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

Optimal. Leaf size=313 \[ -\frac {3 \left (-10 \sqrt {a} \sqrt {b}+7 a+4 b\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{5/2} \sqrt [4]{b} d \left (\sqrt {a}-\sqrt {b}\right )^{5/2}}-\frac {3 \left (10 \sqrt {a} \sqrt {b}+7 a+4 b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{5/2} \sqrt [4]{b} d \left (\sqrt {a}+\sqrt {b}\right )^{5/2}}-\frac {\cos (c+d x) \left ((7 a-3 b) (a+2 b)-6 b (2 a-b) \cos ^2(c+d x)\right )}{32 a^2 d (a-b)^2 \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}-\frac {\cos (c+d x) \left (a-b \cos ^2(c+d x)+b\right )}{8 a d (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2} \]

[Out]

-1/8*cos(d*x+c)*(a+b-b*cos(d*x+c)^2)/a/(a-b)/d/(a-b+2*b*cos(d*x+c)^2-b*cos(d*x+c)^4)^2-1/32*cos(d*x+c)*((7*a-3
*b)*(a+2*b)-6*(2*a-b)*b*cos(d*x+c)^2)/a^2/(a-b)^2/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))*(7*a+4*b-10*a^(1/2)*b^(1/2))/a^(5/2)/b^(1/4)/d/(a^(1/2)-b^(1/2))^(5/2)-3/6
4*arctanh(b^(1/4)*cos(d*x+c)/(a^(1/2)+b^(1/2))^(1/2))*(7*a+4*b+10*a^(1/2)*b^(1/2))/a^(5/2)/b^(1/4)/d/(a^(1/2)+
b^(1/2))^(5/2)

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Rubi [A]  time = 0.46, antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3215, 1092, 1178, 1166, 205, 208} \[ -\frac {\cos (c+d x) \left ((7 a-3 b) (a+2 b)-6 b (2 a-b) \cos ^2(c+d x)\right )}{32 a^2 d (a-b)^2 \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}-\frac {3 \left (-10 \sqrt {a} \sqrt {b}+7 a+4 b\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{5/2} \sqrt [4]{b} d \left (\sqrt {a}-\sqrt {b}\right )^{5/2}}-\frac {3 \left (10 \sqrt {a} \sqrt {b}+7 a+4 b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{5/2} \sqrt [4]{b} d \left (\sqrt {a}+\sqrt {b}\right )^{5/2}}-\frac {\cos (c+d x) \left (a-b \cos ^2(c+d x)+b\right )}{8 a d (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(7*a - 10*Sqrt[a]*Sqrt[b] + 4*b)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(64*a^(5/2)*(Sqrt
[a] - Sqrt[b])^(5/2)*b^(1/4)*d) - (3*(7*a + 10*Sqrt[a]*Sqrt[b] + 4*b)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt
[a] + Sqrt[b]]])/(64*a^(5/2)*(Sqrt[a] + Sqrt[b])^(5/2)*b^(1/4)*d) - (Cos[c + d*x]*(a + b - b*Cos[c + d*x]^2))/
(8*a*(a - b)*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4)^2) - (Cos[c + d*x]*((7*a - 3*b)*(a + 2*b) - 6*(
2*a - b)*b*Cos[c + d*x]^2))/(32*a^2*(a - b)^2*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 1092

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> -Simp[(x*(b^2 - 2*a*c + b*c*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[(b^2 - 2*a*c + 2*(p + 1)
*(b^2 - 4*a*c) + b*c*(4*p + 7)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*
a*c, 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 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 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 (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1}{\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 (a+b-b \cos ^2(c+d x)\right )}{8 a (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 {2 (a-b) b+4 b^2-4 \left (4 (a-b) b+4 b^2\right )+10 b^2 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 (a+b-b \cos ^2(c+d x)\right )}{8 a (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 ((7 a-3 b) (a+2 b)-6 (2 a-b) b \cos ^2(c+d x)\right )}{32 a^2 (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 {12 b^2 \left (7 a^2-5 a b+2 b^2\right )-24 (2 a-b) b^3 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 (a+b-b \cos ^2(c+d x)\right )}{8 a (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 ((7 a-3 b) (a+2 b)-6 (2 a-b) b \cos ^2(c+d x)\right )}{32 a^2 (a-b)^2 d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}+\frac {\left (3 \sqrt {b} \left (7 a-10 \sqrt {a} \sqrt {b}+4 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 a^{5/2} \left (\sqrt {a}-\sqrt {b}\right )^2 d}-\frac {\left (3 \sqrt {b} \left (7 a+10 \sqrt {a} \sqrt {b}+4 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 a^{5/2} \left (\sqrt {a}+\sqrt {b}\right )^2 d}\\ &=-\frac {3 \left (7 a-10 \sqrt {a} \sqrt {b}+4 b\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{5/2} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} \sqrt [4]{b} d}-\frac {3 \left (7 a+10 \sqrt {a} \sqrt {b}+4 b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{5/2} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} \sqrt [4]{b} d}-\frac {\cos (c+d x) \left (a+b-b \cos ^2(c+d x)\right )}{8 a (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 ((7 a-3 b) (a+2 b)-6 (2 a-b) b \cos ^2(c+d x)\right )}{32 a^2 (a-b)^2 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 = 1.28, size = 784, normalized size = 2.50 \[ \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 {-4 \text {$\#$1}^6 a b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+2 \text {$\#$1}^6 b^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+28 \text {$\#$1}^4 a^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-24 \text {$\#$1}^4 a b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+10 \text {$\#$1}^4 b^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+14 i \text {$\#$1}^2 a^2 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )-28 \text {$\#$1}^2 a^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-12 i \text {$\#$1}^2 a b \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )-2 i a b \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )+24 \text {$\#$1}^2 a b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+5 i \text {$\#$1}^2 b^2 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )+i b^2 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )-10 \text {$\#$1}^2 b^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+2 i \text {$\#$1}^6 a b \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )-i \text {$\#$1}^6 b^2 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )-14 i \text {$\#$1}^4 a^2 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )+12 i \text {$\#$1}^4 a b \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )-5 i \text {$\#$1}^4 b^2 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )+4 a b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-2 b^2 \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) \left (7 a^2+3 b (b-2 a) \cos (2 (c+d x))+5 a b-3 b^2\right )}{8 a+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))-3 b}-\frac {512 a (a-b) \cos (c+d x) (2 a-b \cos (2 (c+d x))+b)}{(-8 a-4 b \cos (2 (c+d x))+b \cos (4 (c+d x))+3 b)^2}}{128 a^2 d (a-b)^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-32*Cos[c + d*x]*(7*a^2 + 5*a*b - 3*b^2 + 3*b*(-2*a + 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)*Cos[c + d*x]*(2*a + b - b*Cos[2*(c + d*x)]))/(-8*a + 3*b - 4*b*Cos[2*
(c + d*x)] + b*Cos[4*(c + d*x)])^2 + (3*I)*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & ,
(4*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - 2*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - (2*I)*a*b*
Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + I*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - 28*a^2*ArcTan[Sin[c + d*x]/(Cos[
c + d*x] - #1)]*#1^2 + 24*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - 10*b^2*ArcTan[Sin[c + d*x]/(Cos[
c + d*x] - #1)]*#1^2 + (14*I)*a^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (12*I)*a*b*Log[1 - 2*Cos[c + d*x]*#
1 + #1^2]*#1^2 + (5*I)*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + 28*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] -
#1)]*#1^4 - 24*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + 10*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] -
#1)]*#1^4 - (14*I)*a^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + (12*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*
#1^4 - (5*I)*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 - 4*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6
+ 2*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 + (2*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6 - I*b
^2*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) & ])/(128*a^2*(
a - b)^2*d)

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fricas [B]  time = 1.41, size = 4160, normalized size = 13.29 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/128*(24*(2*a*b^2 - b^3)*cos(d*x + c)^7 - 4*(7*a^2*b + 35*a*b^2 - 18*b^3)*cos(d*x + c)^5 - 8*(a^2*b - 22*a*b
^2 + 9*b^3)*cos(d*x + c)^3 + 3*((a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^8 - 4*(a^4*b^2 - 2*a^3*b^3 + a^
2*b^4)*d*cos(d*x + c)^6 - 2*(a^5*b - 5*a^4*b^2 + 7*a^3*b^3 - 3*a^2*b^4)*d*cos(d*x + c)^4 + 4*(a^5*b - 3*a^4*b^
2 + 3*a^3*b^3 - a^2*b^4)*d*cos(d*x + c)^2 + (a^6 - 4*a^5*b + 6*a^4*b^2 - 4*a^3*b^3 + a^2*b^4)*d)*sqrt(-(105*a^
4 - 210*a^3*b + 189*a^2*b^2 - 84*a*b^3 + 16*b^4 + (a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*
b^5)*d^2*sqrt((2401*a^4 - 5292*a^3*b + 4974*a^2*b^2 - 2268*a*b^3 + 441*b^4)/((a^15*b - 10*a^14*b^2 + 45*a^13*b
^3 - 120*a^12*b^4 + 210*a^11*b^5 - 252*a^10*b^6 + 210*a^9*b^7 - 120*a^8*b^8 + 45*a^7*b^9 - 10*a^6*b^10 + a^5*b
^11)*d^4)))/((a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2))*log(27*(2401*a^4 - 4802*a^
3*b + 4189*a^2*b^2 - 1788*a*b^3 + 336*b^4)*cos(d*x + c) - 27*((11*a^12*b - 66*a^11*b^2 + 169*a^10*b^3 - 240*a^
9*b^4 + 205*a^8*b^5 - 106*a^7*b^6 + 31*a^6*b^7 - 4*a^5*b^8)*d^3*sqrt((2401*a^4 - 5292*a^3*b + 4974*a^2*b^2 - 2
268*a*b^3 + 441*b^4)/((a^15*b - 10*a^14*b^2 + 45*a^13*b^3 - 120*a^12*b^4 + 210*a^11*b^5 - 252*a^10*b^6 + 210*a
^9*b^7 - 120*a^8*b^8 + 45*a^7*b^9 - 10*a^6*b^10 + a^5*b^11)*d^4)) - (343*a^7 - 623*a^6*b + 515*a^5*b^2 - 213*a
^4*b^3 + 42*a^3*b^4)*d)*sqrt(-(105*a^4 - 210*a^3*b + 189*a^2*b^2 - 84*a*b^3 + 16*b^4 + (a^10 - 5*a^9*b + 10*a^
8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2*sqrt((2401*a^4 - 5292*a^3*b + 4974*a^2*b^2 - 2268*a*b^3 + 441*b^
4)/((a^15*b - 10*a^14*b^2 + 45*a^13*b^3 - 120*a^12*b^4 + 210*a^11*b^5 - 252*a^10*b^6 + 210*a^9*b^7 - 120*a^8*b
^8 + 45*a^7*b^9 - 10*a^6*b^10 + a^5*b^11)*d^4)))/((a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*
b^5)*d^2))) - 3*((a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^8 - 4*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*
x + c)^6 - 2*(a^5*b - 5*a^4*b^2 + 7*a^3*b^3 - 3*a^2*b^4)*d*cos(d*x + c)^4 + 4*(a^5*b - 3*a^4*b^2 + 3*a^3*b^3 -
a^2*b^4)*d*cos(d*x + c)^2 + (a^6 - 4*a^5*b + 6*a^4*b^2 - 4*a^3*b^3 + a^2*b^4)*d)*sqrt(-(105*a^4 - 210*a^3*b +
189*a^2*b^2 - 84*a*b^3 + 16*b^4 - (a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2*sqrt((
2401*a^4 - 5292*a^3*b + 4974*a^2*b^2 - 2268*a*b^3 + 441*b^4)/((a^15*b - 10*a^14*b^2 + 45*a^13*b^3 - 120*a^12*b
^4 + 210*a^11*b^5 - 252*a^10*b^6 + 210*a^9*b^7 - 120*a^8*b^8 + 45*a^7*b^9 - 10*a^6*b^10 + a^5*b^11)*d^4)))/((a
^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2))*log(27*(2401*a^4 - 4802*a^3*b + 4189*a^2*
b^2 - 1788*a*b^3 + 336*b^4)*cos(d*x + c) - 27*((11*a^12*b - 66*a^11*b^2 + 169*a^10*b^3 - 240*a^9*b^4 + 205*a^8
*b^5 - 106*a^7*b^6 + 31*a^6*b^7 - 4*a^5*b^8)*d^3*sqrt((2401*a^4 - 5292*a^3*b + 4974*a^2*b^2 - 2268*a*b^3 + 441
*b^4)/((a^15*b - 10*a^14*b^2 + 45*a^13*b^3 - 120*a^12*b^4 + 210*a^11*b^5 - 252*a^10*b^6 + 210*a^9*b^7 - 120*a^
8*b^8 + 45*a^7*b^9 - 10*a^6*b^10 + a^5*b^11)*d^4)) + (343*a^7 - 623*a^6*b + 515*a^5*b^2 - 213*a^4*b^3 + 42*a^3
*b^4)*d)*sqrt(-(105*a^4 - 210*a^3*b + 189*a^2*b^2 - 84*a*b^3 + 16*b^4 - (a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*
b^3 + 5*a^6*b^4 - a^5*b^5)*d^2*sqrt((2401*a^4 - 5292*a^3*b + 4974*a^2*b^2 - 2268*a*b^3 + 441*b^4)/((a^15*b - 1
0*a^14*b^2 + 45*a^13*b^3 - 120*a^12*b^4 + 210*a^11*b^5 - 252*a^10*b^6 + 210*a^9*b^7 - 120*a^8*b^8 + 45*a^7*b^9
- 10*a^6*b^10 + a^5*b^11)*d^4)))/((a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2))) - 3
*((a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^8 - 4*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^6 - 2*(a
^5*b - 5*a^4*b^2 + 7*a^3*b^3 - 3*a^2*b^4)*d*cos(d*x + c)^4 + 4*(a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d*cos
(d*x + c)^2 + (a^6 - 4*a^5*b + 6*a^4*b^2 - 4*a^3*b^3 + a^2*b^4)*d)*sqrt(-(105*a^4 - 210*a^3*b + 189*a^2*b^2 -
84*a*b^3 + 16*b^4 + (a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2*sqrt((2401*a^4 - 5292
*a^3*b + 4974*a^2*b^2 - 2268*a*b^3 + 441*b^4)/((a^15*b - 10*a^14*b^2 + 45*a^13*b^3 - 120*a^12*b^4 + 210*a^11*b
^5 - 252*a^10*b^6 + 210*a^9*b^7 - 120*a^8*b^8 + 45*a^7*b^9 - 10*a^6*b^10 + a^5*b^11)*d^4)))/((a^10 - 5*a^9*b +
10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2))*log(-27*(2401*a^4 - 4802*a^3*b + 4189*a^2*b^2 - 1788*a*b
^3 + 336*b^4)*cos(d*x + c) - 27*((11*a^12*b - 66*a^11*b^2 + 169*a^10*b^3 - 240*a^9*b^4 + 205*a^8*b^5 - 106*a^7
*b^6 + 31*a^6*b^7 - 4*a^5*b^8)*d^3*sqrt((2401*a^4 - 5292*a^3*b + 4974*a^2*b^2 - 2268*a*b^3 + 441*b^4)/((a^15*b
- 10*a^14*b^2 + 45*a^13*b^3 - 120*a^12*b^4 + 210*a^11*b^5 - 252*a^10*b^6 + 210*a^9*b^7 - 120*a^8*b^8 + 45*a^7
*b^9 - 10*a^6*b^10 + a^5*b^11)*d^4)) - (343*a^7 - 623*a^6*b + 515*a^5*b^2 - 213*a^4*b^3 + 42*a^3*b^4)*d)*sqrt(
-(105*a^4 - 210*a^3*b + 189*a^2*b^2 - 84*a*b^3 + 16*b^4 + (a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^
4 - a^5*b^5)*d^2*sqrt((2401*a^4 - 5292*a^3*b + 4974*a^2*b^2 - 2268*a*b^3 + 441*b^4)/((a^15*b - 10*a^14*b^2 + 4
5*a^13*b^3 - 120*a^12*b^4 + 210*a^11*b^5 - 252*a^10*b^6 + 210*a^9*b^7 - 120*a^8*b^8 + 45*a^7*b^9 - 10*a^6*b^10
+ a^5*b^11)*d^4)))/((a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2))) + 3*((a^4*b^2 - 2
*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^8 - 4*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^6 - 2*(a^5*b - 5*a^4*b
^2 + 7*a^3*b^3 - 3*a^2*b^4)*d*cos(d*x + c)^4 + 4*(a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d*cos(d*x + c)^2 +
(a^6 - 4*a^5*b + 6*a^4*b^2 - 4*a^3*b^3 + a^2*b^4)*d)*sqrt(-(105*a^4 - 210*a^3*b + 189*a^2*b^2 - 84*a*b^3 + 16*
b^4 - (a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2*sqrt((2401*a^4 - 5292*a^3*b + 4974*
a^2*b^2 - 2268*a*b^3 + 441*b^4)/((a^15*b - 10*a^14*b^2 + 45*a^13*b^3 - 120*a^12*b^4 + 210*a^11*b^5 - 252*a^10*
b^6 + 210*a^9*b^7 - 120*a^8*b^8 + 45*a^7*b^9 - 10*a^6*b^10 + a^5*b^11)*d^4)))/((a^10 - 5*a^9*b + 10*a^8*b^2 -
10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2))*log(-27*(2401*a^4 - 4802*a^3*b + 4189*a^2*b^2 - 1788*a*b^3 + 336*b^4)*
cos(d*x + c) - 27*((11*a^12*b - 66*a^11*b^2 + 169*a^10*b^3 - 240*a^9*b^4 + 205*a^8*b^5 - 106*a^7*b^6 + 31*a^6*
b^7 - 4*a^5*b^8)*d^3*sqrt((2401*a^4 - 5292*a^3*b + 4974*a^2*b^2 - 2268*a*b^3 + 441*b^4)/((a^15*b - 10*a^14*b^2
+ 45*a^13*b^3 - 120*a^12*b^4 + 210*a^11*b^5 - 252*a^10*b^6 + 210*a^9*b^7 - 120*a^8*b^8 + 45*a^7*b^9 - 10*a^6*
b^10 + a^5*b^11)*d^4)) + (343*a^7 - 623*a^6*b + 515*a^5*b^2 - 213*a^4*b^3 + 42*a^3*b^4)*d)*sqrt(-(105*a^4 - 21
0*a^3*b + 189*a^2*b^2 - 84*a*b^3 + 16*b^4 - (a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d
^2*sqrt((2401*a^4 - 5292*a^3*b + 4974*a^2*b^2 - 2268*a*b^3 + 441*b^4)/((a^15*b - 10*a^14*b^2 + 45*a^13*b^3 - 1
20*a^12*b^4 + 210*a^11*b^5 - 252*a^10*b^6 + 210*a^9*b^7 - 120*a^8*b^8 + 45*a^7*b^9 - 10*a^6*b^10 + a^5*b^11)*d
^4)))/((a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2))) + 4*(11*a^3 + 4*a^2*b - 21*a*b^
2 + 6*b^3)*cos(d*x + c))/((a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^8 - 4*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)
*d*cos(d*x + c)^6 - 2*(a^5*b - 5*a^4*b^2 + 7*a^3*b^3 - 3*a^2*b^4)*d*cos(d*x + c)^4 + 4*(a^5*b - 3*a^4*b^2 + 3*
a^3*b^3 - a^2*b^4)*d*cos(d*x + c)^2 + (a^6 - 4*a^5*b + 6*a^4*b^2 - 4*a^3*b^3 + a^2*b^4)*d)

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giac [B]  time = 1.75, size = 766, normalized size = 2.45 \[ -\frac {3 \, {\left (7 \, a^{3} - 5 \, a^{2} b + 2 \, a b^{2} - {\left (11 \, a^{2} - 11 \, a b + 4 \, b^{2}\right )} \sqrt {a b}\right )} \sqrt {-b^{2} - \sqrt {a b} b} \arctan \left (\frac {\cos \left (d x + c\right )}{d \sqrt {-\frac {a^{4} b d^{2} - 2 \, a^{3} b^{2} d^{2} + a^{2} b^{3} d^{2} + \sqrt {{\left (a^{4} b d^{2} - 2 \, a^{3} b^{2} d^{2} + a^{2} b^{3} d^{2}\right )}^{2} + {\left (a^{4} b d^{4} - 2 \, a^{3} b^{2} d^{4} + a^{2} b^{3} d^{4}\right )} {\left (a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right )}}}{a^{4} b d^{4} - 2 \, a^{3} b^{2} d^{4} + a^{2} b^{3} d^{4}}}}\right )}{64 \, {\left (a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right )} d {\left | b \right |}} - \frac {3 \, {\left (7 \, a^{3} - 5 \, a^{2} b + 2 \, a b^{2} + {\left (11 \, a^{2} - 11 \, a b + 4 \, b^{2}\right )} \sqrt {a b}\right )} \sqrt {-b^{2} + \sqrt {a b} b} \arctan \left (\frac {\cos \left (d x + c\right )}{d \sqrt {-\frac {a^{4} b d^{2} - 2 \, a^{3} b^{2} d^{2} + a^{2} b^{3} d^{2} - \sqrt {{\left (a^{4} b d^{2} - 2 \, a^{3} b^{2} d^{2} + a^{2} b^{3} d^{2}\right )}^{2} + {\left (a^{4} b d^{4} - 2 \, a^{3} b^{2} d^{4} + a^{2} b^{3} d^{4}\right )} {\left (a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right )}}}{a^{4} b d^{4} - 2 \, a^{3} b^{2} d^{4} + a^{2} b^{3} d^{4}}}}\right )}{64 \, {\left (a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right )} d {\left | b \right |}} - \frac {\frac {12 \, a b^{2} \cos \left (d x + c\right )^{7}}{d} - \frac {6 \, b^{3} \cos \left (d x + c\right )^{7}}{d} - \frac {7 \, a^{2} b \cos \left (d x + c\right )^{5}}{d} - \frac {35 \, a b^{2} \cos \left (d x + c\right )^{5}}{d} + \frac {18 \, b^{3} \cos \left (d x + c\right )^{5}}{d} - \frac {2 \, a^{2} b \cos \left (d x + c\right )^{3}}{d} + \frac {44 \, a b^{2} \cos \left (d x + c\right )^{3}}{d} - \frac {18 \, b^{3} \cos \left (d x + c\right )^{3}}{d} + \frac {11 \, a^{3} \cos \left (d x + c\right )}{d} + \frac {4 \, a^{2} b \cos \left (d x + c\right )}{d} - \frac {21 \, a b^{2} \cos \left (d x + c\right )}{d} + \frac {6 \, b^{3} \cos \left (d x + c\right )}{d}}{32 \, {\left (b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} - a + b\right )}^{2} {\left (a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/64*(7*a^3 - 5*a^2*b + 2*a*b^2 - (11*a^2 - 11*a*b + 4*b^2)*sqrt(a*b))*sqrt(-b^2 - sqrt(a*b)*b)*arctan(cos(d*
x + c)/(d*sqrt(-(a^4*b*d^2 - 2*a^3*b^2*d^2 + a^2*b^3*d^2 + sqrt((a^4*b*d^2 - 2*a^3*b^2*d^2 + a^2*b^3*d^2)^2 +
(a^4*b*d^4 - 2*a^3*b^2*d^4 + a^2*b^3*d^4)*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)))/(a^4*b*d^4 - 2*a^3*b^2*d^4 +
a^2*b^3*d^4))))/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d*abs(b)) - 3/64*(7*a^3 - 5*a^2*b + 2*a*b^2 + (11*a^2
- 11*a*b + 4*b^2)*sqrt(a*b))*sqrt(-b^2 + sqrt(a*b)*b)*arctan(cos(d*x + c)/(d*sqrt(-(a^4*b*d^2 - 2*a^3*b^2*d^2
+ a^2*b^3*d^2 - sqrt((a^4*b*d^2 - 2*a^3*b^2*d^2 + a^2*b^3*d^2)^2 + (a^4*b*d^4 - 2*a^3*b^2*d^4 + a^2*b^3*d^4)*(
a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)))/(a^4*b*d^4 - 2*a^3*b^2*d^4 + a^2*b^3*d^4))))/((a^6 - 3*a^5*b + 3*a^4*b^
2 - a^3*b^3)*d*abs(b)) - 1/32*(12*a*b^2*cos(d*x + c)^7/d - 6*b^3*cos(d*x + c)^7/d - 7*a^2*b*cos(d*x + c)^5/d -
35*a*b^2*cos(d*x + c)^5/d + 18*b^3*cos(d*x + c)^5/d - 2*a^2*b*cos(d*x + c)^3/d + 44*a*b^2*cos(d*x + c)^3/d -
18*b^3*cos(d*x + c)^3/d + 11*a^3*cos(d*x + c)/d + 4*a^2*b*cos(d*x + c)/d - 21*a*b^2*cos(d*x + c)/d + 6*b^3*cos
(d*x + c)/d)/((b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 - a + b)^2*(a^4 - 2*a^3*b + a^2*b^2))

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maple [B]  time = 0.64, size = 1281, normalized size = 4.09 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/16/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+3/32/d*b/a^2/(cos(d*x+c)^2+(a*b)^(1/2)
/b-1)^2/(a^2-2*a*b+b^2)*cos(d*x+c)^3-9/64/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+3/64/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)^3-11/64/d/b/a/(cos(
d*x+c)^2+(a*b)^(1/2)/b-1)^2/(a-b)*cos(d*x+c)+3/32/d/a^2/(cos(d*x+c)^2+(a*b)^(1/2)/b-1)^2/(a-b)*cos(d*x+c)-5/64
/d/(a*b)^(1/2)/a/(cos(d*x+c)^2+(a*b)^(1/2)/b-1)^2/(a-b)*cos(d*x+c)-3/16/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+3/32/d*b^2/a^2/(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))-21/64/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))+27/64/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^2-3/16/d*b^3/a^2/(a*b)^(1/2)/(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))-3/16/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+3/32/d*b/a^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+9/64/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)^3-3/64/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)^3-11/64/d/b/a/(cos(d*x+c)^2-1-(a*b)^(1/2)/b)^2/(a-b)*cos(d*x+c)+3/3
2/d/a^2/(cos(d*x+c)^2-1-(a*b)^(1/2)/b)^2/(a-b)*cos(d*x+c)+5/64/d/(a*b)^(1/2)/a/(cos(d*x+c)^2-1-(a*b)^(1/2)/b)^
2/(a-b)*cos(d*x+c)+3/16/d/a/(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))*b-3/32/d*b^2/a^2/(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))-21/64/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))+27/64/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^2-3/16/d*b^3/a^2/(a*b)^(1/2)/(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))

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

mupad [B]  time = 18.91, size = 5753, normalized size = 18.38 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan(((((3*(16384*a^5*b^7 - 73728*a^6*b^6 + 155648*a^7*b^5 - 155648*a^8*b^4 + 57344*a^9*b^3))/(16384*(a^10 -
4*a^9*b + a^6*b^4 - 4*a^7*b^3 + 6*a^8*b^2)) - (cos(c + d*x)*((9*(49*a^2*(a^15*b)^(1/2) + 21*b^2*(a^15*b)^(1/2)
- 105*a^9*b - 16*a^5*b^5 + 84*a^6*b^4 - 189*a^7*b^3 + 210*a^8*b^2 - 54*a*b*(a^15*b)^(1/2)))/(16384*(a^15*b -
a^10*b^6 + 5*a^11*b^5 - 10*a^12*b^4 + 10*a^13*b^3 - 5*a^14*b^2)))^(1/2)*(16384*a^5*b^8 - 65536*a^6*b^7 + 98304
*a^7*b^6 - 65536*a^8*b^5 + 16384*a^9*b^4))/(256*(a^8 - 4*a^7*b + a^4*b^4 - 4*a^5*b^3 + 6*a^6*b^2)))*((9*(49*a^
2*(a^15*b)^(1/2) + 21*b^2*(a^15*b)^(1/2) - 105*a^9*b - 16*a^5*b^5 + 84*a^6*b^4 - 189*a^7*b^3 + 210*a^8*b^2 - 5
4*a*b*(a^15*b)^(1/2)))/(16384*(a^15*b - a^10*b^6 + 5*a^11*b^5 - 10*a^12*b^4 + 10*a^13*b^3 - 5*a^14*b^2)))^(1/2
) + (cos(c + d*x)*(144*b^7 - 612*a*b^6 + 1089*a^2*b^5 - 990*a^3*b^4 + 441*a^4*b^3))/(256*(a^8 - 4*a^7*b + a^4*
b^4 - 4*a^5*b^3 + 6*a^6*b^2)))*((9*(49*a^2*(a^15*b)^(1/2) + 21*b^2*(a^15*b)^(1/2) - 105*a^9*b - 16*a^5*b^5 + 8
4*a^6*b^4 - 189*a^7*b^3 + 210*a^8*b^2 - 54*a*b*(a^15*b)^(1/2)))/(16384*(a^15*b - a^10*b^6 + 5*a^11*b^5 - 10*a^
12*b^4 + 10*a^13*b^3 - 5*a^14*b^2)))^(1/2)*1i - (((3*(16384*a^5*b^7 - 73728*a^6*b^6 + 155648*a^7*b^5 - 155648*
a^8*b^4 + 57344*a^9*b^3))/(16384*(a^10 - 4*a^9*b + a^6*b^4 - 4*a^7*b^3 + 6*a^8*b^2)) + (cos(c + d*x)*((9*(49*a
^2*(a^15*b)^(1/2) + 21*b^2*(a^15*b)^(1/2) - 105*a^9*b - 16*a^5*b^5 + 84*a^6*b^4 - 189*a^7*b^3 + 210*a^8*b^2 -
54*a*b*(a^15*b)^(1/2)))/(16384*(a^15*b - a^10*b^6 + 5*a^11*b^5 - 10*a^12*b^4 + 10*a^13*b^3 - 5*a^14*b^2)))^(1/
2)*(16384*a^5*b^8 - 65536*a^6*b^7 + 98304*a^7*b^6 - 65536*a^8*b^5 + 16384*a^9*b^4))/(256*(a^8 - 4*a^7*b + a^4*
b^4 - 4*a^5*b^3 + 6*a^6*b^2)))*((9*(49*a^2*(a^15*b)^(1/2) + 21*b^2*(a^15*b)^(1/2) - 105*a^9*b - 16*a^5*b^5 + 8
4*a^6*b^4 - 189*a^7*b^3 + 210*a^8*b^2 - 54*a*b*(a^15*b)^(1/2)))/(16384*(a^15*b - a^10*b^6 + 5*a^11*b^5 - 10*a^
12*b^4 + 10*a^13*b^3 - 5*a^14*b^2)))^(1/2) - (cos(c + d*x)*(144*b^7 - 612*a*b^6 + 1089*a^2*b^5 - 990*a^3*b^4 +
441*a^4*b^3))/(256*(a^8 - 4*a^7*b + a^4*b^4 - 4*a^5*b^3 + 6*a^6*b^2)))*((9*(49*a^2*(a^15*b)^(1/2) + 21*b^2*(a
^15*b)^(1/2) - 105*a^9*b - 16*a^5*b^5 + 84*a^6*b^4 - 189*a^7*b^3 + 210*a^8*b^2 - 54*a*b*(a^15*b)^(1/2)))/(1638
4*(a^15*b - a^10*b^6 + 5*a^11*b^5 - 10*a^12*b^4 + 10*a^13*b^3 - 5*a^14*b^2)))^(1/2)*1i)/((3*(684*a*b^5 - 144*b
^6 - 1233*a^2*b^4 + 882*a^3*b^3))/(8192*(a^10 - 4*a^9*b + a^6*b^4 - 4*a^7*b^3 + 6*a^8*b^2)) + (((3*(16384*a^5*
b^7 - 73728*a^6*b^6 + 155648*a^7*b^5 - 155648*a^8*b^4 + 57344*a^9*b^3))/(16384*(a^10 - 4*a^9*b + a^6*b^4 - 4*a
^7*b^3 + 6*a^8*b^2)) - (cos(c + d*x)*((9*(49*a^2*(a^15*b)^(1/2) + 21*b^2*(a^15*b)^(1/2) - 105*a^9*b - 16*a^5*b
^5 + 84*a^6*b^4 - 189*a^7*b^3 + 210*a^8*b^2 - 54*a*b*(a^15*b)^(1/2)))/(16384*(a^15*b - a^10*b^6 + 5*a^11*b^5 -
10*a^12*b^4 + 10*a^13*b^3 - 5*a^14*b^2)))^(1/2)*(16384*a^5*b^8 - 65536*a^6*b^7 + 98304*a^7*b^6 - 65536*a^8*b^
5 + 16384*a^9*b^4))/(256*(a^8 - 4*a^7*b + a^4*b^4 - 4*a^5*b^3 + 6*a^6*b^2)))*((9*(49*a^2*(a^15*b)^(1/2) + 21*b
^2*(a^15*b)^(1/2) - 105*a^9*b - 16*a^5*b^5 + 84*a^6*b^4 - 189*a^7*b^3 + 210*a^8*b^2 - 54*a*b*(a^15*b)^(1/2)))/
(16384*(a^15*b - a^10*b^6 + 5*a^11*b^5 - 10*a^12*b^4 + 10*a^13*b^3 - 5*a^14*b^2)))^(1/2) + (cos(c + d*x)*(144*
b^7 - 612*a*b^6 + 1089*a^2*b^5 - 990*a^3*b^4 + 441*a^4*b^3))/(256*(a^8 - 4*a^7*b + a^4*b^4 - 4*a^5*b^3 + 6*a^6
*b^2)))*((9*(49*a^2*(a^15*b)^(1/2) + 21*b^2*(a^15*b)^(1/2) - 105*a^9*b - 16*a^5*b^5 + 84*a^6*b^4 - 189*a^7*b^3
+ 210*a^8*b^2 - 54*a*b*(a^15*b)^(1/2)))/(16384*(a^15*b - a^10*b^6 + 5*a^11*b^5 - 10*a^12*b^4 + 10*a^13*b^3 -
5*a^14*b^2)))^(1/2) + (((3*(16384*a^5*b^7 - 73728*a^6*b^6 + 155648*a^7*b^5 - 155648*a^8*b^4 + 57344*a^9*b^3))/
(16384*(a^10 - 4*a^9*b + a^6*b^4 - 4*a^7*b^3 + 6*a^8*b^2)) + (cos(c + d*x)*((9*(49*a^2*(a^15*b)^(1/2) + 21*b^2
*(a^15*b)^(1/2) - 105*a^9*b - 16*a^5*b^5 + 84*a^6*b^4 - 189*a^7*b^3 + 210*a^8*b^2 - 54*a*b*(a^15*b)^(1/2)))/(1
6384*(a^15*b - a^10*b^6 + 5*a^11*b^5 - 10*a^12*b^4 + 10*a^13*b^3 - 5*a^14*b^2)))^(1/2)*(16384*a^5*b^8 - 65536*
a^6*b^7 + 98304*a^7*b^6 - 65536*a^8*b^5 + 16384*a^9*b^4))/(256*(a^8 - 4*a^7*b + a^4*b^4 - 4*a^5*b^3 + 6*a^6*b^
2)))*((9*(49*a^2*(a^15*b)^(1/2) + 21*b^2*(a^15*b)^(1/2) - 105*a^9*b - 16*a^5*b^5 + 84*a^6*b^4 - 189*a^7*b^3 +
210*a^8*b^2 - 54*a*b*(a^15*b)^(1/2)))/(16384*(a^15*b - a^10*b^6 + 5*a^11*b^5 - 10*a^12*b^4 + 10*a^13*b^3 - 5*a
^14*b^2)))^(1/2) - (cos(c + d*x)*(144*b^7 - 612*a*b^6 + 1089*a^2*b^5 - 990*a^3*b^4 + 441*a^4*b^3))/(256*(a^8 -
4*a^7*b + a^4*b^4 - 4*a^5*b^3 + 6*a^6*b^2)))*((9*(49*a^2*(a^15*b)^(1/2) + 21*b^2*(a^15*b)^(1/2) - 105*a^9*b -
16*a^5*b^5 + 84*a^6*b^4 - 189*a^7*b^3 + 210*a^8*b^2 - 54*a*b*(a^15*b)^(1/2)))/(16384*(a^15*b - a^10*b^6 + 5*a
^11*b^5 - 10*a^12*b^4 + 10*a^13*b^3 - 5*a^14*b^2)))^(1/2)))*((9*(49*a^2*(a^15*b)^(1/2) + 21*b^2*(a^15*b)^(1/2)
- 105*a^9*b - 16*a^5*b^5 + 84*a^6*b^4 - 189*a^7*b^3 + 210*a^8*b^2 - 54*a*b*(a^15*b)^(1/2)))/(16384*(a^15*b -
a^10*b^6 + 5*a^11*b^5 - 10*a^12*b^4 + 10*a^13*b^3 - 5*a^14*b^2)))^(1/2)*2i)/d - ((cos(c + d*x)*(15*a*b + 11*a^
2 - 6*b^2))/(32*a^2*(a - b)) - (cos(c + d*x)^3*(a^2*b - 22*a*b^2 + 9*b^3))/(16*a^2*(a - b)^2) + (3*b*cos(c + d
*x)^7*(2*a*b - b^2))/(16*a^2*(a^2 - 2*a*b + b^2)) - (b*cos(c + d*x)^5*(35*a*b + 7*a^2 - 18*b^2))/(32*a^2*(a^2
- 2*a*b + b^2)))/(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*(16384*a^5*b^7 - 73728*a^6*b^6 + 155648*a^7*b^5 - 15564
8*a^8*b^4 + 57344*a^9*b^3))/(16384*(a^10 - 4*a^9*b + a^6*b^4 - 4*a^7*b^3 + 6*a^8*b^2)) - (cos(c + d*x)*(-(9*(4
9*a^2*(a^15*b)^(1/2) + 21*b^2*(a^15*b)^(1/2) + 105*a^9*b + 16*a^5*b^5 - 84*a^6*b^4 + 189*a^7*b^3 - 210*a^8*b^2
- 54*a*b*(a^15*b)^(1/2)))/(16384*(a^15*b - a^10*b^6 + 5*a^11*b^5 - 10*a^12*b^4 + 10*a^13*b^3 - 5*a^14*b^2)))^
(1/2)*(16384*a^5*b^8 - 65536*a^6*b^7 + 98304*a^7*b^6 - 65536*a^8*b^5 + 16384*a^9*b^4))/(256*(a^8 - 4*a^7*b + a
^4*b^4 - 4*a^5*b^3 + 6*a^6*b^2)))*(-(9*(49*a^2*(a^15*b)^(1/2) + 21*b^2*(a^15*b)^(1/2) + 105*a^9*b + 16*a^5*b^5
- 84*a^6*b^4 + 189*a^7*b^3 - 210*a^8*b^2 - 54*a*b*(a^15*b)^(1/2)))/(16384*(a^15*b - a^10*b^6 + 5*a^11*b^5 - 1
0*a^12*b^4 + 10*a^13*b^3 - 5*a^14*b^2)))^(1/2) + (cos(c + d*x)*(144*b^7 - 612*a*b^6 + 1089*a^2*b^5 - 990*a^3*b
^4 + 441*a^4*b^3))/(256*(a^8 - 4*a^7*b + a^4*b^4 - 4*a^5*b^3 + 6*a^6*b^2)))*(-(9*(49*a^2*(a^15*b)^(1/2) + 21*b
^2*(a^15*b)^(1/2) + 105*a^9*b + 16*a^5*b^5 - 84*a^6*b^4 + 189*a^7*b^3 - 210*a^8*b^2 - 54*a*b*(a^15*b)^(1/2)))/
(16384*(a^15*b - a^10*b^6 + 5*a^11*b^5 - 10*a^12*b^4 + 10*a^13*b^3 - 5*a^14*b^2)))^(1/2)*1i - (((3*(16384*a^5*
b^7 - 73728*a^6*b^6 + 155648*a^7*b^5 - 155648*a^8*b^4 + 57344*a^9*b^3))/(16384*(a^10 - 4*a^9*b + a^6*b^4 - 4*a
^7*b^3 + 6*a^8*b^2)) + (cos(c + d*x)*(-(9*(49*a^2*(a^15*b)^(1/2) + 21*b^2*(a^15*b)^(1/2) + 105*a^9*b + 16*a^5*
b^5 - 84*a^6*b^4 + 189*a^7*b^3 - 210*a^8*b^2 - 54*a*b*(a^15*b)^(1/2)))/(16384*(a^15*b - a^10*b^6 + 5*a^11*b^5
- 10*a^12*b^4 + 10*a^13*b^3 - 5*a^14*b^2)))^(1/2)*(16384*a^5*b^8 - 65536*a^6*b^7 + 98304*a^7*b^6 - 65536*a^8*b
^5 + 16384*a^9*b^4))/(256*(a^8 - 4*a^7*b + a^4*b^4 - 4*a^5*b^3 + 6*a^6*b^2)))*(-(9*(49*a^2*(a^15*b)^(1/2) + 21
*b^2*(a^15*b)^(1/2) + 105*a^9*b + 16*a^5*b^5 - 84*a^6*b^4 + 189*a^7*b^3 - 210*a^8*b^2 - 54*a*b*(a^15*b)^(1/2))
)/(16384*(a^15*b - a^10*b^6 + 5*a^11*b^5 - 10*a^12*b^4 + 10*a^13*b^3 - 5*a^14*b^2)))^(1/2) - (cos(c + d*x)*(14
4*b^7 - 612*a*b^6 + 1089*a^2*b^5 - 990*a^3*b^4 + 441*a^4*b^3))/(256*(a^8 - 4*a^7*b + a^4*b^4 - 4*a^5*b^3 + 6*a
^6*b^2)))*(-(9*(49*a^2*(a^15*b)^(1/2) + 21*b^2*(a^15*b)^(1/2) + 105*a^9*b + 16*a^5*b^5 - 84*a^6*b^4 + 189*a^7*
b^3 - 210*a^8*b^2 - 54*a*b*(a^15*b)^(1/2)))/(16384*(a^15*b - a^10*b^6 + 5*a^11*b^5 - 10*a^12*b^4 + 10*a^13*b^3
- 5*a^14*b^2)))^(1/2)*1i)/((3*(684*a*b^5 - 144*b^6 - 1233*a^2*b^4 + 882*a^3*b^3))/(8192*(a^10 - 4*a^9*b + a^6
*b^4 - 4*a^7*b^3 + 6*a^8*b^2)) + (((3*(16384*a^5*b^7 - 73728*a^6*b^6 + 155648*a^7*b^5 - 155648*a^8*b^4 + 57344
*a^9*b^3))/(16384*(a^10 - 4*a^9*b + a^6*b^4 - 4*a^7*b^3 + 6*a^8*b^2)) - (cos(c + d*x)*(-(9*(49*a^2*(a^15*b)^(1
/2) + 21*b^2*(a^15*b)^(1/2) + 105*a^9*b + 16*a^5*b^5 - 84*a^6*b^4 + 189*a^7*b^3 - 210*a^8*b^2 - 54*a*b*(a^15*b
)^(1/2)))/(16384*(a^15*b - a^10*b^6 + 5*a^11*b^5 - 10*a^12*b^4 + 10*a^13*b^3 - 5*a^14*b^2)))^(1/2)*(16384*a^5*
b^8 - 65536*a^6*b^7 + 98304*a^7*b^6 - 65536*a^8*b^5 + 16384*a^9*b^4))/(256*(a^8 - 4*a^7*b + a^4*b^4 - 4*a^5*b^
3 + 6*a^6*b^2)))*(-(9*(49*a^2*(a^15*b)^(1/2) + 21*b^2*(a^15*b)^(1/2) + 105*a^9*b + 16*a^5*b^5 - 84*a^6*b^4 + 1
89*a^7*b^3 - 210*a^8*b^2 - 54*a*b*(a^15*b)^(1/2)))/(16384*(a^15*b - a^10*b^6 + 5*a^11*b^5 - 10*a^12*b^4 + 10*a
^13*b^3 - 5*a^14*b^2)))^(1/2) + (cos(c + d*x)*(144*b^7 - 612*a*b^6 + 1089*a^2*b^5 - 990*a^3*b^4 + 441*a^4*b^3)
)/(256*(a^8 - 4*a^7*b + a^4*b^4 - 4*a^5*b^3 + 6*a^6*b^2)))*(-(9*(49*a^2*(a^15*b)^(1/2) + 21*b^2*(a^15*b)^(1/2)
+ 105*a^9*b + 16*a^5*b^5 - 84*a^6*b^4 + 189*a^7*b^3 - 210*a^8*b^2 - 54*a*b*(a^15*b)^(1/2)))/(16384*(a^15*b -
a^10*b^6 + 5*a^11*b^5 - 10*a^12*b^4 + 10*a^13*b^3 - 5*a^14*b^2)))^(1/2) + (((3*(16384*a^5*b^7 - 73728*a^6*b^6
+ 155648*a^7*b^5 - 155648*a^8*b^4 + 57344*a^9*b^3))/(16384*(a^10 - 4*a^9*b + a^6*b^4 - 4*a^7*b^3 + 6*a^8*b^2))
+ (cos(c + d*x)*(-(9*(49*a^2*(a^15*b)^(1/2) + 21*b^2*(a^15*b)^(1/2) + 105*a^9*b + 16*a^5*b^5 - 84*a^6*b^4 + 1
89*a^7*b^3 - 210*a^8*b^2 - 54*a*b*(a^15*b)^(1/2)))/(16384*(a^15*b - a^10*b^6 + 5*a^11*b^5 - 10*a^12*b^4 + 10*a
^13*b^3 - 5*a^14*b^2)))^(1/2)*(16384*a^5*b^8 - 65536*a^6*b^7 + 98304*a^7*b^6 - 65536*a^8*b^5 + 16384*a^9*b^4))
/(256*(a^8 - 4*a^7*b + a^4*b^4 - 4*a^5*b^3 + 6*a^6*b^2)))*(-(9*(49*a^2*(a^15*b)^(1/2) + 21*b^2*(a^15*b)^(1/2)
+ 105*a^9*b + 16*a^5*b^5 - 84*a^6*b^4 + 189*a^7*b^3 - 210*a^8*b^2 - 54*a*b*(a^15*b)^(1/2)))/(16384*(a^15*b - a
^10*b^6 + 5*a^11*b^5 - 10*a^12*b^4 + 10*a^13*b^3 - 5*a^14*b^2)))^(1/2) - (cos(c + d*x)*(144*b^7 - 612*a*b^6 +
1089*a^2*b^5 - 990*a^3*b^4 + 441*a^4*b^3))/(256*(a^8 - 4*a^7*b + a^4*b^4 - 4*a^5*b^3 + 6*a^6*b^2)))*(-(9*(49*a
^2*(a^15*b)^(1/2) + 21*b^2*(a^15*b)^(1/2) + 105*a^9*b + 16*a^5*b^5 - 84*a^6*b^4 + 189*a^7*b^3 - 210*a^8*b^2 -
54*a*b*(a^15*b)^(1/2)))/(16384*(a^15*b - a^10*b^6 + 5*a^11*b^5 - 10*a^12*b^4 + 10*a^13*b^3 - 5*a^14*b^2)))^(1/
2)))*(-(9*(49*a^2*(a^15*b)^(1/2) + 21*b^2*(a^15*b)^(1/2) + 105*a^9*b + 16*a^5*b^5 - 84*a^6*b^4 + 189*a^7*b^3 -
210*a^8*b^2 - 54*a*b*(a^15*b)^(1/2)))/(16384*(a^15*b - a^10*b^6 + 5*a^11*b^5 - 10*a^12*b^4 + 10*a^13*b^3 - 5*
a^14*b^2)))^(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)/(a-b*sin(d*x+c)**4)**3,x)

[Out]

Timed out

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