3.3.19 \(\int \frac {\sin ^6(c+d x)}{(a-b \sin ^4(c+d x))^2} \, dx\) [219]
Optimal. Leaf size=233 \[ -\frac {\left (2 \sqrt {a}-3 \sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} b^{3/2} d}+\frac {\left (2 \sqrt {a}+3 \sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 \sqrt [4]{a} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} b^{3/2} d}-\frac {\tan (c+d x)}{4 (a-b) b d}+\frac {\sec ^2(c+d x) \tan ^3(c+d x)}{4 b d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )} \]
[Out]
-1/8*arctan((a^(1/2)-b^(1/2))^(1/2)*tan(d*x+c)/a^(1/4))*(2*a^(1/2)-3*b^(1/2))/a^(1/4)/b^(3/2)/d/(a^(1/2)-b^(1/
2))^(3/2)+1/8*arctan((a^(1/2)+b^(1/2))^(1/2)*tan(d*x+c)/a^(1/4))*(2*a^(1/2)+3*b^(1/2))/a^(1/4)/b^(3/2)/d/(a^(1
/2)+b^(1/2))^(3/2)-1/4*tan(d*x+c)/(a-b)/b/d+1/4*sec(d*x+c)^2*tan(d*x+c)^3/b/d/(a+2*a*tan(d*x+c)^2+(a-b)*tan(d*
x+c)^4)
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Rubi [A]
time = 0.22, antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3296, 1134,
1293, 1180, 211} \begin {gather*} -\frac {\left (2 \sqrt {a}-3 \sqrt {b}\right ) \text {ArcTan}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 \sqrt [4]{a} b^{3/2} d \left (\sqrt {a}-\sqrt {b}\right )^{3/2}}+\frac {\left (2 \sqrt {a}+3 \sqrt {b}\right ) \text {ArcTan}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 \sqrt [4]{a} b^{3/2} d \left (\sqrt {a}+\sqrt {b}\right )^{3/2}}-\frac {\tan (c+d x)}{4 b d (a-b)}+\frac {\tan ^3(c+d x) \sec ^2(c+d x)}{4 b d \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Sin[c + d*x]^6/(a - b*Sin[c + d*x]^4)^2,x]
[Out]
-1/8*((2*Sqrt[a] - 3*Sqrt[b])*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(a^(1/4)*(Sqrt[a] - Sqrt
[b])^(3/2)*b^(3/2)*d) + ((2*Sqrt[a] + 3*Sqrt[b])*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(8*a^
(1/4)*(Sqrt[a] + Sqrt[b])^(3/2)*b^(3/2)*d) - Tan[c + d*x]/(4*(a - b)*b*d) + (Sec[c + d*x]^2*Tan[c + d*x]^3)/(4
*b*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[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 1134
Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-d^3)*(d*x)^(m - 3)*(2*a
+ b*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*(p + 1)*(b^2 - 4*a*c))), x] + Dist[d^4/(2*(p + 1)*(b^2 - 4*a*c)), Int
[(d*x)^(m - 4)*(2*a*(m - 3) + b*(m + 4*p + 3)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && GtQ[m, 3] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
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 1293
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[e*f*
(f*x)^(m - 1)*((a + b*x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 3))), x] - Dist[f^2/(c*(m + 4*p + 3)), Int[(f*x)^(m -
2)*(a + b*x^2 + c*x^4)^p*Simp[a*e*(m - 1) + (b*e*(m + 2*p + 1) - c*d*(m + 4*p + 3))*x^2, x], x], x] /; FreeQ[
{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[m, 1] && NeQ[m + 4*p + 3, 0] && IntegerQ[2*p] && (Inte
gerQ[p] || IntegerQ[m])
Rule 3296
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[x^m*((a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p/(1 + ff^2*
x^2)^(m/2 + 2*p + 1)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]
Rubi steps
\begin {align*} \int \frac {\sin ^6(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {x^6}{\left (a+2 a x^2+(a-b) x^4\right )^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\sec ^2(c+d x) \tan ^3(c+d x)}{4 b d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {x^2 \left (6 a+2 a x^2\right )}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{8 a b d}\\ &=-\frac {\tan (c+d x)}{4 (a-b) b d}+\frac {\sec ^2(c+d x) \tan ^3(c+d x)}{4 b d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {2 a^2-2 a (a-3 b) x^2}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{8 a (a-b) b d}\\ &=-\frac {\tan (c+d x)}{4 (a-b) b d}+\frac {\sec ^2(c+d x) \tan ^3(c+d x)}{4 b d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}-\frac {\left (a-\frac {2 \sqrt {a} (a-2 b)}{\sqrt {b}}-3 b\right ) \text {Subst}\left (\int \frac {1}{a-\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{8 (a-b) b d}-\frac {\left (a+\frac {2 \sqrt {a} (a-2 b)}{\sqrt {b}}-3 b\right ) \text {Subst}\left (\int \frac {1}{a+\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{8 (a-b) b d}\\ &=-\frac {\left (2 \sqrt {a}-3 \sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} b^{3/2} d}+\frac {\left (2 \sqrt {a}+3 \sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 \sqrt [4]{a} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} b^{3/2} d}-\frac {\tan (c+d x)}{4 (a-b) b d}+\frac {\sec ^2(c+d x) \tan ^3(c+d x)}{4 b d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}\\ \end {align*}
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Mathematica [A]
time = 1.77, size = 238, normalized size = 1.02 \begin {gather*} \frac {\frac {\left (2 a+\sqrt {a} \sqrt {b}-3 b\right ) \sqrt {b} \tan ^{-1}\left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a+\sqrt {a} \sqrt {b}}}-\frac {\sqrt {b} \left (-2 a+\sqrt {a} \sqrt {b}+3 b\right ) \tanh ^{-1}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tan (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {-a+\sqrt {a} \sqrt {b}}}+\frac {4 b (-2 a-b+b \cos (2 (c+d x))) \sin (2 (c+d x))}{8 a-3 b+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))}}{8 (a-b) b^2 d} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Sin[c + d*x]^6/(a - b*Sin[c + d*x]^4)^2,x]
[Out]
(((2*a + Sqrt[a]*Sqrt[b] - 3*b)*Sqrt[b]*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/
Sqrt[a + Sqrt[a]*Sqrt[b]] - (Sqrt[b]*(-2*a + Sqrt[a]*Sqrt[b] + 3*b)*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])
/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/Sqrt[-a + Sqrt[a]*Sqrt[b]] + (4*b*(-2*a - b + b*Cos[2*(c + d*x)])*Sin[2*(c + d*x
)])/(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)]))/(8*(a - b)*b^2*d)
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Maple [A]
time = 0.71, size = 262, normalized size = 1.12
| | |
| method |
result |
size |
| | |
| derivativedivides |
\(\frac {\frac {-\frac {\left (a +b \right ) \left (\tan ^{3}\left (d x +c \right )\right )}{4 b \left (a -b \right )}-\frac {a \tan \left (d x +c \right )}{4 b \left (a -b \right )}}{\left (\tan ^{4}\left (d x +c \right )\right ) a -\left (\tan ^{4}\left (d x +c \right )\right ) b +2 a \left (\tan ^{2}\left (d x +c \right )\right )+a}+\frac {\frac {\left (-a \sqrt {a b}+3 \sqrt {a b}\, b -2 a^{2}+4 a b \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {\left (-a \sqrt {a b}+3 \sqrt {a b}\, b +2 a^{2}-4 a b \right ) \arctanh \left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}}{4 b}}{d}\) |
\(262\) |
| default |
\(\frac {\frac {-\frac {\left (a +b \right ) \left (\tan ^{3}\left (d x +c \right )\right )}{4 b \left (a -b \right )}-\frac {a \tan \left (d x +c \right )}{4 b \left (a -b \right )}}{\left (\tan ^{4}\left (d x +c \right )\right ) a -\left (\tan ^{4}\left (d x +c \right )\right ) b +2 a \left (\tan ^{2}\left (d x +c \right )\right )+a}+\frac {\frac {\left (-a \sqrt {a b}+3 \sqrt {a b}\, b -2 a^{2}+4 a b \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {\left (-a \sqrt {a b}+3 \sqrt {a b}\, b +2 a^{2}-4 a b \right ) \arctanh \left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}}{4 b}}{d}\) |
\(262\) |
| risch |
\(-\frac {i \left (2 a \,{\mathrm e}^{6 i \left (d x +c \right )}-b \,{\mathrm e}^{6 i \left (d x +c \right )}-8 a \,{\mathrm e}^{4 i \left (d x +c \right )}+3 b \,{\mathrm e}^{4 i \left (d x +c \right )}-2 a \,{\mathrm e}^{2 i \left (d x +c \right )}-3 b \,{\mathrm e}^{2 i \left (d x +c \right )}+b \right )}{2 b \left (a -b \right ) d \left (b \,{\mathrm e}^{8 i \left (d x +c \right )}-4 b \,{\mathrm e}^{6 i \left (d x +c \right )}-16 a \,{\mathrm e}^{4 i \left (d x +c \right )}+6 b \,{\mathrm e}^{4 i \left (d x +c \right )}-4 b \,{\mathrm e}^{2 i \left (d x +c \right )}+b \right )}-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (a^{4} b^{6} d^{4}-3 a^{3} b^{7} d^{4}+3 a^{2} b^{8} d^{4}-a \,b^{9} d^{4}\right ) \textit {\_Z}^{4}+\left (128 a^{3} b^{3} d^{2}-480 a^{2} b^{4} d^{2}+480 a \,b^{5} d^{2}\right ) \textit {\_Z}^{2}+4096 a^{2}-18432 a b +20736 b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\frac {i a^{5} b^{5} d^{3}}{640 a^{2} b^{2}-2592 a \,b^{3}+2592 b^{4}}-\frac {6 i a^{4} b^{6} d^{3}}{640 a^{2} b^{2}-2592 a \,b^{3}+2592 b^{4}}+\frac {12 i a^{3} b^{7} d^{3}}{640 a^{2} b^{2}-2592 a \,b^{3}+2592 b^{4}}-\frac {10 i a^{2} b^{8} d^{3}}{640 a^{2} b^{2}-2592 a \,b^{3}+2592 b^{4}}+\frac {3 i a \,b^{9} d^{3}}{640 a^{2} b^{2}-2592 a \,b^{3}+2592 b^{4}}\right ) \textit {\_R}^{3}+\left (\frac {16 d^{2} b^{3} a^{5}}{640 a^{2} b^{2}-2592 a \,b^{3}+2592 b^{4}}-\frac {84 d^{2} b^{4} a^{4}}{640 a^{2} b^{2}-2592 a \,b^{3}+2592 b^{4}}+\frac {156 d^{2} b^{5} a^{3}}{640 a^{2} b^{2}-2592 a \,b^{3}+2592 b^{4}}-\frac {124 d^{2} b^{6} a^{2}}{640 a^{2} b^{2}-2592 a \,b^{3}+2592 b^{4}}+\frac {36 d^{2} b^{7} a}{640 a^{2} b^{2}-2592 a \,b^{3}+2592 b^{4}}\right ) \textit {\_R}^{2}+\left (\frac {64 i a^{4} b^{2} d}{640 a^{2} b^{2}-2592 a \,b^{3}+2592 b^{4}}-\frac {592 i a^{3} b^{3} d}{640 a^{2} b^{2}-2592 a \,b^{3}+2592 b^{4}}+\frac {1568 i a^{2} b^{4} d}{640 a^{2} b^{2}-2592 a \,b^{3}+2592 b^{4}}-\frac {1296 i a \,b^{5} d}{640 a^{2} b^{2}-2592 a \,b^{3}+2592 b^{4}}\right ) \textit {\_R} +\frac {1024 a^{4}}{640 a^{2} b^{2}-2592 a \,b^{3}+2592 b^{4}}-\frac {6144 a^{3} b}{640 a^{2} b^{2}-2592 a \,b^{3}+2592 b^{4}}+\frac {11840 a^{2} b^{2}}{640 a^{2} b^{2}-2592 a \,b^{3}+2592 b^{4}}-\frac {6048 a \,b^{3}}{640 a^{2} b^{2}-2592 a \,b^{3}+2592 b^{4}}-\frac {2592 b^{4}}{640 a^{2} b^{2}-2592 a \,b^{3}+2592 b^{4}}\right )\right )}{64}\) |
\(881\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sin(d*x+c)^6/(a-b*sin(d*x+c)^4)^2,x,method=_RETURNVERBOSE)
[Out]
1/d*((-1/4*(a+b)/b/(a-b)*tan(d*x+c)^3-1/4*a/b/(a-b)*tan(d*x+c))/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^
2+a)+1/4/b*(1/2*(-a*(a*b)^(1/2)+3*(a*b)^(1/2)*b-2*a^2+4*a*b)/(a*b)^(1/2)/(a-b)/(((a*b)^(1/2)+a)*(a-b))^(1/2)*a
rctan((a-b)*tan(d*x+c)/(((a*b)^(1/2)+a)*(a-b))^(1/2))+1/2*(-a*(a*b)^(1/2)+3*(a*b)^(1/2)*b+2*a^2-4*a*b)/(a*b)^(
1/2)/(a-b)/(((a*b)^(1/2)-a)*(a-b))^(1/2)*arctanh((-a+b)*tan(d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1/2))))
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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)^6/(a-b*sin(d*x+c)^4)^2,x, algorithm="maxima")
[Out]
1/2*(2*(16*a^2 + 2*a*b - 3*b^2)*cos(4*d*x + 4*c)*sin(2*d*x + 2*c) + ((2*a*b - b^2)*sin(6*d*x + 6*c) - (8*a*b -
3*b^2)*sin(4*d*x + 4*c) - (2*a*b + 3*b^2)*sin(2*d*x + 2*c))*cos(8*d*x + 8*c) + 2*((16*a^2 + 2*a*b - 3*b^2)*si
n(4*d*x + 4*c) + 4*(2*a*b + b^2)*sin(2*d*x + 2*c))*cos(6*d*x + 6*c) - 2*((a*b^3 - b^4)*d*cos(8*d*x + 8*c)^2 +
16*(a*b^3 - b^4)*d*cos(6*d*x + 6*c)^2 + 4*(64*a^3*b - 112*a^2*b^2 + 57*a*b^3 - 9*b^4)*d*cos(4*d*x + 4*c)^2 + 1
6*(a*b^3 - b^4)*d*cos(2*d*x + 2*c)^2 + (a*b^3 - b^4)*d*sin(8*d*x + 8*c)^2 + 16*(a*b^3 - b^4)*d*sin(6*d*x + 6*c
)^2 + 4*(64*a^3*b - 112*a^2*b^2 + 57*a*b^3 - 9*b^4)*d*sin(4*d*x + 4*c)^2 + 16*(8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d
*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*(a*b^3 - b^4)*d*sin(2*d*x + 2*c)^2 - 8*(a*b^3 - b^4)*d*cos(2*d*x + 2*c
) + (a*b^3 - b^4)*d - 2*(4*(a*b^3 - b^4)*d*cos(6*d*x + 6*c) + 2*(8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*cos(4*d*x + 4
*c) + 4*(a*b^3 - b^4)*d*cos(2*d*x + 2*c) - (a*b^3 - b^4)*d)*cos(8*d*x + 8*c) + 8*(2*(8*a^2*b^2 - 11*a*b^3 + 3*
b^4)*d*cos(4*d*x + 4*c) + 4*(a*b^3 - b^4)*d*cos(2*d*x + 2*c) - (a*b^3 - b^4)*d)*cos(6*d*x + 6*c) + 4*(4*(8*a^2
*b^2 - 11*a*b^3 + 3*b^4)*d*cos(2*d*x + 2*c) - (8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d)*cos(4*d*x + 4*c) - 4*(2*(a*b^3
- b^4)*d*sin(6*d*x + 6*c) + (8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*sin(4*d*x + 4*c) + 2*(a*b^3 - b^4)*d*sin(2*d*x +
2*c))*sin(8*d*x + 8*c) + 16*((8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*sin(4*d*x + 4*c) + 2*(a*b^3 - b^4)*d*sin(2*d*x
+ 2*c))*sin(6*d*x + 6*c))*integrate(-(4*(2*a*b - 3*b^2)*cos(6*d*x + 6*c)^2 + 12*(8*a*b - 3*b^2)*cos(4*d*x + 4*
c)^2 + 4*(2*a*b - 3*b^2)*cos(2*d*x + 2*c)^2 + 4*(2*a*b - 3*b^2)*sin(6*d*x + 6*c)^2 + 12*(8*a*b - 3*b^2)*sin(4*
d*x + 4*c)^2 + 2*(16*a^2 - 30*a*b + 21*b^2)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 4*(2*a*b - 3*b^2)*sin(2*d*x +
2*c)^2 - (6*b^2*cos(4*d*x + 4*c) + (2*a*b - 3*b^2)*cos(6*d*x + 6*c) + (2*a*b - 3*b^2)*cos(2*d*x + 2*c))*cos(8*
d*x + 8*c) - (2*a*b - 3*b^2 - 2*(16*a^2 - 30*a*b + 21*b^2)*cos(4*d*x + 4*c) - 8*(2*a*b - 3*b^2)*cos(2*d*x + 2*
c))*cos(6*d*x + 6*c) - 2*(3*b^2 - (16*a^2 - 30*a*b + 21*b^2)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - (2*a*b - 3*b
^2)*cos(2*d*x + 2*c) - (6*b^2*sin(4*d*x + 4*c) + (2*a*b - 3*b^2)*sin(6*d*x + 6*c) + (2*a*b - 3*b^2)*sin(2*d*x
+ 2*c))*sin(8*d*x + 8*c) + 2*((16*a^2 - 30*a*b + 21*b^2)*sin(4*d*x + 4*c) + 4*(2*a*b - 3*b^2)*sin(2*d*x + 2*c)
)*sin(6*d*x + 6*c))/(a*b^3 - b^4 + (a*b^3 - b^4)*cos(8*d*x + 8*c)^2 + 16*(a*b^3 - b^4)*cos(6*d*x + 6*c)^2 + 4*
(64*a^3*b - 112*a^2*b^2 + 57*a*b^3 - 9*b^4)*cos(4*d*x + 4*c)^2 + 16*(a*b^3 - b^4)*cos(2*d*x + 2*c)^2 + (a*b^3
- b^4)*sin(8*d*x + 8*c)^2 + 16*(a*b^3 - b^4)*sin(6*d*x + 6*c)^2 + 4*(64*a^3*b - 112*a^2*b^2 + 57*a*b^3 - 9*b^4
)*sin(4*d*x + 4*c)^2 + 16*(8*a^2*b^2 - 11*a*b^3 + 3*b^4)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*(a*b^3 - b^4)*
sin(2*d*x + 2*c)^2 + 2*(a*b^3 - b^4 - 4*(a*b^3 - b^4)*cos(6*d*x + 6*c) - 2*(8*a^2*b^2 - 11*a*b^3 + 3*b^4)*cos(
4*d*x + 4*c) - 4*(a*b^3 - b^4)*cos(2*d*x + 2*c))*cos(8*d*x + 8*c) - 8*(a*b^3 - b^4 - 2*(8*a^2*b^2 - 11*a*b^3 +
3*b^4)*cos(4*d*x + 4*c) - 4*(a*b^3 - b^4)*cos(2*d*x + 2*c))*cos(6*d*x + 6*c) - 4*(8*a^2*b^2 - 11*a*b^3 + 3*b^
4 - 4*(8*a^2*b^2 - 11*a*b^3 + 3*b^4)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - 8*(a*b^3 - b^4)*cos(2*d*x + 2*c) - 4
*(2*(a*b^3 - b^4)*sin(6*d*x + 6*c) + (8*a^2*b^2 - 11*a*b^3 + 3*b^4)*sin(4*d*x + 4*c) + 2*(a*b^3 - b^4)*sin(2*d
*x + 2*c))*sin(8*d*x + 8*c) + 16*((8*a^2*b^2 - 11*a*b^3 + 3*b^4)*sin(4*d*x + 4*c) + 2*(a*b^3 - b^4)*sin(2*d*x
+ 2*c))*sin(6*d*x + 6*c)), x) - (b^2 + (2*a*b - b^2)*cos(6*d*x + 6*c) - (8*a*b - 3*b^2)*cos(4*d*x + 4*c) - (2*
a*b + 3*b^2)*cos(2*d*x + 2*c))*sin(8*d*x + 8*c) + (2*a*b + 3*b^2 - 2*(16*a^2 + 2*a*b - 3*b^2)*cos(4*d*x + 4*c)
- 8*(2*a*b + b^2)*cos(2*d*x + 2*c))*sin(6*d*x + 6*c) + (8*a*b - 3*b^2 - 2*(16*a^2 + 2*a*b - 3*b^2)*cos(2*d*x
+ 2*c))*sin(4*d*x + 4*c) - (2*a*b - b^2)*sin(2*d*x + 2*c))/((a*b^3 - b^4)*d*cos(8*d*x + 8*c)^2 + 16*(a*b^3 - b
^4)*d*cos(6*d*x + 6*c)^2 + 4*(64*a^3*b - 112*a^2*b^2 + 57*a*b^3 - 9*b^4)*d*cos(4*d*x + 4*c)^2 + 16*(a*b^3 - b^
4)*d*cos(2*d*x + 2*c)^2 + (a*b^3 - b^4)*d*sin(8*d*x + 8*c)^2 + 16*(a*b^3 - b^4)*d*sin(6*d*x + 6*c)^2 + 4*(64*a
^3*b - 112*a^2*b^2 + 57*a*b^3 - 9*b^4)*d*sin(4*d*x + 4*c)^2 + 16*(8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*sin(4*d*x +
4*c)*sin(2*d*x + 2*c) + 16*(a*b^3 - b^4)*d*sin(2*d*x + 2*c)^2 - 8*(a*b^3 - b^4)*d*cos(2*d*x + 2*c) + (a*b^3 -
b^4)*d - 2*(4*(a*b^3 - b^4)*d*cos(6*d*x + 6*c) + 2*(8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*cos(4*d*x + 4*c) + 4*(a*b^
3 - b^4)*d*cos(2*d*x + 2*c) - (a*b^3 - b^4)*d)*cos(8*d*x + 8*c) + 8*(2*(8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*cos(4*
d*x + 4*c) + 4*(a*b^3 - b^4)*d*cos(2*d*x + 2*c) - (a*b^3 - b^4)*d)*cos(6*d*x + 6*c) + 4*(4*(8*a^2*b^2 - 11*a*b
^3 + 3*b^4)*d*cos(2*d*x + 2*c) - (8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d)*cos(4*d*x + 4*c) - 4*(2*(a*b^3 - b^4)*d*sin
(6*d*x + 6*c) + (8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*sin(4*d*x + 4*c) + 2*(a*b^3 - b^4)*d*sin(2*d*x + 2*c))*sin(8*
d*x + 8*c) + 16*((8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*sin(4*d*x + 4*c) + 2*(a*b^3 - b^4)*d*sin(2*d*x + 2*c))*sin(6
*d*x + 6*c))
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3135 vs.
\(2 (181) = 362\).
time = 1.00, size = 3135, normalized size = 13.45 \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)^6/(a-b*sin(d*x+c)^4)^2,x, algorithm="fricas")
[Out]
-1/32*(((a*b^2 - b^3)*d*cos(d*x + c)^4 - 2*(a*b^2 - b^3)*d*cos(d*x + c)^2 - (a^2*b - 2*a*b^2 + b^3)*d)*sqrt(-(
(a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2*sqrt((25*a^2 - 90*a*b + 81*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 -
20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4)) + 4*a^2 - 15*a*b + 15*b^2)/((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5
- b^6)*d^2))*log(1/4*(20*a^2 - 81*a*b + 81*b^2)*cos(d*x + c)^2 - 5*a^2 + 81/4*a*b - 81/4*b^2 + 1/2*((a^5*b^3
- 6*a^4*b^4 + 12*a^3*b^5 - 10*a^2*b^6 + 3*a*b^7)*d^3*sqrt((25*a^2 - 90*a*b + 81*b^2)/((a^7*b^3 - 6*a^6*b^4 + 1
5*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4))*cos(d*x + c)*sin(d*x + c) + 2*(5*a^3*b - 19*a^2
*b^2 + 18*a*b^3)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2*sqrt((25*a^2 -
90*a*b + 81*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4)) + 4*a
^2 - 15*a*b + 15*b^2)/((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2)) + 1/4*(2*(4*a^5*b - 21*a^4*b^2 + 39*a^3*b^3
- 31*a^2*b^4 + 9*a*b^5)*d^2*cos(d*x + c)^2 - (4*a^5*b - 21*a^4*b^2 + 39*a^3*b^3 - 31*a^2*b^4 + 9*a*b^5)*d^2)*
sqrt((25*a^2 - 90*a*b + 81*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b
^9)*d^4))) - ((a*b^2 - b^3)*d*cos(d*x + c)^4 - 2*(a*b^2 - b^3)*d*cos(d*x + c)^2 - (a^2*b - 2*a*b^2 + b^3)*d)*s
qrt(-((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2*sqrt((25*a^2 - 90*a*b + 81*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5
*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4)) + 4*a^2 - 15*a*b + 15*b^2)/((a^3*b^3 - 3*a^2*b^4 + 3
*a*b^5 - b^6)*d^2))*log(1/4*(20*a^2 - 81*a*b + 81*b^2)*cos(d*x + c)^2 - 5*a^2 + 81/4*a*b - 81/4*b^2 - 1/2*((a^
5*b^3 - 6*a^4*b^4 + 12*a^3*b^5 - 10*a^2*b^6 + 3*a*b^7)*d^3*sqrt((25*a^2 - 90*a*b + 81*b^2)/((a^7*b^3 - 6*a^6*b
^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4))*cos(d*x + c)*sin(d*x + c) + 2*(5*a^3*b -
19*a^2*b^2 + 18*a*b^3)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2*sqrt((25*
a^2 - 90*a*b + 81*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4))
+ 4*a^2 - 15*a*b + 15*b^2)/((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2)) + 1/4*(2*(4*a^5*b - 21*a^4*b^2 + 39*a
^3*b^3 - 31*a^2*b^4 + 9*a*b^5)*d^2*cos(d*x + c)^2 - (4*a^5*b - 21*a^4*b^2 + 39*a^3*b^3 - 31*a^2*b^4 + 9*a*b^5)
*d^2)*sqrt((25*a^2 - 90*a*b + 81*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8
+ a*b^9)*d^4))) + ((a*b^2 - b^3)*d*cos(d*x + c)^4 - 2*(a*b^2 - b^3)*d*cos(d*x + c)^2 - (a^2*b - 2*a*b^2 + b^3
)*d)*sqrt(((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2*sqrt((25*a^2 - 90*a*b + 81*b^2)/((a^7*b^3 - 6*a^6*b^4 + 1
5*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4)) - 4*a^2 + 15*a*b - 15*b^2)/((a^3*b^3 - 3*a^2*b^
4 + 3*a*b^5 - b^6)*d^2))*log(-1/4*(20*a^2 - 81*a*b + 81*b^2)*cos(d*x + c)^2 + 5*a^2 - 81/4*a*b + 81/4*b^2 + 1/
2*((a^5*b^3 - 6*a^4*b^4 + 12*a^3*b^5 - 10*a^2*b^6 + 3*a*b^7)*d^3*sqrt((25*a^2 - 90*a*b + 81*b^2)/((a^7*b^3 - 6
*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4))*cos(d*x + c)*sin(d*x + c) - 2*(5*a^
3*b - 19*a^2*b^2 + 18*a*b^3)*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2*sqrt
((25*a^2 - 90*a*b + 81*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*
d^4)) - 4*a^2 + 15*a*b - 15*b^2)/((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2)) + 1/4*(2*(4*a^5*b - 21*a^4*b^2 +
39*a^3*b^3 - 31*a^2*b^4 + 9*a*b^5)*d^2*cos(d*x + c)^2 - (4*a^5*b - 21*a^4*b^2 + 39*a^3*b^3 - 31*a^2*b^4 + 9*a
*b^5)*d^2)*sqrt((25*a^2 - 90*a*b + 81*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^
2*b^8 + a*b^9)*d^4))) - ((a*b^2 - b^3)*d*cos(d*x + c)^4 - 2*(a*b^2 - b^3)*d*cos(d*x + c)^2 - (a^2*b - 2*a*b^2
+ b^3)*d)*sqrt(((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2*sqrt((25*a^2 - 90*a*b + 81*b^2)/((a^7*b^3 - 6*a^6*b^
4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4)) - 4*a^2 + 15*a*b - 15*b^2)/((a^3*b^3 - 3*a
^2*b^4 + 3*a*b^5 - b^6)*d^2))*log(-1/4*(20*a^2 - 81*a*b + 81*b^2)*cos(d*x + c)^2 + 5*a^2 - 81/4*a*b + 81/4*b^2
- 1/2*((a^5*b^3 - 6*a^4*b^4 + 12*a^3*b^5 - 10*a^2*b^6 + 3*a*b^7)*d^3*sqrt((25*a^2 - 90*a*b + 81*b^2)/((a^7*b^
3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4))*cos(d*x + c)*sin(d*x + c) - 2*
(5*a^3*b - 19*a^2*b^2 + 18*a*b^3)*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2
*sqrt((25*a^2 - 90*a*b + 81*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*
b^9)*d^4)) - 4*a^2 + 15*a*b - 15*b^2)/((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2)) + 1/4*(2*(4*a^5*b - 21*a^4*
b^2 + 39*a^3*b^3 - 31*a^2*b^4 + 9*a*b^5)*d^2*cos(d*x + c)^2 - (4*a^5*b - 21*a^4*b^2 + 39*a^3*b^3 - 31*a^2*b^4
+ 9*a*b^5)*d^2)*sqrt((25*a^2 - 90*a*b + 81*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 -
6*a^2*b^8 + a*b^9)*d^4))) + 8*(b*cos(d*x + c)^3 - (a + b)*cos(d*x + c))*sin(d*x + c))/((a*b^2 - b^3)*d*cos(d*
x + c)^4 - 2*(a*b^2 - b^3)*d*cos(d*x + c)^2 - (...
________________________________________________________________________________________
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)**6/(a-b*sin(d*x+c)**4)**2,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1481 vs.
\(2 (181) = 362\).
time = 0.87, size = 1481, normalized size = 6.36 \begin {gather*} \frac {\frac {{\left ({\left (3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{3} - 15 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} b + 17 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b^{2} + 3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} b^{3}\right )} {\left (a b - b^{2}\right )}^{2} {\left | -a + b \right |} + {\left (3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{5} b - 12 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{4} b^{2} + 14 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{3} b^{3} - 4 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{2} b^{4} - \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a b^{5}\right )} {\left | -a b + b^{2} \right |} {\left | -a + b \right |} - 2 \, {\left (3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{6} b - 18 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{5} b^{2} + 38 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{4} b^{3} - 32 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{3} b^{4} + 7 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} b^{5} + 2 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b^{6}\right )} {\left | -a + b \right |}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\tan \left (d x + c\right )}{\sqrt {\frac {a^{2} b - a b^{2} + \sqrt {{\left (a^{2} b - a b^{2}\right )}^{2} - {\left (a^{2} b - a b^{2}\right )} {\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )}}}{a^{2} b - 2 \, a b^{2} + b^{3}}}}\right )\right )}}{{\left (3 \, a^{8} b^{2} - 21 \, a^{7} b^{3} + 59 \, a^{6} b^{4} - 85 \, a^{5} b^{5} + 65 \, a^{4} b^{6} - 23 \, a^{3} b^{7} + a^{2} b^{8} + a b^{9}\right )} {\left | -a b + b^{2} \right |}} - \frac {{\left ({\left (3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{3} - 15 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} b + 17 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b^{2} + 3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} b^{3}\right )} {\left (a b - b^{2}\right )}^{2} {\left | -a + b \right |} - {\left (3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{5} b - 12 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{4} b^{2} + 14 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{3} b^{3} - 4 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{2} b^{4} - \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a b^{5}\right )} {\left | -a b + b^{2} \right |} {\left | -a + b \right |} - 2 \, {\left (3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{6} b - 18 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{5} b^{2} + 38 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{4} b^{3} - 32 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{3} b^{4} + 7 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} b^{5} + 2 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b^{6}\right )} {\left | -a + b \right |}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\tan \left (d x + c\right )}{\sqrt {\frac {a^{2} b - a b^{2} - \sqrt {{\left (a^{2} b - a b^{2}\right )}^{2} - {\left (a^{2} b - a b^{2}\right )} {\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )}}}{a^{2} b - 2 \, a b^{2} + b^{3}}}}\right )\right )}}{{\left (3 \, a^{8} b^{2} - 21 \, a^{7} b^{3} + 59 \, a^{6} b^{4} - 85 \, a^{5} b^{5} + 65 \, a^{4} b^{6} - 23 \, a^{3} b^{7} + a^{2} b^{8} + a b^{9}\right )} {\left | -a b + b^{2} \right |}} - \frac {2 \, {\left (a \tan \left (d x + c\right )^{3} + b \tan \left (d x + c\right )^{3} + a \tan \left (d x + c\right )\right )}}{{\left (a \tan \left (d x + c\right )^{4} - b \tan \left (d x + c\right )^{4} + 2 \, a \tan \left (d x + c\right )^{2} + a\right )} {\left (a b - b^{2}\right )}}}{8 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(d*x+c)^6/(a-b*sin(d*x+c)^4)^2,x, algorithm="giac")
[Out]
1/8*(((3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3 - 15*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*
a^2*b + 17*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^2 + 3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*
b)*b^3)*(a*b - b^2)^2*abs(-a + b) + (3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^5*b - 12*sqrt(a^2 - a*b - sqrt(a*
b)*(a - b))*a^4*b^2 + 14*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^3*b^3 - 4*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a
^2*b^4 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a*b^5)*abs(-a*b + b^2)*abs(-a + b) - 2*(3*sqrt(a^2 - a*b - sqrt(a
*b)*(a - b))*sqrt(a*b)*a^6*b - 18*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^5*b^2 + 38*sqrt(a^2 - a*b -
sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b^3 - 32*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^4 + 7*sqrt(a^2 -
a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^5 + 2*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^6)*abs(-a +
b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^2*b - a*b^2 + sqrt((a^2*b - a*b^2)^2 - (a^2*b
- a*b^2)*(a^2*b - 2*a*b^2 + b^3)))/(a^2*b - 2*a*b^2 + b^3))))/((3*a^8*b^2 - 21*a^7*b^3 + 59*a^6*b^4 - 85*a^5*b
^5 + 65*a^4*b^6 - 23*a^3*b^7 + a^2*b^8 + a*b^9)*abs(-a*b + b^2)) - ((3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqr
t(a*b)*a^3 - 15*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b + 17*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*s
qrt(a*b)*a*b^2 + 3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^3)*(a*b - b^2)^2*abs(-a + b) - (3*sqrt(a^2
- a*b + sqrt(a*b)*(a - b))*a^5*b - 12*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^4*b^2 + 14*sqrt(a^2 - a*b + sqrt(a
*b)*(a - b))*a^3*b^3 - 4*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^2*b^4 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a*b
^5)*abs(-a*b + b^2)*abs(-a + b) - 2*(3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^6*b - 18*sqrt(a^2 - a*b
+ sqrt(a*b)*(a - b))*sqrt(a*b)*a^5*b^2 + 38*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b^3 - 32*sqrt(a
^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^4 + 7*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^5 + 2*
sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^6)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d
*x + c)/sqrt((a^2*b - a*b^2 - sqrt((a^2*b - a*b^2)^2 - (a^2*b - a*b^2)*(a^2*b - 2*a*b^2 + b^3)))/(a^2*b - 2*a*
b^2 + b^3))))/((3*a^8*b^2 - 21*a^7*b^3 + 59*a^6*b^4 - 85*a^5*b^5 + 65*a^4*b^6 - 23*a^3*b^7 + a^2*b^8 + a*b^9)*
abs(-a*b + b^2)) - 2*(a*tan(d*x + c)^3 + b*tan(d*x + c)^3 + a*tan(d*x + c))/((a*tan(d*x + c)^4 - b*tan(d*x + c
)^4 + 2*a*tan(d*x + c)^2 + a)*(a*b - b^2)))/d
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Mupad [B]
time = 16.54, size = 2500, normalized size = 10.73 \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)^6/(a - b*sin(c + d*x)^4)^2,x)
[Out]
(atan(((((256*a^2*b^5 - 512*a^3*b^4 + 256*a^4*b^3)/(64*(a*b^3 - b^4)) - (tan(c + d*x)*((15*a*b^5 - 5*a*(a*b^9)
^(1/2) + 9*b*(a*b^9)^(1/2) - 15*a^2*b^4 + 4*a^3*b^3)/(256*(a*b^9 - 3*a^2*b^8 + 3*a^3*b^7 - a^4*b^6)))^(1/2)*(2
56*a^2*b^6 - 768*a^3*b^5 + 768*a^4*b^4 - 256*a^5*b^3))/(4*(a*b^2 - b^3)))*((15*a*b^5 - 5*a*(a*b^9)^(1/2) + 9*b
*(a*b^9)^(1/2) - 15*a^2*b^4 + 4*a^3*b^3)/(256*(a*b^9 - 3*a^2*b^8 + 3*a^3*b^7 - a^4*b^6)))^(1/2) + (tan(c + d*x
)*(9*a*b^3 - 15*a^3*b + 4*a^4 + 10*a^2*b^2))/(4*(a*b^2 - b^3)))*((15*a*b^5 - 5*a*(a*b^9)^(1/2) + 9*b*(a*b^9)^(
1/2) - 15*a^2*b^4 + 4*a^3*b^3)/(256*(a*b^9 - 3*a^2*b^8 + 3*a^3*b^7 - a^4*b^6)))^(1/2)*1i - (((256*a^2*b^5 - 51
2*a^3*b^4 + 256*a^4*b^3)/(64*(a*b^3 - b^4)) + (tan(c + d*x)*((15*a*b^5 - 5*a*(a*b^9)^(1/2) + 9*b*(a*b^9)^(1/2)
- 15*a^2*b^4 + 4*a^3*b^3)/(256*(a*b^9 - 3*a^2*b^8 + 3*a^3*b^7 - a^4*b^6)))^(1/2)*(256*a^2*b^6 - 768*a^3*b^5 +
768*a^4*b^4 - 256*a^5*b^3))/(4*(a*b^2 - b^3)))*((15*a*b^5 - 5*a*(a*b^9)^(1/2) + 9*b*(a*b^9)^(1/2) - 15*a^2*b^
4 + 4*a^3*b^3)/(256*(a*b^9 - 3*a^2*b^8 + 3*a^3*b^7 - a^4*b^6)))^(1/2) - (tan(c + d*x)*(9*a*b^3 - 15*a^3*b + 4*
a^4 + 10*a^2*b^2))/(4*(a*b^2 - b^3)))*((15*a*b^5 - 5*a*(a*b^9)^(1/2) + 9*b*(a*b^9)^(1/2) - 15*a^2*b^4 + 4*a^3*
b^3)/(256*(a*b^9 - 3*a^2*b^8 + 3*a^3*b^7 - a^4*b^6)))^(1/2)*1i)/((27*a*b^2 - 21*a^2*b + 4*a^3)/(32*(a*b^3 - b^
4)) + (((256*a^2*b^5 - 512*a^3*b^4 + 256*a^4*b^3)/(64*(a*b^3 - b^4)) - (tan(c + d*x)*((15*a*b^5 - 5*a*(a*b^9)^
(1/2) + 9*b*(a*b^9)^(1/2) - 15*a^2*b^4 + 4*a^3*b^3)/(256*(a*b^9 - 3*a^2*b^8 + 3*a^3*b^7 - a^4*b^6)))^(1/2)*(25
6*a^2*b^6 - 768*a^3*b^5 + 768*a^4*b^4 - 256*a^5*b^3))/(4*(a*b^2 - b^3)))*((15*a*b^5 - 5*a*(a*b^9)^(1/2) + 9*b*
(a*b^9)^(1/2) - 15*a^2*b^4 + 4*a^3*b^3)/(256*(a*b^9 - 3*a^2*b^8 + 3*a^3*b^7 - a^4*b^6)))^(1/2) + (tan(c + d*x)
*(9*a*b^3 - 15*a^3*b + 4*a^4 + 10*a^2*b^2))/(4*(a*b^2 - b^3)))*((15*a*b^5 - 5*a*(a*b^9)^(1/2) + 9*b*(a*b^9)^(1
/2) - 15*a^2*b^4 + 4*a^3*b^3)/(256*(a*b^9 - 3*a^2*b^8 + 3*a^3*b^7 - a^4*b^6)))^(1/2) + (((256*a^2*b^5 - 512*a^
3*b^4 + 256*a^4*b^3)/(64*(a*b^3 - b^4)) + (tan(c + d*x)*((15*a*b^5 - 5*a*(a*b^9)^(1/2) + 9*b*(a*b^9)^(1/2) - 1
5*a^2*b^4 + 4*a^3*b^3)/(256*(a*b^9 - 3*a^2*b^8 + 3*a^3*b^7 - a^4*b^6)))^(1/2)*(256*a^2*b^6 - 768*a^3*b^5 + 768
*a^4*b^4 - 256*a^5*b^3))/(4*(a*b^2 - b^3)))*((15*a*b^5 - 5*a*(a*b^9)^(1/2) + 9*b*(a*b^9)^(1/2) - 15*a^2*b^4 +
4*a^3*b^3)/(256*(a*b^9 - 3*a^2*b^8 + 3*a^3*b^7 - a^4*b^6)))^(1/2) - (tan(c + d*x)*(9*a*b^3 - 15*a^3*b + 4*a^4
+ 10*a^2*b^2))/(4*(a*b^2 - b^3)))*((15*a*b^5 - 5*a*(a*b^9)^(1/2) + 9*b*(a*b^9)^(1/2) - 15*a^2*b^4 + 4*a^3*b^3)
/(256*(a*b^9 - 3*a^2*b^8 + 3*a^3*b^7 - a^4*b^6)))^(1/2)))*((15*a*b^5 - 5*a*(a*b^9)^(1/2) + 9*b*(a*b^9)^(1/2) -
15*a^2*b^4 + 4*a^3*b^3)/(256*(a*b^9 - 3*a^2*b^8 + 3*a^3*b^7 - a^4*b^6)))^(1/2)*2i)/d + (atan(((((256*a^2*b^5
- 512*a^3*b^4 + 256*a^4*b^3)/(64*(a*b^3 - b^4)) - (tan(c + d*x)*((15*a*b^5 + 5*a*(a*b^9)^(1/2) - 9*b*(a*b^9)^(
1/2) - 15*a^2*b^4 + 4*a^3*b^3)/(256*(a*b^9 - 3*a^2*b^8 + 3*a^3*b^7 - a^4*b^6)))^(1/2)*(256*a^2*b^6 - 768*a^3*b
^5 + 768*a^4*b^4 - 256*a^5*b^3))/(4*(a*b^2 - b^3)))*((15*a*b^5 + 5*a*(a*b^9)^(1/2) - 9*b*(a*b^9)^(1/2) - 15*a^
2*b^4 + 4*a^3*b^3)/(256*(a*b^9 - 3*a^2*b^8 + 3*a^3*b^7 - a^4*b^6)))^(1/2) + (tan(c + d*x)*(9*a*b^3 - 15*a^3*b
+ 4*a^4 + 10*a^2*b^2))/(4*(a*b^2 - b^3)))*((15*a*b^5 + 5*a*(a*b^9)^(1/2) - 9*b*(a*b^9)^(1/2) - 15*a^2*b^4 + 4*
a^3*b^3)/(256*(a*b^9 - 3*a^2*b^8 + 3*a^3*b^7 - a^4*b^6)))^(1/2)*1i - (((256*a^2*b^5 - 512*a^3*b^4 + 256*a^4*b^
3)/(64*(a*b^3 - b^4)) + (tan(c + d*x)*((15*a*b^5 + 5*a*(a*b^9)^(1/2) - 9*b*(a*b^9)^(1/2) - 15*a^2*b^4 + 4*a^3*
b^3)/(256*(a*b^9 - 3*a^2*b^8 + 3*a^3*b^7 - a^4*b^6)))^(1/2)*(256*a^2*b^6 - 768*a^3*b^5 + 768*a^4*b^4 - 256*a^5
*b^3))/(4*(a*b^2 - b^3)))*((15*a*b^5 + 5*a*(a*b^9)^(1/2) - 9*b*(a*b^9)^(1/2) - 15*a^2*b^4 + 4*a^3*b^3)/(256*(a
*b^9 - 3*a^2*b^8 + 3*a^3*b^7 - a^4*b^6)))^(1/2) - (tan(c + d*x)*(9*a*b^3 - 15*a^3*b + 4*a^4 + 10*a^2*b^2))/(4*
(a*b^2 - b^3)))*((15*a*b^5 + 5*a*(a*b^9)^(1/2) - 9*b*(a*b^9)^(1/2) - 15*a^2*b^4 + 4*a^3*b^3)/(256*(a*b^9 - 3*a
^2*b^8 + 3*a^3*b^7 - a^4*b^6)))^(1/2)*1i)/((27*a*b^2 - 21*a^2*b + 4*a^3)/(32*(a*b^3 - b^4)) + (((256*a^2*b^5 -
512*a^3*b^4 + 256*a^4*b^3)/(64*(a*b^3 - b^4)) - (tan(c + d*x)*((15*a*b^5 + 5*a*(a*b^9)^(1/2) - 9*b*(a*b^9)^(1
/2) - 15*a^2*b^4 + 4*a^3*b^3)/(256*(a*b^9 - 3*a^2*b^8 + 3*a^3*b^7 - a^4*b^6)))^(1/2)*(256*a^2*b^6 - 768*a^3*b^
5 + 768*a^4*b^4 - 256*a^5*b^3))/(4*(a*b^2 - b^3)))*((15*a*b^5 + 5*a*(a*b^9)^(1/2) - 9*b*(a*b^9)^(1/2) - 15*a^2
*b^4 + 4*a^3*b^3)/(256*(a*b^9 - 3*a^2*b^8 + 3*a^3*b^7 - a^4*b^6)))^(1/2) + (tan(c + d*x)*(9*a*b^3 - 15*a^3*b +
4*a^4 + 10*a^2*b^2))/(4*(a*b^2 - b^3)))*((15*a*b^5 + 5*a*(a*b^9)^(1/2) - 9*b*(a*b^9)^(1/2) - 15*a^2*b^4 + 4*a
^3*b^3)/(256*(a*b^9 - 3*a^2*b^8 + 3*a^3*b^7 - a^4*b^6)))^(1/2) + (((256*a^2*b^5 - 512*a^3*b^4 + 256*a^4*b^3)/(
64*(a*b^3 - b^4)) + (tan(c + d*x)*((15*a*b^5 + 5*a*(a*b^9)^(1/2) - 9*b*(a*b^9)^(1/2) - 15*a^2*b^4 + 4*a^3*b^3)
/(256*(a*b^9 - 3*a^2*b^8 + 3*a^3*b^7 - a^4*b^6)))^(1/2)*(256*a^2*b^6 - 768*a^3*b^5 + 768*a^4*b^4 - 256*a^5*b^3
))/(4*(a*b^2 - b^3)))*((15*a*b^5 + 5*a*(a*b^9)^...
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