3.225 \(\int \frac {\sin ^7(c+d x)}{(a-b \sin ^4(c+d x))^3} \, dx\)

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

[Out]

-1/8*a*cos(d*x+c)*(2-cos(d*x+c)^2)/(a-b)/b/d/(a-b+2*b*cos(d*x+c)^2-b*cos(d*x+c)^4)^2+1/32*cos(d*x+c)*(5*a-17*b
-3*(a-3*b)*cos(d*x+c)^2)/(a-b)^2/b/d/(a-b+2*b*cos(d*x+c)^2-b*cos(d*x+c)^4)+3/64*arctan(b^(1/4)*cos(d*x+c)/(a^(
1/2)-b^(1/2))^(1/2))*(a^(1/2)-2*b^(1/2))/b^(7/4)/d/a^(1/2)/(a^(1/2)-b^(1/2))^(5/2)-3/64*arctanh(b^(1/4)*cos(d*
x+c)/(a^(1/2)+b^(1/2))^(1/2))*(a^(1/2)+2*b^(1/2))/b^(7/4)/d/a^(1/2)/(a^(1/2)+b^(1/2))^(5/2)

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Rubi [A]  time = 0.43, antiderivative size = 290, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3215, 1205, 1178, 1166, 205, 208} \[ \frac {3 \left (\sqrt {a}-2 \sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 \sqrt {a} b^{7/4} d \left (\sqrt {a}-\sqrt {b}\right )^{5/2}}-\frac {3 \left (\sqrt {a}+2 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 \sqrt {a} b^{7/4} d \left (\sqrt {a}+\sqrt {b}\right )^{5/2}}+\frac {\cos (c+d x) \left (-3 (a-3 b) \cos ^2(c+d x)+5 a-17 b\right )}{32 b d (a-b)^2 \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}-\frac {a \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 b d (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(Sqrt[a] - 2*Sqrt[b])*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(64*Sqrt[a]*(Sqrt[a] - Sqrt[b
])^(5/2)*b^(7/4)*d) - (3*(Sqrt[a] + 2*Sqrt[b])*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(64*Sq
rt[a]*(Sqrt[a] + Sqrt[b])^(5/2)*b^(7/4)*d) - (a*Cos[c + d*x]*(2 - Cos[c + d*x]^2))/(8*(a - b)*b*d*(a - b + 2*b
*Cos[c + d*x]^2 - b*Cos[c + d*x]^4)^2) + (Cos[c + d*x]*(5*a - 17*b - 3*(a - 3*b)*Cos[c + d*x]^2))/(32*(a - b)^
2*b*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4))

Rule 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]

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 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 1205

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 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]

Rubi steps

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

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Mathematica [C]  time = 1.15, size = 630, normalized size = 2.17 \[ \frac {-3 i \text {RootSum}\left [\text {$\#$1}^8 b-4 \text {$\#$1}^6 b-16 \text {$\#$1}^4 a+6 \text {$\#$1}^4 b-4 \text {$\#$1}^2 b+b\& ,\frac {-2 \text {$\#$1}^6 a \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+6 \text {$\#$1}^6 b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+6 \text {$\#$1}^4 a \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-34 \text {$\#$1}^4 b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+3 i \text {$\#$1}^2 a \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )-i a \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )-6 \text {$\#$1}^2 a \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-17 i \text {$\#$1}^2 b \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )+3 i b \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )+34 \text {$\#$1}^2 b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+i \text {$\#$1}^6 a \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )-3 i \text {$\#$1}^6 b \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )-3 i \text {$\#$1}^4 a \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )+17 i \text {$\#$1}^4 b \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )+2 a \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-6 b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )}{\text {$\#$1}^7 b-3 \text {$\#$1}^5 b-8 \text {$\#$1}^3 a+3 \text {$\#$1}^3 b-\text {$\#$1} b}\& \right ]-\frac {32 \cos (c+d x) (3 (a-3 b) \cos (2 (c+d x))-7 a+25 b)}{8 a+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))-3 b}+\frac {512 a (a-b) (\cos (3 (c+d x))-5 \cos (c+d x))}{(-8 a-4 b \cos (2 (c+d x))+b \cos (4 (c+d x))+3 b)^2}}{256 b d (a-b)^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-32*Cos[c + d*x]*(-7*a + 25*b + 3*(a - 3*b)*Cos[2*(c + d*x)]))/(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(
c + d*x)]) + (512*a*(a - b)*(-5*Cos[c + d*x] + Cos[3*(c + d*x)]))/(-8*a + 3*b - 4*b*Cos[2*(c + d*x)] + b*Cos[4
*(c + d*x)])^2 - (3*I)*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (2*a*ArcTan[Sin[c +
d*x]/(Cos[c + d*x] - #1)] - 6*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - I*a*Log[1 - 2*Cos[c + d*x]*#1 + #1
^2] + (3*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - 6*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + 34*b*Arc
Tan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (3*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (17*I)*b*Log[1 -
2*Cos[c + d*x]*#1 + #1^2]*#1^2 + 6*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - 34*b*ArcTan[Sin[c + d*x]
/(Cos[c + d*x] - #1)]*#1^4 - (3*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + (17*I)*b*Log[1 - 2*Cos[c + d*x]*
#1 + #1^2]*#1^4 - 2*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 + 6*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] -
#1)]*#1^6 + I*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6 - (3*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6)/(-(b*
#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(256*(a - b)^2*b*d)

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fricas [B]  time = 1.13, size = 4185, normalized size = 14.43 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/128*(12*(a*b - 3*b^2)*cos(d*x + c)^7 - 4*(11*a*b - 35*b^2)*cos(d*x + c)^5 + 4*(a^2 + 18*a*b - 43*b^2)*cos(d*
x + c)^3 + 3*((a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^8 - 4*(a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^6 - 2*(a
^3*b^2 - 5*a^2*b^3 + 7*a*b^4 - 3*b^5)*d*cos(d*x + c)^4 + 4*(a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*d*cos(d*x + c
)^2 + (a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*d)*sqrt(-((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6
+ 5*a^2*b^7 - a*b^8)*d^2*sqrt((a^6 - 12*a^5*b + 46*a^4*b^2 - 28*a^3*b^3 - 167*a^2*b^4 + 160*a*b^5 + 256*b^6)/
((a^11*b^7 - 10*a^10*b^8 + 45*a^9*b^9 - 120*a^8*b^10 + 210*a^7*b^11 - 252*a^6*b^12 + 210*a^5*b^13 - 120*a^4*b^
14 + 45*a^3*b^15 - 10*a^2*b^16 + a*b^17)*d^4)) + a^3 - 10*a^2*b + 21*a*b^2 + 4*b^3)/((a^6*b^3 - 5*a^5*b^4 + 10
*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2))*log(27*(a^4 - 10*a^3*b + 29*a^2*b^2 - 4*a*b^3 - 64*b^4)*cos(d
*x + c) + 27*((a^8*b^5 - 8*a^7*b^6 + 23*a^6*b^7 - 30*a^5*b^8 + 15*a^4*b^9 + 4*a^3*b^10 - 7*a^2*b^11 + 2*a*b^12
)*d^3*sqrt((a^6 - 12*a^5*b + 46*a^4*b^2 - 28*a^3*b^3 - 167*a^2*b^4 + 160*a*b^5 + 256*b^6)/((a^11*b^7 - 10*a^10
*b^8 + 45*a^9*b^9 - 120*a^8*b^10 + 210*a^7*b^11 - 252*a^6*b^12 + 210*a^5*b^13 - 120*a^4*b^14 + 45*a^3*b^15 - 1
0*a^2*b^16 + a*b^17)*d^4)) - (a^5*b^2 - 11*a^4*b^3 + 35*a^3*b^4 - 9*a^2*b^5 - 80*a*b^6)*d)*sqrt(-((a^6*b^3 - 5
*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2*sqrt((a^6 - 12*a^5*b + 46*a^4*b^2 - 28*a^3*b^3 - 1
67*a^2*b^4 + 160*a*b^5 + 256*b^6)/((a^11*b^7 - 10*a^10*b^8 + 45*a^9*b^9 - 120*a^8*b^10 + 210*a^7*b^11 - 252*a^
6*b^12 + 210*a^5*b^13 - 120*a^4*b^14 + 45*a^3*b^15 - 10*a^2*b^16 + a*b^17)*d^4)) + a^3 - 10*a^2*b + 21*a*b^2 +
4*b^3)/((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2))) - 3*((a^2*b^3 - 2*a*b^4 +
b^5)*d*cos(d*x + c)^8 - 4*(a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^6 - 2*(a^3*b^2 - 5*a^2*b^3 + 7*a*b^4 - 3*b^
5)*d*cos(d*x + c)^4 + 4*(a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*d*cos(d*x + c)^2 + (a^4*b - 4*a^3*b^2 + 6*a^2*b^
3 - 4*a*b^4 + b^5)*d)*sqrt(((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2*sqrt((a^6
- 12*a^5*b + 46*a^4*b^2 - 28*a^3*b^3 - 167*a^2*b^4 + 160*a*b^5 + 256*b^6)/((a^11*b^7 - 10*a^10*b^8 + 45*a^9*b^
9 - 120*a^8*b^10 + 210*a^7*b^11 - 252*a^6*b^12 + 210*a^5*b^13 - 120*a^4*b^14 + 45*a^3*b^15 - 10*a^2*b^16 + a*b
^17)*d^4)) - a^3 + 10*a^2*b - 21*a*b^2 - 4*b^3)/((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 -
a*b^8)*d^2))*log(27*(a^4 - 10*a^3*b + 29*a^2*b^2 - 4*a*b^3 - 64*b^4)*cos(d*x + c) + 27*((a^8*b^5 - 8*a^7*b^6 +
23*a^6*b^7 - 30*a^5*b^8 + 15*a^4*b^9 + 4*a^3*b^10 - 7*a^2*b^11 + 2*a*b^12)*d^3*sqrt((a^6 - 12*a^5*b + 46*a^4*
b^2 - 28*a^3*b^3 - 167*a^2*b^4 + 160*a*b^5 + 256*b^6)/((a^11*b^7 - 10*a^10*b^8 + 45*a^9*b^9 - 120*a^8*b^10 + 2
10*a^7*b^11 - 252*a^6*b^12 + 210*a^5*b^13 - 120*a^4*b^14 + 45*a^3*b^15 - 10*a^2*b^16 + a*b^17)*d^4)) + (a^5*b^
2 - 11*a^4*b^3 + 35*a^3*b^4 - 9*a^2*b^5 - 80*a*b^6)*d)*sqrt(((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 +
5*a^2*b^7 - a*b^8)*d^2*sqrt((a^6 - 12*a^5*b + 46*a^4*b^2 - 28*a^3*b^3 - 167*a^2*b^4 + 160*a*b^5 + 256*b^6)/((a
^11*b^7 - 10*a^10*b^8 + 45*a^9*b^9 - 120*a^8*b^10 + 210*a^7*b^11 - 252*a^6*b^12 + 210*a^5*b^13 - 120*a^4*b^14
+ 45*a^3*b^15 - 10*a^2*b^16 + a*b^17)*d^4)) - a^3 + 10*a^2*b - 21*a*b^2 - 4*b^3)/((a^6*b^3 - 5*a^5*b^4 + 10*a^
4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2))) - 3*((a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^8 - 4*(a^2*b^3 -
2*a*b^4 + b^5)*d*cos(d*x + c)^6 - 2*(a^3*b^2 - 5*a^2*b^3 + 7*a*b^4 - 3*b^5)*d*cos(d*x + c)^4 + 4*(a^3*b^2 - 3*
a^2*b^3 + 3*a*b^4 - b^5)*d*cos(d*x + c)^2 + (a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*d)*sqrt(-((a^6*b^3
- 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2*sqrt((a^6 - 12*a^5*b + 46*a^4*b^2 - 28*a^3*b^3
- 167*a^2*b^4 + 160*a*b^5 + 256*b^6)/((a^11*b^7 - 10*a^10*b^8 + 45*a^9*b^9 - 120*a^8*b^10 + 210*a^7*b^11 - 25
2*a^6*b^12 + 210*a^5*b^13 - 120*a^4*b^14 + 45*a^3*b^15 - 10*a^2*b^16 + a*b^17)*d^4)) + a^3 - 10*a^2*b + 21*a*b
^2 + 4*b^3)/((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2))*log(-27*(a^4 - 10*a^3*b
+ 29*a^2*b^2 - 4*a*b^3 - 64*b^4)*cos(d*x + c) + 27*((a^8*b^5 - 8*a^7*b^6 + 23*a^6*b^7 - 30*a^5*b^8 + 15*a^4*b
^9 + 4*a^3*b^10 - 7*a^2*b^11 + 2*a*b^12)*d^3*sqrt((a^6 - 12*a^5*b + 46*a^4*b^2 - 28*a^3*b^3 - 167*a^2*b^4 + 16
0*a*b^5 + 256*b^6)/((a^11*b^7 - 10*a^10*b^8 + 45*a^9*b^9 - 120*a^8*b^10 + 210*a^7*b^11 - 252*a^6*b^12 + 210*a^
5*b^13 - 120*a^4*b^14 + 45*a^3*b^15 - 10*a^2*b^16 + a*b^17)*d^4)) - (a^5*b^2 - 11*a^4*b^3 + 35*a^3*b^4 - 9*a^2
*b^5 - 80*a*b^6)*d)*sqrt(-((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2*sqrt((a^6 -
12*a^5*b + 46*a^4*b^2 - 28*a^3*b^3 - 167*a^2*b^4 + 160*a*b^5 + 256*b^6)/((a^11*b^7 - 10*a^10*b^8 + 45*a^9*b^9
- 120*a^8*b^10 + 210*a^7*b^11 - 252*a^6*b^12 + 210*a^5*b^13 - 120*a^4*b^14 + 45*a^3*b^15 - 10*a^2*b^16 + a*b^
17)*d^4)) + a^3 - 10*a^2*b + 21*a*b^2 + 4*b^3)/((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a
*b^8)*d^2))) + 3*((a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^8 - 4*(a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^6 -
2*(a^3*b^2 - 5*a^2*b^3 + 7*a*b^4 - 3*b^5)*d*cos(d*x + c)^4 + 4*(a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*d*cos(d*x
+ c)^2 + (a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*d)*sqrt(((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*
b^6 + 5*a^2*b^7 - a*b^8)*d^2*sqrt((a^6 - 12*a^5*b + 46*a^4*b^2 - 28*a^3*b^3 - 167*a^2*b^4 + 160*a*b^5 + 256*b^
6)/((a^11*b^7 - 10*a^10*b^8 + 45*a^9*b^9 - 120*a^8*b^10 + 210*a^7*b^11 - 252*a^6*b^12 + 210*a^5*b^13 - 120*a^4
*b^14 + 45*a^3*b^15 - 10*a^2*b^16 + a*b^17)*d^4)) - a^3 + 10*a^2*b - 21*a*b^2 - 4*b^3)/((a^6*b^3 - 5*a^5*b^4 +
10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2))*log(-27*(a^4 - 10*a^3*b + 29*a^2*b^2 - 4*a*b^3 - 64*b^4)*c
os(d*x + c) + 27*((a^8*b^5 - 8*a^7*b^6 + 23*a^6*b^7 - 30*a^5*b^8 + 15*a^4*b^9 + 4*a^3*b^10 - 7*a^2*b^11 + 2*a*
b^12)*d^3*sqrt((a^6 - 12*a^5*b + 46*a^4*b^2 - 28*a^3*b^3 - 167*a^2*b^4 + 160*a*b^5 + 256*b^6)/((a^11*b^7 - 10*
a^10*b^8 + 45*a^9*b^9 - 120*a^8*b^10 + 210*a^7*b^11 - 252*a^6*b^12 + 210*a^5*b^13 - 120*a^4*b^14 + 45*a^3*b^15
- 10*a^2*b^16 + a*b^17)*d^4)) + (a^5*b^2 - 11*a^4*b^3 + 35*a^3*b^4 - 9*a^2*b^5 - 80*a*b^6)*d)*sqrt(((a^6*b^3
- 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2*sqrt((a^6 - 12*a^5*b + 46*a^4*b^2 - 28*a^3*b^3
- 167*a^2*b^4 + 160*a*b^5 + 256*b^6)/((a^11*b^7 - 10*a^10*b^8 + 45*a^9*b^9 - 120*a^8*b^10 + 210*a^7*b^11 - 252
*a^6*b^12 + 210*a^5*b^13 - 120*a^4*b^14 + 45*a^3*b^15 - 10*a^2*b^16 + a*b^17)*d^4)) - a^3 + 10*a^2*b - 21*a*b^
2 - 4*b^3)/((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2))) - 4*(3*a^2 + 14*a*b - 1
7*b^2)*cos(d*x + c))/((a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^8 - 4*(a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^
6 - 2*(a^3*b^2 - 5*a^2*b^3 + 7*a*b^4 - 3*b^5)*d*cos(d*x + c)^4 + 4*(a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*d*cos
(d*x + c)^2 + (a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*d)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Warning, need to choose a branch for the
root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[61,-66]Warning,
need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done
assuming [a,b]=[30,-29]-2/d*(-3*((1-cos(c+d*x))/(1+cos(c+d*x)))^7*a^2*b+3*((1-cos(c+d*x))/(1+cos(c+d*x)))^6*a
^3-30*((1-cos(c+d*x))/(1+cos(c+d*x)))^6*a^2*b+16*((1-cos(c+d*x))/(1+cos(c+d*x)))^5*a^3-111*((1-cos(c+d*x))/(1+
cos(c+d*x)))^5*a^2*b+80*((1-cos(c+d*x))/(1+cos(c+d*x)))^5*a*b^2+35*((1-cos(c+d*x))/(1+cos(c+d*x)))^4*a^3-26*((
1-cos(c+d*x))/(1+cos(c+d*x)))^4*a^2*b-64*((1-cos(c+d*x))/(1+cos(c+d*x)))^4*a*b^2+256*((1-cos(c+d*x))/(1+cos(c+
d*x)))^4*b^3+40*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*a^3+95*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*a^2*b-336*((1-cos(c
+d*x))/(1+cos(c+d*x)))^3*a*b^2+25*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*a^3+54*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*a
^2*b-64*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*a*b^2+8*(1-cos(c+d*x))/(1+cos(c+d*x))*a^3+19*(1-cos(c+d*x))/(1+cos(c
+d*x))*a^2*b+a^3+2*a^2*b)/(16*a^3*b-32*a^2*b^2+16*a*b^3)/(((1-cos(c+d*x))/(1+cos(c+d*x)))^4*a+4*((1-cos(c+d*x)
)/(1+cos(c+d*x)))^3*a+6*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*a-16*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*b+4*(1-cos(c+
d*x))/(1+cos(c+d*x))*a+a)^2-2/d/(16*a^2*b-32*a*b^2+16*b^3)*2/d*(-(-3*a+9*b)/(4*b*2)*(c+d*x)+((6*a^5*b-42*a^4*b
^2-6*a^4*a*b+9*a^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+114*a^3*b^3+42*a^3*b*a*b-63*a^3*b*sqrt(a*b)*sqrt(a^
2-a*b+sqrt(a*b)*(a-b))-126*a^2*b^4-114*a^2*b^2*a*b+159*a^2*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+48*a*b^
5+126*a*b^3*a*b-129*a*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-48*b^4*a*b-24*b^4*sqrt(a*b)*sqrt(a^2-a*b+sqr
t(a*b)*(a-b)))*abs(a-b)*b^2+(-12*a^5*b^2-9*a^5*b*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+60*a^4*b^3+54*a^4*b^2*sqrt(a^2-
a*b+sqrt(a*b)*(a-b))+12*a^4*b*a*b-132*a^3*b^4-132*a^3*b^3*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-60*a^3*b^2*a*b+132*a^2
*b^5+150*a^2*b^4*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+132*a^2*b^3*a*b-48*a*b^6-51*a*b^5*sqrt(a^2-a*b+sqrt(a*b)*(a-b))
-132*a*b^4*a*b-12*b^6*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+48*b^5*a*b)*abs(a-b)*abs(b)+(6*a^5*b^3-18*a^4*b^4-6*a^4*b^
2*a*b+9*a^4*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-6*a^3*b^5+18*a^3*b^3*a*b-27*a^3*b^3*sqrt(a*b)*sqrt(a^2
-a*b+sqrt(a*b)*(a-b))+42*a^2*b^6+6*a^2*b^4*a*b-21*a^2*b^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-24*a*b^7-42*
a*b^5*a*b+75*a*b^5*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+24*b^6*a*b+12*b^6*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*
(a-b)))*abs(a-b))/((48*a^6*b^2-240*a^5*b^3+416*a^4*b^4-288*a^3*b^5+48*a^2*b^6+16*a*b^7)*abs(b))*(atan(tan(c+d*
x)/sqrt(-(16*a*b+sqrt(16*a*b*16*a*b+4*(-8*a*b+8*b^2)*8*a*b))/2/(-8*a*b+8*b^2)))+pi*floor((c+d*x)/pi+1/2))-((6*
a^5*b-42*a^4*b^2-6*a^4*a*b+9*a^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+114*a^3*b^3+42*a^3*b*a*b-63*a^3*b*sq
rt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-126*a^2*b^4-114*a^2*b^2*a*b+159*a^2*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b
)*(-a+b))+48*a*b^5+126*a*b^3*a*b-129*a*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-48*b^4*a*b-24*b^4*sqrt(a*b
)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b)))*abs(a-b)*b^2+(-12*a^5*b^2+9*a^5*b*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+60*a^4*b^3-
54*a^4*b^2*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+12*a^4*b*a*b-132*a^3*b^4+132*a^3*b^3*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-
60*a^3*b^2*a*b+132*a^2*b^5-150*a^2*b^4*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+132*a^2*b^3*a*b-48*a*b^6+51*a*b^5*sqrt(a
^2-a*b+sqrt(a*b)*(-a+b))-132*a*b^4*a*b+12*b^6*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+48*b^5*a*b)*abs(a-b)*abs(b)+(6*a^
5*b^3-18*a^4*b^4-6*a^4*b^2*a*b+9*a^4*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-6*a^3*b^5+18*a^3*b^3*a*b-27*
a^3*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+42*a^2*b^6+6*a^2*b^4*a*b-21*a^2*b^4*sqrt(a*b)*sqrt(a^2-a*b+sq
rt(a*b)*(-a+b))-24*a*b^7-42*a*b^5*a*b+75*a*b^5*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+24*b^6*a*b+12*b^6*sqrt
(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b)))*abs(a-b))/((48*a^6*b^2-240*a^5*b^3+416*a^4*b^4-288*a^3*b^5+48*a^2*b^6+16
*a*b^7)*abs(b))*(atan(tan(c+d*x)/sqrt(-(16*a*b-sqrt(16*a*b*16*a*b+4*(-8*a*b+8*b^2)*8*a*b))/2/(-8*a*b+8*b^2)))+
pi*floor((c+d*x)/pi+1/2)))

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maple [B]  time = 0.36, size = 814, normalized size = 2.81 \[ \frac {3 \left (\cos ^{7}\left (d x +c \right )\right ) a}{32 d \left (b \left (\cos ^{4}\left (d x +c \right )\right )-2 b \left (\cos ^{2}\left (d x +c \right )\right )-a +b \right )^{2} \left (a^{2}-2 a b +b^{2}\right )}-\frac {9 \left (\cos ^{7}\left (d x +c \right )\right ) b}{32 d \left (b \left (\cos ^{4}\left (d x +c \right )\right )-2 b \left (\cos ^{2}\left (d x +c \right )\right )-a +b \right )^{2} \left (a^{2}-2 a b +b^{2}\right )}-\frac {11 \left (\cos ^{5}\left (d x +c \right )\right ) a}{32 d \left (b \left (\cos ^{4}\left (d x +c \right )\right )-2 b \left (\cos ^{2}\left (d x +c \right )\right )-a +b \right )^{2} \left (a^{2}-2 a b +b^{2}\right )}+\frac {35 b \left (\cos ^{5}\left (d x +c \right )\right )}{32 d \left (b \left (\cos ^{4}\left (d x +c \right )\right )-2 b \left (\cos ^{2}\left (d x +c \right )\right )-a +b \right )^{2} \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (\cos ^{3}\left (d x +c \right )\right ) a^{2}}{32 d \left (b \left (\cos ^{4}\left (d x +c \right )\right )-2 b \left (\cos ^{2}\left (d x +c \right )\right )-a +b \right )^{2} b \left (a^{2}-2 a b +b^{2}\right )}+\frac {9 \left (\cos ^{3}\left (d x +c \right )\right ) a}{16 d \left (b \left (\cos ^{4}\left (d x +c \right )\right )-2 b \left (\cos ^{2}\left (d x +c \right )\right )-a +b \right )^{2} \left (a^{2}-2 a b +b^{2}\right )}-\frac {43 b \left (\cos ^{3}\left (d x +c \right )\right )}{32 d \left (b \left (\cos ^{4}\left (d x +c \right )\right )-2 b \left (\cos ^{2}\left (d x +c \right )\right )-a +b \right )^{2} \left (a^{2}-2 a b +b^{2}\right )}-\frac {3 \cos \left (d x +c \right ) a}{32 d \left (b \left (\cos ^{4}\left (d x +c \right )\right )-2 b \left (\cos ^{2}\left (d x +c \right )\right )-a +b \right )^{2} b \left (a -b \right )}-\frac {17 \cos \left (d x +c \right )}{32 d \left (b \left (\cos ^{4}\left (d x +c \right )\right )-2 b \left (\cos ^{2}\left (d x +c \right )\right )-a +b \right )^{2} \left (a -b \right )}-\frac {3 \arctanh \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right ) a}{64 d \left (a^{2}-2 a b +b^{2}\right ) b \sqrt {\left (\sqrt {a b}+b \right ) b}}+\frac {9 \arctanh \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{64 d \left (a^{2}-2 a b +b^{2}\right ) \sqrt {\left (\sqrt {a b}+b \right ) b}}-\frac {3 b \arctanh \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{32 d \left (a^{2}-2 a b +b^{2}\right ) \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+b \right ) b}}+\frac {3 \arctan \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right ) a}{64 d \left (a^{2}-2 a b +b^{2}\right ) b \sqrt {\left (\sqrt {a b}-b \right ) b}}-\frac {9 \arctan \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{64 d \left (a^{2}-2 a b +b^{2}\right ) \sqrt {\left (\sqrt {a b}-b \right ) b}}-\frac {3 b \arctan \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{32 d \left (a^{2}-2 a b +b^{2}\right ) \sqrt {a b}\, \sqrt {\left (\sqrt {a b}-b \right ) b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/32/d/(b*cos(d*x+c)^4-2*b*cos(d*x+c)^2-a+b)^2/(a^2-2*a*b+b^2)*cos(d*x+c)^7*a-9/32/d/(b*cos(d*x+c)^4-2*b*cos(d
*x+c)^2-a+b)^2/(a^2-2*a*b+b^2)*cos(d*x+c)^7*b-11/32/d/(b*cos(d*x+c)^4-2*b*cos(d*x+c)^2-a+b)^2/(a^2-2*a*b+b^2)*
cos(d*x+c)^5*a+35/32/d/(b*cos(d*x+c)^4-2*b*cos(d*x+c)^2-a+b)^2*b/(a^2-2*a*b+b^2)*cos(d*x+c)^5+1/32/d/(b*cos(d*
x+c)^4-2*b*cos(d*x+c)^2-a+b)^2/b/(a^2-2*a*b+b^2)*cos(d*x+c)^3*a^2+9/16/d/(b*cos(d*x+c)^4-2*b*cos(d*x+c)^2-a+b)
^2/(a^2-2*a*b+b^2)*cos(d*x+c)^3*a-43/32/d/(b*cos(d*x+c)^4-2*b*cos(d*x+c)^2-a+b)^2*b/(a^2-2*a*b+b^2)*cos(d*x+c)
^3-3/32/d/(b*cos(d*x+c)^4-2*b*cos(d*x+c)^2-a+b)^2/b/(a-b)*cos(d*x+c)*a-17/32/d/(b*cos(d*x+c)^4-2*b*cos(d*x+c)^
2-a+b)^2/(a-b)*cos(d*x+c)-3/64/d/(a^2-2*a*b+b^2)/b/(((a*b)^(1/2)+b)*b)^(1/2)*arctanh(cos(d*x+c)*b/(((a*b)^(1/2
)+b)*b)^(1/2))*a+9/64/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))-3/32/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))+3/64/d/(a^2-2*a*b+b^2)/b/(((a*b)^(1/2)-b)*b)^(1/2)*arctan(cos(d*x+c)*b/(((a*b)^(1/2)-b)*b)^(1/2))*a-9/64/
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))-3/32/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))

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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mupad [B]  time = 19.62, size = 5824, normalized size = 20.08 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((3*cos(c + d*x)^7*(a - 3*b))/(32*(a^2 - 2*a*b + b^2)) - (cos(c + d*x)^5*(11*a - 35*b))/(32*(a^2 - 2*a*b + b^2
)) + (cos(c + d*x)^3*(18*a*b + a^2 - 43*b^2))/(32*b*(a - b)^2) - (cos(c + d*x)*(3*a + 17*b))/(32*b*(a - b)))/(
d*(a^2 - 2*a*b + b^2 + cos(c + d*x)^2*(4*a*b - 4*b^2) - cos(c + d*x)^4*(2*a*b - 6*b^2) - 4*b^2*cos(c + d*x)^6
+ b^2*cos(c + d*x)^8)) + (atan(((((3*(81920*a*b^7 - 180224*a^2*b^6 + 114688*a^3*b^5 - 16384*a^4*b^4))/(32768*(
b^6 - 4*a*b^5 + 6*a^2*b^4 - 4*a^3*b^3 + a^4*b^2)) - (cos(c + d*x)*((9*(a^3*(a^3*b^7)^(1/2) + 16*b^3*(a^3*b^7)^
(1/2) + 4*a*b^7 + 21*a^2*b^6 - 10*a^3*b^5 + a^4*b^4 + 5*a*b^2*(a^3*b^7)^(1/2) - 6*a^2*b*(a^3*b^7)^(1/2)))/(163
84*(a^2*b^12 - 5*a^3*b^11 + 10*a^4*b^10 - 10*a^5*b^9 + 5*a^6*b^8 - a^7*b^7)))^(1/2)*(16384*a*b^8 - 65536*a^2*b
^7 + 98304*a^3*b^6 - 65536*a^4*b^5 + 16384*a^5*b^4))/(256*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)))*((9*(a
^3*(a^3*b^7)^(1/2) + 16*b^3*(a^3*b^7)^(1/2) + 4*a*b^7 + 21*a^2*b^6 - 10*a^3*b^5 + a^4*b^4 + 5*a*b^2*(a^3*b^7)^
(1/2) - 6*a^2*b*(a^3*b^7)^(1/2)))/(16384*(a^2*b^12 - 5*a^3*b^11 + 10*a^4*b^10 - 10*a^5*b^9 + 5*a^6*b^8 - a^7*b
^7)))^(1/2) + (cos(c + d*x)*(81*a*b^2 - 54*a^2*b + 9*a^3 + 36*b^3))/(256*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^
2*b^2)))*((9*(a^3*(a^3*b^7)^(1/2) + 16*b^3*(a^3*b^7)^(1/2) + 4*a*b^7 + 21*a^2*b^6 - 10*a^3*b^5 + a^4*b^4 + 5*a
*b^2*(a^3*b^7)^(1/2) - 6*a^2*b*(a^3*b^7)^(1/2)))/(16384*(a^2*b^12 - 5*a^3*b^11 + 10*a^4*b^10 - 10*a^5*b^9 + 5*
a^6*b^8 - a^7*b^7)))^(1/2)*1i - (((3*(81920*a*b^7 - 180224*a^2*b^6 + 114688*a^3*b^5 - 16384*a^4*b^4))/(32768*(
b^6 - 4*a*b^5 + 6*a^2*b^4 - 4*a^3*b^3 + a^4*b^2)) + (cos(c + d*x)*((9*(a^3*(a^3*b^7)^(1/2) + 16*b^3*(a^3*b^7)^
(1/2) + 4*a*b^7 + 21*a^2*b^6 - 10*a^3*b^5 + a^4*b^4 + 5*a*b^2*(a^3*b^7)^(1/2) - 6*a^2*b*(a^3*b^7)^(1/2)))/(163
84*(a^2*b^12 - 5*a^3*b^11 + 10*a^4*b^10 - 10*a^5*b^9 + 5*a^6*b^8 - a^7*b^7)))^(1/2)*(16384*a*b^8 - 65536*a^2*b
^7 + 98304*a^3*b^6 - 65536*a^4*b^5 + 16384*a^5*b^4))/(256*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)))*((9*(a
^3*(a^3*b^7)^(1/2) + 16*b^3*(a^3*b^7)^(1/2) + 4*a*b^7 + 21*a^2*b^6 - 10*a^3*b^5 + a^4*b^4 + 5*a*b^2*(a^3*b^7)^
(1/2) - 6*a^2*b*(a^3*b^7)^(1/2)))/(16384*(a^2*b^12 - 5*a^3*b^11 + 10*a^4*b^10 - 10*a^5*b^9 + 5*a^6*b^8 - a^7*b
^7)))^(1/2) - (cos(c + d*x)*(81*a*b^2 - 54*a^2*b + 9*a^3 + 36*b^3))/(256*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^
2*b^2)))*((9*(a^3*(a^3*b^7)^(1/2) + 16*b^3*(a^3*b^7)^(1/2) + 4*a*b^7 + 21*a^2*b^6 - 10*a^3*b^5 + a^4*b^4 + 5*a
*b^2*(a^3*b^7)^(1/2) - 6*a^2*b*(a^3*b^7)^(1/2)))/(16384*(a^2*b^12 - 5*a^3*b^11 + 10*a^4*b^10 - 10*a^5*b^9 + 5*
a^6*b^8 - a^7*b^7)))^(1/2)*1i)/((((3*(81920*a*b^7 - 180224*a^2*b^6 + 114688*a^3*b^5 - 16384*a^4*b^4))/(32768*(
b^6 - 4*a*b^5 + 6*a^2*b^4 - 4*a^3*b^3 + a^4*b^2)) - (cos(c + d*x)*((9*(a^3*(a^3*b^7)^(1/2) + 16*b^3*(a^3*b^7)^
(1/2) + 4*a*b^7 + 21*a^2*b^6 - 10*a^3*b^5 + a^4*b^4 + 5*a*b^2*(a^3*b^7)^(1/2) - 6*a^2*b*(a^3*b^7)^(1/2)))/(163
84*(a^2*b^12 - 5*a^3*b^11 + 10*a^4*b^10 - 10*a^5*b^9 + 5*a^6*b^8 - a^7*b^7)))^(1/2)*(16384*a*b^8 - 65536*a^2*b
^7 + 98304*a^3*b^6 - 65536*a^4*b^5 + 16384*a^5*b^4))/(256*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)))*((9*(a
^3*(a^3*b^7)^(1/2) + 16*b^3*(a^3*b^7)^(1/2) + 4*a*b^7 + 21*a^2*b^6 - 10*a^3*b^5 + a^4*b^4 + 5*a*b^2*(a^3*b^7)^
(1/2) - 6*a^2*b*(a^3*b^7)^(1/2)))/(16384*(a^2*b^12 - 5*a^3*b^11 + 10*a^4*b^10 - 10*a^5*b^9 + 5*a^6*b^8 - a^7*b
^7)))^(1/2) + (cos(c + d*x)*(81*a*b^2 - 54*a^2*b + 9*a^3 + 36*b^3))/(256*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^
2*b^2)))*((9*(a^3*(a^3*b^7)^(1/2) + 16*b^3*(a^3*b^7)^(1/2) + 4*a*b^7 + 21*a^2*b^6 - 10*a^3*b^5 + a^4*b^4 + 5*a
*b^2*(a^3*b^7)^(1/2) - 6*a^2*b*(a^3*b^7)^(1/2)))/(16384*(a^2*b^12 - 5*a^3*b^11 + 10*a^4*b^10 - 10*a^5*b^9 + 5*
a^6*b^8 - a^7*b^7)))^(1/2) + (((3*(81920*a*b^7 - 180224*a^2*b^6 + 114688*a^3*b^5 - 16384*a^4*b^4))/(32768*(b^6
- 4*a*b^5 + 6*a^2*b^4 - 4*a^3*b^3 + a^4*b^2)) + (cos(c + d*x)*((9*(a^3*(a^3*b^7)^(1/2) + 16*b^3*(a^3*b^7)^(1/
2) + 4*a*b^7 + 21*a^2*b^6 - 10*a^3*b^5 + a^4*b^4 + 5*a*b^2*(a^3*b^7)^(1/2) - 6*a^2*b*(a^3*b^7)^(1/2)))/(16384*
(a^2*b^12 - 5*a^3*b^11 + 10*a^4*b^10 - 10*a^5*b^9 + 5*a^6*b^8 - a^7*b^7)))^(1/2)*(16384*a*b^8 - 65536*a^2*b^7
+ 98304*a^3*b^6 - 65536*a^4*b^5 + 16384*a^5*b^4))/(256*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)))*((9*(a^3*
(a^3*b^7)^(1/2) + 16*b^3*(a^3*b^7)^(1/2) + 4*a*b^7 + 21*a^2*b^6 - 10*a^3*b^5 + a^4*b^4 + 5*a*b^2*(a^3*b^7)^(1/
2) - 6*a^2*b*(a^3*b^7)^(1/2)))/(16384*(a^2*b^12 - 5*a^3*b^11 + 10*a^4*b^10 - 10*a^5*b^9 + 5*a^6*b^8 - a^7*b^7)
))^(1/2) - (cos(c + d*x)*(81*a*b^2 - 54*a^2*b + 9*a^3 + 36*b^3))/(256*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b
^2)))*((9*(a^3*(a^3*b^7)^(1/2) + 16*b^3*(a^3*b^7)^(1/2) + 4*a*b^7 + 21*a^2*b^6 - 10*a^3*b^5 + a^4*b^4 + 5*a*b^
2*(a^3*b^7)^(1/2) - 6*a^2*b*(a^3*b^7)^(1/2)))/(16384*(a^2*b^12 - 5*a^3*b^11 + 10*a^4*b^10 - 10*a^5*b^9 + 5*a^6
*b^8 - a^7*b^7)))^(1/2) + (3*(9*a^2 - 63*a*b + 108*b^2))/(16384*(b^6 - 4*a*b^5 + 6*a^2*b^4 - 4*a^3*b^3 + a^4*b
^2))))*((9*(a^3*(a^3*b^7)^(1/2) + 16*b^3*(a^3*b^7)^(1/2) + 4*a*b^7 + 21*a^2*b^6 - 10*a^3*b^5 + a^4*b^4 + 5*a*b
^2*(a^3*b^7)^(1/2) - 6*a^2*b*(a^3*b^7)^(1/2)))/(16384*(a^2*b^12 - 5*a^3*b^11 + 10*a^4*b^10 - 10*a^5*b^9 + 5*a^
6*b^8 - a^7*b^7)))^(1/2)*2i)/d + (atan(((((3*(81920*a*b^7 - 180224*a^2*b^6 + 114688*a^3*b^5 - 16384*a^4*b^4))/
(32768*(b^6 - 4*a*b^5 + 6*a^2*b^4 - 4*a^3*b^3 + a^4*b^2)) - (cos(c + d*x)*(-(9*(a^3*(a^3*b^7)^(1/2) + 16*b^3*(
a^3*b^7)^(1/2) - 4*a*b^7 - 21*a^2*b^6 + 10*a^3*b^5 - a^4*b^4 + 5*a*b^2*(a^3*b^7)^(1/2) - 6*a^2*b*(a^3*b^7)^(1/
2)))/(16384*(a^2*b^12 - 5*a^3*b^11 + 10*a^4*b^10 - 10*a^5*b^9 + 5*a^6*b^8 - a^7*b^7)))^(1/2)*(16384*a*b^8 - 65
536*a^2*b^7 + 98304*a^3*b^6 - 65536*a^4*b^5 + 16384*a^5*b^4))/(256*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)
))*(-(9*(a^3*(a^3*b^7)^(1/2) + 16*b^3*(a^3*b^7)^(1/2) - 4*a*b^7 - 21*a^2*b^6 + 10*a^3*b^5 - a^4*b^4 + 5*a*b^2*
(a^3*b^7)^(1/2) - 6*a^2*b*(a^3*b^7)^(1/2)))/(16384*(a^2*b^12 - 5*a^3*b^11 + 10*a^4*b^10 - 10*a^5*b^9 + 5*a^6*b
^8 - a^7*b^7)))^(1/2) + (cos(c + d*x)*(81*a*b^2 - 54*a^2*b + 9*a^3 + 36*b^3))/(256*(a^4 - 4*a^3*b - 4*a*b^3 +
b^4 + 6*a^2*b^2)))*(-(9*(a^3*(a^3*b^7)^(1/2) + 16*b^3*(a^3*b^7)^(1/2) - 4*a*b^7 - 21*a^2*b^6 + 10*a^3*b^5 - a^
4*b^4 + 5*a*b^2*(a^3*b^7)^(1/2) - 6*a^2*b*(a^3*b^7)^(1/2)))/(16384*(a^2*b^12 - 5*a^3*b^11 + 10*a^4*b^10 - 10*a
^5*b^9 + 5*a^6*b^8 - a^7*b^7)))^(1/2)*1i - (((3*(81920*a*b^7 - 180224*a^2*b^6 + 114688*a^3*b^5 - 16384*a^4*b^4
))/(32768*(b^6 - 4*a*b^5 + 6*a^2*b^4 - 4*a^3*b^3 + a^4*b^2)) + (cos(c + d*x)*(-(9*(a^3*(a^3*b^7)^(1/2) + 16*b^
3*(a^3*b^7)^(1/2) - 4*a*b^7 - 21*a^2*b^6 + 10*a^3*b^5 - a^4*b^4 + 5*a*b^2*(a^3*b^7)^(1/2) - 6*a^2*b*(a^3*b^7)^
(1/2)))/(16384*(a^2*b^12 - 5*a^3*b^11 + 10*a^4*b^10 - 10*a^5*b^9 + 5*a^6*b^8 - a^7*b^7)))^(1/2)*(16384*a*b^8 -
65536*a^2*b^7 + 98304*a^3*b^6 - 65536*a^4*b^5 + 16384*a^5*b^4))/(256*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b
^2)))*(-(9*(a^3*(a^3*b^7)^(1/2) + 16*b^3*(a^3*b^7)^(1/2) - 4*a*b^7 - 21*a^2*b^6 + 10*a^3*b^5 - a^4*b^4 + 5*a*b
^2*(a^3*b^7)^(1/2) - 6*a^2*b*(a^3*b^7)^(1/2)))/(16384*(a^2*b^12 - 5*a^3*b^11 + 10*a^4*b^10 - 10*a^5*b^9 + 5*a^
6*b^8 - a^7*b^7)))^(1/2) - (cos(c + d*x)*(81*a*b^2 - 54*a^2*b + 9*a^3 + 36*b^3))/(256*(a^4 - 4*a^3*b - 4*a*b^3
+ b^4 + 6*a^2*b^2)))*(-(9*(a^3*(a^3*b^7)^(1/2) + 16*b^3*(a^3*b^7)^(1/2) - 4*a*b^7 - 21*a^2*b^6 + 10*a^3*b^5 -
a^4*b^4 + 5*a*b^2*(a^3*b^7)^(1/2) - 6*a^2*b*(a^3*b^7)^(1/2)))/(16384*(a^2*b^12 - 5*a^3*b^11 + 10*a^4*b^10 - 1
0*a^5*b^9 + 5*a^6*b^8 - a^7*b^7)))^(1/2)*1i)/((((3*(81920*a*b^7 - 180224*a^2*b^6 + 114688*a^3*b^5 - 16384*a^4*
b^4))/(32768*(b^6 - 4*a*b^5 + 6*a^2*b^4 - 4*a^3*b^3 + a^4*b^2)) - (cos(c + d*x)*(-(9*(a^3*(a^3*b^7)^(1/2) + 16
*b^3*(a^3*b^7)^(1/2) - 4*a*b^7 - 21*a^2*b^6 + 10*a^3*b^5 - a^4*b^4 + 5*a*b^2*(a^3*b^7)^(1/2) - 6*a^2*b*(a^3*b^
7)^(1/2)))/(16384*(a^2*b^12 - 5*a^3*b^11 + 10*a^4*b^10 - 10*a^5*b^9 + 5*a^6*b^8 - a^7*b^7)))^(1/2)*(16384*a*b^
8 - 65536*a^2*b^7 + 98304*a^3*b^6 - 65536*a^4*b^5 + 16384*a^5*b^4))/(256*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^
2*b^2)))*(-(9*(a^3*(a^3*b^7)^(1/2) + 16*b^3*(a^3*b^7)^(1/2) - 4*a*b^7 - 21*a^2*b^6 + 10*a^3*b^5 - a^4*b^4 + 5*
a*b^2*(a^3*b^7)^(1/2) - 6*a^2*b*(a^3*b^7)^(1/2)))/(16384*(a^2*b^12 - 5*a^3*b^11 + 10*a^4*b^10 - 10*a^5*b^9 + 5
*a^6*b^8 - a^7*b^7)))^(1/2) + (cos(c + d*x)*(81*a*b^2 - 54*a^2*b + 9*a^3 + 36*b^3))/(256*(a^4 - 4*a^3*b - 4*a*
b^3 + b^4 + 6*a^2*b^2)))*(-(9*(a^3*(a^3*b^7)^(1/2) + 16*b^3*(a^3*b^7)^(1/2) - 4*a*b^7 - 21*a^2*b^6 + 10*a^3*b^
5 - a^4*b^4 + 5*a*b^2*(a^3*b^7)^(1/2) - 6*a^2*b*(a^3*b^7)^(1/2)))/(16384*(a^2*b^12 - 5*a^3*b^11 + 10*a^4*b^10
- 10*a^5*b^9 + 5*a^6*b^8 - a^7*b^7)))^(1/2) + (((3*(81920*a*b^7 - 180224*a^2*b^6 + 114688*a^3*b^5 - 16384*a^4*
b^4))/(32768*(b^6 - 4*a*b^5 + 6*a^2*b^4 - 4*a^3*b^3 + a^4*b^2)) + (cos(c + d*x)*(-(9*(a^3*(a^3*b^7)^(1/2) + 16
*b^3*(a^3*b^7)^(1/2) - 4*a*b^7 - 21*a^2*b^6 + 10*a^3*b^5 - a^4*b^4 + 5*a*b^2*(a^3*b^7)^(1/2) - 6*a^2*b*(a^3*b^
7)^(1/2)))/(16384*(a^2*b^12 - 5*a^3*b^11 + 10*a^4*b^10 - 10*a^5*b^9 + 5*a^6*b^8 - a^7*b^7)))^(1/2)*(16384*a*b^
8 - 65536*a^2*b^7 + 98304*a^3*b^6 - 65536*a^4*b^5 + 16384*a^5*b^4))/(256*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^
2*b^2)))*(-(9*(a^3*(a^3*b^7)^(1/2) + 16*b^3*(a^3*b^7)^(1/2) - 4*a*b^7 - 21*a^2*b^6 + 10*a^3*b^5 - a^4*b^4 + 5*
a*b^2*(a^3*b^7)^(1/2) - 6*a^2*b*(a^3*b^7)^(1/2)))/(16384*(a^2*b^12 - 5*a^3*b^11 + 10*a^4*b^10 - 10*a^5*b^9 + 5
*a^6*b^8 - a^7*b^7)))^(1/2) - (cos(c + d*x)*(81*a*b^2 - 54*a^2*b + 9*a^3 + 36*b^3))/(256*(a^4 - 4*a^3*b - 4*a*
b^3 + b^4 + 6*a^2*b^2)))*(-(9*(a^3*(a^3*b^7)^(1/2) + 16*b^3*(a^3*b^7)^(1/2) - 4*a*b^7 - 21*a^2*b^6 + 10*a^3*b^
5 - a^4*b^4 + 5*a*b^2*(a^3*b^7)^(1/2) - 6*a^2*b*(a^3*b^7)^(1/2)))/(16384*(a^2*b^12 - 5*a^3*b^11 + 10*a^4*b^10
- 10*a^5*b^9 + 5*a^6*b^8 - a^7*b^7)))^(1/2) + (3*(9*a^2 - 63*a*b + 108*b^2))/(16384*(b^6 - 4*a*b^5 + 6*a^2*b^4
- 4*a^3*b^3 + a^4*b^2))))*(-(9*(a^3*(a^3*b^7)^(1/2) + 16*b^3*(a^3*b^7)^(1/2) - 4*a*b^7 - 21*a^2*b^6 + 10*a^3*
b^5 - a^4*b^4 + 5*a*b^2*(a^3*b^7)^(1/2) - 6*a^2*b*(a^3*b^7)^(1/2)))/(16384*(a^2*b^12 - 5*a^3*b^11 + 10*a^4*b^1
0 - 10*a^5*b^9 + 5*a^6*b^8 - a^7*b^7)))^(1/2)*2i)/d

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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