3.3.20 \(\int \frac {\sin ^4(c+d x)}{(a-b \sin ^4(c+d x))^2} \, dx\) [220]

Optimal. Leaf size=195 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{3/4} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} \sqrt {b} d}-\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{3/4} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} \sqrt {b} d}-\frac {\tan (c+d x)}{4 a (a-b) d}+\frac {\tan ^5(c+d x)}{4 a d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )} \]

[Out]

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

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Rubi [A]
time = 0.16, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3296, 1289, 12, 1136, 1180, 211} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{3/4} \sqrt {b} d \left (\sqrt {a}-\sqrt {b}\right )^{3/2}}-\frac {\text {ArcTan}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{3/4} \sqrt {b} d \left (\sqrt {a}+\sqrt {b}\right )^{3/2}}+\frac {\tan ^5(c+d x)}{4 a d \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}-\frac {\tan (c+d x)}{4 a d (a-b)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)]/(8*a^(3/4)*(Sqrt[a] - Sqrt[b])^(3/2)*Sqrt[b]*d) - ArcTa
n[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)]/(8*a^(3/4)*(Sqrt[a] + Sqrt[b])^(3/2)*Sqrt[b]*d) - Tan[c + d*
x]/(4*a*(a - b)*d) + Tan[c + d*x]^5/(4*a*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 1136

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[d^3*(d*x)^(m - 3)*((a + b*
x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 1))), x] - Dist[d^4/(c*(m + 4*p + 1)), Int[(d*x)^(m - 4)*Simp[a*(m - 3) + b
*(m + 2*p - 1)*x^2, x]*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 - 4*a*c, 0] && Gt
Q[m, 3] && NeQ[m + 4*p + 1, 0] && 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 1289

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[f*(
f*x)^(m - 1)*(a + b*x^2 + c*x^4)^(p + 1)*((b*d - 2*a*e - (b*e - 2*c*d)*x^2)/(2*(p + 1)*(b^2 - 4*a*c))), x] - D
ist[f^2/(2*(p + 1)*(b^2 - 4*a*c)), Int[(f*x)^(m - 2)*(a + b*x^2 + c*x^4)^(p + 1)*Simp[(m - 1)*(b*d - 2*a*e) -
(4*p + 4 + m + 1)*(b*e - 2*c*d)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[
p, -1] && GtQ[m, 1] && IntegerQ[2*p] && (IntegerQ[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 ^4(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {x^4 \left (1+x^2\right )}{\left (a+2 a x^2+(a-b) x^4\right )^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\tan ^5(c+d x)}{4 a 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 b x^4}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{8 a b d}\\ &=\frac {\tan ^5(c+d x)}{4 a 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^4}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{4 a d}\\ &=-\frac {\tan (c+d x)}{4 a (a-b) d}+\frac {\tan ^5(c+d x)}{4 a d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {a+2 a x^2}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{4 a (a-b) d}\\ &=-\frac {\tan (c+d x)}{4 a (a-b) d}+\frac {\tan ^5(c+d x)}{4 a d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}+\frac {\left (2 \sqrt {a}-\frac {a+b}{\sqrt {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 \sqrt {a} (a-b) d}+\frac {\left (2 \sqrt {a}+\frac {a+b}{\sqrt {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 \sqrt {a} (a-b) d}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{3/4} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} \sqrt {b} d}-\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{3/4} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} \sqrt {b} d}-\frac {\tan (c+d x)}{4 a (a-b) d}+\frac {\tan ^5(c+d x)}{4 a 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 = 2.98, size = 225, normalized size = 1.15 \begin {gather*} -\frac {\frac {\left (\sqrt {a}-\sqrt {b}\right ) \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 {a} \sqrt {b}} \sqrt {b}}+\frac {\left (\sqrt {a}+\sqrt {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 {a} \sqrt {b}} \sqrt {b}}-\frac {2 (-6 \sin (2 (c+d x))+\sin (4 (c+d x)))}{8 a-3 b+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))}}{8 (a-b) d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*(((Sqrt[a] - Sqrt[b])*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt
[a + Sqrt[a]*Sqrt[b]]*Sqrt[b]) + ((Sqrt[a] + Sqrt[b])*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqr
t[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt[-a + Sqrt[a]*Sqrt[b]]*Sqrt[b]) - (2*(-6*Sin[2*(c + d*x)] + Sin[4*(c + d*x)]))/(8
*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)]))/((a - b)*d)

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Maple [A]
time = 0.55, size = 218, normalized size = 1.12

method	result	size
derivativedivides	\(\frac {\frac {-\frac {\tan ^{3}\left (d x +c \right )}{2 \left (a -b \right )}-\frac {\tan \left (d x +c \right )}{4 \left (a -b \right )}}{\left (\tan ^{4}\left (d x +c \right )\right ) a -\left (\tan ^{4}\left (d x +c \right )\right ) b +2 a \left (\tan ^{2}\left (d x +c \right )\right )+a}+\frac {\left (-a -b +2 \sqrt {a b}\right ) \arctanh \left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{8 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}+\frac {\left (a +b +2 \sqrt {a b}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{8 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}}{d}\)	\(218\)
default	\(\frac {\frac {-\frac {\tan ^{3}\left (d x +c \right )}{2 \left (a -b \right )}-\frac {\tan \left (d x +c \right )}{4 \left (a -b \right )}}{\left (\tan ^{4}\left (d x +c \right )\right ) a -\left (\tan ^{4}\left (d x +c \right )\right ) b +2 a \left (\tan ^{2}\left (d x +c \right )\right )+a}+\frac {\left (-a -b +2 \sqrt {a b}\right ) \arctanh \left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{8 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}+\frac {\left (a +b +2 \sqrt {a b}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{8 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}}{d}\)	\(218\)
risch	\(-\frac {i \left (b \,{\mathrm e}^{6 i \left (d x +c \right )}-8 a \,{\mathrm e}^{4 i \left (d x +c \right )}+3 b \,{\mathrm e}^{4 i \left (d x +c \right )}-5 b \,{\mathrm e}^{2 i \left (d x +c \right )}+b \right )}{2 b \left (a -b \right ) d \left (b \,{\mathrm e}^{8 i \left (d x +c \right )}-4 b \,{\mathrm e}^{6 i \left (d x +c \right )}-16 a \,{\mathrm e}^{4 i \left (d x +c \right )}+6 b \,{\mathrm e}^{4 i \left (d x +c \right )}-4 b \,{\mathrm e}^{2 i \left (d x +c \right )}+b \right )}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (1+\left (a^{6} b^{2} d^{4}-3 a^{5} b^{3} d^{4}+3 a^{4} b^{4} d^{4}-a^{3} b^{5} d^{4}\right ) \textit {\_Z}^{4}+\left (2 a^{3} b \,d^{2}+6 a^{2} b^{2} d^{2}\right ) \textit {\_Z}^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (-\frac {4 i d^{3} a^{6} b^{2}}{3 a b +b^{2}}+\frac {12 i d^{3} a^{5} b^{3}}{3 a b +b^{2}}-\frac {12 i d^{3} a^{4} b^{4}}{3 a b +b^{2}}+\frac {4 i d^{3} a^{3} b^{5}}{3 a b +b^{2}}\right ) \textit {\_R}^{3}+\left (-\frac {2 d^{2} a^{5} b}{3 a b +b^{2}}+\frac {6 d^{2} b^{2} a^{4}}{3 a b +b^{2}}-\frac {6 d^{2} b^{3} a^{3}}{3 a b +b^{2}}+\frac {2 d^{2} b^{4} a^{2}}{3 a b +b^{2}}\right ) \textit {\_R}^{2}+\left (-\frac {10 i d \,a^{3} b}{3 a b +b^{2}}-\frac {20 i d \,a^{2} b^{2}}{3 a b +b^{2}}-\frac {2 i d \,b^{3} a}{3 a b +b^{2}}\right ) \textit {\_R} -\frac {2 a^{2}}{3 a b +b^{2}}-\frac {9 a b}{3 a b +b^{2}}-\frac {b^{2}}{3 a b +b^{2}}\right )\right )}{16}\)	\(507\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*((-1/2/(a-b)*tan(d*x+c)^3-1/4/(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)+1/8*(-a
-b+2*(a*b)^(1/2))/(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))+1/8*(a+b+2*(a*b)^(1/2))/(a*b)^(1/2)/(a-b)/(((a*b)^(1/2)+a)*(a-b))^(1/2)*arctan((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)^4/(a-b*sin(d*x+c)^4)^2,x, algorithm="maxima")

[Out]

1/2*(6*(8*a - 3*b)*cos(4*d*x + 4*c)*sin(2*d*x + 2*c) + (b*sin(6*d*x + 6*c) - (8*a - 3*b)*sin(4*d*x + 4*c) - 5*
b*sin(2*d*x + 2*c))*cos(8*d*x + 8*c) + 6*((8*a - 3*b)*sin(4*d*x + 4*c) + 4*b*sin(2*d*x + 2*c))*cos(6*d*x + 6*c
) - 2*((a*b^2 - b^3)*d*cos(8*d*x + 8*c)^2 + 16*(a*b^2 - b^3)*d*cos(6*d*x + 6*c)^2 + 4*(64*a^3 - 112*a^2*b + 57
*a*b^2 - 9*b^3)*d*cos(4*d*x + 4*c)^2 + 16*(a*b^2 - b^3)*d*cos(2*d*x + 2*c)^2 + (a*b^2 - b^3)*d*sin(8*d*x + 8*c
)^2 + 16*(a*b^2 - b^3)*d*sin(6*d*x + 6*c)^2 + 4*(64*a^3 - 112*a^2*b + 57*a*b^2 - 9*b^3)*d*sin(4*d*x + 4*c)^2 +
16*(8*a^2*b - 11*a*b^2 + 3*b^3)*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*(a*b^2 - b^3)*d*sin(2*d*x + 2*c)^2 -
8*(a*b^2 - b^3)*d*cos(2*d*x + 2*c) + (a*b^2 - b^3)*d - 2*(4*(a*b^2 - b^3)*d*cos(6*d*x + 6*c) + 2*(8*a^2*b - 1
1*a*b^2 + 3*b^3)*d*cos(4*d*x + 4*c) + 4*(a*b^2 - b^3)*d*cos(2*d*x + 2*c) - (a*b^2 - b^3)*d)*cos(8*d*x + 8*c) +
8*(2*(8*a^2*b - 11*a*b^2 + 3*b^3)*d*cos(4*d*x + 4*c) + 4*(a*b^2 - b^3)*d*cos(2*d*x + 2*c) - (a*b^2 - b^3)*d)*
cos(6*d*x + 6*c) + 4*(4*(8*a^2*b - 11*a*b^2 + 3*b^3)*d*cos(2*d*x + 2*c) - (8*a^2*b - 11*a*b^2 + 3*b^3)*d)*cos(
4*d*x + 4*c) - 4*(2*(a*b^2 - b^3)*d*sin(6*d*x + 6*c) + (8*a^2*b - 11*a*b^2 + 3*b^3)*d*sin(4*d*x + 4*c) + 2*(a*
b^2 - b^3)*d*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 16*((8*a^2*b - 11*a*b^2 + 3*b^3)*d*sin(4*d*x + 4*c) + 2*(a*b
^2 - b^3)*d*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*integrate((4*b*cos(6*d*x + 6*c)^2 - 12*(8*a - 3*b)*cos(4*d*x +
4*c)^2 + 4*b*cos(2*d*x + 2*c)^2 + 4*b*sin(6*d*x + 6*c)^2 - 12*(8*a - 3*b)*sin(4*d*x + 4*c)^2 + 2*(8*a - 15*b)
*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 4*b*sin(2*d*x + 2*c)^2 - (b*cos(6*d*x + 6*c) - 6*b*cos(4*d*x + 4*c) + b*c
os(2*d*x + 2*c))*cos(8*d*x + 8*c) + (2*(8*a - 15*b)*cos(4*d*x + 4*c) + 8*b*cos(2*d*x + 2*c) - b)*cos(6*d*x + 6
*c) + 2*((8*a - 15*b)*cos(2*d*x + 2*c) + 3*b)*cos(4*d*x + 4*c) - b*cos(2*d*x + 2*c) - (b*sin(6*d*x + 6*c) - 6*
b*sin(4*d*x + 4*c) + b*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 2*((8*a - 15*b)*sin(4*d*x + 4*c) + 4*b*sin(2*d*x +
2*c))*sin(6*d*x + 6*c))/(a*b^2 - b^3 + (a*b^2 - b^3)*cos(8*d*x + 8*c)^2 + 16*(a*b^2 - b^3)*cos(6*d*x + 6*c)^2
+ 4*(64*a^3 - 112*a^2*b + 57*a*b^2 - 9*b^3)*cos(4*d*x + 4*c)^2 + 16*(a*b^2 - b^3)*cos(2*d*x + 2*c)^2 + (a*b^2
- b^3)*sin(8*d*x + 8*c)^2 + 16*(a*b^2 - b^3)*sin(6*d*x + 6*c)^2 + 4*(64*a^3 - 112*a^2*b + 57*a*b^2 - 9*b^3)*s
in(4*d*x + 4*c)^2 + 16*(8*a^2*b - 11*a*b^2 + 3*b^3)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*(a*b^2 - b^3)*sin(2
*d*x + 2*c)^2 + 2*(a*b^2 - b^3 - 4*(a*b^2 - b^3)*cos(6*d*x + 6*c) - 2*(8*a^2*b - 11*a*b^2 + 3*b^3)*cos(4*d*x +
4*c) - 4*(a*b^2 - b^3)*cos(2*d*x + 2*c))*cos(8*d*x + 8*c) - 8*(a*b^2 - b^3 - 2*(8*a^2*b - 11*a*b^2 + 3*b^3)*c
os(4*d*x + 4*c) - 4*(a*b^2 - b^3)*cos(2*d*x + 2*c))*cos(6*d*x + 6*c) - 4*(8*a^2*b - 11*a*b^2 + 3*b^3 - 4*(8*a^
2*b - 11*a*b^2 + 3*b^3)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - 8*(a*b^2 - b^3)*cos(2*d*x + 2*c) - 4*(2*(a*b^2 -
b^3)*sin(6*d*x + 6*c) + (8*a^2*b - 11*a*b^2 + 3*b^3)*sin(4*d*x + 4*c) + 2*(a*b^2 - b^3)*sin(2*d*x + 2*c))*sin(
8*d*x + 8*c) + 16*((8*a^2*b - 11*a*b^2 + 3*b^3)*sin(4*d*x + 4*c) + 2*(a*b^2 - b^3)*sin(2*d*x + 2*c))*sin(6*d*x
+ 6*c)), x) - (b*cos(6*d*x + 6*c) - (8*a - 3*b)*cos(4*d*x + 4*c) - 5*b*cos(2*d*x + 2*c) + b)*sin(8*d*x + 8*c)
- (6*(8*a - 3*b)*cos(4*d*x + 4*c) + 24*b*cos(2*d*x + 2*c) - 5*b)*sin(6*d*x + 6*c) - (6*(8*a - 3*b)*cos(2*d*x
+ 2*c) - 8*a + 3*b)*sin(4*d*x + 4*c) - b*sin(2*d*x + 2*c))/((a*b^2 - b^3)*d*cos(8*d*x + 8*c)^2 + 16*(a*b^2 - b
^3)*d*cos(6*d*x + 6*c)^2 + 4*(64*a^3 - 112*a^2*b + 57*a*b^2 - 9*b^3)*d*cos(4*d*x + 4*c)^2 + 16*(a*b^2 - b^3)*d
*cos(2*d*x + 2*c)^2 + (a*b^2 - b^3)*d*sin(8*d*x + 8*c)^2 + 16*(a*b^2 - b^3)*d*sin(6*d*x + 6*c)^2 + 4*(64*a^3 -
112*a^2*b + 57*a*b^2 - 9*b^3)*d*sin(4*d*x + 4*c)^2 + 16*(8*a^2*b - 11*a*b^2 + 3*b^3)*d*sin(4*d*x + 4*c)*sin(2
*d*x + 2*c) + 16*(a*b^2 - b^3)*d*sin(2*d*x + 2*c)^2 - 8*(a*b^2 - b^3)*d*cos(2*d*x + 2*c) + (a*b^2 - b^3)*d - 2
*(4*(a*b^2 - b^3)*d*cos(6*d*x + 6*c) + 2*(8*a^2*b - 11*a*b^2 + 3*b^3)*d*cos(4*d*x + 4*c) + 4*(a*b^2 - b^3)*d*c
os(2*d*x + 2*c) - (a*b^2 - b^3)*d)*cos(8*d*x + 8*c) + 8*(2*(8*a^2*b - 11*a*b^2 + 3*b^3)*d*cos(4*d*x + 4*c) + 4
*(a*b^2 - b^3)*d*cos(2*d*x + 2*c) - (a*b^2 - b^3)*d)*cos(6*d*x + 6*c) + 4*(4*(8*a^2*b - 11*a*b^2 + 3*b^3)*d*co
s(2*d*x + 2*c) - (8*a^2*b - 11*a*b^2 + 3*b^3)*d)*cos(4*d*x + 4*c) - 4*(2*(a*b^2 - b^3)*d*sin(6*d*x + 6*c) + (8
*a^2*b - 11*a*b^2 + 3*b^3)*d*sin(4*d*x + 4*c) + 2*(a*b^2 - b^3)*d*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 16*((8*
a^2*b - 11*a*b^2 + 3*b^3)*d*sin(4*d*x + 4*c) + 2*(a*b^2 - b^3)*d*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2796 vs. \(2 (151) = 302\).
time = 0.76, size = 2796, normalized size = 14.34 \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)^4/(a-b*sin(d*x+c)^4)^2,x, algorithm="fricas")

[Out]

-1/32*(((a*b - b^2)*d*cos(d*x + c)^4 - 2*(a*b - b^2)*d*cos(d*x + c)^2 - (a^2 - 2*a*b + b^2)*d)*sqrt(-((a^4*b -
3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2*sqrt((9*a^2 + 6*a*b + b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 +
15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4)) + a + 3*b)/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2))*log(1/4*(3*
a + b)*cos(d*x + c)^2 + 1/2*(2*(a^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*d^3*sqrt((9*a^2 + 6*a*b + b^2)/((a^9*
b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4))*cos(d*x + c)*sin(d*x + c) -
(3*a^3 + 4*a^2*b + a*b^2)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2*sqrt
((9*a^2 + 6*a*b + b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4))
+ a + 3*b)/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2)) - 1/4*(2*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d^2*
cos(d*x + c)^2 - (a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d^2)*sqrt((9*a^2 + 6*a*b + b^2)/((a^9*b - 6*a^8*b^2 + 1
5*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4)) - 3/4*a - 1/4*b) - ((a*b - b^2)*d*cos(d*x + c
)^4 - 2*(a*b - b^2)*d*cos(d*x + c)^2 - (a^2 - 2*a*b + b^2)*d)*sqrt(-((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d
^2*sqrt((9*a^2 + 6*a*b + b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7
)*d^4)) + a + 3*b)/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2))*log(1/4*(3*a + b)*cos(d*x + c)^2 - 1/2*(2*(a
^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*d^3*sqrt((9*a^2 + 6*a*b + b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a
^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4))*cos(d*x + c)*sin(d*x + c) - (3*a^3 + 4*a^2*b + a*b^2)*d*cos(d
*x + c)*sin(d*x + c))*sqrt(-((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2*sqrt((9*a^2 + 6*a*b + b^2)/((a^9*b -
6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4)) + a + 3*b)/((a^4*b - 3*a^3*b^2 +
3*a^2*b^3 - a*b^4)*d^2)) - 1/4*(2*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d^2*cos(d*x + c)^2 - (a^5 - 3*a^4*b +
3*a^3*b^2 - a^2*b^3)*d^2)*sqrt((9*a^2 + 6*a*b + b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b
^5 - 6*a^4*b^6 + a^3*b^7)*d^4)) - 3/4*a - 1/4*b) + ((a*b - b^2)*d*cos(d*x + c)^4 - 2*(a*b - b^2)*d*cos(d*x + c
)^2 - (a^2 - 2*a*b + b^2)*d)*sqrt(((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2*sqrt((9*a^2 + 6*a*b + b^2)/((a^
9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4)) - a - 3*b)/((a^4*b - 3*a^3
*b^2 + 3*a^2*b^3 - a*b^4)*d^2))*log(-1/4*(3*a + b)*cos(d*x + c)^2 + 1/2*(2*(a^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^
3*b^4)*d^3*sqrt((9*a^2 + 6*a*b + b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 +
a^3*b^7)*d^4))*cos(d*x + c)*sin(d*x + c) + (3*a^3 + 4*a^2*b + a*b^2)*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^4*
b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2*sqrt((9*a^2 + 6*a*b + b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^
4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4)) - a - 3*b)/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2)) - 1/4*(2
*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d^2*cos(d*x + c)^2 - (a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d^2)*sqrt((9
*a^2 + 6*a*b + b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4)) +
3/4*a + 1/4*b) - ((a*b - b^2)*d*cos(d*x + c)^4 - 2*(a*b - b^2)*d*cos(d*x + c)^2 - (a^2 - 2*a*b + b^2)*d)*sqrt(
((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2*sqrt((9*a^2 + 6*a*b + b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*
a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4)) - a - 3*b)/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2))*lo
g(-1/4*(3*a + b)*cos(d*x + c)^2 - 1/2*(2*(a^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*d^3*sqrt((9*a^2 + 6*a*b + b
^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4))*cos(d*x + c)*sin(d
*x + c) + (3*a^3 + 4*a^2*b + a*b^2)*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)
*d^2*sqrt((9*a^2 + 6*a*b + b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b
^7)*d^4)) - a - 3*b)/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2)) - 1/4*(2*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*
b^3)*d^2*cos(d*x + c)^2 - (a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d^2)*sqrt((9*a^2 + 6*a*b + b^2)/((a^9*b - 6*a^
8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4)) + 3/4*a + 1/4*b) + 8*(cos(d*x + c)^3
- 2*cos(d*x + c))*sin(d*x + c))/((a*b - b^2)*d*cos(d*x + c)^4 - 2*(a*b - b^2)*d*cos(d*x + c)^2 - (a^2 - 2*a*b
+ b^2)*d)

________________________________________________________________________________________

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)**4/(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. 1262 vs. \(2 (151) = 302\).
time = 0.89, size = 1262, normalized size = 6.47 \begin {gather*} \frac {\frac {{\left (3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{5} - 9 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{4} b + 2 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{3} b^{2} + 10 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} b^{3} - 5 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b^{4} - \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} b^{5} - 2 \, {\left (3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} b - 6 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b^{2} - \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} b^{3}\right )} {\left (a - b\right )}^{2} + {\left (3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{4} b - 12 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{3} b^{2} + 14 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{2} b^{3} - 4 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a b^{4} - \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} b^{5}\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} - a b + \sqrt {{\left (a^{2} - a b\right )}^{2} - {\left (a^{2} - a b\right )} {\left (a^{2} - 2 \, a b + b^{2}\right )}}}{a^{2} - 2 \, a b + b^{2}}}}\right )\right )}}{3 \, a^{8} b - 21 \, a^{7} b^{2} + 59 \, a^{6} b^{3} - 85 \, a^{5} b^{4} + 65 \, a^{4} b^{5} - 23 \, a^{3} b^{6} + a^{2} b^{7} + a b^{8}} + \frac {{\left (3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{5} - 9 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{4} b + 2 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{3} b^{2} + 10 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} b^{3} - 5 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b^{4} - \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} b^{5} - 2 \, {\left (3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} b - 6 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b^{2} - \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} b^{3}\right )} {\left (a - b\right )}^{2} + {\left (3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{4} b - 12 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{3} b^{2} + 14 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{2} b^{3} - 4 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a b^{4} - \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} b^{5}\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} - a b - \sqrt {{\left (a^{2} - a b\right )}^{2} - {\left (a^{2} - a b\right )} {\left (a^{2} - 2 \, a b + b^{2}\right )}}}{a^{2} - 2 \, a b + b^{2}}}}\right )\right )}}{3 \, a^{8} b - 21 \, a^{7} b^{2} + 59 \, a^{6} b^{3} - 85 \, a^{5} b^{4} + 65 \, a^{4} b^{5} - 23 \, a^{3} b^{6} + a^{2} b^{7} + a b^{8}} - \frac {2 \, {\left (2 \, \tan \left (d x + c\right )^{3} + \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\right )}}}{8 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^4/(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^5 - 9*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^
4*b + 2*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^2 + 10*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*
b)*a^2*b^3 - 5*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^4 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(
a*b)*b^5 - 2*(3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b - 6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sq
rt(a*b)*a*b^2 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^3)*(a - b)^2 + (3*sqrt(a^2 - a*b - sqrt(a*b)*(
a - b))*a^4*b - 12*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^3*b^2 + 14*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b^
3 - 4*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a*b^4 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*b^5)*abs(-a + b))*(pi*fl
oor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^2 - a*b + sqrt((a^2 - a*b)^2 - (a^2 - a*b)*(a^2 - 2*a*b
+ b^2)))/(a^2 - 2*a*b + b^2))))/(3*a^8*b - 21*a^7*b^2 + 59*a^6*b^3 - 85*a^5*b^4 + 65*a^4*b^5 - 23*a^3*b^6 + a^
2*b^7 + a*b^8) + (3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^5 - 9*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*
sqrt(a*b)*a^4*b + 2*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^2 + 10*sqrt(a^2 - a*b + sqrt(a*b)*(a -
b))*sqrt(a*b)*a^2*b^3 - 5*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^4 - sqrt(a^2 - a*b + sqrt(a*b)*(a
- b))*sqrt(a*b)*b^5 - 2*(3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b - 6*sqrt(a^2 - a*b + sqrt(a*b)
*(a - b))*sqrt(a*b)*a*b^2 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^3)*(a - b)^2 + (3*sqrt(a^2 - a*b +
sqrt(a*b)*(a - b))*a^4*b - 12*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^3*b^2 + 14*sqrt(a^2 - a*b + sqrt(a*b)*(a
- b))*a^2*b^3 - 4*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a*b^4 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*b^5)*abs(-a
+ b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^2 - a*b - sqrt((a^2 - a*b)^2 - (a^2 - a*b)*(
a^2 - 2*a*b + b^2)))/(a^2 - 2*a*b + b^2))))/(3*a^8*b - 21*a^7*b^2 + 59*a^6*b^3 - 85*a^5*b^4 + 65*a^4*b^5 - 23*
a^3*b^6 + a^2*b^7 + a*b^8) - 2*(2*tan(d*x + c)^3 + tan(d*x + c))/((a*tan(d*x + c)^4 - b*tan(d*x + c)^4 + 2*a*t
an(d*x + c)^2 + a)*(a - b)))/d

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Mupad [B]
time = 15.99, size = 2980, normalized size = 15.28 \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)^4/(a - b*sin(c + d*x)^4)^2,x)

[Out]

- (atan(((((128*a*b^3 + 128*a^3*b - 256*a^2*b^2)/(32*(a - b)) - (tan(c + d*x)*((3*a*(a^3*b^3)^(1/2) + b*(a^3*b
^3)^(1/2) + a^3*b + 3*a^2*b^2)/(256*(a^3*b^5 - 3*a^4*b^4 + 3*a^5*b^3 - a^6*b^2)))^(1/2)*(256*a^5*b - 256*a^2*b
^4 + 768*a^3*b^3 - 768*a^4*b^2))/(4*(a - b)))*((3*a*(a^3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) + a^3*b + 3*a^2*b^2)/(
256*(a^3*b^5 - 3*a^4*b^4 + 3*a^5*b^3 - a^6*b^2)))^(1/2) - (tan(c + d*x)*(6*a*b + a^2 + b^2))/(4*(a - b)))*((3*
a*(a^3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) + a^3*b + 3*a^2*b^2)/(256*(a^3*b^5 - 3*a^4*b^4 + 3*a^5*b^3 - a^6*b^2)))^
(1/2)*1i - (((128*a*b^3 + 128*a^3*b - 256*a^2*b^2)/(32*(a - b)) + (tan(c + d*x)*((3*a*(a^3*b^3)^(1/2) + b*(a^3
*b^3)^(1/2) + a^3*b + 3*a^2*b^2)/(256*(a^3*b^5 - 3*a^4*b^4 + 3*a^5*b^3 - a^6*b^2)))^(1/2)*(256*a^5*b - 256*a^2
*b^4 + 768*a^3*b^3 - 768*a^4*b^2))/(4*(a - b)))*((3*a*(a^3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) + a^3*b + 3*a^2*b^2)
/(256*(a^3*b^5 - 3*a^4*b^4 + 3*a^5*b^3 - a^6*b^2)))^(1/2) + (tan(c + d*x)*(6*a*b + a^2 + b^2))/(4*(a - b)))*((
3*a*(a^3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) + a^3*b + 3*a^2*b^2)/(256*(a^3*b^5 - 3*a^4*b^4 + 3*a^5*b^3 - a^6*b^2))
)^(1/2)*1i)/((((128*a*b^3 + 128*a^3*b - 256*a^2*b^2)/(32*(a - b)) - (tan(c + d*x)*((3*a*(a^3*b^3)^(1/2) + b*(a
^3*b^3)^(1/2) + a^3*b + 3*a^2*b^2)/(256*(a^3*b^5 - 3*a^4*b^4 + 3*a^5*b^3 - a^6*b^2)))^(1/2)*(256*a^5*b - 256*a
^2*b^4 + 768*a^3*b^3 - 768*a^4*b^2))/(4*(a - b)))*((3*a*(a^3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) + a^3*b + 3*a^2*b^
2)/(256*(a^3*b^5 - 3*a^4*b^4 + 3*a^5*b^3 - a^6*b^2)))^(1/2) - (tan(c + d*x)*(6*a*b + a^2 + b^2))/(4*(a - b)))*
((3*a*(a^3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) + a^3*b + 3*a^2*b^2)/(256*(a^3*b^5 - 3*a^4*b^4 + 3*a^5*b^3 - a^6*b^2
)))^(1/2) + (((128*a*b^3 + 128*a^3*b - 256*a^2*b^2)/(32*(a - b)) + (tan(c + d*x)*((3*a*(a^3*b^3)^(1/2) + b*(a^
3*b^3)^(1/2) + a^3*b + 3*a^2*b^2)/(256*(a^3*b^5 - 3*a^4*b^4 + 3*a^5*b^3 - a^6*b^2)))^(1/2)*(256*a^5*b - 256*a^
2*b^4 + 768*a^3*b^3 - 768*a^4*b^2))/(4*(a - b)))*((3*a*(a^3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) + a^3*b + 3*a^2*b^2
)/(256*(a^3*b^5 - 3*a^4*b^4 + 3*a^5*b^3 - a^6*b^2)))^(1/2) + (tan(c + d*x)*(6*a*b + a^2 + b^2))/(4*(a - b)))*(
(3*a*(a^3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) + a^3*b + 3*a^2*b^2)/(256*(a^3*b^5 - 3*a^4*b^4 + 3*a^5*b^3 - a^6*b^2)
))^(1/2) - 1/(16*(a - b))))*((3*a*(a^3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) + a^3*b + 3*a^2*b^2)/(256*(a^3*b^5 - 3*a
^4*b^4 + 3*a^5*b^3 - a^6*b^2)))^(1/2)*2i)/d - (atan(((((128*a*b^3 + 128*a^3*b - 256*a^2*b^2)/(32*(a - b)) - (t
an(c + d*x)*(-(3*a*(a^3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) - a^3*b - 3*a^2*b^2)/(256*(a^3*b^5 - 3*a^4*b^4 + 3*a^5*
b^3 - a^6*b^2)))^(1/2)*(256*a^5*b - 256*a^2*b^4 + 768*a^3*b^3 - 768*a^4*b^2))/(4*(a - b)))*(-(3*a*(a^3*b^3)^(1
/2) + b*(a^3*b^3)^(1/2) - a^3*b - 3*a^2*b^2)/(256*(a^3*b^5 - 3*a^4*b^4 + 3*a^5*b^3 - a^6*b^2)))^(1/2) - (tan(c
+ d*x)*(6*a*b + a^2 + b^2))/(4*(a - b)))*(-(3*a*(a^3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) - a^3*b - 3*a^2*b^2)/(256
*(a^3*b^5 - 3*a^4*b^4 + 3*a^5*b^3 - a^6*b^2)))^(1/2)*1i - (((128*a*b^3 + 128*a^3*b - 256*a^2*b^2)/(32*(a - b))
+ (tan(c + d*x)*(-(3*a*(a^3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) - a^3*b - 3*a^2*b^2)/(256*(a^3*b^5 - 3*a^4*b^4 + 3
*a^5*b^3 - a^6*b^2)))^(1/2)*(256*a^5*b - 256*a^2*b^4 + 768*a^3*b^3 - 768*a^4*b^2))/(4*(a - b)))*(-(3*a*(a^3*b^
3)^(1/2) + b*(a^3*b^3)^(1/2) - a^3*b - 3*a^2*b^2)/(256*(a^3*b^5 - 3*a^4*b^4 + 3*a^5*b^3 - a^6*b^2)))^(1/2) + (
tan(c + d*x)*(6*a*b + a^2 + b^2))/(4*(a - b)))*(-(3*a*(a^3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) - a^3*b - 3*a^2*b^2)
/(256*(a^3*b^5 - 3*a^4*b^4 + 3*a^5*b^3 - a^6*b^2)))^(1/2)*1i)/((((128*a*b^3 + 128*a^3*b - 256*a^2*b^2)/(32*(a
- b)) - (tan(c + d*x)*(-(3*a*(a^3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) - a^3*b - 3*a^2*b^2)/(256*(a^3*b^5 - 3*a^4*b^
4 + 3*a^5*b^3 - a^6*b^2)))^(1/2)*(256*a^5*b - 256*a^2*b^4 + 768*a^3*b^3 - 768*a^4*b^2))/(4*(a - b)))*(-(3*a*(a
^3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) - a^3*b - 3*a^2*b^2)/(256*(a^3*b^5 - 3*a^4*b^4 + 3*a^5*b^3 - a^6*b^2)))^(1/2
) - (tan(c + d*x)*(6*a*b + a^2 + b^2))/(4*(a - b)))*(-(3*a*(a^3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) - a^3*b - 3*a^2
*b^2)/(256*(a^3*b^5 - 3*a^4*b^4 + 3*a^5*b^3 - a^6*b^2)))^(1/2) + (((128*a*b^3 + 128*a^3*b - 256*a^2*b^2)/(32*(
a - b)) + (tan(c + d*x)*(-(3*a*(a^3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) - a^3*b - 3*a^2*b^2)/(256*(a^3*b^5 - 3*a^4*
b^4 + 3*a^5*b^3 - a^6*b^2)))^(1/2)*(256*a^5*b - 256*a^2*b^4 + 768*a^3*b^3 - 768*a^4*b^2))/(4*(a - b)))*(-(3*a*
(a^3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) - a^3*b - 3*a^2*b^2)/(256*(a^3*b^5 - 3*a^4*b^4 + 3*a^5*b^3 - a^6*b^2)))^(1
/2) + (tan(c + d*x)*(6*a*b + a^2 + b^2))/(4*(a - b)))*(-(3*a*(a^3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) - a^3*b - 3*a
^2*b^2)/(256*(a^3*b^5 - 3*a^4*b^4 + 3*a^5*b^3 - a^6*b^2)))^(1/2) - 1/(16*(a - b))))*(-(3*a*(a^3*b^3)^(1/2) + b
*(a^3*b^3)^(1/2) - a^3*b - 3*a^2*b^2)/(256*(a^3*b^5 - 3*a^4*b^4 + 3*a^5*b^3 - a^6*b^2)))^(1/2)*2i)/d - (tan(c
+ d*x)^3/(2*(a - b)) + tan(c + d*x)/(4*(a - b)))/(d*(a + 2*a*tan(c + d*x)^2 + tan(c + d*x)^4*(a - b)))

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