3.412 \(\int \frac{\cos ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx\)

Optimal. Leaf size=186 \[ \frac{\left (\sqrt{a}-\sqrt{b}\right )^{7/2} \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^2 d}+\frac{\left (\sqrt{a}+\sqrt{b}\right )^{7/2} \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^2 d}-\frac{x (a+3 b)}{b^2}-\frac{\sin (c+d x) \cos ^3(c+d x)}{4 b d}-\frac{11 \sin (c+d x) \cos (c+d x)}{8 b d}-\frac{11 x}{8 b} \]

[Out]

(-11*x)/(8*b) - ((a + 3*b)*x)/b^2 + ((Sqrt[a] - Sqrt[b])^(7/2)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a
^(1/4)])/(2*a^(3/4)*b^2*d) + ((Sqrt[a] + Sqrt[b])^(7/2)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)]
)/(2*a^(3/4)*b^2*d) - (11*Cos[c + d*x]*Sin[c + d*x])/(8*b*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(4*b*d)

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Rubi [A]  time = 0.326102, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3224, 1170, 199, 203, 1166, 205} \[ \frac{\left (\sqrt{a}-\sqrt{b}\right )^{7/2} \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^2 d}+\frac{\left (\sqrt{a}+\sqrt{b}\right )^{7/2} \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^2 d}-\frac{x (a+3 b)}{b^2}-\frac{\sin (c+d x) \cos ^3(c+d x)}{4 b d}-\frac{11 \sin (c+d x) \cos (c+d x)}{8 b d}-\frac{11 x}{8 b} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^8/(a - b*Sin[c + d*x]^4),x]

[Out]

(-11*x)/(8*b) - ((a + 3*b)*x)/b^2 + ((Sqrt[a] - Sqrt[b])^(7/2)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a
^(1/4)])/(2*a^(3/4)*b^2*d) + ((Sqrt[a] + Sqrt[b])^(7/2)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)]
)/(2*a^(3/4)*b^2*d) - (11*Cos[c + d*x]*Sin[c + d*x])/(8*b*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(4*b*d)

Rule 3224

Int[cos[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p/(1 + ff^2*x^2)^(m/2 + 2
*p + 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]

Rule 1170

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(d + e*x
^2)^q/(a + b*x^2 + c*x^4), 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] && IntegerQ[q]

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 203

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

Rule 1166

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 205

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

Rubi steps

\begin{align*} \int \frac{\cos ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right )^3 \left (a+2 a x^2+(a-b) x^4\right )} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{1}{b \left (1+x^2\right )^3}-\frac{2}{b \left (1+x^2\right )^2}+\frac{-a-3 b}{b^2 \left (1+x^2\right )}+\frac{a^2+6 a b+b^2+(a-b) (a+3 b) x^2}{b^2 \left (a+2 a x^2+(a-b) x^4\right )}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{a^2+6 a b+b^2+(a-b) (a+3 b) x^2}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{b^2 d}-\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right )^3} \, dx,x,\tan (c+d x)\right )}{b d}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{b d}-\frac{(a+3 b) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{b^2 d}\\ &=-\frac{(a+3 b) x}{b^2}-\frac{\cos (c+d x) \sin (c+d x)}{b d}-\frac{\cos ^3(c+d x) \sin (c+d x)}{4 b d}+\frac{\left (\left (\sqrt{a}-\sqrt{b}\right )^4 \left (\sqrt{a}+\sqrt{b}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+\sqrt{a} \sqrt{b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{2 \sqrt{a} b^2 d}+\frac{\left (\left (\sqrt{a}-\sqrt{b}\right ) \left (\sqrt{a}+\sqrt{b}\right )^4\right ) \operatorname{Subst}\left (\int \frac{1}{a-\sqrt{a} \sqrt{b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{2 \sqrt{a} b^2 d}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{4 b d}-\frac{\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{b d}\\ &=-\frac{x}{b}-\frac{(a+3 b) x}{b^2}+\frac{\left (\sqrt{a}-\sqrt{b}\right )^{7/2} \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^2 d}+\frac{\left (\sqrt{a}+\sqrt{b}\right )^{7/2} \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^2 d}-\frac{11 \cos (c+d x) \sin (c+d x)}{8 b d}-\frac{\cos ^3(c+d x) \sin (c+d x)}{4 b d}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{8 b d}\\ &=-\frac{11 x}{8 b}-\frac{(a+3 b) x}{b^2}+\frac{\left (\sqrt{a}-\sqrt{b}\right )^{7/2} \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^2 d}+\frac{\left (\sqrt{a}+\sqrt{b}\right )^{7/2} \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^2 d}-\frac{11 \cos (c+d x) \sin (c+d x)}{8 b d}-\frac{\cos ^3(c+d x) \sin (c+d x)}{4 b d}\\ \end{align*}

Mathematica [A]  time = 0.681536, size = 200, normalized size = 1.08 \[ -\frac{4 (8 a+35 b) (c+d x)-\frac{16 \left (\sqrt{a}+\sqrt{b}\right )^4 \tan ^{-1}\left (\frac{\left (\sqrt{a}+\sqrt{b}\right ) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right )}{\sqrt{a} \sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{16 \left (\sqrt{a}-\sqrt{b}\right )^4 \tanh ^{-1}\left (\frac{\left (\sqrt{a}-\sqrt{b}\right ) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right )}{\sqrt{a} \sqrt{\sqrt{a} \sqrt{b}-a}}+24 b \sin (2 (c+d x))+b \sin (4 (c+d x))}{32 b^2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^8/(a - b*Sin[c + d*x]^4),x]

[Out]

-(4*(8*a + 35*b)*(c + d*x) - (16*(Sqrt[a] + Sqrt[b])^4*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt
[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt[a + Sqrt[a]*Sqrt[b]]) + (16*(Sqrt[a] - Sqrt[b])^4*ArcTanh[((Sqrt[a] - Sqrt[b])*Ta
n[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt[-a + Sqrt[a]*Sqrt[b]]) + 24*b*Sin[2*(c + d*x)] + b*Sin[
4*(c + d*x)])/(32*b^2*d)

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Maple [B]  time = 0.127, size = 750, normalized size = 4. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^8/(a-b*sin(d*x+c)^4),x)

[Out]

1/2/d/b^2/(((a*b)^(1/2)+a)*(a-b))^(1/2)*arctan((a-b)*tan(d*x+c)/(((a*b)^(1/2)+a)*(a-b))^(1/2))*a^2+1/d/b*a/(((
a*b)^(1/2)+a)*(a-b))^(1/2)*arctan((a-b)*tan(d*x+c)/(((a*b)^(1/2)+a)*(a-b))^(1/2))-3/2/d/b/(a*b)^(1/2)/(((a*b)^
(1/2)+a)*(a-b))^(1/2)*arctan((a-b)*tan(d*x+c)/(((a*b)^(1/2)+a)*(a-b))^(1/2))*a^2+1/d*a/(a*b)^(1/2)/(((a*b)^(1/
2)+a)*(a-b))^(1/2)*arctan((a-b)*tan(d*x+c)/(((a*b)^(1/2)+a)*(a-b))^(1/2))+1/2/d/b^2/(((a*b)^(1/2)-a)*(a-b))^(1
/2)*arctanh((-a+b)*tan(d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1/2))*a^2+1/d/b*a/(((a*b)^(1/2)-a)*(a-b))^(1/2)*arctanh
((-a+b)*tan(d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1/2))+3/2/d/b/(a*b)^(1/2)/(((a*b)^(1/2)-a)*(a-b))^(1/2)*arctanh((-
a+b)*tan(d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1/2))*a^2-1/d*a/(a*b)^(1/2)/(((a*b)^(1/2)-a)*(a-b))^(1/2)*arctanh((-a
+b)*tan(d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1/2))-3/2/d/(((a*b)^(1/2)+a)*(a-b))^(1/2)*arctan((a-b)*tan(d*x+c)/(((a
*b)^(1/2)+a)*(a-b))^(1/2))+1/2/d*b/(a*b)^(1/2)/(((a*b)^(1/2)+a)*(a-b))^(1/2)*arctan((a-b)*tan(d*x+c)/(((a*b)^(
1/2)+a)*(a-b))^(1/2))-3/2/d/(((a*b)^(1/2)-a)*(a-b))^(1/2)*arctanh((-a+b)*tan(d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1
/2))-1/2/d*b/(a*b)^(1/2)/(((a*b)^(1/2)-a)*(a-b))^(1/2)*arctanh((-a+b)*tan(d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1/2)
)-11/8/d/b/(tan(d*x+c)^2+1)^2*tan(d*x+c)^3-13/8/d/b/(tan(d*x+c)^2+1)^2*tan(d*x+c)-35/8/d/b*arctan(tan(d*x+c))-
1/d/b^2*arctan(tan(d*x+c))*a

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8/(a-b*sin(d*x+c)^4),x, algorithm="maxima")

[Out]

-1/32*(32*b^2*d*integrate(-16*(4*(a*b^2 + b^3)*cos(6*d*x + 6*c)^2 + 2*(8*a^3 + 29*a^2*b - 20*a*b^2 + 3*b^3)*co
s(4*d*x + 4*c)^2 + 4*(a*b^2 + b^3)*cos(2*d*x + 2*c)^2 + 4*(a*b^2 + b^3)*sin(6*d*x + 6*c)^2 + 2*(8*a^3 + 29*a^2
*b - 20*a*b^2 + 3*b^3)*sin(4*d*x + 4*c)^2 + 2*(10*a^2*b + 13*a*b^2 - 5*b^3)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c)
+ 4*(a*b^2 + b^3)*sin(2*d*x + 2*c)^2 - ((a*b^2 + b^3)*cos(6*d*x + 6*c) + (a^2*b + 4*a*b^2 - b^3)*cos(4*d*x + 4
*c) + (a*b^2 + b^3)*cos(2*d*x + 2*c))*cos(8*d*x + 8*c) - (a*b^2 + b^3 - 2*(10*a^2*b + 13*a*b^2 - 5*b^3)*cos(4*
d*x + 4*c) - 8*(a*b^2 + b^3)*cos(2*d*x + 2*c))*cos(6*d*x + 6*c) - (a^2*b + 4*a*b^2 - b^3 - 2*(10*a^2*b + 13*a*
b^2 - 5*b^3)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - (a*b^2 + b^3)*cos(2*d*x + 2*c) - ((a*b^2 + b^3)*sin(6*d*x +
6*c) + (a^2*b + 4*a*b^2 - b^3)*sin(4*d*x + 4*c) + (a*b^2 + b^3)*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 2*((10*a^
2*b + 13*a*b^2 - 5*b^3)*sin(4*d*x + 4*c) + 4*(a*b^2 + b^3)*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))/(b^4*cos(8*d*x
+ 8*c)^2 + 16*b^4*cos(6*d*x + 6*c)^2 + 16*b^4*cos(2*d*x + 2*c)^2 + b^4*sin(8*d*x + 8*c)^2 + 16*b^4*sin(6*d*x +
6*c)^2 + 16*b^4*sin(2*d*x + 2*c)^2 - 8*b^4*cos(2*d*x + 2*c) + b^4 + 4*(64*a^2*b^2 - 48*a*b^3 + 9*b^4)*cos(4*d
*x + 4*c)^2 + 4*(64*a^2*b^2 - 48*a*b^3 + 9*b^4)*sin(4*d*x + 4*c)^2 + 16*(8*a*b^3 - 3*b^4)*sin(4*d*x + 4*c)*sin
(2*d*x + 2*c) - 2*(4*b^4*cos(6*d*x + 6*c) + 4*b^4*cos(2*d*x + 2*c) - b^4 + 2*(8*a*b^3 - 3*b^4)*cos(4*d*x + 4*c
))*cos(8*d*x + 8*c) + 8*(4*b^4*cos(2*d*x + 2*c) - b^4 + 2*(8*a*b^3 - 3*b^4)*cos(4*d*x + 4*c))*cos(6*d*x + 6*c)
- 4*(8*a*b^3 - 3*b^4 - 4*(8*a*b^3 - 3*b^4)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - 4*(2*b^4*sin(6*d*x + 6*c) + 2
*b^4*sin(2*d*x + 2*c) + (8*a*b^3 - 3*b^4)*sin(4*d*x + 4*c))*sin(8*d*x + 8*c) + 16*(2*b^4*sin(2*d*x + 2*c) + (8
*a*b^3 - 3*b^4)*sin(4*d*x + 4*c))*sin(6*d*x + 6*c)), x) + 4*(8*a + 35*b)*d*x + b*sin(4*d*x + 4*c) + 24*b*sin(2
*d*x + 2*c))/(b^2*d)

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Fricas [B]  time = 8.86691, size = 5565, normalized size = 29.92 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8/(a-b*sin(d*x+c)^4),x, algorithm="fricas")

[Out]

1/8*(b^2*d*sqrt(-(a*b^4*d^2*sqrt((49*a^6 + 490*a^5*b + 1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 +
b^6)/(a^3*b^7*d^4)) + a^3 + 21*a^2*b + 35*a*b^2 + 7*b^3)/(a*b^4*d^2))*log(7/4*a^6 + 7/2*a^5*b - 63/4*a^4*b^2 +
9*a^3*b^3 + 25/4*a^2*b^4 - 9/2*a*b^5 - 1/4*b^6 - 1/4*(7*a^6 + 14*a^5*b - 63*a^4*b^2 + 36*a^3*b^3 + 25*a^2*b^4
- 18*a*b^5 - b^6)*cos(d*x + c)^2 + 1/2*((a^4*b^5 + 3*a^3*b^6)*d^3*sqrt((49*a^6 + 490*a^5*b + 1519*a^4*b^2 + 1
484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4))*cos(d*x + c)*sin(d*x + c) - (21*a^5*b^2 + 112*a^4*b
^3 + 98*a^3*b^4 + 24*a^2*b^5 + a*b^6)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(a*b^4*d^2*sqrt((49*a^6 + 490*a^5*b +
1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4)) + a^3 + 21*a^2*b + 35*a*b^2 + 7*b^
3)/(a*b^4*d^2)) - 1/4*(2*(a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*d^2*cos(d*x + c)^2 - (a^5*b^3 - 3*a^4*b^4
+ 3*a^3*b^5 - a^2*b^6)*d^2)*sqrt((49*a^6 + 490*a^5*b + 1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 +
b^6)/(a^3*b^7*d^4))) - b^2*d*sqrt(-(a*b^4*d^2*sqrt((49*a^6 + 490*a^5*b + 1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^
2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4)) + a^3 + 21*a^2*b + 35*a*b^2 + 7*b^3)/(a*b^4*d^2))*log(7/4*a^6 + 7/2*a^5
*b - 63/4*a^4*b^2 + 9*a^3*b^3 + 25/4*a^2*b^4 - 9/2*a*b^5 - 1/4*b^6 - 1/4*(7*a^6 + 14*a^5*b - 63*a^4*b^2 + 36*a
^3*b^3 + 25*a^2*b^4 - 18*a*b^5 - b^6)*cos(d*x + c)^2 - 1/2*((a^4*b^5 + 3*a^3*b^6)*d^3*sqrt((49*a^6 + 490*a^5*b
+ 1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4))*cos(d*x + c)*sin(d*x + c) - (21*
a^5*b^2 + 112*a^4*b^3 + 98*a^3*b^4 + 24*a^2*b^5 + a*b^6)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(a*b^4*d^2*sqrt((4
9*a^6 + 490*a^5*b + 1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4)) + a^3 + 21*a^2*
b + 35*a*b^2 + 7*b^3)/(a*b^4*d^2)) - 1/4*(2*(a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*d^2*cos(d*x + c)^2 - (
a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*d^2)*sqrt((49*a^6 + 490*a^5*b + 1519*a^4*b^2 + 1484*a^3*b^3 + 511*a
^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4))) + b^2*d*sqrt((a*b^4*d^2*sqrt((49*a^6 + 490*a^5*b + 1519*a^4*b^2 + 148
4*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4)) - a^3 - 21*a^2*b - 35*a*b^2 - 7*b^3)/(a*b^4*d^2))*log
(-7/4*a^6 - 7/2*a^5*b + 63/4*a^4*b^2 - 9*a^3*b^3 - 25/4*a^2*b^4 + 9/2*a*b^5 + 1/4*b^6 + 1/4*(7*a^6 + 14*a^5*b
- 63*a^4*b^2 + 36*a^3*b^3 + 25*a^2*b^4 - 18*a*b^5 - b^6)*cos(d*x + c)^2 + 1/2*((a^4*b^5 + 3*a^3*b^6)*d^3*sqrt(
(49*a^6 + 490*a^5*b + 1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4))*cos(d*x + c)*
sin(d*x + c) + (21*a^5*b^2 + 112*a^4*b^3 + 98*a^3*b^4 + 24*a^2*b^5 + a*b^6)*d*cos(d*x + c)*sin(d*x + c))*sqrt(
(a*b^4*d^2*sqrt((49*a^6 + 490*a^5*b + 1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4
)) - a^3 - 21*a^2*b - 35*a*b^2 - 7*b^3)/(a*b^4*d^2)) - 1/4*(2*(a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*d^2*
cos(d*x + c)^2 - (a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*d^2)*sqrt((49*a^6 + 490*a^5*b + 1519*a^4*b^2 + 14
84*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4))) - b^2*d*sqrt((a*b^4*d^2*sqrt((49*a^6 + 490*a^5*b +
1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4)) - a^3 - 21*a^2*b - 35*a*b^2 - 7*b^3
)/(a*b^4*d^2))*log(-7/4*a^6 - 7/2*a^5*b + 63/4*a^4*b^2 - 9*a^3*b^3 - 25/4*a^2*b^4 + 9/2*a*b^5 + 1/4*b^6 + 1/4*
(7*a^6 + 14*a^5*b - 63*a^4*b^2 + 36*a^3*b^3 + 25*a^2*b^4 - 18*a*b^5 - b^6)*cos(d*x + c)^2 - 1/2*((a^4*b^5 + 3*
a^3*b^6)*d^3*sqrt((49*a^6 + 490*a^5*b + 1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d
^4))*cos(d*x + c)*sin(d*x + c) + (21*a^5*b^2 + 112*a^4*b^3 + 98*a^3*b^4 + 24*a^2*b^5 + a*b^6)*d*cos(d*x + c)*s
in(d*x + c))*sqrt((a*b^4*d^2*sqrt((49*a^6 + 490*a^5*b + 1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 +
b^6)/(a^3*b^7*d^4)) - a^3 - 21*a^2*b - 35*a*b^2 - 7*b^3)/(a*b^4*d^2)) - 1/4*(2*(a^5*b^3 - 3*a^4*b^4 + 3*a^3*b
^5 - a^2*b^6)*d^2*cos(d*x + c)^2 - (a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*d^2)*sqrt((49*a^6 + 490*a^5*b +
1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4))) - (8*a + 35*b)*d*x - (2*b*cos(d*x
+ c)^3 + 11*b*cos(d*x + c))*sin(d*x + c))/(b^2*d)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**8/(a-b*sin(d*x+c)**4),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8/(a-b*sin(d*x+c)^4),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError