3.227 \(\int \frac{\sin ^3(c+d x)}{(a-b \sin ^4(c+d x))^3} \, dx\)
Optimal. Leaf size=288 \[ -\frac{\left (5 \sqrt{a}-2 \sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{64 a^{3/2} b^{3/4} d \left (\sqrt{a}-\sqrt{b}\right )^{5/2}}+\frac{\left (5 \sqrt{a}+2 \sqrt{b}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{64 a^{3/2} b^{3/4} d \left (\sqrt{a}+\sqrt{b}\right )^{5/2}}-\frac{\cos (c+d x) \left (-(5 a+b) \cos ^2(c+d x)+11 a+b\right )}{32 a d (a-b)^2 \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}-\frac{\cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 d (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2} \]
[Out]
-((5*Sqrt[a] - 2*Sqrt[b])*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(64*a^(3/2)*(Sqrt[a] - Sqrt[
b])^(5/2)*b^(3/4)*d) + ((5*Sqrt[a] + 2*Sqrt[b])*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(64*a
^(3/2)*(Sqrt[a] + Sqrt[b])^(5/2)*b^(3/4)*d) - (Cos[c + d*x]*(2 - Cos[c + d*x]^2))/(8*(a - b)*d*(a - b + 2*b*Co
s[c + d*x]^2 - b*Cos[c + d*x]^4)^2) - (Cos[c + d*x]*(11*a + b - (5*a + b)*Cos[c + d*x]^2))/(32*a*(a - b)^2*d*(
a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4))
________________________________________________________________________________________
Rubi [A] time = 0.497525, antiderivative size = 288, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used =
{3215, 1178, 1166, 205, 208} \[ -\frac{\left (5 \sqrt{a}-2 \sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{64 a^{3/2} b^{3/4} d \left (\sqrt{a}-\sqrt{b}\right )^{5/2}}+\frac{\left (5 \sqrt{a}+2 \sqrt{b}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{64 a^{3/2} b^{3/4} d \left (\sqrt{a}+\sqrt{b}\right )^{5/2}}-\frac{\cos (c+d x) \left (-(5 a+b) \cos ^2(c+d x)+11 a+b\right )}{32 a d (a-b)^2 \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}-\frac{\cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 d (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2} \]
Antiderivative was successfully verified.
[In]
Int[Sin[c + d*x]^3/(a - b*Sin[c + d*x]^4)^3,x]
[Out]
-((5*Sqrt[a] - 2*Sqrt[b])*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(64*a^(3/2)*(Sqrt[a] - Sqrt[
b])^(5/2)*b^(3/4)*d) + ((5*Sqrt[a] + 2*Sqrt[b])*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(64*a
^(3/2)*(Sqrt[a] + Sqrt[b])^(5/2)*b^(3/4)*d) - (Cos[c + d*x]*(2 - Cos[c + d*x]^2))/(8*(a - b)*d*(a - b + 2*b*Co
s[c + d*x]^2 - b*Cos[c + d*x]^4)^2) - (Cos[c + d*x]*(11*a + b - (5*a + b)*Cos[c + d*x]^2))/(32*a*(a - b)^2*d*(
a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4))
Rule 3215
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]
Rule 1178
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 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]
Rule 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rubi steps
\begin{align*} \int \frac{\sin ^3(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1-x^2}{\left (a-b+2 b x^2-b x^4\right )^3} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{\cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{-12 a b+10 a 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{\cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 (a-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 (11 a+b-(5 a+b) \cos ^2(c+d x)\right )}{32 a (a-b)^2 d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{4 a (13 a-b) b^2-4 a b^2 (5 a+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{\cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 (a-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 (11 a+b-(5 a+b) \cos ^2(c+d x)\right )}{32 a (a-b)^2 d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}+\frac{\left (5 \sqrt{a}-2 \sqrt{b}\right ) \operatorname{Subst}\left (\int \frac{1}{-\sqrt{a} \sqrt{b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{64 a^{3/2} \left (\sqrt{a}-\sqrt{b}\right )^2 d}+\frac{\left (5 \sqrt{a}+2 \sqrt{b}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a} \sqrt{b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{64 a^{3/2} \left (\sqrt{a}+\sqrt{b}\right )^2 d}\\ &=-\frac{\left (5 \sqrt{a}-2 \sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{64 a^{3/2} \left (\sqrt{a}-\sqrt{b}\right )^{5/2} b^{3/4} d}+\frac{\left (5 \sqrt{a}+2 \sqrt{b}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{64 a^{3/2} \left (\sqrt{a}+\sqrt{b}\right )^{5/2} b^{3/4} d}-\frac{\cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 (a-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 (11 a+b-(5 a+b) \cos ^2(c+d x)\right )}{32 a (a-b)^2 d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}\\ \end{align*}
Mathematica [C] time = 1.12044, size = 631, normalized size = 2.19 \[ \frac{\frac{i \text{RootSum}\left [-16 \text{$\#$1}^4 a+\text{$\#$1}^8 b-4 \text{$\#$1}^6 b+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b\& ,\frac{5 i \text{$\#$1}^6 a \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )-47 i \text{$\#$1}^4 a \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )+47 i \text{$\#$1}^2 a \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )-5 i a \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )-10 \text{$\#$1}^6 a \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )+94 \text{$\#$1}^4 a \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )-94 \text{$\#$1}^2 a \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )+i \text{$\#$1}^6 b \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )+5 i \text{$\#$1}^4 b \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )-5 i \text{$\#$1}^2 b \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )-i b \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )-2 \text{$\#$1}^6 b \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )-10 \text{$\#$1}^4 b \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )+10 \text{$\#$1}^2 b \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )+10 a \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )+2 b \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )}{-8 \text{$\#$1}^3 a+\text{$\#$1}^7 b-3 \text{$\#$1}^5 b+3 \text{$\#$1}^3 b-\text{$\#$1} b}\& \right ]}{a}+\frac{32 \cos (c+d x) ((5 a+b) \cos (2 (c+d x))-17 a-b)}{a (8 a+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))-3 b)}+\frac{512 (a-b) (\cos (3 (c+d x))-5 \cos (c+d x))}{(-8 a-4 b \cos (2 (c+d x))+b \cos (4 (c+d x))+3 b)^2}}{256 d (a-b)^2} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[Sin[c + d*x]^3/(a - b*Sin[c + d*x]^4)^3,x]
[Out]
((32*Cos[c + d*x]*(-17*a - b + (5*a + b)*Cos[2*(c + d*x)]))/(a*(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c
+ d*x)])) + (512*(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 + (I*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (10*a*ArcTan[Sin[c + d*x]
/(Cos[c + d*x] - #1)] + 2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - (5*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^
2] - I*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - 94*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + 10*b*ArcTan[
Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (47*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (5*I)*b*Log[1 - 2*C
os[c + d*x]*#1 + #1^2]*#1^2 + 94*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - 10*b*ArcTan[Sin[c + d*x]/(C
os[c + d*x] - #1)]*#1^4 - (47*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + (5*I)*b*Log[1 - 2*Cos[c + d*x]*#1
+ #1^2]*#1^4 - 10*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 - 2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1
)]*#1^6 + (5*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#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)/(256*(a - b)^2*d)
________________________________________________________________________________________
Maple [B] time = 0.181, size = 1153, 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(sin(d*x+c)^3/(a-b*sin(d*x+c)^4)^3,x)
[Out]
-5/64/d/b/(cos(d*x+c)^2-1-(a*b)^(1/2)/b)^2/(a^2-2*a*b+b^2)*cos(d*x+c)^3-1/64/d/a/(cos(d*x+c)^2-1-(a*b)^(1/2)/b
)^2/(a^2-2*a*b+b^2)*cos(d*x+c)^3+1/8/d/(a*b)^(1/2)/(cos(d*x+c)^2-1-(a*b)^(1/2)/b)^2/(a^2-2*a*b+b^2)*cos(d*x+c)
^3-1/32/d*b/(a*b)^(1/2)/a/(cos(d*x+c)^2-1-(a*b)^(1/2)/b)^2/(a^2-2*a*b+b^2)*cos(d*x+c)^3+7/64/d/b/(a*b)^(1/2)/(
cos(d*x+c)^2-1-(a*b)^(1/2)/b)^2/(a-b)*cos(d*x+c)-5/64/d/b/a/(cos(d*x+c)^2-1-(a*b)^(1/2)/b)^2/(a-b)*cos(d*x+c)-
1/32/d/(a*b)^(1/2)/a/(cos(d*x+c)^2-1-(a*b)^(1/2)/b)^2/(a-b)*cos(d*x+c)+5/64/d/(a^2-2*a*b+b^2)/(((a*b)^(1/2)+b)
*b)^(1/2)*arctanh(cos(d*x+c)*b/(((a*b)^(1/2)+b)*b)^(1/2))+1/64/d/a/(a^2-2*a*b+b^2)/(((a*b)^(1/2)+b)*b)^(1/2)*a
rctanh(cos(d*x+c)*b/(((a*b)^(1/2)+b)*b)^(1/2))*b-1/8/d/(a^2-2*a*b+b^2)*b/(a*b)^(1/2)/(((a*b)^(1/2)+b)*b)^(1/2)
*arctanh(cos(d*x+c)*b/(((a*b)^(1/2)+b)*b)^(1/2))+1/32/d/a/(a^2-2*a*b+b^2)/(a*b)^(1/2)/(((a*b)^(1/2)+b)*b)^(1/2
)*arctanh(cos(d*x+c)*b/(((a*b)^(1/2)+b)*b)^(1/2))*b^2-5/64/d/b/(cos(d*x+c)^2+(a*b)^(1/2)/b-1)^2/(a^2-2*a*b+b^2
)*cos(d*x+c)^3-1/64/d/a/(cos(d*x+c)^2+(a*b)^(1/2)/b-1)^2/(a^2-2*a*b+b^2)*cos(d*x+c)^3-1/8/d/(a*b)^(1/2)/(cos(d
*x+c)^2+(a*b)^(1/2)/b-1)^2/(a^2-2*a*b+b^2)*cos(d*x+c)^3+1/32/d*b/(a*b)^(1/2)/a/(cos(d*x+c)^2+(a*b)^(1/2)/b-1)^
2/(a^2-2*a*b+b^2)*cos(d*x+c)^3-7/64/d/b/(a*b)^(1/2)/(cos(d*x+c)^2+(a*b)^(1/2)/b-1)^2/(a-b)*cos(d*x+c)-5/64/d/b
/a/(cos(d*x+c)^2+(a*b)^(1/2)/b-1)^2/(a-b)*cos(d*x+c)+1/32/d/(a*b)^(1/2)/a/(cos(d*x+c)^2+(a*b)^(1/2)/b-1)^2/(a-
b)*cos(d*x+c)-5/64/d/(a^2-2*a*b+b^2)/(((a*b)^(1/2)-b)*b)^(1/2)*arctan(cos(d*x+c)*b/(((a*b)^(1/2)-b)*b)^(1/2))-
1/64/d/a/(a^2-2*a*b+b^2)/(((a*b)^(1/2)-b)*b)^(1/2)*arctan(cos(d*x+c)*b/(((a*b)^(1/2)-b)*b)^(1/2))*b-1/8/d/(a^2
-2*a*b+b^2)*b/(a*b)^(1/2)/(((a*b)^(1/2)-b)*b)^(1/2)*arctan(cos(d*x+c)*b/(((a*b)^(1/2)-b)*b)^(1/2))+1/32/d/a/(a
^2-2*a*b+b^2)/(a*b)^(1/2)/(((a*b)^(1/2)-b)*b)^(1/2)*arctan(cos(d*x+c)*b/(((a*b)^(1/2)-b)*b)^(1/2))*b^2
________________________________________________________________________________________
Maxima [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(sin(d*x+c)^3/(a-b*sin(d*x+c)^4)^3,x, algorithm="maxima")
[Out]
Timed out
________________________________________________________________________________________
Fricas [B] time = 8.15395, size = 9106, normalized size = 31.62 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(d*x+c)^3/(a-b*sin(d*x+c)^4)^3,x, algorithm="fricas")
[Out]
-1/128*(4*(5*a*b + b^2)*cos(d*x + c)^7 - 12*(7*a*b + b^2)*cos(d*x + c)^5 - 12*(3*a^2 - 10*a*b - b^2)*cos(d*x +
c)^3 + ((a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cos(d*x + c)^8 - 4*(a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cos(d*x + c)^6 - 2
*(a^4*b - 5*a^3*b^2 + 7*a^2*b^3 - 3*a*b^4)*d*cos(d*x + c)^4 + 4*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*cos(
d*x + c)^2 + (a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d)*sqrt(-((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^
5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2*sqrt((625*a^4 + 7700*a^3*b + 21966*a^2*b^2 - 10780*a*b^3 + 1225*b^4)/((a^13*b
^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^
5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4)) + 105*a^3 + 70*a^2*b - 35*a*b^2 + 4*b^3)/((a^8*b - 5*a^7*b^2 + 10*a^6*b
^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2))*log((625*a^3 + 3750*a^2*b - 1491*a*b^2 + 140*b^3)*cos(d*x + c) +
((5*a^10*b^2 - 16*a^9*b^3 + 3*a^8*b^4 + 50*a^7*b^5 - 85*a^6*b^6 + 60*a^5*b^7 - 19*a^4*b^8 + 2*a^3*b^9)*d^3*sqr
t((625*a^4 + 7700*a^3*b + 21966*a^2*b^2 - 10780*a*b^3 + 1225*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120
*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4
)) - (325*a^5*b + 1977*a^4*b^2 - 609*a^3*b^3 + 35*a^2*b^4)*d)*sqrt(-((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*
b^4 + 5*a^4*b^5 - a^3*b^6)*d^2*sqrt((625*a^4 + 7700*a^3*b + 21966*a^2*b^2 - 10780*a*b^3 + 1225*b^4)/((a^13*b^3
- 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*
b^11 - 10*a^4*b^12 + a^3*b^13)*d^4)) + 105*a^3 + 70*a^2*b - 35*a*b^2 + 4*b^3)/((a^8*b - 5*a^7*b^2 + 10*a^6*b^3
- 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2))) - ((a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cos(d*x + c)^8 - 4*(a^3*b^2 - 2
*a^2*b^3 + a*b^4)*d*cos(d*x + c)^6 - 2*(a^4*b - 5*a^3*b^2 + 7*a^2*b^3 - 3*a*b^4)*d*cos(d*x + c)^4 + 4*(a^4*b -
3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*cos(d*x + c)^2 + (a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d)*sqrt(((a
^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2*sqrt((625*a^4 + 7700*a^3*b + 21966*a^2*b
^2 - 10780*a*b^3 + 1225*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8
+ 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4)) - 105*a^3 - 70*a^2*b + 35*a*b^2 -
4*b^3)/((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2))*log((625*a^3 + 3750*a^2*b -
1491*a*b^2 + 140*b^3)*cos(d*x + c) + ((5*a^10*b^2 - 16*a^9*b^3 + 3*a^8*b^4 + 50*a^7*b^5 - 85*a^6*b^6 + 60*a^5*
b^7 - 19*a^4*b^8 + 2*a^3*b^9)*d^3*sqrt((625*a^4 + 7700*a^3*b + 21966*a^2*b^2 - 10780*a*b^3 + 1225*b^4)/((a^13*
b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a
^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4)) + (325*a^5*b + 1977*a^4*b^2 - 609*a^3*b^3 + 35*a^2*b^4)*d)*sqrt(((a^8*
b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2*sqrt((625*a^4 + 7700*a^3*b + 21966*a^2*b^2
- 10780*a*b^3 + 1225*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 +
210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4)) - 105*a^3 - 70*a^2*b + 35*a*b^2 - 4*b
^3)/((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2))) - ((a^3*b^2 - 2*a^2*b^3 + a*b^
4)*d*cos(d*x + c)^8 - 4*(a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cos(d*x + c)^6 - 2*(a^4*b - 5*a^3*b^2 + 7*a^2*b^3 - 3*
a*b^4)*d*cos(d*x + c)^4 + 4*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*cos(d*x + c)^2 + (a^5 - 4*a^4*b + 6*a^3*
b^2 - 4*a^2*b^3 + a*b^4)*d)*sqrt(-((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2*sqr
t((625*a^4 + 7700*a^3*b + 21966*a^2*b^2 - 10780*a*b^3 + 1225*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120
*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4
)) + 105*a^3 + 70*a^2*b - 35*a*b^2 + 4*b^3)/((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^
6)*d^2))*log(-(625*a^3 + 3750*a^2*b - 1491*a*b^2 + 140*b^3)*cos(d*x + c) + ((5*a^10*b^2 - 16*a^9*b^3 + 3*a^8*b
^4 + 50*a^7*b^5 - 85*a^6*b^6 + 60*a^5*b^7 - 19*a^4*b^8 + 2*a^3*b^9)*d^3*sqrt((625*a^4 + 7700*a^3*b + 21966*a^2
*b^2 - 10780*a*b^3 + 1225*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b
^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4)) - (325*a^5*b + 1977*a^4*b^2 - 60
9*a^3*b^3 + 35*a^2*b^4)*d)*sqrt(-((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2*sqrt
((625*a^4 + 7700*a^3*b + 21966*a^2*b^2 - 10780*a*b^3 + 1225*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*
a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4)
) + 105*a^3 + 70*a^2*b - 35*a*b^2 + 4*b^3)/((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6
)*d^2))) + ((a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cos(d*x + c)^8 - 4*(a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cos(d*x + c)^6
- 2*(a^4*b - 5*a^3*b^2 + 7*a^2*b^3 - 3*a*b^4)*d*cos(d*x + c)^4 + 4*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*c
os(d*x + c)^2 + (a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d)*sqrt(((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*
a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2*sqrt((625*a^4 + 7700*a^3*b + 21966*a^2*b^2 - 10780*a*b^3 + 1225*b^4)/((a^13
*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*
a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4)) - 105*a^3 - 70*a^2*b + 35*a*b^2 - 4*b^3)/((a^8*b - 5*a^7*b^2 + 10*a^6
*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2))*log(-(625*a^3 + 3750*a^2*b - 1491*a*b^2 + 140*b^3)*cos(d*x + c)
+ ((5*a^10*b^2 - 16*a^9*b^3 + 3*a^8*b^4 + 50*a^7*b^5 - 85*a^6*b^6 + 60*a^5*b^7 - 19*a^4*b^8 + 2*a^3*b^9)*d^3*
sqrt((625*a^4 + 7700*a^3*b + 21966*a^2*b^2 - 10780*a*b^3 + 1225*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 -
120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*
d^4)) + (325*a^5*b + 1977*a^4*b^2 - 609*a^3*b^3 + 35*a^2*b^4)*d)*sqrt(((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^
5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2*sqrt((625*a^4 + 7700*a^3*b + 21966*a^2*b^2 - 10780*a*b^3 + 1225*b^4)/((a^13*b
^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^
5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4)) - 105*a^3 - 70*a^2*b + 35*a*b^2 - 4*b^3)/((a^8*b - 5*a^7*b^2 + 10*a^6*b
^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2))) + 4*(19*a^2 - 18*a*b - b^2)*cos(d*x + c))/((a^3*b^2 - 2*a^2*b^3
+ a*b^4)*d*cos(d*x + c)^8 - 4*(a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cos(d*x + c)^6 - 2*(a^4*b - 5*a^3*b^2 + 7*a^2*b^
3 - 3*a*b^4)*d*cos(d*x + c)^4 + 4*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*cos(d*x + c)^2 + (a^5 - 4*a^4*b +
6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d)
________________________________________________________________________________________
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(sin(d*x+c)**3/(a-b*sin(d*x+c)**4)**3,x)
[Out]
Timed out
________________________________________________________________________________________
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(sin(d*x+c)^3/(a-b*sin(d*x+c)^4)^3,x, algorithm="giac")
[Out]
Exception raised: NotImplementedError