d + 324*a^3*b^5*d + 120*a^4*b^2 - 48*a^2*b^4
- 8*b^6 - 64*a^6, d, k)^2*a^2*b^4 + 12*a*b*sin(c + d*x) + 6*root(729*a^5*b^7*d^3 + 648*a^5*b^3*d + 324*a^3*b^
5*d + 120*a^4*b^2 - 48*a^2*b^4 - 8*b^6 - 64*a^6, d, k)*b^4*sin(c + d*x) + 12*root(729*a^5*b^7*d^3 + 648*a^5*b^
3*d + 324*a^3*b^5*d + 120*a^4*b^2 - 48*a^2*b^4 - 8*b^6 - 64*a^6, d, k)*a^2*b^2*sin(c + d*x))/(3*a*b^2))*root(7
29*a^5*b^7*d^3 + 648*a^5*b^3*d + 324*a^3*b^5*d + 120*a^4*b^2 - 48*a^2*b^4 - 8*b^6 - 64*a^6, d, k), k, 1, 3)/d
- sin(c + d*x)/(b^2*d) - (b*sin(c + d*x)^2 - b + (sin(c + d*x)*(a^2 - b^2))/(3*a))/(d*(a*b^2 + b^3*sin(c + d*x
)^3))

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