3.212 \(\int \frac {\sin ^9(c+d x)}{(a-b \sin ^4(c+d x))^2} \, dx\)

Optimal. Leaf size=236 \[ \frac {\sqrt {a} \left (5 \sqrt {a}-6 \sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 b^{9/4} d \left (\sqrt {a}-\sqrt {b}\right )^{3/2}}+\frac {\sqrt {a} \left (5 \sqrt {a}+6 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 b^{9/4} d \left (\sqrt {a}+\sqrt {b}\right )^{3/2}}-\frac {a \cos (c+d x) \left (a-b \cos ^2(c+d x)+b\right )}{4 b^2 d (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}-\frac {\cos (c+d x)}{b^2 d} \]

[Out]

-cos(d*x+c)/b^2/d-1/4*a*cos(d*x+c)*(a+b-b*cos(d*x+c)^2)/(a-b)/b^2/d/(a-b+2*b*cos(d*x+c)^2-b*cos(d*x+c)^4)+1/8*
arctan(b^(1/4)*cos(d*x+c)/(a^(1/2)-b^(1/2))^(1/2))*a^(1/2)*(5*a^(1/2)-6*b^(1/2))/b^(9/4)/d/(a^(1/2)-b^(1/2))^(
3/2)+1/8*arctanh(b^(1/4)*cos(d*x+c)/(a^(1/2)+b^(1/2))^(1/2))*a^(1/2)*(5*a^(1/2)+6*b^(1/2))/b^(9/4)/d/(a^(1/2)+
b^(1/2))^(3/2)

________________________________________________________________________________________

Rubi [A]  time = 0.48, antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3215, 1205, 1676, 1166, 205, 208} \[ -\frac {a \cos (c+d x) \left (a-b \cos ^2(c+d x)+b\right )}{4 b^2 d (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}+\frac {\sqrt {a} \left (5 \sqrt {a}-6 \sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 b^{9/4} d \left (\sqrt {a}-\sqrt {b}\right )^{3/2}}+\frac {\sqrt {a} \left (5 \sqrt {a}+6 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 b^{9/4} d \left (\sqrt {a}+\sqrt {b}\right )^{3/2}}-\frac {\cos (c+d x)}{b^2 d} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]^9/(a - b*Sin[c + d*x]^4)^2,x]

[Out]

(Sqrt[a]*(5*Sqrt[a] - 6*Sqrt[b])*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(8*(Sqrt[a] - Sqrt[b]
)^(3/2)*b^(9/4)*d) + (Sqrt[a]*(5*Sqrt[a] + 6*Sqrt[b])*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])
/(8*(Sqrt[a] + Sqrt[b])^(3/2)*b^(9/4)*d) - Cos[c + d*x]/(b^2*d) - (a*Cos[c + d*x]*(a + b - b*Cos[c + d*x]^2))/
(4*(a - b)*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 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 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 1676

Int[(Pq_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^2 + c*x^4), x], x
] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && Expon[Pq, x^2] > 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 ^9(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^4}{\left (a-b+2 b x^2-b x^4\right )^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a \cos (c+d x) \left (a+b-b \cos ^2(c+d x)\right )}{4 (a-b) 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 {2 a \left (a+\frac {a^2}{b}-4 b\right )-2 a (7 a-8 b) x^2+8 a (a-b) x^4}{a-b+2 b x^2-b x^4} \, dx,x,\cos (c+d x)\right )}{8 a (a-b) b d}\\ &=-\frac {a \cos (c+d x) \left (a+b-b \cos ^2(c+d x)\right )}{4 (a-b) 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 \left (-\frac {8 a (a-b)}{b}+\frac {2 \left (a^2 (5 a-7 b)+a^2 b x^2\right )}{b \left (a-b+2 b x^2-b x^4\right )}\right ) \, dx,x,\cos (c+d x)\right )}{8 a (a-b) b d}\\ &=-\frac {\cos (c+d x)}{b^2 d}-\frac {a \cos (c+d x) \left (a+b-b \cos ^2(c+d x)\right )}{4 (a-b) 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 {a^2 (5 a-7 b)+a^2 b x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cos (c+d x)\right )}{4 a (a-b) b^2 d}\\ &=-\frac {\cos (c+d x)}{b^2 d}-\frac {a \cos (c+d x) \left (a+b-b \cos ^2(c+d x)\right )}{4 (a-b) b^2 d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}-\frac {\left (\sqrt {a} \left (5 \sqrt {a}-6 \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 )}{8 \left (\sqrt {a}-\sqrt {b}\right ) b^{3/2} d}+\frac {\left (\sqrt {a} \left (5 a+\sqrt {a} \sqrt {b}-6 b\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{8 (a-b) b^{3/2} d}\\ &=\frac {\sqrt {a} \left (5 \sqrt {a}-6 \sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 \left (\sqrt {a}-\sqrt {b}\right )^{3/2} b^{9/4} d}+\frac {\sqrt {a} \left (5 \sqrt {a}+6 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 \left (\sqrt {a}+\sqrt {b}\right )^{3/2} b^{9/4} d}-\frac {\cos (c+d x)}{b^2 d}-\frac {a \cos (c+d x) \left (a+b-b \cos ^2(c+d x)\right )}{4 (a-b) 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 = 1.19, size = 486, normalized size = 2.06 \[ -\frac {\frac {i a \text {RootSum}\left [\text {$\#$1}^8 b-4 \text {$\#$1}^6 b-16 \text {$\#$1}^4 a+6 \text {$\#$1}^4 b-4 \text {$\#$1}^2 b+b\& ,\frac {2 \text {$\#$1}^6 b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+40 \text {$\#$1}^4 a \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-54 \text {$\#$1}^4 b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+20 i \text {$\#$1}^2 a \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )-40 \text {$\#$1}^2 a \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-27 i \text {$\#$1}^2 b \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )+i b \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )+54 \text {$\#$1}^2 b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-i \text {$\#$1}^6 b \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )-20 i \text {$\#$1}^4 a \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )+27 i \text {$\#$1}^4 b \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )-2 b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )}{\text {$\#$1}^7 b-3 \text {$\#$1}^5 b-8 \text {$\#$1}^3 a+3 \text {$\#$1}^3 b-\text {$\#$1} b}\& \right ]}{a-b}+\frac {32 a \cos (c+d x) (2 a-b \cos (2 (c+d x))+b)}{(a-b) (8 a+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))-3 b)}+32 \cos (c+d x)}{32 b^2 d} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sin[c + d*x]^9/(a - b*Sin[c + d*x]^4)^2,x]

[Out]

-1/32*(32*Cos[c + d*x] + (32*a*Cos[c + d*x]*(2*a + b - b*Cos[2*(c + d*x)]))/((a - b)*(8*a - 3*b + 4*b*Cos[2*(c
+ d*x)] - b*Cos[4*(c + d*x)])) + (I*a*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-2
*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + I*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - 40*a*ArcTan[Sin[c + d*x]
/(Cos[c + d*x] - #1)]*#1^2 + 54*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (20*I)*a*Log[1 - 2*Cos[c + d
*x]*#1 + #1^2]*#1^2 - (27*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + 40*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x]
- #1)]*#1^4 - 54*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - (20*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]
*#1^4 + (27*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + 2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#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 - b))/
(b^2*d)

________________________________________________________________________________________

fricas [B]  time = 0.90, size = 2649, normalized size = 11.22 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^9/(a-b*sin(d*x+c)^4)^2,x, algorithm="fricas")

[Out]

-1/16*(16*(a*b - b^2)*cos(d*x + c)^5 - 4*(7*a*b - 8*b^2)*cos(d*x + c)^3 + ((a*b^3 - b^4)*d*cos(d*x + c)^4 - 2*
(a*b^3 - b^4)*d*cos(d*x + c)^2 - (a^2*b^2 - 2*a*b^3 + b^4)*d)*sqrt(-((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2
*sqrt((625*a^7 - 3450*a^6*b + 7161*a^5*b^2 - 6624*a^4*b^3 + 2304*a^3*b^4)/((a^6*b^9 - 6*a^5*b^10 + 15*a^4*b^11
- 20*a^3*b^12 + 15*a^2*b^13 - 6*a*b^14 + b^15)*d^4)) + 15*a^3 - 47*a^2*b + 36*a*b^2)/((a^3*b^4 - 3*a^2*b^5 +
3*a*b^6 - b^7)*d^2))*log((625*a^5 - 2625*a^4*b + 3684*a^3*b^2 - 1728*a^2*b^3)*cos(d*x + c) + (2*(2*a^4*b^7 - 9
*a^3*b^8 + 15*a^2*b^9 - 11*a*b^10 + 3*b^11)*d^3*sqrt((625*a^7 - 3450*a^6*b + 7161*a^5*b^2 - 6624*a^4*b^3 + 230
4*a^3*b^4)/((a^6*b^9 - 6*a^5*b^10 + 15*a^4*b^11 - 20*a^3*b^12 + 15*a^2*b^13 - 6*a*b^14 + b^15)*d^4)) - (125*a^
5*b^2 - 520*a^4*b^3 + 723*a^3*b^4 - 336*a^2*b^5)*d)*sqrt(-((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2*sqrt((625
*a^7 - 3450*a^6*b + 7161*a^5*b^2 - 6624*a^4*b^3 + 2304*a^3*b^4)/((a^6*b^9 - 6*a^5*b^10 + 15*a^4*b^11 - 20*a^3*
b^12 + 15*a^2*b^13 - 6*a*b^14 + b^15)*d^4)) + 15*a^3 - 47*a^2*b + 36*a*b^2)/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 -
b^7)*d^2))) - ((a*b^3 - b^4)*d*cos(d*x + c)^4 - 2*(a*b^3 - b^4)*d*cos(d*x + c)^2 - (a^2*b^2 - 2*a*b^3 + b^4)*d
)*sqrt(((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2*sqrt((625*a^7 - 3450*a^6*b + 7161*a^5*b^2 - 6624*a^4*b^3 + 2
304*a^3*b^4)/((a^6*b^9 - 6*a^5*b^10 + 15*a^4*b^11 - 20*a^3*b^12 + 15*a^2*b^13 - 6*a*b^14 + b^15)*d^4)) - 15*a^
3 + 47*a^2*b - 36*a*b^2)/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2))*log((625*a^5 - 2625*a^4*b + 3684*a^3*b^2
- 1728*a^2*b^3)*cos(d*x + c) + (2*(2*a^4*b^7 - 9*a^3*b^8 + 15*a^2*b^9 - 11*a*b^10 + 3*b^11)*d^3*sqrt((625*a^7
- 3450*a^6*b + 7161*a^5*b^2 - 6624*a^4*b^3 + 2304*a^3*b^4)/((a^6*b^9 - 6*a^5*b^10 + 15*a^4*b^11 - 20*a^3*b^12
+ 15*a^2*b^13 - 6*a*b^14 + b^15)*d^4)) + (125*a^5*b^2 - 520*a^4*b^3 + 723*a^3*b^4 - 336*a^2*b^5)*d)*sqrt(((a^
3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2*sqrt((625*a^7 - 3450*a^6*b + 7161*a^5*b^2 - 6624*a^4*b^3 + 2304*a^3*b^4
)/((a^6*b^9 - 6*a^5*b^10 + 15*a^4*b^11 - 20*a^3*b^12 + 15*a^2*b^13 - 6*a*b^14 + b^15)*d^4)) - 15*a^3 + 47*a^2*
b - 36*a*b^2)/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2))) - ((a*b^3 - b^4)*d*cos(d*x + c)^4 - 2*(a*b^3 - b^4
)*d*cos(d*x + c)^2 - (a^2*b^2 - 2*a*b^3 + b^4)*d)*sqrt(-((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2*sqrt((625*a
^7 - 3450*a^6*b + 7161*a^5*b^2 - 6624*a^4*b^3 + 2304*a^3*b^4)/((a^6*b^9 - 6*a^5*b^10 + 15*a^4*b^11 - 20*a^3*b^
12 + 15*a^2*b^13 - 6*a*b^14 + b^15)*d^4)) + 15*a^3 - 47*a^2*b + 36*a*b^2)/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^
7)*d^2))*log(-(625*a^5 - 2625*a^4*b + 3684*a^3*b^2 - 1728*a^2*b^3)*cos(d*x + c) + (2*(2*a^4*b^7 - 9*a^3*b^8 +
15*a^2*b^9 - 11*a*b^10 + 3*b^11)*d^3*sqrt((625*a^7 - 3450*a^6*b + 7161*a^5*b^2 - 6624*a^4*b^3 + 2304*a^3*b^4)/
((a^6*b^9 - 6*a^5*b^10 + 15*a^4*b^11 - 20*a^3*b^12 + 15*a^2*b^13 - 6*a*b^14 + b^15)*d^4)) - (125*a^5*b^2 - 520
*a^4*b^3 + 723*a^3*b^4 - 336*a^2*b^5)*d)*sqrt(-((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2*sqrt((625*a^7 - 3450
*a^6*b + 7161*a^5*b^2 - 6624*a^4*b^3 + 2304*a^3*b^4)/((a^6*b^9 - 6*a^5*b^10 + 15*a^4*b^11 - 20*a^3*b^12 + 15*a
^2*b^13 - 6*a*b^14 + b^15)*d^4)) + 15*a^3 - 47*a^2*b + 36*a*b^2)/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2)))
+ ((a*b^3 - b^4)*d*cos(d*x + c)^4 - 2*(a*b^3 - b^4)*d*cos(d*x + c)^2 - (a^2*b^2 - 2*a*b^3 + b^4)*d)*sqrt(((a^
3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2*sqrt((625*a^7 - 3450*a^6*b + 7161*a^5*b^2 - 6624*a^4*b^3 + 2304*a^3*b^4
)/((a^6*b^9 - 6*a^5*b^10 + 15*a^4*b^11 - 20*a^3*b^12 + 15*a^2*b^13 - 6*a*b^14 + b^15)*d^4)) - 15*a^3 + 47*a^2*
b - 36*a*b^2)/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2))*log(-(625*a^5 - 2625*a^4*b + 3684*a^3*b^2 - 1728*a^
2*b^3)*cos(d*x + c) + (2*(2*a^4*b^7 - 9*a^3*b^8 + 15*a^2*b^9 - 11*a*b^10 + 3*b^11)*d^3*sqrt((625*a^7 - 3450*a^
6*b + 7161*a^5*b^2 - 6624*a^4*b^3 + 2304*a^3*b^4)/((a^6*b^9 - 6*a^5*b^10 + 15*a^4*b^11 - 20*a^3*b^12 + 15*a^2*
b^13 - 6*a*b^14 + b^15)*d^4)) + (125*a^5*b^2 - 520*a^4*b^3 + 723*a^3*b^4 - 336*a^2*b^5)*d)*sqrt(((a^3*b^4 - 3*
a^2*b^5 + 3*a*b^6 - b^7)*d^2*sqrt((625*a^7 - 3450*a^6*b + 7161*a^5*b^2 - 6624*a^4*b^3 + 2304*a^3*b^4)/((a^6*b^
9 - 6*a^5*b^10 + 15*a^4*b^11 - 20*a^3*b^12 + 15*a^2*b^13 - 6*a*b^14 + b^15)*d^4)) - 15*a^3 + 47*a^2*b - 36*a*b
^2)/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2))) - 4*(5*a^2 - 7*a*b + 4*b^2)*cos(d*x + c))/((a*b^3 - b^4)*d*c
os(d*x + c)^4 - 2*(a*b^3 - b^4)*d*cos(d*x + c)^2 - (a^2*b^2 - 2*a*b^3 + b^4)*d)

________________________________________________________________________________________

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)^9/(a-b*sin(d*x+c)^4)^2,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]=[-54,3]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]=[65,-69]2/d*(5*((1-cos(c+d*x))/(1+cos(c+d*x)))^4*a^2-6*((1-cos(c+d*x))/(1+cos(c+d*x)))^4*a*b+20
*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*a^2-26*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*a*b+30*((1-cos(c+d*x))/(1+cos(c+d*
x)))^2*a^2-94*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*a*b+64*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*b^2+20*(1-cos(c+d*x))
/(1+cos(c+d*x))*a^2-14*(1-cos(c+d*x))/(1+cos(c+d*x))*a*b+5*a^2-4*a*b)/(-4*a*b^2+4*b^3)/(((1-cos(c+d*x))/(1+cos
(c+d*x)))^5*a+5*((1-cos(c+d*x))/(1+cos(c+d*x)))^4*a+10*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*a-16*((1-cos(c+d*x))/
(1+cos(c+d*x)))^3*b+10*((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+5*(1-cos(c+d
*x))/(1+cos(c+d*x))*a+a)+2/d/(-4*a*b^2+4*b^3)*2/d*(-a/4*(c+d*x)+(-10*a^4*b+72*a^3*b^2+33*a^3*b*sqrt(a^2-a*b+sq
rt(a*b)*(a-b))+10*a^3*a*b-15*a^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-118*a^2*b^3-102*a^2*b^2*sqrt(a^2-a*b+
sqrt(a*b)*(a-b))-72*a^2*b*a*b+42*a^2*b*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+56*a*b^4+61*a*b^3*sqrt(a^2-a*b+
sqrt(a*b)*(a-b))+118*a*b^2*a*b-19*a*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+12*b^4*sqrt(a^2-a*b+sqrt(a*b)*
(a-b))-56*b^3*a*b-4*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b)))*abs(a-b)/(24*a^4*b-96*a^3*b^2+112*a^2*b^3-32*
a*b^4-8*b^5)*(atan(tan(c+d*x)/sqrt(-(8*a+sqrt(8*a*8*a+4*(-4*a+4*b)*4*a))/2/(-4*a+4*b)))+pi*floor((c+d*x)/pi+1/
2))-(-10*a^4*b+72*a^3*b^2-33*a^3*b*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+10*a^3*a*b-15*a^3*sqrt(a*b)*sqrt(a^2-a*b+sqr
t(a*b)*(-a+b))-118*a^2*b^3+102*a^2*b^2*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-72*a^2*b*a*b+42*a^2*b*sqrt(a*b)*sqrt(a^2
-a*b+sqrt(a*b)*(-a+b))+56*a*b^4-61*a*b^3*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+118*a*b^2*a*b-19*a*b^2*sqrt(a*b)*sqrt(
a^2-a*b+sqrt(a*b)*(-a+b))-12*b^4*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-56*b^3*a*b-4*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a
*b)*(-a+b)))*abs(a-b)/(24*a^4*b-96*a^3*b^2+112*a^2*b^3-32*a*b^4-8*b^5)*(atan(tan(c+d*x)/sqrt(-(8*a-sqrt(8*a*8*
a+4*(-4*a+4*b)*4*a))/2/(-4*a+4*b)))+pi*floor((c+d*x)/pi+1/2)))

________________________________________________________________________________________

maple [B]  time = 0.34, size = 482, normalized size = 2.04 \[ -\frac {\cos \left (d x +c \right )}{b^{2} d}-\frac {a \left (\cos ^{3}\left (d x +c \right )\right )}{4 d b \left (b \left (\cos ^{4}\left (d x +c \right )\right )-2 b \left (\cos ^{2}\left (d x +c \right )\right )-a +b \right ) \left (a -b \right )}+\frac {a^{2} \cos \left (d x +c \right )}{4 d \,b^{2} \left (b \left (\cos ^{4}\left (d x +c \right )\right )-2 b \left (\cos ^{2}\left (d x +c \right )\right )-a +b \right ) \left (a -b \right )}+\frac {a \cos \left (d x +c \right )}{4 d b \left (b \left (\cos ^{4}\left (d x +c \right )\right )-2 b \left (\cos ^{2}\left (d x +c \right )\right )-a +b \right ) \left (a -b \right )}+\frac {a \arctanh \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{8 d b \left (a -b \right ) \sqrt {\left (\sqrt {a b}+b \right ) b}}-\frac {3 a \arctanh \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{4 d \left (a -b \right ) \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+b \right ) b}}+\frac {5 a^{2} \arctanh \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{8 d b \left (a -b \right ) \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+b \right ) b}}-\frac {a \arctan \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{8 d b \left (a -b \right ) \sqrt {\left (\sqrt {a b}-b \right ) b}}-\frac {3 a \arctan \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{4 d \left (a -b \right ) \sqrt {a b}\, \sqrt {\left (\sqrt {a b}-b \right ) b}}+\frac {5 a^{2} \arctan \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{8 d b \left (a -b \right ) \sqrt {a b}\, \sqrt {\left (\sqrt {a b}-b \right ) b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)^9/(a-b*sin(d*x+c)^4)^2,x)

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^9/(a-b*sin(d*x+c)^4)^2,x, algorithm="maxima")

[Out]

1/2*((2*a*b^2 - 3*b^3)*cos(2*d*x + 2*c)*cos(d*x + c) - 4*(2*a*b^2 - 3*b^3)*sin(3*d*x + 3*c)*sin(2*d*x + 2*c) +
(2*a*b^2 - 3*b^3)*sin(2*d*x + 2*c)*sin(d*x + c) - ((a*b^2 - b^3)*cos(9*d*x + 9*c) - 4*(a*b^2 - b^3)*cos(7*d*x
+ 7*c) - 2*(8*a^2*b - 11*a*b^2 + 3*b^3)*cos(5*d*x + 5*c) - 4*(a*b^2 - b^3)*cos(3*d*x + 3*c) + (a*b^2 - b^3)*c
os(d*x + c))*cos(10*d*x + 10*c) - (a*b^2 - b^3 - (2*a*b^2 - 3*b^3)*cos(8*d*x + 8*c) - (20*a^2*b - 17*a*b^2 + 2
*b^3)*cos(6*d*x + 6*c) - (20*a^2*b - 17*a*b^2 + 2*b^3)*cos(4*d*x + 4*c) - (2*a*b^2 - 3*b^3)*cos(2*d*x + 2*c))*
cos(9*d*x + 9*c) - (4*(2*a*b^2 - 3*b^3)*cos(7*d*x + 7*c) + 2*(16*a^2*b - 30*a*b^2 + 9*b^3)*cos(5*d*x + 5*c) +
4*(2*a*b^2 - 3*b^3)*cos(3*d*x + 3*c) - (2*a*b^2 - 3*b^3)*cos(d*x + c))*cos(8*d*x + 8*c) + 4*(a*b^2 - b^3 - (20
*a^2*b - 17*a*b^2 + 2*b^3)*cos(6*d*x + 6*c) - (20*a^2*b - 17*a*b^2 + 2*b^3)*cos(4*d*x + 4*c) - (2*a*b^2 - 3*b^
3)*cos(2*d*x + 2*c))*cos(7*d*x + 7*c) - (2*(160*a^3 - 196*a^2*b + 67*a*b^2 - 6*b^3)*cos(5*d*x + 5*c) + 4*(20*a
^2*b - 17*a*b^2 + 2*b^3)*cos(3*d*x + 3*c) - (20*a^2*b - 17*a*b^2 + 2*b^3)*cos(d*x + c))*cos(6*d*x + 6*c) + 2*(
8*a^2*b - 11*a*b^2 + 3*b^3 - (160*a^3 - 196*a^2*b + 67*a*b^2 - 6*b^3)*cos(4*d*x + 4*c) - (16*a^2*b - 30*a*b^2
+ 9*b^3)*cos(2*d*x + 2*c))*cos(5*d*x + 5*c) - (4*(20*a^2*b - 17*a*b^2 + 2*b^3)*cos(3*d*x + 3*c) - (20*a^2*b -
17*a*b^2 + 2*b^3)*cos(d*x + c))*cos(4*d*x + 4*c) + 4*(a*b^2 - b^3 - (2*a*b^2 - 3*b^3)*cos(2*d*x + 2*c))*cos(3*
d*x + 3*c) - (a*b^2 - b^3)*cos(d*x + c) + 2*((a*b^4 - b^5)*d*cos(9*d*x + 9*c)^2 + 16*(a*b^4 - b^5)*d*cos(7*d*x
+ 7*c)^2 + 4*(64*a^3*b^2 - 112*a^2*b^3 + 57*a*b^4 - 9*b^5)*d*cos(5*d*x + 5*c)^2 + 16*(a*b^4 - b^5)*d*cos(3*d*
x + 3*c)^2 - 8*(a*b^4 - b^5)*d*cos(3*d*x + 3*c)*cos(d*x + c) + (a*b^4 - b^5)*d*cos(d*x + c)^2 + (a*b^4 - b^5)*
d*sin(9*d*x + 9*c)^2 + 16*(a*b^4 - b^5)*d*sin(7*d*x + 7*c)^2 + 4*(64*a^3*b^2 - 112*a^2*b^3 + 57*a*b^4 - 9*b^5)
*d*sin(5*d*x + 5*c)^2 + 16*(a*b^4 - b^5)*d*sin(3*d*x + 3*c)^2 - 8*(a*b^4 - b^5)*d*sin(3*d*x + 3*c)*sin(d*x + c
) + (a*b^4 - b^5)*d*sin(d*x + c)^2 - 2*(4*(a*b^4 - b^5)*d*cos(7*d*x + 7*c) + 2*(8*a^2*b^3 - 11*a*b^4 + 3*b^5)*
d*cos(5*d*x + 5*c) + 4*(a*b^4 - b^5)*d*cos(3*d*x + 3*c) - (a*b^4 - b^5)*d*cos(d*x + c))*cos(9*d*x + 9*c) + 8*(
2*(8*a^2*b^3 - 11*a*b^4 + 3*b^5)*d*cos(5*d*x + 5*c) + 4*(a*b^4 - b^5)*d*cos(3*d*x + 3*c) - (a*b^4 - b^5)*d*cos
(d*x + c))*cos(7*d*x + 7*c) + 4*(4*(8*a^2*b^3 - 11*a*b^4 + 3*b^5)*d*cos(3*d*x + 3*c) - (8*a^2*b^3 - 11*a*b^4 +
3*b^5)*d*cos(d*x + c))*cos(5*d*x + 5*c) - 2*(4*(a*b^4 - b^5)*d*sin(7*d*x + 7*c) + 2*(8*a^2*b^3 - 11*a*b^4 + 3
*b^5)*d*sin(5*d*x + 5*c) + 4*(a*b^4 - b^5)*d*sin(3*d*x + 3*c) - (a*b^4 - b^5)*d*sin(d*x + c))*sin(9*d*x + 9*c)
+ 8*(2*(8*a^2*b^3 - 11*a*b^4 + 3*b^5)*d*sin(5*d*x + 5*c) + 4*(a*b^4 - b^5)*d*sin(3*d*x + 3*c) - (a*b^4 - b^5)
*d*sin(d*x + c))*sin(7*d*x + 7*c) + 4*(4*(8*a^2*b^3 - 11*a*b^4 + 3*b^5)*d*sin(3*d*x + 3*c) - (8*a^2*b^3 - 11*a
*b^4 + 3*b^5)*d*sin(d*x + c))*sin(5*d*x + 5*c))*integrate(-1/2*(4*a*b^2*cos(d*x + c)*sin(2*d*x + 2*c) - 4*a*b^
2*cos(2*d*x + 2*c)*sin(d*x + c) + a*b^2*sin(d*x + c) + 4*(20*a^2*b - 27*a*b^2)*cos(3*d*x + 3*c)*sin(2*d*x + 2*
c) - (a*b^2*sin(7*d*x + 7*c) - a*b^2*sin(d*x + c) + (20*a^2*b - 27*a*b^2)*sin(5*d*x + 5*c) - (20*a^2*b - 27*a*
b^2)*sin(3*d*x + 3*c))*cos(8*d*x + 8*c) - 2*(2*a*b^2*sin(6*d*x + 6*c) + 2*a*b^2*sin(2*d*x + 2*c) + (8*a^2*b -
3*a*b^2)*sin(4*d*x + 4*c))*cos(7*d*x + 7*c) - 4*(a*b^2*sin(d*x + c) - (20*a^2*b - 27*a*b^2)*sin(5*d*x + 5*c) +
(20*a^2*b - 27*a*b^2)*sin(3*d*x + 3*c))*cos(6*d*x + 6*c) - 2*((160*a^3 - 276*a^2*b + 81*a*b^2)*sin(4*d*x + 4*
c) + 2*(20*a^2*b - 27*a*b^2)*sin(2*d*x + 2*c))*cos(5*d*x + 5*c) - 2*((160*a^3 - 276*a^2*b + 81*a*b^2)*sin(3*d*
x + 3*c) + (8*a^2*b - 3*a*b^2)*sin(d*x + c))*cos(4*d*x + 4*c) + (a*b^2*cos(7*d*x + 7*c) - a*b^2*cos(d*x + c) +
(20*a^2*b - 27*a*b^2)*cos(5*d*x + 5*c) - (20*a^2*b - 27*a*b^2)*cos(3*d*x + 3*c))*sin(8*d*x + 8*c) + (4*a*b^2*
cos(6*d*x + 6*c) + 4*a*b^2*cos(2*d*x + 2*c) - a*b^2 + 2*(8*a^2*b - 3*a*b^2)*cos(4*d*x + 4*c))*sin(7*d*x + 7*c)
+ 4*(a*b^2*cos(d*x + c) - (20*a^2*b - 27*a*b^2)*cos(5*d*x + 5*c) + (20*a^2*b - 27*a*b^2)*cos(3*d*x + 3*c))*si
n(6*d*x + 6*c) - (20*a^2*b - 27*a*b^2 - 2*(160*a^3 - 276*a^2*b + 81*a*b^2)*cos(4*d*x + 4*c) - 4*(20*a^2*b - 27
*a*b^2)*cos(2*d*x + 2*c))*sin(5*d*x + 5*c) + 2*((160*a^3 - 276*a^2*b + 81*a*b^2)*cos(3*d*x + 3*c) + (8*a^2*b -
3*a*b^2)*cos(d*x + c))*sin(4*d*x + 4*c) + (20*a^2*b - 27*a*b^2 - 4*(20*a^2*b - 27*a*b^2)*cos(2*d*x + 2*c))*si
n(3*d*x + 3*c))/(a*b^4 - b^5 + (a*b^4 - b^5)*cos(8*d*x + 8*c)^2 + 16*(a*b^4 - b^5)*cos(6*d*x + 6*c)^2 + 4*(64*
a^3*b^2 - 112*a^2*b^3 + 57*a*b^4 - 9*b^5)*cos(4*d*x + 4*c)^2 + 16*(a*b^4 - b^5)*cos(2*d*x + 2*c)^2 + (a*b^4 -
b^5)*sin(8*d*x + 8*c)^2 + 16*(a*b^4 - b^5)*sin(6*d*x + 6*c)^2 + 4*(64*a^3*b^2 - 112*a^2*b^3 + 57*a*b^4 - 9*b^5
)*sin(4*d*x + 4*c)^2 + 16*(8*a^2*b^3 - 11*a*b^4 + 3*b^5)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*(a*b^4 - b^5)*
sin(2*d*x + 2*c)^2 + 2*(a*b^4 - b^5 - 4*(a*b^4 - b^5)*cos(6*d*x + 6*c) - 2*(8*a^2*b^3 - 11*a*b^4 + 3*b^5)*cos(
4*d*x + 4*c) - 4*(a*b^4 - b^5)*cos(2*d*x + 2*c))*cos(8*d*x + 8*c) - 8*(a*b^4 - b^5 - 2*(8*a^2*b^3 - 11*a*b^4 +
3*b^5)*cos(4*d*x + 4*c) - 4*(a*b^4 - b^5)*cos(2*d*x + 2*c))*cos(6*d*x + 6*c) - 4*(8*a^2*b^3 - 11*a*b^4 + 3*b^
5 - 4*(8*a^2*b^3 - 11*a*b^4 + 3*b^5)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - 8*(a*b^4 - b^5)*cos(2*d*x + 2*c) - 4
*(2*(a*b^4 - b^5)*sin(6*d*x + 6*c) + (8*a^2*b^3 - 11*a*b^4 + 3*b^5)*sin(4*d*x + 4*c) + 2*(a*b^4 - b^5)*sin(2*d
*x + 2*c))*sin(8*d*x + 8*c) + 16*((8*a^2*b^3 - 11*a*b^4 + 3*b^5)*sin(4*d*x + 4*c) + 2*(a*b^4 - b^5)*sin(2*d*x
+ 2*c))*sin(6*d*x + 6*c)), x) - ((a*b^2 - b^3)*sin(9*d*x + 9*c) - 4*(a*b^2 - b^3)*sin(7*d*x + 7*c) - 2*(8*a^2*
b - 11*a*b^2 + 3*b^3)*sin(5*d*x + 5*c) - 4*(a*b^2 - b^3)*sin(3*d*x + 3*c) + (a*b^2 - b^3)*sin(d*x + c))*sin(10
*d*x + 10*c) + ((2*a*b^2 - 3*b^3)*sin(8*d*x + 8*c) + (20*a^2*b - 17*a*b^2 + 2*b^3)*sin(6*d*x + 6*c) + (20*a^2*
b - 17*a*b^2 + 2*b^3)*sin(4*d*x + 4*c) + (2*a*b^2 - 3*b^3)*sin(2*d*x + 2*c))*sin(9*d*x + 9*c) - (4*(2*a*b^2 -
3*b^3)*sin(7*d*x + 7*c) + 2*(16*a^2*b - 30*a*b^2 + 9*b^3)*sin(5*d*x + 5*c) + 4*(2*a*b^2 - 3*b^3)*sin(3*d*x + 3
*c) - (2*a*b^2 - 3*b^3)*sin(d*x + c))*sin(8*d*x + 8*c) - 4*((20*a^2*b - 17*a*b^2 + 2*b^3)*sin(6*d*x + 6*c) + (
20*a^2*b - 17*a*b^2 + 2*b^3)*sin(4*d*x + 4*c) + (2*a*b^2 - 3*b^3)*sin(2*d*x + 2*c))*sin(7*d*x + 7*c) - (2*(160
*a^3 - 196*a^2*b + 67*a*b^2 - 6*b^3)*sin(5*d*x + 5*c) + 4*(20*a^2*b - 17*a*b^2 + 2*b^3)*sin(3*d*x + 3*c) - (20
*a^2*b - 17*a*b^2 + 2*b^3)*sin(d*x + c))*sin(6*d*x + 6*c) - 2*((160*a^3 - 196*a^2*b + 67*a*b^2 - 6*b^3)*sin(4*
d*x + 4*c) + (16*a^2*b - 30*a*b^2 + 9*b^3)*sin(2*d*x + 2*c))*sin(5*d*x + 5*c) - (4*(20*a^2*b - 17*a*b^2 + 2*b^
3)*sin(3*d*x + 3*c) - (20*a^2*b - 17*a*b^2 + 2*b^3)*sin(d*x + c))*sin(4*d*x + 4*c))/((a*b^4 - b^5)*d*cos(9*d*x
+ 9*c)^2 + 16*(a*b^4 - b^5)*d*cos(7*d*x + 7*c)^2 + 4*(64*a^3*b^2 - 112*a^2*b^3 + 57*a*b^4 - 9*b^5)*d*cos(5*d*
x + 5*c)^2 + 16*(a*b^4 - b^5)*d*cos(3*d*x + 3*c)^2 - 8*(a*b^4 - b^5)*d*cos(3*d*x + 3*c)*cos(d*x + c) + (a*b^4
- b^5)*d*cos(d*x + c)^2 + (a*b^4 - b^5)*d*sin(9*d*x + 9*c)^2 + 16*(a*b^4 - b^5)*d*sin(7*d*x + 7*c)^2 + 4*(64*a
^3*b^2 - 112*a^2*b^3 + 57*a*b^4 - 9*b^5)*d*sin(5*d*x + 5*c)^2 + 16*(a*b^4 - b^5)*d*sin(3*d*x + 3*c)^2 - 8*(a*b
^4 - b^5)*d*sin(3*d*x + 3*c)*sin(d*x + c) + (a*b^4 - b^5)*d*sin(d*x + c)^2 - 2*(4*(a*b^4 - b^5)*d*cos(7*d*x +
7*c) + 2*(8*a^2*b^3 - 11*a*b^4 + 3*b^5)*d*cos(5*d*x + 5*c) + 4*(a*b^4 - b^5)*d*cos(3*d*x + 3*c) - (a*b^4 - b^5
)*d*cos(d*x + c))*cos(9*d*x + 9*c) + 8*(2*(8*a^2*b^3 - 11*a*b^4 + 3*b^5)*d*cos(5*d*x + 5*c) + 4*(a*b^4 - b^5)*
d*cos(3*d*x + 3*c) - (a*b^4 - b^5)*d*cos(d*x + c))*cos(7*d*x + 7*c) + 4*(4*(8*a^2*b^3 - 11*a*b^4 + 3*b^5)*d*co
s(3*d*x + 3*c) - (8*a^2*b^3 - 11*a*b^4 + 3*b^5)*d*cos(d*x + c))*cos(5*d*x + 5*c) - 2*(4*(a*b^4 - b^5)*d*sin(7*
d*x + 7*c) + 2*(8*a^2*b^3 - 11*a*b^4 + 3*b^5)*d*sin(5*d*x + 5*c) + 4*(a*b^4 - b^5)*d*sin(3*d*x + 3*c) - (a*b^4
- b^5)*d*sin(d*x + c))*sin(9*d*x + 9*c) + 8*(2*(8*a^2*b^3 - 11*a*b^4 + 3*b^5)*d*sin(5*d*x + 5*c) + 4*(a*b^4 -
b^5)*d*sin(3*d*x + 3*c) - (a*b^4 - b^5)*d*sin(d*x + c))*sin(7*d*x + 7*c) + 4*(4*(8*a^2*b^3 - 11*a*b^4 + 3*b^5
)*d*sin(3*d*x + 3*c) - (8*a^2*b^3 - 11*a*b^4 + 3*b^5)*d*sin(d*x + c))*sin(5*d*x + 5*c))

________________________________________________________________________________________

mupad [B]  time = 16.00, size = 3941, normalized size = 16.70 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(c + d*x)^9/(a - b*sin(c + d*x)^4)^2,x)

[Out]

- cos(c + d*x)/(b^2*d) - ((cos(c + d*x)*(a*b + a^2))/(4*(a - b)) - (a*b*cos(c + d*x)^3)/(4*(a - b)))/(d*(a*b^2
- b^3 + 2*b^3*cos(c + d*x)^2 - b^3*cos(c + d*x)^4)) - (atan(((((1792*a^2*b^6 - 3072*a^3*b^5 + 1280*a^4*b^4)/(
64*(b^5 - 2*a*b^4 + a^2*b^3)) - (cos(c + d*x)*((25*a^2*(a^3*b^9)^(1/2) + 48*b^2*(a^3*b^9)^(1/2) - 36*a*b^7 + 4
7*a^2*b^6 - 15*a^3*b^5 - 69*a*b*(a^3*b^9)^(1/2))/(256*(3*a*b^11 - b^12 - 3*a^2*b^10 + a^3*b^9)))^(1/2)*(256*a*
b^7 - 512*a^2*b^6 + 256*a^3*b^5))/(4*(a^2*b - 2*a*b^2 + b^3)))*((25*a^2*(a^3*b^9)^(1/2) + 48*b^2*(a^3*b^9)^(1/
2) - 36*a*b^7 + 47*a^2*b^6 - 15*a^3*b^5 - 69*a*b*(a^3*b^9)^(1/2))/(256*(3*a*b^11 - b^12 - 3*a^2*b^10 + a^3*b^9
)))^(1/2) + (cos(c + d*x)*(25*a^4 - 59*a^3*b + 36*a^2*b^2))/(4*(a^2*b - 2*a*b^2 + b^3)))*((25*a^2*(a^3*b^9)^(1
/2) + 48*b^2*(a^3*b^9)^(1/2) - 36*a*b^7 + 47*a^2*b^6 - 15*a^3*b^5 - 69*a*b*(a^3*b^9)^(1/2))/(256*(3*a*b^11 - b
^12 - 3*a^2*b^10 + a^3*b^9)))^(1/2)*1i - (((1792*a^2*b^6 - 3072*a^3*b^5 + 1280*a^4*b^4)/(64*(b^5 - 2*a*b^4 + a
^2*b^3)) + (cos(c + d*x)*((25*a^2*(a^3*b^9)^(1/2) + 48*b^2*(a^3*b^9)^(1/2) - 36*a*b^7 + 47*a^2*b^6 - 15*a^3*b^
5 - 69*a*b*(a^3*b^9)^(1/2))/(256*(3*a*b^11 - b^12 - 3*a^2*b^10 + a^3*b^9)))^(1/2)*(256*a*b^7 - 512*a^2*b^6 + 2
56*a^3*b^5))/(4*(a^2*b - 2*a*b^2 + b^3)))*((25*a^2*(a^3*b^9)^(1/2) + 48*b^2*(a^3*b^9)^(1/2) - 36*a*b^7 + 47*a^
2*b^6 - 15*a^3*b^5 - 69*a*b*(a^3*b^9)^(1/2))/(256*(3*a*b^11 - b^12 - 3*a^2*b^10 + a^3*b^9)))^(1/2) - (cos(c +
d*x)*(25*a^4 - 59*a^3*b + 36*a^2*b^2))/(4*(a^2*b - 2*a*b^2 + b^3)))*((25*a^2*(a^3*b^9)^(1/2) + 48*b^2*(a^3*b^9
)^(1/2) - 36*a*b^7 + 47*a^2*b^6 - 15*a^3*b^5 - 69*a*b*(a^3*b^9)^(1/2))/(256*(3*a*b^11 - b^12 - 3*a^2*b^10 + a^
3*b^9)))^(1/2)*1i)/((36*a^3*b - 25*a^4)/(32*(b^5 - 2*a*b^4 + a^2*b^3)) + (((1792*a^2*b^6 - 3072*a^3*b^5 + 1280
*a^4*b^4)/(64*(b^5 - 2*a*b^4 + a^2*b^3)) - (cos(c + d*x)*((25*a^2*(a^3*b^9)^(1/2) + 48*b^2*(a^3*b^9)^(1/2) - 3
6*a*b^7 + 47*a^2*b^6 - 15*a^3*b^5 - 69*a*b*(a^3*b^9)^(1/2))/(256*(3*a*b^11 - b^12 - 3*a^2*b^10 + a^3*b^9)))^(1
/2)*(256*a*b^7 - 512*a^2*b^6 + 256*a^3*b^5))/(4*(a^2*b - 2*a*b^2 + b^3)))*((25*a^2*(a^3*b^9)^(1/2) + 48*b^2*(a
^3*b^9)^(1/2) - 36*a*b^7 + 47*a^2*b^6 - 15*a^3*b^5 - 69*a*b*(a^3*b^9)^(1/2))/(256*(3*a*b^11 - b^12 - 3*a^2*b^1
0 + a^3*b^9)))^(1/2) + (cos(c + d*x)*(25*a^4 - 59*a^3*b + 36*a^2*b^2))/(4*(a^2*b - 2*a*b^2 + b^3)))*((25*a^2*(
a^3*b^9)^(1/2) + 48*b^2*(a^3*b^9)^(1/2) - 36*a*b^7 + 47*a^2*b^6 - 15*a^3*b^5 - 69*a*b*(a^3*b^9)^(1/2))/(256*(3
*a*b^11 - b^12 - 3*a^2*b^10 + a^3*b^9)))^(1/2) + (((1792*a^2*b^6 - 3072*a^3*b^5 + 1280*a^4*b^4)/(64*(b^5 - 2*a
*b^4 + a^2*b^3)) + (cos(c + d*x)*((25*a^2*(a^3*b^9)^(1/2) + 48*b^2*(a^3*b^9)^(1/2) - 36*a*b^7 + 47*a^2*b^6 - 1
5*a^3*b^5 - 69*a*b*(a^3*b^9)^(1/2))/(256*(3*a*b^11 - b^12 - 3*a^2*b^10 + a^3*b^9)))^(1/2)*(256*a*b^7 - 512*a^2
*b^6 + 256*a^3*b^5))/(4*(a^2*b - 2*a*b^2 + b^3)))*((25*a^2*(a^3*b^9)^(1/2) + 48*b^2*(a^3*b^9)^(1/2) - 36*a*b^7
+ 47*a^2*b^6 - 15*a^3*b^5 - 69*a*b*(a^3*b^9)^(1/2))/(256*(3*a*b^11 - b^12 - 3*a^2*b^10 + a^3*b^9)))^(1/2) - (
cos(c + d*x)*(25*a^4 - 59*a^3*b + 36*a^2*b^2))/(4*(a^2*b - 2*a*b^2 + b^3)))*((25*a^2*(a^3*b^9)^(1/2) + 48*b^2*
(a^3*b^9)^(1/2) - 36*a*b^7 + 47*a^2*b^6 - 15*a^3*b^5 - 69*a*b*(a^3*b^9)^(1/2))/(256*(3*a*b^11 - b^12 - 3*a^2*b
^10 + a^3*b^9)))^(1/2)))*((25*a^2*(a^3*b^9)^(1/2) + 48*b^2*(a^3*b^9)^(1/2) - 36*a*b^7 + 47*a^2*b^6 - 15*a^3*b^
5 - 69*a*b*(a^3*b^9)^(1/2))/(256*(3*a*b^11 - b^12 - 3*a^2*b^10 + a^3*b^9)))^(1/2)*2i)/d - (atan(((((1792*a^2*b
^6 - 3072*a^3*b^5 + 1280*a^4*b^4)/(64*(b^5 - 2*a*b^4 + a^2*b^3)) - (cos(c + d*x)*(-(25*a^2*(a^3*b^9)^(1/2) + 4
8*b^2*(a^3*b^9)^(1/2) + 36*a*b^7 - 47*a^2*b^6 + 15*a^3*b^5 - 69*a*b*(a^3*b^9)^(1/2))/(256*(3*a*b^11 - b^12 - 3
*a^2*b^10 + a^3*b^9)))^(1/2)*(256*a*b^7 - 512*a^2*b^6 + 256*a^3*b^5))/(4*(a^2*b - 2*a*b^2 + b^3)))*(-(25*a^2*(
a^3*b^9)^(1/2) + 48*b^2*(a^3*b^9)^(1/2) + 36*a*b^7 - 47*a^2*b^6 + 15*a^3*b^5 - 69*a*b*(a^3*b^9)^(1/2))/(256*(3
*a*b^11 - b^12 - 3*a^2*b^10 + a^3*b^9)))^(1/2) + (cos(c + d*x)*(25*a^4 - 59*a^3*b + 36*a^2*b^2))/(4*(a^2*b - 2
*a*b^2 + b^3)))*(-(25*a^2*(a^3*b^9)^(1/2) + 48*b^2*(a^3*b^9)^(1/2) + 36*a*b^7 - 47*a^2*b^6 + 15*a^3*b^5 - 69*a
*b*(a^3*b^9)^(1/2))/(256*(3*a*b^11 - b^12 - 3*a^2*b^10 + a^3*b^9)))^(1/2)*1i - (((1792*a^2*b^6 - 3072*a^3*b^5
+ 1280*a^4*b^4)/(64*(b^5 - 2*a*b^4 + a^2*b^3)) + (cos(c + d*x)*(-(25*a^2*(a^3*b^9)^(1/2) + 48*b^2*(a^3*b^9)^(1
/2) + 36*a*b^7 - 47*a^2*b^6 + 15*a^3*b^5 - 69*a*b*(a^3*b^9)^(1/2))/(256*(3*a*b^11 - b^12 - 3*a^2*b^10 + a^3*b^
9)))^(1/2)*(256*a*b^7 - 512*a^2*b^6 + 256*a^3*b^5))/(4*(a^2*b - 2*a*b^2 + b^3)))*(-(25*a^2*(a^3*b^9)^(1/2) + 4
8*b^2*(a^3*b^9)^(1/2) + 36*a*b^7 - 47*a^2*b^6 + 15*a^3*b^5 - 69*a*b*(a^3*b^9)^(1/2))/(256*(3*a*b^11 - b^12 - 3
*a^2*b^10 + a^3*b^9)))^(1/2) - (cos(c + d*x)*(25*a^4 - 59*a^3*b + 36*a^2*b^2))/(4*(a^2*b - 2*a*b^2 + b^3)))*(-
(25*a^2*(a^3*b^9)^(1/2) + 48*b^2*(a^3*b^9)^(1/2) + 36*a*b^7 - 47*a^2*b^6 + 15*a^3*b^5 - 69*a*b*(a^3*b^9)^(1/2)
)/(256*(3*a*b^11 - b^12 - 3*a^2*b^10 + a^3*b^9)))^(1/2)*1i)/((36*a^3*b - 25*a^4)/(32*(b^5 - 2*a*b^4 + a^2*b^3)
) + (((1792*a^2*b^6 - 3072*a^3*b^5 + 1280*a^4*b^4)/(64*(b^5 - 2*a*b^4 + a^2*b^3)) - (cos(c + d*x)*(-(25*a^2*(a
^3*b^9)^(1/2) + 48*b^2*(a^3*b^9)^(1/2) + 36*a*b^7 - 47*a^2*b^6 + 15*a^3*b^5 - 69*a*b*(a^3*b^9)^(1/2))/(256*(3*
a*b^11 - b^12 - 3*a^2*b^10 + a^3*b^9)))^(1/2)*(256*a*b^7 - 512*a^2*b^6 + 256*a^3*b^5))/(4*(a^2*b - 2*a*b^2 + b
^3)))*(-(25*a^2*(a^3*b^9)^(1/2) + 48*b^2*(a^3*b^9)^(1/2) + 36*a*b^7 - 47*a^2*b^6 + 15*a^3*b^5 - 69*a*b*(a^3*b^
9)^(1/2))/(256*(3*a*b^11 - b^12 - 3*a^2*b^10 + a^3*b^9)))^(1/2) + (cos(c + d*x)*(25*a^4 - 59*a^3*b + 36*a^2*b^
2))/(4*(a^2*b - 2*a*b^2 + b^3)))*(-(25*a^2*(a^3*b^9)^(1/2) + 48*b^2*(a^3*b^9)^(1/2) + 36*a*b^7 - 47*a^2*b^6 +
15*a^3*b^5 - 69*a*b*(a^3*b^9)^(1/2))/(256*(3*a*b^11 - b^12 - 3*a^2*b^10 + a^3*b^9)))^(1/2) + (((1792*a^2*b^6 -
3072*a^3*b^5 + 1280*a^4*b^4)/(64*(b^5 - 2*a*b^4 + a^2*b^3)) + (cos(c + d*x)*(-(25*a^2*(a^3*b^9)^(1/2) + 48*b^
2*(a^3*b^9)^(1/2) + 36*a*b^7 - 47*a^2*b^6 + 15*a^3*b^5 - 69*a*b*(a^3*b^9)^(1/2))/(256*(3*a*b^11 - b^12 - 3*a^2
*b^10 + a^3*b^9)))^(1/2)*(256*a*b^7 - 512*a^2*b^6 + 256*a^3*b^5))/(4*(a^2*b - 2*a*b^2 + b^3)))*(-(25*a^2*(a^3*
b^9)^(1/2) + 48*b^2*(a^3*b^9)^(1/2) + 36*a*b^7 - 47*a^2*b^6 + 15*a^3*b^5 - 69*a*b*(a^3*b^9)^(1/2))/(256*(3*a*b
^11 - b^12 - 3*a^2*b^10 + a^3*b^9)))^(1/2) - (cos(c + d*x)*(25*a^4 - 59*a^3*b + 36*a^2*b^2))/(4*(a^2*b - 2*a*b
^2 + b^3)))*(-(25*a^2*(a^3*b^9)^(1/2) + 48*b^2*(a^3*b^9)^(1/2) + 36*a*b^7 - 47*a^2*b^6 + 15*a^3*b^5 - 69*a*b*(
a^3*b^9)^(1/2))/(256*(3*a*b^11 - b^12 - 3*a^2*b^10 + a^3*b^9)))^(1/2)))*(-(25*a^2*(a^3*b^9)^(1/2) + 48*b^2*(a^
3*b^9)^(1/2) + 36*a*b^7 - 47*a^2*b^6 + 15*a^3*b^5 - 69*a*b*(a^3*b^9)^(1/2))/(256*(3*a*b^11 - b^12 - 3*a^2*b^10
+ a^3*b^9)))^(1/2)*2i)/d

________________________________________________________________________________________

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)**9/(a-b*sin(d*x+c)**4)**2,x)

[Out]

Timed out

________________________________________________________________________________________