3.3.24 \(\int \frac {\sin ^9(c+d x)}{(a-b \sin ^4(c+d x))^3} \, dx\) [224]

Optimal. Leaf size=315 \[ -\frac {\left (5 a-14 \sqrt {a} \sqrt {b}+12 b\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 \sqrt {a} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} b^{9/4} d}-\frac {\left (5 a+14 \sqrt {a} \sqrt {b}+12 b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 \sqrt {a} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} b^{9/4} d}-\frac {a \cos (c+d x) \left (a+b-b \cos ^2(c+d x)\right )}{8 (a-b) b^2 d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )^2}+\frac {\cos (c+d x) \left (9 a^2-11 a b-10 b^2-2 (2 a-5 b) b \cos ^2(c+d x)\right )}{32 (a-b)^2 b^2 d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )} \]

[Out]

-1/8*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)^2+1/32*cos(d*x+c)*(9*
a^2-11*a*b-10*b^2-2*(2*a-5*b)*b*cos(d*x+c)^2)/(a-b)^2/b^2/d/(a-b+2*b*cos(d*x+c)^2-b*cos(d*x+c)^4)-1/64*arctan(
b^(1/4)*cos(d*x+c)/(a^(1/2)-b^(1/2))^(1/2))*(5*a+12*b-14*a^(1/2)*b^(1/2))/b^(9/4)/d/a^(1/2)/(a^(1/2)-b^(1/2))^
(5/2)-1/64*arctanh(b^(1/4)*cos(d*x+c)/(a^(1/2)+b^(1/2))^(1/2))*(5*a+12*b+14*a^(1/2)*b^(1/2))/b^(9/4)/d/a^(1/2)
/(a^(1/2)+b^(1/2))^(5/2)

________________________________________________________________________________________

Rubi [A]
time = 0.39, antiderivative size = 315, 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 = {3294, 1219, 1692, 1180, 211, 214} \begin {gather*} \frac {\cos (c+d x) \left (9 a^2-2 b (2 a-5 b) \cos ^2(c+d x)-11 a b-10 b^2\right )}{32 b^2 d (a-b)^2 \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}-\frac {\left (-14 \sqrt {a} \sqrt {b}+5 a+12 b\right ) \text {ArcTan}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 \sqrt {a} b^{9/4} d \left (\sqrt {a}-\sqrt {b}\right )^{5/2}}-\frac {\left (14 \sqrt {a} \sqrt {b}+5 a+12 b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 \sqrt {a} b^{9/4} d \left (\sqrt {a}+\sqrt {b}\right )^{5/2}}-\frac {a \cos (c+d x) \left (a-b \cos ^2(c+d x)+b\right )}{8 b^2 d (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]^9/(a - b*Sin[c + d*x]^4)^3,x]

[Out]

-1/64*((5*a - 14*Sqrt[a]*Sqrt[b] + 12*b)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(Sqrt[a]*(Sqr
t[a] - Sqrt[b])^(5/2)*b^(9/4)*d) - ((5*a + 14*Sqrt[a]*Sqrt[b] + 12*b)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt
[a] + Sqrt[b]]])/(64*Sqrt[a]*(Sqrt[a] + Sqrt[b])^(5/2)*b^(9/4)*d) - (a*Cos[c + d*x]*(a + b - b*Cos[c + d*x]^2)
)/(8*(a - b)*b^2*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4)^2) + (Cos[c + d*x]*(9*a^2 - 11*a*b - 10*b^2
- 2*(2*a - 5*b)*b*Cos[c + d*x]^2))/(32*(a - b)^2*b^2*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4))

Rule 211

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

Rule 214

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

Rule 1180

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 1219

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 1692

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Pq, a +
b*x^2 + c*x^4, x], x, 0], e = Coeff[PolynomialRemainder[Pq, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 +
c*x^4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*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)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuot
ient[Pq, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4*p + 7)*(b*d - 2*a*e)*x^2,
x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && Expon[Pq, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1
]

Rule 3294

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 )^3} \, dx &=-\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^4}{\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 (a+b-b \cos ^2(c+d x)\right )}{8 (a-b) b^2 d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )^2}+\frac {\text {Subst}\left (\int \frac {\frac {2 a \left (a^2+a b-8 b^2\right )}{b}-2 a (11 a-16 b) x^2+16 a (a-b) x^4}{\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 (a+b-b \cos ^2(c+d x)\right )}{8 (a-b) b^2 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 (9 a^2-11 a b-10 b^2-2 (2 a-5 b) b \cos ^2(c+d x)\right )}{32 (a-b)^2 b^2 d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {4 a^2 \left (5 a^2-15 a b+22 b^2\right )+8 a^2 (2 a-5 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 (a+b-b \cos ^2(c+d x)\right )}{8 (a-b) b^2 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 (9 a^2-11 a b-10 b^2-2 (2 a-5 b) b \cos ^2(c+d x)\right )}{32 (a-b)^2 b^2 d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}+\frac {\left (5 a-14 \sqrt {a} \sqrt {b}+12 b\right ) \text {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^{3/2} d}-\frac {\left (5 a+14 \sqrt {a} \sqrt {b}+12 b\right ) \text {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^{3/2} d}\\ &=-\frac {\left (5 a-14 \sqrt {a} \sqrt {b}+12 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^{9/4} d}-\frac {\left (5 a+14 \sqrt {a} \sqrt {b}+12 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^{9/4} d}-\frac {a \cos (c+d x) \left (a+b-b \cos ^2(c+d x)\right )}{8 (a-b) b^2 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 (9 a^2-11 a b-10 b^2-2 (2 a-5 b) b \cos ^2(c+d x)\right )}{32 (a-b)^2 b^2 d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 1.07, size = 785, normalized size = 2.49 \begin {gather*} \frac {-\frac {32 \cos (c+d x) \left (-9 a^2+13 a b+5 b^2+(2 a-5 b) b \cos (2 (c+d x))\right )}{8 a-3 b+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))}-\frac {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}+i \text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {-4 a b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+10 b^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+2 i a b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )-5 i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )-20 a^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+56 a b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-78 b^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+10 i a^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-28 i a b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2+39 i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2+20 a^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-56 a b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4+78 b^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-10 i a^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4+28 i a b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4-39 i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4+4 a b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^6-10 b^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^6-2 i a b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^6+5 i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^6}{-b \text {$\#$1}-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-3 b \text {$\#$1}^5+b \text {$\#$1}^7}\&\right ]}{128 (a-b)^2 b^2 d} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Sin[c + d*x]^9/(a - b*Sin[c + d*x]^4)^3,x]

[Out]

((-32*Cos[c + d*x]*(-9*a^2 + 13*a*b + 5*b^2 + (2*a - 5*b)*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 + 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)] + 10*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + (2*I)*a*b*L
og[1 - 2*Cos[c + d*x]*#1 + #1^2] - (5*I)*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - 20*a^2*ArcTan[Sin[c + d*x]/(C
os[c + d*x] - #1)]*#1^2 + 56*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - 78*b^2*ArcTan[Sin[c + d*x]/(C
os[c + d*x] - #1)]*#1^2 + (10*I)*a^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (28*I)*a*b*Log[1 - 2*Cos[c + d*x
]*#1 + #1^2]*#1^2 + (39*I)*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + 20*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*
x] - #1)]*#1^4 - 56*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + 78*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*
x] - #1)]*#1^4 - (10*I)*a^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + (28*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1
^2]*#1^4 - (39*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 - 10*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
+ (5*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 - b)^2*b^2*d)

________________________________________________________________________________________

Maple [A]
time = 1.75, size = 342, normalized size = 1.09

method	result	size
derivativedivides	\(\frac {-\frac {-\frac {\left (2 a -5 b \right ) \left (\cos ^{7}\left (d x +c \right )\right )}{16 \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 \left (3 a^{2}-a b -10 b^{2}\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{32 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {3 \left (3 a^{2}-2 a b -5 b^{2}\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{16 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {5 \left (a^{2}-3 a b -2 b^{2}\right ) \cos \left (d x +c \right )}{32 b^{2} \left (a -b \right )}}{\left (a -b +2 b \left (\cos ^{2}\left (d x +c \right )\right )-b \left (\cos ^{4}\left (d x +c \right )\right )\right )^{2}}-\frac {\frac {\left (-4 a \sqrt {a b}+10 \sqrt {a b}\, b +5 a^{2}-11 a b +12 b^{2}\right ) \arctan \left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}-b \right ) b}}-\frac {\left (-4 a \sqrt {a b}+10 \sqrt {a b}\, b -5 a^{2}+11 a b -12 b^{2}\right ) \arctanh \left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+b \right ) b}}}{32 \left (a^{2}-2 a b +b^{2}\right ) b}}{d}\)	\(342\)
default	\(\frac {-\frac {-\frac {\left (2 a -5 b \right ) \left (\cos ^{7}\left (d x +c \right )\right )}{16 \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 \left (3 a^{2}-a b -10 b^{2}\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{32 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {3 \left (3 a^{2}-2 a b -5 b^{2}\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{16 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {5 \left (a^{2}-3 a b -2 b^{2}\right ) \cos \left (d x +c \right )}{32 b^{2} \left (a -b \right )}}{\left (a -b +2 b \left (\cos ^{2}\left (d x +c \right )\right )-b \left (\cos ^{4}\left (d x +c \right )\right )\right )^{2}}-\frac {\frac {\left (-4 a \sqrt {a b}+10 \sqrt {a b}\, b +5 a^{2}-11 a b +12 b^{2}\right ) \arctan \left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}-b \right ) b}}-\frac {\left (-4 a \sqrt {a b}+10 \sqrt {a b}\, b -5 a^{2}+11 a b -12 b^{2}\right ) \arctanh \left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+b \right ) b}}}{32 \left (a^{2}-2 a b +b^{2}\right ) b}}{d}\)	\(342\)
risch	\(\text {Expression too large to display}\)	\(1583\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)^9/(a-b*sin(d*x+c)^4)^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(-(-1/16*(2*a-5*b)/(a^2-2*a*b+b^2)*cos(d*x+c)^7+3/32*(3*a^2-a*b-10*b^2)/b/(a^2-2*a*b+b^2)*cos(d*x+c)^5-3/1
6*(3*a^2-2*a*b-5*b^2)/b/(a^2-2*a*b+b^2)*cos(d*x+c)^3-5/32*(a^2-3*a*b-2*b^2)/b^2/(a-b)*cos(d*x+c))/(a-b+2*b*cos
(d*x+c)^2-b*cos(d*x+c)^4)^2-1/32/(a^2-2*a*b+b^2)/b*(1/2*(-4*a*(a*b)^(1/2)+10*(a*b)^(1/2)*b+5*a^2-11*a*b+12*b^2
)/(a*b)^(1/2)/(((a*b)^(1/2)-b)*b)^(1/2)*arctan(b*cos(d*x+c)/(((a*b)^(1/2)-b)*b)^(1/2))-1/2*(-4*a*(a*b)^(1/2)+1
0*(a*b)^(1/2)*b-5*a^2+11*a*b-12*b^2)/(a*b)^(1/2)/(((a*b)^(1/2)+b)*b)^(1/2)*arctanh(b*cos(d*x+c)/(((a*b)^(1/2)+
b)*b)^(1/2))))

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^9/(a-b*sin(d*x+c)^4)^3,x, algorithm="maxima")

[Out]

-1/8*(8*(2*a*b^4 - 5*b^5)*cos(2*d*x + 2*c)*cos(d*x + c) - 8*(18*a^2*b^3 - 20*a*b^4 - 25*b^5)*sin(3*d*x + 3*c)*
sin(2*d*x + 2*c) + 8*(2*a*b^4 - 5*b^5)*sin(2*d*x + 2*c)*sin(d*x + c) - ((2*a*b^4 - 5*b^5)*cos(15*d*x + 15*c) -
(18*a^2*b^3 - 20*a*b^4 - 25*b^5)*cos(13*d*x + 13*c) + 3*(18*a^2*b^3 - 8*a*b^4 - 15*b^5)*cos(11*d*x + 11*c) +
(160*a^3*b^2 - 388*a^2*b^3 + 2*a*b^4 + 25*b^5)*cos(9*d*x + 9*c) + (160*a^3*b^2 - 388*a^2*b^3 + 2*a*b^4 + 25*b^
5)*cos(7*d*x + 7*c) + 3*(18*a^2*b^3 - 8*a*b^4 - 15*b^5)*cos(5*d*x + 5*c) - (18*a^2*b^3 - 20*a*b^4 - 25*b^5)*co
s(3*d*x + 3*c) + (2*a*b^4 - 5*b^5)*cos(d*x + c))*cos(16*d*x + 16*c) - (2*a*b^4 - 5*b^5 - 8*(2*a*b^4 - 5*b^5)*c
os(14*d*x + 14*c) - 4*(16*a^2*b^3 - 54*a*b^4 + 35*b^5)*cos(12*d*x + 12*c) + 8*(32*a^2*b^3 - 94*a*b^4 + 35*b^5)
*cos(10*d*x + 10*c) + 2*(256*a^3*b^2 - 832*a^2*b^3 + 550*a*b^4 - 175*b^5)*cos(8*d*x + 8*c) + 8*(32*a^2*b^3 - 9
4*a*b^4 + 35*b^5)*cos(6*d*x + 6*c) - 4*(16*a^2*b^3 - 54*a*b^4 + 35*b^5)*cos(4*d*x + 4*c) - 8*(2*a*b^4 - 5*b^5)
*cos(2*d*x + 2*c))*cos(15*d*x + 15*c) - 8*((18*a^2*b^3 - 20*a*b^4 - 25*b^5)*cos(13*d*x + 13*c) - 3*(18*a^2*b^3
- 8*a*b^4 - 15*b^5)*cos(11*d*x + 11*c) - (160*a^3*b^2 - 388*a^2*b^3 + 2*a*b^4 + 25*b^5)*cos(9*d*x + 9*c) - (1
60*a^3*b^2 - 388*a^2*b^3 + 2*a*b^4 + 25*b^5)*cos(7*d*x + 7*c) - 3*(18*a^2*b^3 - 8*a*b^4 - 15*b^5)*cos(5*d*x +
5*c) + (18*a^2*b^3 - 20*a*b^4 - 25*b^5)*cos(3*d*x + 3*c) - (2*a*b^4 - 5*b^5)*cos(d*x + c))*cos(14*d*x + 14*c)
+ (18*a^2*b^3 - 20*a*b^4 - 25*b^5 - 4*(144*a^3*b^2 - 286*a^2*b^3 - 60*a*b^4 + 175*b^5)*cos(12*d*x + 12*c) + 8*
(288*a^3*b^2 - 446*a^2*b^3 - 260*a*b^4 + 175*b^5)*cos(10*d*x + 10*c) + 2*(2304*a^4*b - 4288*a^3*b^2 - 650*a^2*
b^3 + 1700*a*b^4 - 875*b^5)*cos(8*d*x + 8*c) + 8*(288*a^3*b^2 - 446*a^2*b^3 - 260*a*b^4 + 175*b^5)*cos(6*d*x +
6*c) - 4*(144*a^3*b^2 - 286*a^2*b^3 - 60*a*b^4 + 175*b^5)*cos(4*d*x + 4*c) - 8*(18*a^2*b^3 - 20*a*b^4 - 25*b^
5)*cos(2*d*x + 2*c))*cos(13*d*x + 13*c) + 4*(3*(144*a^3*b^2 - 190*a^2*b^3 - 64*a*b^4 + 105*b^5)*cos(11*d*x + 1
1*c) + (1280*a^4*b - 4224*a^3*b^2 + 2732*a^2*b^3 + 186*a*b^4 - 175*b^5)*cos(9*d*x + 9*c) + (1280*a^4*b - 4224*
a^3*b^2 + 2732*a^2*b^3 + 186*a*b^4 - 175*b^5)*cos(7*d*x + 7*c) + 3*(144*a^3*b^2 - 190*a^2*b^3 - 64*a*b^4 + 105
*b^5)*cos(5*d*x + 5*c) - (144*a^3*b^2 - 286*a^2*b^3 - 60*a*b^4 + 175*b^5)*cos(3*d*x + 3*c) + (16*a^2*b^3 - 54*
a*b^4 + 35*b^5)*cos(d*x + c))*cos(12*d*x + 12*c) - 3*(18*a^2*b^3 - 8*a*b^4 - 15*b^5 + 8*(288*a^3*b^2 - 254*a^2
*b^3 - 184*a*b^4 + 105*b^5)*cos(10*d*x + 10*c) + 2*(2304*a^4*b - 2752*a^3*b^2 - 522*a^2*b^3 + 1160*a*b^4 - 525
*b^5)*cos(8*d*x + 8*c) + 8*(288*a^3*b^2 - 254*a^2*b^3 - 184*a*b^4 + 105*b^5)*cos(6*d*x + 6*c) - 4*(144*a^3*b^2
- 190*a^2*b^3 - 64*a*b^4 + 105*b^5)*cos(4*d*x + 4*c) - 8*(18*a^2*b^3 - 8*a*b^4 - 15*b^5)*cos(2*d*x + 2*c))*co
s(11*d*x + 11*c) - 8*((2560*a^4*b - 7328*a^3*b^2 + 2748*a^2*b^3 + 386*a*b^4 - 175*b^5)*cos(9*d*x + 9*c) + (256
0*a^4*b - 7328*a^3*b^2 + 2748*a^2*b^3 + 386*a*b^4 - 175*b^5)*cos(7*d*x + 7*c) + 3*(288*a^3*b^2 - 254*a^2*b^3 -
184*a*b^4 + 105*b^5)*cos(5*d*x + 5*c) - (288*a^3*b^2 - 446*a^2*b^3 - 260*a*b^4 + 175*b^5)*cos(3*d*x + 3*c) +
(32*a^2*b^3 - 94*a*b^4 + 35*b^5)*cos(d*x + c))*cos(10*d*x + 10*c) - (160*a^3*b^2 - 388*a^2*b^3 + 2*a*b^4 + 25*
b^5 + 2*(20480*a^5 - 65024*a^4*b + 43104*a^3*b^2 - 10572*a^2*b^3 - 2330*a*b^4 + 875*b^5)*cos(8*d*x + 8*c) + 8*
(2560*a^4*b - 7328*a^3*b^2 + 2748*a^2*b^3 + 386*a*b^4 - 175*b^5)*cos(6*d*x + 6*c) - 4*(1280*a^4*b - 4224*a^3*b
^2 + 2732*a^2*b^3 + 186*a*b^4 - 175*b^5)*cos(4*d*x + 4*c) - 8*(160*a^3*b^2 - 388*a^2*b^3 + 2*a*b^4 + 25*b^5)*c
os(2*d*x + 2*c))*cos(9*d*x + 9*c) - 2*((20480*a^5 - 65024*a^4*b + 43104*a^3*b^2 - 10572*a^2*b^3 - 2330*a*b^4 +
875*b^5)*cos(7*d*x + 7*c) + 3*(2304*a^4*b - 2752*a^3*b^2 - 522*a^2*b^3 + 1160*a*b^4 - 525*b^5)*cos(5*d*x + 5*
c) - (2304*a^4*b - 4288*a^3*b^2 - 650*a^2*b^3 + 1700*a*b^4 - 875*b^5)*cos(3*d*x + 3*c) + (256*a^3*b^2 - 832*a^
2*b^3 + 550*a*b^4 - 175*b^5)*cos(d*x + c))*cos(8*d*x + 8*c) - (160*a^3*b^2 - 388*a^2*b^3 + 2*a*b^4 + 25*b^5 +
8*(2560*a^4*b - 7328*a^3*b^2 + 2748*a^2*b^3 + 386*a*b^4 - 175*b^5)*cos(6*d*x + 6*c) - 4*(1280*a^4*b - 4224*a^3
*b^2 + 2732*a^2*b^3 + 186*a*b^4 - 175*b^5)*cos(4*d*x + 4*c) - 8*(160*a^3*b^2 - 388*a^2*b^3 + 2*a*b^4 + 25*b^5)
*cos(2*d*x + 2*c))*cos(7*d*x + 7*c) - 8*(3*(288*a^3*b^2 - 254*a^2*b^3 - 184*a*b^4 + 105*b^5)*cos(5*d*x + 5*c)
- (288*a^3*b^2 - 446*a^2*b^3 - 260*a*b^4 + 175*b^5)*cos(3*d*x + 3*c) + (32*a^2*b^3 - 94*a*b^4 + 35*b^5)*cos(d*
x + c))*cos(6*d*x + 6*c) - 3*(18*a^2*b^3 - 8*a*b^4 - 15*b^5 - 4*(144*a^3*b^2 - 190*a^2*b^3 - 64*a*b^4 + 105*b^
5)*cos(4*d*x + 4*c) - 8*(18*a^2*b^3 - 8*a*b^4 - 15*b^5)*cos(2*d*x + 2*c))*cos(5*d*x + 5*c) - 4*((144*a^3*b^2 -
286*a^2*b^3 - 60*a*b^4 + 175*b^5)*cos(3*d*x + 3*c) - (16*a^2*b^3 - 54*a*b^4 + 35*b^5)*cos(d*x + c))*cos(4*d*x
+ 4*c) + (18*a^2*b^3 - 20*a*b^4 - 25*b^5 - 8*(18*a^2*b^3 - 20*a*b^4 - 25*b^5)*cos(2*d*x + 2*c))*cos(3*d*x + 3
*c) - (2*a*b^4 - 5*b^5)*cos(d*x + c) + 8*((a^2*b^6 - 2*a*b^7 + b^8)*d*cos(16*d*x + 16*c)^2 + 64*(a^2*b^6 - 2*a
*b^7 + b^8)*d*cos(14*d*x + 14*c)^2 + 16*(64*a^4...

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4640 vs. \(2 (264) = 528\).
time = 1.14, size = 4640, normalized size = 14.73 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^9/(a-b*sin(d*x+c)^4)^3,x, algorithm="fricas")

[Out]

1/128*(8*(2*a*b^2 - 5*b^3)*cos(d*x + c)^7 - 12*(3*a^2*b - a*b^2 - 10*b^3)*cos(d*x + c)^5 + 24*(3*a^2*b - 2*a*b
^2 - 5*b^3)*cos(d*x + c)^3 + ((a^2*b^4 - 2*a*b^5 + b^6)*d*cos(d*x + c)^8 - 4*(a^2*b^4 - 2*a*b^5 + b^6)*d*cos(d
*x + c)^6 - 2*(a^3*b^3 - 5*a^2*b^4 + 7*a*b^5 - 3*b^6)*d*cos(d*x + c)^4 + 4*(a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^
6)*d*cos(d*x + c)^2 + (a^4*b^2 - 4*a^3*b^3 + 6*a^2*b^4 - 4*a*b^5 + b^6)*d)*sqrt((15*a^4 - 94*a^3*b + 155*a^2*b
^2 - 76*a*b^3 - 144*b^4 + (a^6*b^4 - 5*a^5*b^5 + 10*a^4*b^6 - 10*a^3*b^7 + 5*a^2*b^8 - a*b^9)*d^2*sqrt((625*a^
8 - 6700*a^7*b + 35406*a^6*b^2 - 117532*a^5*b^3 + 269641*a^4*b^4 - 437952*a^3*b^5 + 498432*a^2*b^6 - 368640*a*
b^7 + 147456*b^8)/((a^11*b^9 - 10*a^10*b^10 + 45*a^9*b^11 - 120*a^8*b^12 + 210*a^7*b^13 - 252*a^6*b^14 + 210*a
^5*b^15 - 120*a^4*b^16 + 45*a^3*b^17 - 10*a^2*b^18 + a*b^19)*d^4)))/((a^6*b^4 - 5*a^5*b^5 + 10*a^4*b^6 - 10*a^
3*b^7 + 5*a^2*b^8 - a*b^9)*d^2))*log((625*a^6 - 5250*a^5*b + 22509*a^4*b^2 - 57820*a^3*b^3 + 96336*a^2*b^4 - 9
8304*a*b^5 + 55296*b^6)*cos(d*x + c) - ((a^8*b^7 - 6*a^7*b^8 + 27*a^6*b^9 - 80*a^5*b^10 + 135*a^4*b^11 - 126*a
^3*b^12 + 61*a^2*b^13 - 12*a*b^14)*d^3*sqrt((625*a^8 - 6700*a^7*b + 35406*a^6*b^2 - 117532*a^5*b^3 + 269641*a^
4*b^4 - 437952*a^3*b^5 + 498432*a^2*b^6 - 368640*a*b^7 + 147456*b^8)/((a^11*b^9 - 10*a^10*b^10 + 45*a^9*b^11 -
120*a^8*b^12 + 210*a^7*b^13 - 252*a^6*b^14 + 210*a^5*b^15 - 120*a^4*b^16 + 45*a^3*b^17 - 10*a^2*b^18 + a*b^19
)*d^4)) + (125*a^7*b^2 - 1045*a^6*b^3 + 4305*a^5*b^4 - 10583*a^4*b^5 + 16798*a^3*b^6 - 16320*a^2*b^7 + 8448*a*
b^8)*d)*sqrt((15*a^4 - 94*a^3*b + 155*a^2*b^2 - 76*a*b^3 - 144*b^4 + (a^6*b^4 - 5*a^5*b^5 + 10*a^4*b^6 - 10*a^
3*b^7 + 5*a^2*b^8 - a*b^9)*d^2*sqrt((625*a^8 - 6700*a^7*b + 35406*a^6*b^2 - 117532*a^5*b^3 + 269641*a^4*b^4 -
437952*a^3*b^5 + 498432*a^2*b^6 - 368640*a*b^7 + 147456*b^8)/((a^11*b^9 - 10*a^10*b^10 + 45*a^9*b^11 - 120*a^8
*b^12 + 210*a^7*b^13 - 252*a^6*b^14 + 210*a^5*b^15 - 120*a^4*b^16 + 45*a^3*b^17 - 10*a^2*b^18 + a*b^19)*d^4)))
/((a^6*b^4 - 5*a^5*b^5 + 10*a^4*b^6 - 10*a^3*b^7 + 5*a^2*b^8 - a*b^9)*d^2))) - ((a^2*b^4 - 2*a*b^5 + b^6)*d*co
s(d*x + c)^8 - 4*(a^2*b^4 - 2*a*b^5 + b^6)*d*cos(d*x + c)^6 - 2*(a^3*b^3 - 5*a^2*b^4 + 7*a*b^5 - 3*b^6)*d*cos(
d*x + c)^4 + 4*(a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d*cos(d*x + c)^2 + (a^4*b^2 - 4*a^3*b^3 + 6*a^2*b^4 - 4*a
*b^5 + b^6)*d)*sqrt((15*a^4 - 94*a^3*b + 155*a^2*b^2 - 76*a*b^3 - 144*b^4 - (a^6*b^4 - 5*a^5*b^5 + 10*a^4*b^6
- 10*a^3*b^7 + 5*a^2*b^8 - a*b^9)*d^2*sqrt((625*a^8 - 6700*a^7*b + 35406*a^6*b^2 - 117532*a^5*b^3 + 269641*a^4
*b^4 - 437952*a^3*b^5 + 498432*a^2*b^6 - 368640*a*b^7 + 147456*b^8)/((a^11*b^9 - 10*a^10*b^10 + 45*a^9*b^11 -
120*a^8*b^12 + 210*a^7*b^13 - 252*a^6*b^14 + 210*a^5*b^15 - 120*a^4*b^16 + 45*a^3*b^17 - 10*a^2*b^18 + a*b^19)
*d^4)))/((a^6*b^4 - 5*a^5*b^5 + 10*a^4*b^6 - 10*a^3*b^7 + 5*a^2*b^8 - a*b^9)*d^2))*log((625*a^6 - 5250*a^5*b +
22509*a^4*b^2 - 57820*a^3*b^3 + 96336*a^2*b^4 - 98304*a*b^5 + 55296*b^6)*cos(d*x + c) - ((a^8*b^7 - 6*a^7*b^8
+ 27*a^6*b^9 - 80*a^5*b^10 + 135*a^4*b^11 - 126*a^3*b^12 + 61*a^2*b^13 - 12*a*b^14)*d^3*sqrt((625*a^8 - 6700*
a^7*b + 35406*a^6*b^2 - 117532*a^5*b^3 + 269641*a^4*b^4 - 437952*a^3*b^5 + 498432*a^2*b^6 - 368640*a*b^7 + 147
456*b^8)/((a^11*b^9 - 10*a^10*b^10 + 45*a^9*b^11 - 120*a^8*b^12 + 210*a^7*b^13 - 252*a^6*b^14 + 210*a^5*b^15 -
120*a^4*b^16 + 45*a^3*b^17 - 10*a^2*b^18 + a*b^19)*d^4)) - (125*a^7*b^2 - 1045*a^6*b^3 + 4305*a^5*b^4 - 10583
*a^4*b^5 + 16798*a^3*b^6 - 16320*a^2*b^7 + 8448*a*b^8)*d)*sqrt((15*a^4 - 94*a^3*b + 155*a^2*b^2 - 76*a*b^3 - 1
44*b^4 - (a^6*b^4 - 5*a^5*b^5 + 10*a^4*b^6 - 10*a^3*b^7 + 5*a^2*b^8 - a*b^9)*d^2*sqrt((625*a^8 - 6700*a^7*b +
35406*a^6*b^2 - 117532*a^5*b^3 + 269641*a^4*b^4 - 437952*a^3*b^5 + 498432*a^2*b^6 - 368640*a*b^7 + 147456*b^8)
/((a^11*b^9 - 10*a^10*b^10 + 45*a^9*b^11 - 120*a^8*b^12 + 210*a^7*b^13 - 252*a^6*b^14 + 210*a^5*b^15 - 120*a^4
*b^16 + 45*a^3*b^17 - 10*a^2*b^18 + a*b^19)*d^4)))/((a^6*b^4 - 5*a^5*b^5 + 10*a^4*b^6 - 10*a^3*b^7 + 5*a^2*b^8
- a*b^9)*d^2))) - ((a^2*b^4 - 2*a*b^5 + b^6)*d*cos(d*x + c)^8 - 4*(a^2*b^4 - 2*a*b^5 + b^6)*d*cos(d*x + c)^6
- 2*(a^3*b^3 - 5*a^2*b^4 + 7*a*b^5 - 3*b^6)*d*cos(d*x + c)^4 + 4*(a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d*cos(d
*x + c)^2 + (a^4*b^2 - 4*a^3*b^3 + 6*a^2*b^4 - 4*a*b^5 + b^6)*d)*sqrt((15*a^4 - 94*a^3*b + 155*a^2*b^2 - 76*a*
b^3 - 144*b^4 + (a^6*b^4 - 5*a^5*b^5 + 10*a^4*b^6 - 10*a^3*b^7 + 5*a^2*b^8 - a*b^9)*d^2*sqrt((625*a^8 - 6700*a
^7*b + 35406*a^6*b^2 - 117532*a^5*b^3 + 269641*a^4*b^4 - 437952*a^3*b^5 + 498432*a^2*b^6 - 368640*a*b^7 + 1474
56*b^8)/((a^11*b^9 - 10*a^10*b^10 + 45*a^9*b^11 - 120*a^8*b^12 + 210*a^7*b^13 - 252*a^6*b^14 + 210*a^5*b^15 -
120*a^4*b^16 + 45*a^3*b^17 - 10*a^2*b^18 + a*b^19)*d^4)))/((a^6*b^4 - 5*a^5*b^5 + 10*a^4*b^6 - 10*a^3*b^7 + 5*
a^2*b^8 - a*b^9)*d^2))*log(-(625*a^6 - 5250*a^5*b + 22509*a^4*b^2 - 57820*a^3*b^3 + 96336*a^2*b^4 - 98304*a*b^
5 + 55296*b^6)*cos(d*x + c) - ((a^8*b^7 - 6*a^7*b^8 + 27*a^6*b^9 - 80*a^5*b^10 + 135*a^4*b^11 - 126*a^3*b^12 +
61*a^2*b^13 - 12*a*b^14)*d^3*sqrt((625*a^8 - 6...

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)**9/(a-b*sin(d*x+c)**4)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^9/(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 [sageVARa,sageVARb]=[-
82,8]Warning, need to

________________________________________________________________________________________

Mupad [B]
time = 19.29, size = 2500, normalized size = 7.94 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(c + d*x)^9/(a - b*sin(c + d*x)^4)^3,x)

[Out]

((cos(c + d*x)^7*(2*a - 5*b))/(16*(a^2 - 2*a*b + b^2)) + (3*cos(c + d*x)^5*(a*b - 3*a^2 + 10*b^2))/(32*b*(a^2
- 2*a*b + b^2)) - (5*cos(c + d*x)*(3*a*b - a^2 + 2*b^2))/(32*b^2*(a - b)) - (3*cos(c + d*x)^3*(2*a*b - 3*a^2 +
5*b^2))/(16*b*(a - 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(((((180224*a*b^8 - 483328*a^2*b^7 + 466944*a^3*b^6
- 204800*a^4*b^5 + 40960*a^5*b^4)/(16384*(b^7 - 4*a*b^6 + 6*a^2*b^5 - 4*a^3*b^4 + a^4*b^3)) - (cos(c + d*x)*(-
(25*a^4*(a^3*b^9)^(1/2) + 384*b^4*(a^3*b^9)^(1/2) - 144*a*b^9 - 76*a^2*b^8 + 155*a^3*b^7 - 94*a^4*b^6 + 15*a^5
*b^5 + 349*a^2*b^2*(a^3*b^9)^(1/2) - 480*a*b^3*(a^3*b^9)^(1/2) - 134*a^3*b*(a^3*b^9)^(1/2))/(16384*(a^2*b^14 -
5*a^3*b^13 + 10*a^4*b^12 - 10*a^5*b^11 + 5*a^6*b^10 - a^7*b^9)))^(1/2)*(16384*a*b^9 - 65536*a^2*b^8 + 98304*a
^3*b^7 - 65536*a^4*b^6 + 16384*a^5*b^5))/(256*(a^4*b - 4*a*b^4 + b^5 + 6*a^2*b^3 - 4*a^3*b^2)))*(-(25*a^4*(a^3
*b^9)^(1/2) + 384*b^4*(a^3*b^9)^(1/2) - 144*a*b^9 - 76*a^2*b^8 + 155*a^3*b^7 - 94*a^4*b^6 + 15*a^5*b^5 + 349*a
^2*b^2*(a^3*b^9)^(1/2) - 480*a*b^3*(a^3*b^9)^(1/2) - 134*a^3*b*(a^3*b^9)^(1/2))/(16384*(a^2*b^14 - 5*a^3*b^13
+ 10*a^4*b^12 - 10*a^5*b^11 + 5*a^6*b^10 - a^7*b^9)))^(1/2) + (cos(c + d*x)*(25*a^4 - 94*a^3*b - 164*a*b^3 + 1
44*b^4 + 161*a^2*b^2))/(256*(a^4*b - 4*a*b^4 + b^5 + 6*a^2*b^3 - 4*a^3*b^2)))*(-(25*a^4*(a^3*b^9)^(1/2) + 384*
b^4*(a^3*b^9)^(1/2) - 144*a*b^9 - 76*a^2*b^8 + 155*a^3*b^7 - 94*a^4*b^6 + 15*a^5*b^5 + 349*a^2*b^2*(a^3*b^9)^(
1/2) - 480*a*b^3*(a^3*b^9)^(1/2) - 134*a^3*b*(a^3*b^9)^(1/2))/(16384*(a^2*b^14 - 5*a^3*b^13 + 10*a^4*b^12 - 10
*a^5*b^11 + 5*a^6*b^10 - a^7*b^9)))^(1/2)*1i - (((180224*a*b^8 - 483328*a^2*b^7 + 466944*a^3*b^6 - 204800*a^4*
b^5 + 40960*a^5*b^4)/(16384*(b^7 - 4*a*b^6 + 6*a^2*b^5 - 4*a^3*b^4 + a^4*b^3)) + (cos(c + d*x)*(-(25*a^4*(a^3*
b^9)^(1/2) + 384*b^4*(a^3*b^9)^(1/2) - 144*a*b^9 - 76*a^2*b^8 + 155*a^3*b^7 - 94*a^4*b^6 + 15*a^5*b^5 + 349*a^
2*b^2*(a^3*b^9)^(1/2) - 480*a*b^3*(a^3*b^9)^(1/2) - 134*a^3*b*(a^3*b^9)^(1/2))/(16384*(a^2*b^14 - 5*a^3*b^13 +
10*a^4*b^12 - 10*a^5*b^11 + 5*a^6*b^10 - a^7*b^9)))^(1/2)*(16384*a*b^9 - 65536*a^2*b^8 + 98304*a^3*b^7 - 6553
6*a^4*b^6 + 16384*a^5*b^5))/(256*(a^4*b - 4*a*b^4 + b^5 + 6*a^2*b^3 - 4*a^3*b^2)))*(-(25*a^4*(a^3*b^9)^(1/2) +
384*b^4*(a^3*b^9)^(1/2) - 144*a*b^9 - 76*a^2*b^8 + 155*a^3*b^7 - 94*a^4*b^6 + 15*a^5*b^5 + 349*a^2*b^2*(a^3*b
^9)^(1/2) - 480*a*b^3*(a^3*b^9)^(1/2) - 134*a^3*b*(a^3*b^9)^(1/2))/(16384*(a^2*b^14 - 5*a^3*b^13 + 10*a^4*b^12
- 10*a^5*b^11 + 5*a^6*b^10 - a^7*b^9)))^(1/2) - (cos(c + d*x)*(25*a^4 - 94*a^3*b - 164*a*b^3 + 144*b^4 + 161*
a^2*b^2))/(256*(a^4*b - 4*a*b^4 + b^5 + 6*a^2*b^3 - 4*a^3*b^2)))*(-(25*a^4*(a^3*b^9)^(1/2) + 384*b^4*(a^3*b^9)
^(1/2) - 144*a*b^9 - 76*a^2*b^8 + 155*a^3*b^7 - 94*a^4*b^6 + 15*a^5*b^5 + 349*a^2*b^2*(a^3*b^9)^(1/2) - 480*a*
b^3*(a^3*b^9)^(1/2) - 134*a^3*b*(a^3*b^9)^(1/2))/(16384*(a^2*b^14 - 5*a^3*b^13 + 10*a^4*b^12 - 10*a^5*b^11 + 5
*a^6*b^10 - a^7*b^9)))^(1/2)*1i)/((((180224*a*b^8 - 483328*a^2*b^7 + 466944*a^3*b^6 - 204800*a^4*b^5 + 40960*a
^5*b^4)/(16384*(b^7 - 4*a*b^6 + 6*a^2*b^5 - 4*a^3*b^4 + a^4*b^3)) - (cos(c + d*x)*(-(25*a^4*(a^3*b^9)^(1/2) +
384*b^4*(a^3*b^9)^(1/2) - 144*a*b^9 - 76*a^2*b^8 + 155*a^3*b^7 - 94*a^4*b^6 + 15*a^5*b^5 + 349*a^2*b^2*(a^3*b^
9)^(1/2) - 480*a*b^3*(a^3*b^9)^(1/2) - 134*a^3*b*(a^3*b^9)^(1/2))/(16384*(a^2*b^14 - 5*a^3*b^13 + 10*a^4*b^12
- 10*a^5*b^11 + 5*a^6*b^10 - a^7*b^9)))^(1/2)*(16384*a*b^9 - 65536*a^2*b^8 + 98304*a^3*b^7 - 65536*a^4*b^6 + 1
6384*a^5*b^5))/(256*(a^4*b - 4*a*b^4 + b^5 + 6*a^2*b^3 - 4*a^3*b^2)))*(-(25*a^4*(a^3*b^9)^(1/2) + 384*b^4*(a^3
*b^9)^(1/2) - 144*a*b^9 - 76*a^2*b^8 + 155*a^3*b^7 - 94*a^4*b^6 + 15*a^5*b^5 + 349*a^2*b^2*(a^3*b^9)^(1/2) - 4
80*a*b^3*(a^3*b^9)^(1/2) - 134*a^3*b*(a^3*b^9)^(1/2))/(16384*(a^2*b^14 - 5*a^3*b^13 + 10*a^4*b^12 - 10*a^5*b^1
1 + 5*a^6*b^10 - a^7*b^9)))^(1/2) + (cos(c + d*x)*(25*a^4 - 94*a^3*b - 164*a*b^3 + 144*b^4 + 161*a^2*b^2))/(25
6*(a^4*b - 4*a*b^4 + b^5 + 6*a^2*b^3 - 4*a^3*b^2)))*(-(25*a^4*(a^3*b^9)^(1/2) + 384*b^4*(a^3*b^9)^(1/2) - 144*
a*b^9 - 76*a^2*b^8 + 155*a^3*b^7 - 94*a^4*b^6 + 15*a^5*b^5 + 349*a^2*b^2*(a^3*b^9)^(1/2) - 480*a*b^3*(a^3*b^9)
^(1/2) - 134*a^3*b*(a^3*b^9)^(1/2))/(16384*(a^2*b^14 - 5*a^3*b^13 + 10*a^4*b^12 - 10*a^5*b^11 + 5*a^6*b^10 - a
^7*b^9)))^(1/2) + (((180224*a*b^8 - 483328*a^2*b^7 + 466944*a^3*b^6 - 204800*a^4*b^5 + 40960*a^5*b^4)/(16384*(
b^7 - 4*a*b^6 + 6*a^2*b^5 - 4*a^3*b^4 + a^4*b^3)) + (cos(c + d*x)*(-(25*a^4*(a^3*b^9)^(1/2) + 384*b^4*(a^3*b^9
)^(1/2) - 144*a*b^9 - 76*a^2*b^8 + 155*a^3*b^7 - 94*a^4*b^6 + 15*a^5*b^5 + 349*a^2*b^2*(a^3*b^9)^(1/2) - 480*a
*b^3*(a^3*b^9)^(1/2) - 134*a^3*b*(a^3*b^9)^(1/2))/(16384*(a^2*b^14 - 5*a^3*b^13 + 10*a^4*b^12 - 10*a^5*b^11 +
5*a^6*b^10 - a^7*b^9)))^(1/2)*(16384*a*b^9 - 65536*a^2*b^8 + 98304*a^3*b^7 - 65536*a^4*b^6 + 16384*a^5*b^5))/(
256*(a^4*b - 4*a*b^4 + b^5 + 6*a^2*b^3 - 4*a^3*b^2)))*(-(25*a^4*(a^3*b^9)^(1/2) + 384*b^4*(a^3*b^9)^(1/2) - 14
4*a*b^9 - 76*a^2*b^8 + 155*a^3*b^7 - 94*a^4*b^6...

________________________________________________________________________________________