3.3.28 \(\int \frac {\sin (c+d x)}{(a-b \sin ^4(c+d x))^3} \, dx\) [228]
Optimal. Leaf size=313 \[ -\frac {3 \left (7 a-10 \sqrt {a} \sqrt {b}+4 b\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{5/2} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} \sqrt [4]{b} d}-\frac {3 \left (7 a+10 \sqrt {a} \sqrt {b}+4 b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{5/2} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} \sqrt [4]{b} d}-\frac {\cos (c+d x) \left (a+b-b \cos ^2(c+d x)\right )}{8 a (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )^2}-\frac {\cos (c+d x) \left ((7 a-3 b) (a+2 b)-6 (2 a-b) b \cos ^2(c+d x)\right )}{32 a^2 (a-b)^2 d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )} \]
[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)
________________________________________________________________________________________
Rubi [A]
time = 0.32, 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 = {3294, 1106,
1192, 1180, 211, 214} \begin {gather*} -\frac {3 \left (-10 \sqrt {a} \sqrt {b}+7 a+4 b\right ) \text {ArcTan}\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} \end {gather*}
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 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 1106
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 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 1192
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 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 (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx &=-\frac {\text {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 {\text {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 {\text {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 ) \text {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 ) \text {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*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.85, size = 784, normalized size = 2.50 \begin {gather*} \frac {-\frac {32 \cos (c+d x) \left (7 a^2+5 a b-3 b^2+3 b (-2 a+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}+3 i \text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {4 a b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-2 b^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-2 i a b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )-28 a^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+24 a b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-10 b^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+14 i a^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-12 i a b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2+5 i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2+28 a^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-24 a b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4+10 b^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-14 i a^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4+12 i a b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4-5 i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4-4 a b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^6+2 b^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^6+2 i a b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^6-i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^6}{-b \text {$\#$1}-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-3 b \text {$\#$1}^5+b \text {$\#$1}^7}\&\right ]}{128 a^2 (a-b)^2 d} \end {gather*}
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)
________________________________________________________________________________________
Maple [A]
time = 1.78, size = 430, normalized size = 1.37
| | |
| method |
result |
size |
| | |
| derivativedivides |
\(\frac {b^{3} \left (\frac {\frac {\frac {3 \left (-4 a \sqrt {a b}+2 \sqrt {a b}\, b +3 a^{2}-a b \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4 b^{2} \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (-11 a \sqrt {a b}+6 \sqrt {a b}\, b +5 a b \right ) \cos \left (d x +c \right )}{4 b^{3} \left (a -b \right )}}{\left (\cos ^{2}\left (d x +c \right )-1-\frac {\sqrt {a b}}{b}\right )^{2}}+\frac {3 \left (4 a \sqrt {a b}-2 \sqrt {a b}\, b -7 a^{2}+9 a b -4 b^{2}\right ) \arctanh \left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{4 b \left (a^{2}-2 a b +b^{2}\right ) \sqrt {\left (\sqrt {a b}+b \right ) b}}}{16 b \,a^{2} \sqrt {a b}}-\frac {\frac {\frac {3 \left (4 a \sqrt {a b}-2 \sqrt {a b}\, b +3 a^{2}-a b \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4 b^{2} \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (11 a \sqrt {a b}-6 \sqrt {a b}\, b +5 a b \right ) \cos \left (d x +c \right )}{4 b^{3} \left (a -b \right )}}{\left (\cos ^{2}\left (d x +c \right )+\frac {\sqrt {a b}}{b}-1\right )^{2}}+\frac {3 \left (4 a \sqrt {a b}-2 \sqrt {a b}\, b +7 a^{2}-9 a b +4 b^{2}\right ) \arctan \left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{4 b \left (a^{2}-2 a b +b^{2}\right ) \sqrt {\left (\sqrt {a b}-b \right ) b}}}{16 b \,a^{2} \sqrt {a b}}\right )}{d}\) |
\(430\) |
| default |
\(\frac {b^{3} \left (\frac {\frac {\frac {3 \left (-4 a \sqrt {a b}+2 \sqrt {a b}\, b +3 a^{2}-a b \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4 b^{2} \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (-11 a \sqrt {a b}+6 \sqrt {a b}\, b +5 a b \right ) \cos \left (d x +c \right )}{4 b^{3} \left (a -b \right )}}{\left (\cos ^{2}\left (d x +c \right )-1-\frac {\sqrt {a b}}{b}\right )^{2}}+\frac {3 \left (4 a \sqrt {a b}-2 \sqrt {a b}\, b -7 a^{2}+9 a b -4 b^{2}\right ) \arctanh \left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{4 b \left (a^{2}-2 a b +b^{2}\right ) \sqrt {\left (\sqrt {a b}+b \right ) b}}}{16 b \,a^{2} \sqrt {a b}}-\frac {\frac {\frac {3 \left (4 a \sqrt {a b}-2 \sqrt {a b}\, b +3 a^{2}-a b \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4 b^{2} \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (11 a \sqrt {a b}-6 \sqrt {a b}\, b +5 a b \right ) \cos \left (d x +c \right )}{4 b^{3} \left (a -b \right )}}{\left (\cos ^{2}\left (d x +c \right )+\frac {\sqrt {a b}}{b}-1\right )^{2}}+\frac {3 \left (4 a \sqrt {a b}-2 \sqrt {a b}\, b +7 a^{2}-9 a b +4 b^{2}\right ) \arctan \left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{4 b \left (a^{2}-2 a b +b^{2}\right ) \sqrt {\left (\sqrt {a b}-b \right ) b}}}{16 b \,a^{2} \sqrt {a b}}\right )}{d}\) |
\(430\) |
| risch |
\(\text {Expression too large to display}\) |
\(1321\) |
| | |
|
|
|
|
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,method=_RETURNVERBOSE)
[Out]
1/d*b^3*(1/16/b/a^2/(a*b)^(1/2)*((3/4*(-4*a*(a*b)^(1/2)+2*(a*b)^(1/2)*b+3*a^2-a*b)/b^2/(a^2-2*a*b+b^2)*cos(d*x
+c)^3+1/4*(-11*a*(a*b)^(1/2)+6*(a*b)^(1/2)*b+5*a*b)/b^3/(a-b)*cos(d*x+c))/(cos(d*x+c)^2-1-(a*b)^(1/2)/b)^2+3/4
*(4*a*(a*b)^(1/2)-2*(a*b)^(1/2)*b-7*a^2+9*a*b-4*b^2)/b/(a^2-2*a*b+b^2)/(((a*b)^(1/2)+b)*b)^(1/2)*arctanh(b*cos
(d*x+c)/(((a*b)^(1/2)+b)*b)^(1/2)))-1/16/b/a^2/(a*b)^(1/2)*((3/4*(4*a*(a*b)^(1/2)-2*(a*b)^(1/2)*b+3*a^2-a*b)/b
^2/(a^2-2*a*b+b^2)*cos(d*x+c)^3+1/4*(11*a*(a*b)^(1/2)-6*(a*b)^(1/2)*b+5*a*b)/b^3/(a-b)*cos(d*x+c))/(cos(d*x+c)
^2+(a*b)^(1/2)/b-1)^2+3/4*(4*a*(a*b)^(1/2)-2*(a*b)^(1/2)*b+7*a^2-9*a*b+4*b^2)/b/(a^2-2*a*b+b^2)/(((a*b)^(1/2)-
b)*b)^(1/2)*arctan(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)/(a-b*sin(d*x+c)^4)^3,x, algorithm="maxima")
[Out]
1/8*(24*(2*a*b^4 - b^5)*cos(2*d*x + 2*c)*cos(d*x + c) - 8*(14*a^2*b^3 + 28*a*b^4 - 15*b^5)*sin(3*d*x + 3*c)*si
n(2*d*x + 2*c) + 24*(2*a*b^4 - b^5)*sin(2*d*x + 2*c)*sin(d*x + c) - (3*(2*a*b^4 - b^5)*cos(15*d*x + 15*c) - (1
4*a^2*b^3 + 28*a*b^4 - 15*b^5)*cos(13*d*x + 13*c) - (86*a^2*b^3 - 128*a*b^4 + 27*b^5)*cos(11*d*x + 11*c) + (35
2*a^3*b^2 - 60*a^2*b^3 - 106*a*b^4 + 15*b^5)*cos(9*d*x + 9*c) + (352*a^3*b^2 - 60*a^2*b^3 - 106*a*b^4 + 15*b^5
)*cos(7*d*x + 7*c) - (86*a^2*b^3 - 128*a*b^4 + 27*b^5)*cos(5*d*x + 5*c) - (14*a^2*b^3 + 28*a*b^4 - 15*b^5)*cos
(3*d*x + 3*c) + 3*(2*a*b^4 - b^5)*cos(d*x + c))*cos(16*d*x + 16*c) - 3*(2*a*b^4 - b^5 - 8*(2*a*b^4 - b^5)*cos(
14*d*x + 14*c) - 4*(16*a^2*b^3 - 22*a*b^4 + 7*b^5)*cos(12*d*x + 12*c) + 8*(32*a^2*b^3 - 30*a*b^4 + 7*b^5)*cos(
10*d*x + 10*c) + 2*(256*a^3*b^2 - 320*a^2*b^3 + 166*a*b^4 - 35*b^5)*cos(8*d*x + 8*c) + 8*(32*a^2*b^3 - 30*a*b^
4 + 7*b^5)*cos(6*d*x + 6*c) - 4*(16*a^2*b^3 - 22*a*b^4 + 7*b^5)*cos(4*d*x + 4*c) - 8*(2*a*b^4 - b^5)*cos(2*d*x
+ 2*c))*cos(15*d*x + 15*c) - 8*((14*a^2*b^3 + 28*a*b^4 - 15*b^5)*cos(13*d*x + 13*c) + (86*a^2*b^3 - 128*a*b^4
+ 27*b^5)*cos(11*d*x + 11*c) - (352*a^3*b^2 - 60*a^2*b^3 - 106*a*b^4 + 15*b^5)*cos(9*d*x + 9*c) - (352*a^3*b^
2 - 60*a^2*b^3 - 106*a*b^4 + 15*b^5)*cos(7*d*x + 7*c) + (86*a^2*b^3 - 128*a*b^4 + 27*b^5)*cos(5*d*x + 5*c) + (
14*a^2*b^3 + 28*a*b^4 - 15*b^5)*cos(3*d*x + 3*c) - 3*(2*a*b^4 - b^5)*cos(d*x + c))*cos(14*d*x + 14*c) + (14*a^
2*b^3 + 28*a*b^4 - 15*b^5 - 4*(112*a^3*b^2 + 126*a^2*b^3 - 316*a*b^4 + 105*b^5)*cos(12*d*x + 12*c) + 8*(224*a^
3*b^2 + 350*a^2*b^3 - 436*a*b^4 + 105*b^5)*cos(10*d*x + 10*c) + 2*(1792*a^4*b + 2240*a^3*b^2 - 4118*a^2*b^3 +
2420*a*b^4 - 525*b^5)*cos(8*d*x + 8*c) + 8*(224*a^3*b^2 + 350*a^2*b^3 - 436*a*b^4 + 105*b^5)*cos(6*d*x + 6*c)
- 4*(112*a^3*b^2 + 126*a^2*b^3 - 316*a*b^4 + 105*b^5)*cos(4*d*x + 4*c) - 8*(14*a^2*b^3 + 28*a*b^4 - 15*b^5)*co
s(2*d*x + 2*c))*cos(13*d*x + 13*c) - 4*((688*a^3*b^2 - 1626*a^2*b^3 + 1112*a*b^4 - 189*b^5)*cos(11*d*x + 11*c)
- (2816*a^4*b - 2944*a^3*b^2 - 428*a^2*b^3 + 862*a*b^4 - 105*b^5)*cos(9*d*x + 9*c) - (2816*a^4*b - 2944*a^3*b
^2 - 428*a^2*b^3 + 862*a*b^4 - 105*b^5)*cos(7*d*x + 7*c) + (688*a^3*b^2 - 1626*a^2*b^3 + 1112*a*b^4 - 189*b^5)
*cos(5*d*x + 5*c) + (112*a^3*b^2 + 126*a^2*b^3 - 316*a*b^4 + 105*b^5)*cos(3*d*x + 3*c) - 3*(16*a^2*b^3 - 22*a*
b^4 + 7*b^5)*cos(d*x + c))*cos(12*d*x + 12*c) + (86*a^2*b^3 - 128*a*b^4 + 27*b^5 + 8*(1376*a^3*b^2 - 2650*a^2*
b^3 + 1328*a*b^4 - 189*b^5)*cos(10*d*x + 10*c) + 2*(11008*a^4*b - 24640*a^3*b^2 + 18754*a^2*b^3 - 7072*a*b^4 +
945*b^5)*cos(8*d*x + 8*c) + 8*(1376*a^3*b^2 - 2650*a^2*b^3 + 1328*a*b^4 - 189*b^5)*cos(6*d*x + 6*c) - 4*(688*
a^3*b^2 - 1626*a^2*b^3 + 1112*a*b^4 - 189*b^5)*cos(4*d*x + 4*c) - 8*(86*a^2*b^3 - 128*a*b^4 + 27*b^5)*cos(2*d*
x + 2*c))*cos(11*d*x + 11*c) - 8*((5632*a^4*b - 3424*a^3*b^2 - 1276*a^2*b^3 + 982*a*b^4 - 105*b^5)*cos(9*d*x +
9*c) + (5632*a^4*b - 3424*a^3*b^2 - 1276*a^2*b^3 + 982*a*b^4 - 105*b^5)*cos(7*d*x + 7*c) - (1376*a^3*b^2 - 26
50*a^2*b^3 + 1328*a*b^4 - 189*b^5)*cos(5*d*x + 5*c) - (224*a^3*b^2 + 350*a^2*b^3 - 436*a*b^4 + 105*b^5)*cos(3*
d*x + 3*c) + 3*(32*a^2*b^3 - 30*a*b^4 + 7*b^5)*cos(d*x + c))*cos(10*d*x + 10*c) - (352*a^3*b^2 - 60*a^2*b^3 -
106*a*b^4 + 15*b^5 + 2*(45056*a^5 - 41472*a^4*b + 4512*a^3*b^2 + 9996*a^2*b^3 - 5150*a*b^4 + 525*b^5)*cos(8*d*
x + 8*c) + 8*(5632*a^4*b - 3424*a^3*b^2 - 1276*a^2*b^3 + 982*a*b^4 - 105*b^5)*cos(6*d*x + 6*c) - 4*(2816*a^4*b
- 2944*a^3*b^2 - 428*a^2*b^3 + 862*a*b^4 - 105*b^5)*cos(4*d*x + 4*c) - 8*(352*a^3*b^2 - 60*a^2*b^3 - 106*a*b^
4 + 15*b^5)*cos(2*d*x + 2*c))*cos(9*d*x + 9*c) - 2*((45056*a^5 - 41472*a^4*b + 4512*a^3*b^2 + 9996*a^2*b^3 - 5
150*a*b^4 + 525*b^5)*cos(7*d*x + 7*c) - (11008*a^4*b - 24640*a^3*b^2 + 18754*a^2*b^3 - 7072*a*b^4 + 945*b^5)*c
os(5*d*x + 5*c) - (1792*a^4*b + 2240*a^3*b^2 - 4118*a^2*b^3 + 2420*a*b^4 - 525*b^5)*cos(3*d*x + 3*c) + 3*(256*
a^3*b^2 - 320*a^2*b^3 + 166*a*b^4 - 35*b^5)*cos(d*x + c))*cos(8*d*x + 8*c) - (352*a^3*b^2 - 60*a^2*b^3 - 106*a
*b^4 + 15*b^5 + 8*(5632*a^4*b - 3424*a^3*b^2 - 1276*a^2*b^3 + 982*a*b^4 - 105*b^5)*cos(6*d*x + 6*c) - 4*(2816*
a^4*b - 2944*a^3*b^2 - 428*a^2*b^3 + 862*a*b^4 - 105*b^5)*cos(4*d*x + 4*c) - 8*(352*a^3*b^2 - 60*a^2*b^3 - 106
*a*b^4 + 15*b^5)*cos(2*d*x + 2*c))*cos(7*d*x + 7*c) + 8*((1376*a^3*b^2 - 2650*a^2*b^3 + 1328*a*b^4 - 189*b^5)*
cos(5*d*x + 5*c) + (224*a^3*b^2 + 350*a^2*b^3 - 436*a*b^4 + 105*b^5)*cos(3*d*x + 3*c) - 3*(32*a^2*b^3 - 30*a*b
^4 + 7*b^5)*cos(d*x + c))*cos(6*d*x + 6*c) + (86*a^2*b^3 - 128*a*b^4 + 27*b^5 - 4*(688*a^3*b^2 - 1626*a^2*b^3
+ 1112*a*b^4 - 189*b^5)*cos(4*d*x + 4*c) - 8*(86*a^2*b^3 - 128*a*b^4 + 27*b^5)*cos(2*d*x + 2*c))*cos(5*d*x + 5
*c) - 4*((112*a^3*b^2 + 126*a^2*b^3 - 316*a*b^4 + 105*b^5)*cos(3*d*x + 3*c) - 3*(16*a^2*b^3 - 22*a*b^4 + 7*b^5
)*cos(d*x + c))*cos(4*d*x + 4*c) + (14*a^2*b^3 + 28*a*b^4 - 15*b^5 - 8*(14*a^2*b^3 + 28*a*b^4 - 15*b^5)*cos(2*
d*x + 2*c))*cos(3*d*x + 3*c) - 3*(2*a*b^4 - b^5)*cos(d*x + c) + 8*((a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d*cos(16*d*
x + 16*c)^2 + 64*(a^4*b^4 - 2*a^3*b^5 + a^2*b^6...
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4160 vs.
\(2 (263) = 526\).
time = 1.04, size = 4160, normalized size = 13.29 \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)/(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...
________________________________________________________________________________________
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)/(a-b*sin(d*x+c)**4)**3,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 766 vs.
\(2 (263) = 526\).
time = 1.20, size = 766, normalized size = 2.45 \begin {gather*} -\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 )}} \end {gather*}
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))
________________________________________________________________________________________
Mupad [B]
time = 18.91, size = 2500, normalized size = 7.99 \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)/(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*...
________________________________________________________________________________________