3.3.22 \(\int \frac {1}{(a-b \sin ^4(c+d x))^2} \, dx\) [222]

Optimal. Leaf size=210 \[ \frac {\left (4 \sqrt {a}-3 \sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{7/4} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} d}+\frac {\left (4 \sqrt {a}+3 \sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{7/4} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} d}-\frac {b \tan (c+d x) \left (1+2 \tan ^2(c+d x)\right )}{4 a (a-b) d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )} \]

[Out]

1/8*arctan((a^(1/2)-b^(1/2))^(1/2)*tan(d*x+c)/a^(1/4))*(4*a^(1/2)-3*b^(1/2))/a^(7/4)/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))*(4*a^(1/2)+3*b^(1/2))/a^(7/4)/d/(a^(1/2)+b^(1/2))^(3/2
)-1/4*b*tan(d*x+c)*(1+2*tan(d*x+c)^2)/a/(a-b)/d/(a+2*a*tan(d*x+c)^2+(a-b)*tan(d*x+c)^4)

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Rubi [A]
time = 0.17, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {3288, 1219, 1180, 211} \begin {gather*} \frac {\left (4 \sqrt {a}-3 \sqrt {b}\right ) \text {ArcTan}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{7/4} d \left (\sqrt {a}-\sqrt {b}\right )^{3/2}}+\frac {\left (4 \sqrt {a}+3 \sqrt {b}\right ) \text {ArcTan}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{7/4} d \left (\sqrt {a}+\sqrt {b}\right )^{3/2}}-\frac {b \tan (c+d x) \left (2 \tan ^2(c+d x)+1\right )}{4 a d (a-b) \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[(a - b*Sin[c + d*x]^4)^(-2),x]

[Out]

((4*Sqrt[a] - 3*Sqrt[b])*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(8*a^(7/4)*(Sqrt[a] - Sqrt[b]
)^(3/2)*d) + ((4*Sqrt[a] + 3*Sqrt[b])*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(8*a^(7/4)*(Sqrt
[a] + Sqrt[b])^(3/2)*d) - (b*Tan[c + d*x]*(1 + 2*Tan[c + d*x]^2))/(4*a*(a - 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 1180

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 1219

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{f = Coeff[Polynom
ialRemainder[(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x
^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4)^(p + 1)*((a*b*g - f*(b^2 - 2*a*c) - c*(b*f - 2*a*g)*x^2)/(2
*a*(p + 1)*(b^2 - 4*a*c))), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToS
um[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[(d + e*x^2)^q, a + b*x^2 + c*x^4, x] + b^2*f*(2*p + 3) - 2*a*c
*f*(4*p + 5) - a*b*g + c*(4*p + 7)*(b*f - 2*a*g)*x^2, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*
a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[q, 1] && LtQ[p, -1]

Rule 3288

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dis
t[ff/f, Subst[Int[(a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p/(1 + ff^2*x^2)^(2*p + 1), x], x, Tan[e + f*x]/ff], x
]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {1}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1+x^2\right )^3}{\left (a+2 a x^2+(a-b) x^4\right )^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac {b \tan (c+d x) \left (1+2 \tan ^2(c+d x)\right )}{4 a (a-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 {-\frac {2 a (4 a-3 b) b}{a-b}-\frac {4 a (2 a-b) b x^2}{a-b}}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{8 a^2 b d}\\ &=-\frac {b \tan (c+d x) \left (1+2 \tan ^2(c+d x)\right )}{4 a (a-b) d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}+\frac {\left (4 a-\sqrt {a} \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^{3/2} \left (\sqrt {a}+\sqrt {b}\right ) d}+\frac {\left (4 a+\sqrt {a} \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^{3/2} \left (\sqrt {a}-\sqrt {b}\right ) d}\\ &=\frac {\left (4 \sqrt {a}-3 \sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{7/4} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} d}+\frac {\left (4 \sqrt {a}+3 \sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{7/4} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} d}-\frac {b \tan (c+d x) \left (1+2 \tan ^2(c+d x)\right )}{4 a (a-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 = 1.96, size = 230, normalized size = 1.10 \begin {gather*} \frac {\frac {\left (4 a-\sqrt {a} \sqrt {b}-3 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 {b}}}-\frac {\left (4 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 {2 \sqrt {a} b (-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^{3/2} (a-b) d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.82, size = 260, normalized size = 1.24

method	result	size
derivativedivides	\(\frac {\frac {-\frac {b \left (\tan ^{3}\left (d x +c \right )\right )}{2 a \left (a -b \right )}-\frac {b \tan \left (d x +c \right )}{4 a \left (a -b \right )}}{\left (\tan ^{4}\left (d x +c \right )\right ) a -\left (\tan ^{4}\left (d x +c \right )\right ) b +2 a \left (\tan ^{2}\left (d x +c \right )\right )+a}+\frac {\frac {\left (4 a \sqrt {a b}-2 \sqrt {a b}\, b -5 a b +3 b^{2}\right ) \arctanh \left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}+\frac {\left (4 a \sqrt {a b}-2 \sqrt {a b}\, b +5 a b -3 b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}}{4 a}}{d}\)	\(260\)
default	\(\frac {\frac {-\frac {b \left (\tan ^{3}\left (d x +c \right )\right )}{2 a \left (a -b \right )}-\frac {b \tan \left (d x +c \right )}{4 a \left (a -b \right )}}{\left (\tan ^{4}\left (d x +c \right )\right ) a -\left (\tan ^{4}\left (d x +c \right )\right ) b +2 a \left (\tan ^{2}\left (d x +c \right )\right )+a}+\frac {\frac {\left (4 a \sqrt {a b}-2 \sqrt {a b}\, b -5 a b +3 b^{2}\right ) \arctanh \left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}+\frac {\left (4 a \sqrt {a b}-2 \sqrt {a b}\, b +5 a b -3 b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}}{4 a}}{d}\)	\(260\)
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 a \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 )}+\left (\munderset {\textit {\_R} =\RootOf \left (\left (65536 a^{10} d^{4}-196608 a^{9} b \,d^{4}+196608 a^{8} b^{2} d^{4}-65536 a^{7} b^{3} d^{4}\right ) \textit {\_Z}^{4}+\left (8192 a^{6} d^{2}-7680 a^{5} b \,d^{2}+1536 a^{4} b^{2} d^{2}\right ) \textit {\_Z}^{2}+256 a^{2}-288 a b +81 b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\frac {32768 i d^{3} a^{10}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}-\frac {114688 i a^{9} b \,d^{3}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}+\frac {147456 i a^{8} b^{2} d^{3}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}-\frac {81920 i a^{7} b^{3} d^{3}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}+\frac {16384 i a^{6} b^{4} d^{3}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}\right ) \textit {\_R}^{3}+\left (-\frac {8192 d^{2} a^{8}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}+\frac {29184 a^{7} b \,d^{2}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}-\frac {38400 a^{6} b^{2} d^{2}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}+\frac {22016 a^{5} b^{3} d^{2}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}-\frac {4608 a^{4} b^{4} d^{2}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}\right ) \textit {\_R}^{2}+\left (\frac {2048 i d \,a^{6}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}+\frac {896 i a^{5} b d}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}-\frac {5600 i a^{4} b^{2} d}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}+\frac {4032 i a^{3} b^{3} d}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}-\frac {864 i d \,b^{4} a^{2}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}\right ) \textit {\_R} -\frac {512 a^{4}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}+\frac {384 a^{3} b}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}+\frac {314 a^{2} b^{2}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}-\frac {351 a \,b^{3}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}+\frac {81 b^{4}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}\right )\right )\)	\(992\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a-b*sin(d*x+c)^4)^2,x,method=_RETURNVERBOSE)

[Out]

1/d*((-1/2*b/a/(a-b)*tan(d*x+c)^3-1/4*b/a/(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/a*(1/2*(4*a*(a*b)^(1/2)-2*(a*b)^(1/2)*b-5*a*b+3*b^2)/(a*b)^(1/2)/(a-b)/(((a*b)^(1/2)-a)*(a-b))^(1/2)*arct
anh((-a+b)*tan(d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1/2))+1/2*(4*a*(a*b)^(1/2)-2*(a*b)^(1/2)*b+5*a*b-3*b^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(1/(a-b*sin(d*x+c)^4)^2,x, algorithm="maxima")

[Out]

-1/2*(b^2*sin(2*d*x + 2*c) - 6*(8*a*b - 3*b^2)*cos(4*d*x + 4*c)*sin(2*d*x + 2*c) - (b^2*sin(6*d*x + 6*c) - 5*b
^2*sin(2*d*x + 2*c) - (8*a*b - 3*b^2)*sin(4*d*x + 4*c))*cos(8*d*x + 8*c) - 6*(4*b^2*sin(2*d*x + 2*c) + (8*a*b
- 3*b^2)*sin(4*d*x + 4*c))*cos(6*d*x + 6*c) + 2*((a^2*b^2 - a*b^3)*d*cos(8*d*x + 8*c)^2 + 16*(a^2*b^2 - a*b^3)
*d*cos(6*d*x + 6*c)^2 + 4*(64*a^4 - 112*a^3*b + 57*a^2*b^2 - 9*a*b^3)*d*cos(4*d*x + 4*c)^2 + 16*(a^2*b^2 - a*b
^3)*d*cos(2*d*x + 2*c)^2 + (a^2*b^2 - a*b^3)*d*sin(8*d*x + 8*c)^2 + 16*(a^2*b^2 - a*b^3)*d*sin(6*d*x + 6*c)^2
+ 4*(64*a^4 - 112*a^3*b + 57*a^2*b^2 - 9*a*b^3)*d*sin(4*d*x + 4*c)^2 + 16*(8*a^3*b - 11*a^2*b^2 + 3*a*b^3)*d*s
in(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*(a^2*b^2 - a*b^3)*d*sin(2*d*x + 2*c)^2 - 8*(a^2*b^2 - a*b^3)*d*cos(2*d*x
+ 2*c) + (a^2*b^2 - a*b^3)*d - 2*(4*(a^2*b^2 - a*b^3)*d*cos(6*d*x + 6*c) + 2*(8*a^3*b - 11*a^2*b^2 + 3*a*b^3)
*d*cos(4*d*x + 4*c) + 4*(a^2*b^2 - a*b^3)*d*cos(2*d*x + 2*c) - (a^2*b^2 - a*b^3)*d)*cos(8*d*x + 8*c) + 8*(2*(8
*a^3*b - 11*a^2*b^2 + 3*a*b^3)*d*cos(4*d*x + 4*c) + 4*(a^2*b^2 - a*b^3)*d*cos(2*d*x + 2*c) - (a^2*b^2 - a*b^3)
*d)*cos(6*d*x + 6*c) + 4*(4*(8*a^3*b - 11*a^2*b^2 + 3*a*b^3)*d*cos(2*d*x + 2*c) - (8*a^3*b - 11*a^2*b^2 + 3*a*
b^3)*d)*cos(4*d*x + 4*c) - 4*(2*(a^2*b^2 - a*b^3)*d*sin(6*d*x + 6*c) + (8*a^3*b - 11*a^2*b^2 + 3*a*b^3)*d*sin(
4*d*x + 4*c) + 2*(a^2*b^2 - a*b^3)*d*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 16*((8*a^3*b - 11*a^2*b^2 + 3*a*b^3)
*d*sin(4*d*x + 4*c) + 2*(a^2*b^2 - a*b^3)*d*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*integrate((4*b^2*cos(6*d*x + 6
*c)^2 + 4*b^2*cos(2*d*x + 2*c)^2 + 4*b^2*sin(6*d*x + 6*c)^2 + 4*b^2*sin(2*d*x + 2*c)^2 - 4*(64*a^2 - 64*a*b +
15*b^2)*cos(4*d*x + 4*c)^2 - b^2*cos(2*d*x + 2*c) - 4*(64*a^2 - 64*a*b + 15*b^2)*sin(4*d*x + 4*c)^2 - 2*(24*a*
b - 17*b^2)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) - (b^2*cos(6*d*x + 6*c) + b^2*cos(2*d*x + 2*c) - 2*(8*a*b - 5*b^
2)*cos(4*d*x + 4*c))*cos(8*d*x + 8*c) + (8*b^2*cos(2*d*x + 2*c) - b^2 - 2*(24*a*b - 17*b^2)*cos(4*d*x + 4*c))*
cos(6*d*x + 6*c) + 2*(8*a*b - 5*b^2 - (24*a*b - 17*b^2)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - (b^2*sin(6*d*x +
6*c) + b^2*sin(2*d*x + 2*c) - 2*(8*a*b - 5*b^2)*sin(4*d*x + 4*c))*sin(8*d*x + 8*c) + 2*(4*b^2*sin(2*d*x + 2*c)
- (24*a*b - 17*b^2)*sin(4*d*x + 4*c))*sin(6*d*x + 6*c))/(a^2*b^2 - a*b^3 + (a^2*b^2 - a*b^3)*cos(8*d*x + 8*c)
^2 + 16*(a^2*b^2 - a*b^3)*cos(6*d*x + 6*c)^2 + 4*(64*a^4 - 112*a^3*b + 57*a^2*b^2 - 9*a*b^3)*cos(4*d*x + 4*c)^
2 + 16*(a^2*b^2 - a*b^3)*cos(2*d*x + 2*c)^2 + (a^2*b^2 - a*b^3)*sin(8*d*x + 8*c)^2 + 16*(a^2*b^2 - a*b^3)*sin(
6*d*x + 6*c)^2 + 4*(64*a^4 - 112*a^3*b + 57*a^2*b^2 - 9*a*b^3)*sin(4*d*x + 4*c)^2 + 16*(8*a^3*b - 11*a^2*b^2 +
3*a*b^3)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*(a^2*b^2 - a*b^3)*sin(2*d*x + 2*c)^2 + 2*(a^2*b^2 - a*b^3 - 4
*(a^2*b^2 - a*b^3)*cos(6*d*x + 6*c) - 2*(8*a^3*b - 11*a^2*b^2 + 3*a*b^3)*cos(4*d*x + 4*c) - 4*(a^2*b^2 - a*b^3
)*cos(2*d*x + 2*c))*cos(8*d*x + 8*c) - 8*(a^2*b^2 - a*b^3 - 2*(8*a^3*b - 11*a^2*b^2 + 3*a*b^3)*cos(4*d*x + 4*c
) - 4*(a^2*b^2 - a*b^3)*cos(2*d*x + 2*c))*cos(6*d*x + 6*c) - 4*(8*a^3*b - 11*a^2*b^2 + 3*a*b^3 - 4*(8*a^3*b -
11*a^2*b^2 + 3*a*b^3)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - 8*(a^2*b^2 - a*b^3)*cos(2*d*x + 2*c) - 4*(2*(a^2*b^
2 - a*b^3)*sin(6*d*x + 6*c) + (8*a^3*b - 11*a^2*b^2 + 3*a*b^3)*sin(4*d*x + 4*c) + 2*(a^2*b^2 - a*b^3)*sin(2*d*
x + 2*c))*sin(8*d*x + 8*c) + 16*((8*a^3*b - 11*a^2*b^2 + 3*a*b^3)*sin(4*d*x + 4*c) + 2*(a^2*b^2 - a*b^3)*sin(2
*d*x + 2*c))*sin(6*d*x + 6*c)), x) + (b^2*cos(6*d*x + 6*c) - 5*b^2*cos(2*d*x + 2*c) + b^2 - (8*a*b - 3*b^2)*co
s(4*d*x + 4*c))*sin(8*d*x + 8*c) + (24*b^2*cos(2*d*x + 2*c) - 5*b^2 + 6*(8*a*b - 3*b^2)*cos(4*d*x + 4*c))*sin(
6*d*x + 6*c) - (8*a*b - 3*b^2 - 6*(8*a*b - 3*b^2)*cos(2*d*x + 2*c))*sin(4*d*x + 4*c))/((a^2*b^2 - a*b^3)*d*cos
(8*d*x + 8*c)^2 + 16*(a^2*b^2 - a*b^3)*d*cos(6*d*x + 6*c)^2 + 4*(64*a^4 - 112*a^3*b + 57*a^2*b^2 - 9*a*b^3)*d*
cos(4*d*x + 4*c)^2 + 16*(a^2*b^2 - a*b^3)*d*cos(2*d*x + 2*c)^2 + (a^2*b^2 - a*b^3)*d*sin(8*d*x + 8*c)^2 + 16*(
a^2*b^2 - a*b^3)*d*sin(6*d*x + 6*c)^2 + 4*(64*a^4 - 112*a^3*b + 57*a^2*b^2 - 9*a*b^3)*d*sin(4*d*x + 4*c)^2 + 1
6*(8*a^3*b - 11*a^2*b^2 + 3*a*b^3)*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*(a^2*b^2 - a*b^3)*d*sin(2*d*x + 2*
c)^2 - 8*(a^2*b^2 - a*b^3)*d*cos(2*d*x + 2*c) + (a^2*b^2 - a*b^3)*d - 2*(4*(a^2*b^2 - a*b^3)*d*cos(6*d*x + 6*c
) + 2*(8*a^3*b - 11*a^2*b^2 + 3*a*b^3)*d*cos(4*d*x + 4*c) + 4*(a^2*b^2 - a*b^3)*d*cos(2*d*x + 2*c) - (a^2*b^2
- a*b^3)*d)*cos(8*d*x + 8*c) + 8*(2*(8*a^3*b - 11*a^2*b^2 + 3*a*b^3)*d*cos(4*d*x + 4*c) + 4*(a^2*b^2 - a*b^3)*
d*cos(2*d*x + 2*c) - (a^2*b^2 - a*b^3)*d)*cos(6*d*x + 6*c) + 4*(4*(8*a^3*b - 11*a^2*b^2 + 3*a*b^3)*d*cos(2*d*x
+ 2*c) - (8*a^3*b - 11*a^2*b^2 + 3*a*b^3)*d)*cos(4*d*x + 4*c) - 4*(2*(a^2*b^2 - a*b^3)*d*sin(6*d*x + 6*c) + (
8*a^3*b - 11*a^2*b^2 + 3*a*b^3)*d*sin(4*d*x + 4*c) + 2*(a^2*b^2 - a*b^3)*d*sin(2*d*x + 2*c))*sin(8*d*x + 8*c)
+ 16*((8*a^3*b - 11*a^2*b^2 + 3*a*b^3)*d*sin(4*d*x + 4*c) + 2*(a^2*b^2 - a*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. 3477 vs. \(2 (164) = 328\).
time = 1.05, size = 3477, normalized size = 16.56 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-b*sin(d*x+c)^4)^2,x, algorithm="fricas")

[Out]

-1/32*(((a^2*b - a*b^2)*d*cos(d*x + c)^4 - 2*(a^2*b - a*b^2)*d*cos(d*x + c)^2 - (a^3 - 2*a^2*b + a*b^2)*d)*sqr
t(-((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b
^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) + 16*a^2 - 15*a*b
+ 3*b^2)/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2))*log(96*a^3*b - 170*a^2*b^2 + 405/4*a*b^3 - 81/4*b^4 - 1/
4*(384*a^3*b - 680*a^2*b^2 + 405*a*b^3 - 81*b^4)*cos(d*x + c)^2 + 1/2*(2*(2*a^10 - 7*a^9*b + 9*a^8*b^2 - 5*a^7
*b^3 + a^6*b^4)*d^3*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15
*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4))*cos(d*x + c)*sin(d*x + c) - (120*a^5*b - 217
*a^4*b^2 + 132*a^3*b^3 - 27*a^2*b^4)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)
*d^2*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*
a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) + 16*a^2 - 15*a*b + 3*b^2)/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^
3*b^3)*d^2)) + 1/4*(2*(16*a^8 - 57*a^7*b + 75*a^6*b^2 - 43*a^5*b^3 + 9*a^4*b^4)*d^2*cos(d*x + c)^2 - (16*a^8 -
57*a^7*b + 75*a^6*b^2 - 43*a^5*b^3 + 9*a^4*b^4)*d^2)*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^
4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4))) - ((a^2*b
- a*b^2)*d*cos(d*x + c)^4 - 2*(a^2*b - a*b^2)*d*cos(d*x + c)^2 - (a^3 - 2*a^2*b + a*b^2)*d)*sqrt(-((a^6 - 3*a
^5*b + 3*a^4*b^2 - a^3*b^3)*d^2*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6
*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) + 16*a^2 - 15*a*b + 3*b^2)/((a^6
- 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2))*log(96*a^3*b - 170*a^2*b^2 + 405/4*a*b^3 - 81/4*b^4 - 1/4*(384*a^3*b -
680*a^2*b^2 + 405*a*b^3 - 81*b^4)*cos(d*x + c)^2 - 1/2*(2*(2*a^10 - 7*a^9*b + 9*a^8*b^2 - 5*a^7*b^3 + a^6*b^4
)*d^3*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20
*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4))*cos(d*x + c)*sin(d*x + c) - (120*a^5*b - 217*a^4*b^2 + 132
*a^3*b^3 - 27*a^2*b^4)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2*sqrt((576
*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*
a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) + 16*a^2 - 15*a*b + 3*b^2)/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2)) +
1/4*(2*(16*a^8 - 57*a^7*b + 75*a^6*b^2 - 43*a^5*b^3 + 9*a^4*b^4)*d^2*cos(d*x + c)^2 - (16*a^8 - 57*a^7*b + 75
*a^6*b^2 - 43*a^5*b^3 + 9*a^4*b^4)*d^2)*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((
a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4))) + ((a^2*b - a*b^2)*d*co
s(d*x + c)^4 - 2*(a^2*b - a*b^2)*d*cos(d*x + c)^2 - (a^3 - 2*a^2*b + a*b^2)*d)*sqrt(((a^6 - 3*a^5*b + 3*a^4*b^
2 - a^3*b^3)*d^2*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^
11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) - 16*a^2 + 15*a*b - 3*b^2)/((a^6 - 3*a^5*b + 3*
a^4*b^2 - a^3*b^3)*d^2))*log(-96*a^3*b + 170*a^2*b^2 - 405/4*a*b^3 + 81/4*b^4 + 1/4*(384*a^3*b - 680*a^2*b^2 +
405*a*b^3 - 81*b^4)*cos(d*x + c)^2 + 1/2*(2*(2*a^10 - 7*a^9*b + 9*a^8*b^2 - 5*a^7*b^3 + a^6*b^4)*d^3*sqrt((57
6*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15
*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4))*cos(d*x + c)*sin(d*x + c) + (120*a^5*b - 217*a^4*b^2 + 132*a^3*b^3 - 27*
a^2*b^4)*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2*sqrt((576*a^4*b - 1392*a
^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8
*b^5 + a^7*b^6)*d^4)) - 16*a^2 + 15*a*b - 3*b^2)/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2)) + 1/4*(2*(16*a^8
- 57*a^7*b + 75*a^6*b^2 - 43*a^5*b^3 + 9*a^4*b^4)*d^2*cos(d*x + c)^2 - (16*a^8 - 57*a^7*b + 75*a^6*b^2 - 43*a
^5*b^3 + 9*a^4*b^4)*d^2)*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b
+ 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4))) - ((a^2*b - a*b^2)*d*cos(d*x + c)^4 -
2*(a^2*b - a*b^2)*d*cos(d*x + c)^2 - (a^3 - 2*a^2*b + a*b^2)*d)*sqrt(((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^
2*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^1
0*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) - 16*a^2 + 15*a*b - 3*b^2)/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b
^3)*d^2))*log(-96*a^3*b + 170*a^2*b^2 - 405/4*a*b^3 + 81/4*b^4 + 1/4*(384*a^3*b - 680*a^2*b^2 + 405*a*b^3 - 81
*b^4)*cos(d*x + c)^2 - 1/2*(2*(2*a^10 - 7*a^9*b + 9*a^8*b^2 - 5*a^7*b^3 + a^6*b^4)*d^3*sqrt((576*a^4*b - 1392*
a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^
8*b^5 + a^7*b^6)*d^4))*cos(d*x + c)*sin(d*x + c...

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1506 vs. \(2 (164) = 328\).
time = 0.56, size = 1506, normalized size = 7.17 \begin {gather*} -\frac {\frac {{\left (2 \, {\left (6 \, \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 + 4 \, \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^{2} - a b\right )}^{2} {\left | -a + b \right |} - {\left (12 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{6} - 57 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{5} b + 92 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{4} b^{2} - 58 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{3} b^{3} + 8 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{2} b^{4} + 3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a b^{5}\right )} {\left | -a^{2} + a b \right |} {\left | -a + b \right |} - {\left (15 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{7} - 69 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{6} b + 106 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{5} b^{2} - 62 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{4} b^{3} + 7 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{3} b^{4} + 3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} 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^{3} - a^{2} b + \sqrt {{\left (a^{3} - a^{2} b\right )}^{2} - {\left (a^{3} - a^{2} b\right )} {\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )}}}{a^{3} - 2 \, a^{2} b + a b^{2}}}}\right )\right )}}{{\left (3 \, a^{10} - 21 \, a^{9} b + 59 \, a^{8} b^{2} - 85 \, a^{7} b^{3} + 65 \, a^{6} b^{4} - 23 \, a^{5} b^{5} + a^{4} b^{6} + a^{3} b^{7}\right )} {\left | -a^{2} + a b \right |}} - \frac {{\left (2 \, {\left (6 \, \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 + 4 \, \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^{2} - a b\right )}^{2} {\left | -a + b \right |} + {\left (12 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{6} - 57 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{5} b + 92 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{4} b^{2} - 58 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{3} b^{3} + 8 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{2} b^{4} + 3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a b^{5}\right )} {\left | -a^{2} + a b \right |} {\left | -a + b \right |} - {\left (15 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{7} - 69 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{6} b + 106 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{5} b^{2} - 62 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{4} b^{3} + 7 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{3} b^{4} + 3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} 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^{3} - a^{2} b - \sqrt {{\left (a^{3} - a^{2} b\right )}^{2} - {\left (a^{3} - a^{2} b\right )} {\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )}}}{a^{3} - 2 \, a^{2} b + a b^{2}}}}\right )\right )}}{{\left (3 \, a^{10} - 21 \, a^{9} b + 59 \, a^{8} b^{2} - 85 \, a^{7} b^{3} + 65 \, a^{6} b^{4} - 23 \, a^{5} b^{5} + a^{4} b^{6} + a^{3} b^{7}\right )} {\left | -a^{2} + a b \right |}} + \frac {2 \, {\left (2 \, b \tan \left (d x + c\right )^{3} + b \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^{2} - a b\right )}}}{8 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-b*sin(d*x+c)^4)^2,x, algorithm="giac")

[Out]

-1/8*((2*(6*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 + 4*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^2 - a*b)^2*abs(-a + b) - (12*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^6 - 57*sqrt(a^2 - a*b - sqrt(a*b
)*(a - b))*a^5*b + 92*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^4*b^2 - 58*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^3
*b^3 + 8*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b^4 + 3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a*b^5)*abs(-a^2 +
a*b)*abs(-a + b) - (15*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^7 - 69*sqrt(a^2 - a*b - sqrt(a*b)*(a -
b))*sqrt(a*b)*a^6*b + 106*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^5*b^2 - 62*sqrt(a^2 - a*b - sqrt(a*
b)*(a - b))*sqrt(a*b)*a^4*b^3 + 7*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^4 + 3*sqrt(a^2 - a*b - s
qrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^5)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^
3 - a^2*b + sqrt((a^3 - a^2*b)^2 - (a^3 - a^2*b)*(a^3 - 2*a^2*b + a*b^2)))/(a^3 - 2*a^2*b + a*b^2))))/((3*a^10
- 21*a^9*b + 59*a^8*b^2 - 85*a^7*b^3 + 65*a^6*b^4 - 23*a^5*b^5 + a^4*b^6 + a^3*b^7)*abs(-a^2 + a*b)) - (2*(6*
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 + 4
*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^2
- a*b)^2*abs(-a + b) + (12*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^6 - 57*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a
^5*b + 92*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^4*b^2 - 58*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^3*b^3 + 8*sqr
t(a^2 - a*b + sqrt(a*b)*(a - b))*a^2*b^4 + 3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a*b^5)*abs(-a^2 + a*b)*abs(-a
+ b) - (15*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^7 - 69*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*
b)*a^6*b + 106*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^5*b^2 - 62*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*
sqrt(a*b)*a^4*b^3 + 7*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^4 + 3*sqrt(a^2 - a*b + sqrt(a*b)*(a
- b))*sqrt(a*b)*a^2*b^5)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^3 - a^2*b -
sqrt((a^3 - a^2*b)^2 - (a^3 - a^2*b)*(a^3 - 2*a^2*b + a*b^2)))/(a^3 - 2*a^2*b + a*b^2))))/((3*a^10 - 21*a^9*b
+ 59*a^8*b^2 - 85*a^7*b^3 + 65*a^6*b^4 - 23*a^5*b^5 + a^4*b^6 + a^3*b^7)*abs(-a^2 + a*b)) + 2*(2*b*tan(d*x + c
)^3 + b*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^2 - a*b)))/d

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Mupad [B]
time = 16.52, size = 2500, normalized size = 11.90 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a - b*sin(c + d*x)^4)^2,x)

[Out]

- (atan(((((512*a^6*b - 384*a^3*b^4 + 1280*a^4*b^3 - 1408*a^5*b^2)/(32*(a^3*b - a^4)) - (tan(c + d*x)*((24*a^2
*(a^7*b)^(1/2) + 9*b^2*(a^7*b)^(1/2) - 15*a^5*b + 16*a^6 + 3*a^4*b^2 - 29*a*b*(a^7*b)^(1/2))/(256*(3*a^9*b - a
^10 + a^7*b^3 - 3*a^8*b^2)))^(1/2)*(256*a^7*b - 256*a^4*b^4 + 768*a^5*b^3 - 768*a^6*b^2))/(4*(a^2*b - a^3)))*(
(24*a^2*(a^7*b)^(1/2) + 9*b^2*(a^7*b)^(1/2) - 15*a^5*b + 16*a^6 + 3*a^4*b^2 - 29*a*b*(a^7*b)^(1/2))/(256*(3*a^
9*b - a^10 + a^7*b^3 - 3*a^8*b^2)))^(1/2) - (tan(c + d*x)*(16*a^3*b - 26*a*b^3 + 9*b^4 + 9*a^2*b^2))/(4*(a^2*b
- a^3)))*((24*a^2*(a^7*b)^(1/2) + 9*b^2*(a^7*b)^(1/2) - 15*a^5*b + 16*a^6 + 3*a^4*b^2 - 29*a*b*(a^7*b)^(1/2))
/(256*(3*a^9*b - a^10 + a^7*b^3 - 3*a^8*b^2)))^(1/2)*1i - (((512*a^6*b - 384*a^3*b^4 + 1280*a^4*b^3 - 1408*a^5
*b^2)/(32*(a^3*b - a^4)) + (tan(c + d*x)*((24*a^2*(a^7*b)^(1/2) + 9*b^2*(a^7*b)^(1/2) - 15*a^5*b + 16*a^6 + 3*
a^4*b^2 - 29*a*b*(a^7*b)^(1/2))/(256*(3*a^9*b - a^10 + a^7*b^3 - 3*a^8*b^2)))^(1/2)*(256*a^7*b - 256*a^4*b^4 +
768*a^5*b^3 - 768*a^6*b^2))/(4*(a^2*b - a^3)))*((24*a^2*(a^7*b)^(1/2) + 9*b^2*(a^7*b)^(1/2) - 15*a^5*b + 16*a
^6 + 3*a^4*b^2 - 29*a*b*(a^7*b)^(1/2))/(256*(3*a^9*b - a^10 + a^7*b^3 - 3*a^8*b^2)))^(1/2) + (tan(c + d*x)*(16
*a^3*b - 26*a*b^3 + 9*b^4 + 9*a^2*b^2))/(4*(a^2*b - a^3)))*((24*a^2*(a^7*b)^(1/2) + 9*b^2*(a^7*b)^(1/2) - 15*a
^5*b + 16*a^6 + 3*a^4*b^2 - 29*a*b*(a^7*b)^(1/2))/(256*(3*a^9*b - a^10 + a^7*b^3 - 3*a^8*b^2)))^(1/2)*1i)/((32
*a^2*b - 34*a*b^2 + 9*b^3)/(16*(a^3*b - a^4)) + (((512*a^6*b - 384*a^3*b^4 + 1280*a^4*b^3 - 1408*a^5*b^2)/(32*
(a^3*b - a^4)) - (tan(c + d*x)*((24*a^2*(a^7*b)^(1/2) + 9*b^2*(a^7*b)^(1/2) - 15*a^5*b + 16*a^6 + 3*a^4*b^2 -
29*a*b*(a^7*b)^(1/2))/(256*(3*a^9*b - a^10 + a^7*b^3 - 3*a^8*b^2)))^(1/2)*(256*a^7*b - 256*a^4*b^4 + 768*a^5*b
^3 - 768*a^6*b^2))/(4*(a^2*b - a^3)))*((24*a^2*(a^7*b)^(1/2) + 9*b^2*(a^7*b)^(1/2) - 15*a^5*b + 16*a^6 + 3*a^4
*b^2 - 29*a*b*(a^7*b)^(1/2))/(256*(3*a^9*b - a^10 + a^7*b^3 - 3*a^8*b^2)))^(1/2) - (tan(c + d*x)*(16*a^3*b - 2
6*a*b^3 + 9*b^4 + 9*a^2*b^2))/(4*(a^2*b - a^3)))*((24*a^2*(a^7*b)^(1/2) + 9*b^2*(a^7*b)^(1/2) - 15*a^5*b + 16*
a^6 + 3*a^4*b^2 - 29*a*b*(a^7*b)^(1/2))/(256*(3*a^9*b - a^10 + a^7*b^3 - 3*a^8*b^2)))^(1/2) + (((512*a^6*b - 3
84*a^3*b^4 + 1280*a^4*b^3 - 1408*a^5*b^2)/(32*(a^3*b - a^4)) + (tan(c + d*x)*((24*a^2*(a^7*b)^(1/2) + 9*b^2*(a
^7*b)^(1/2) - 15*a^5*b + 16*a^6 + 3*a^4*b^2 - 29*a*b*(a^7*b)^(1/2))/(256*(3*a^9*b - a^10 + a^7*b^3 - 3*a^8*b^2
)))^(1/2)*(256*a^7*b - 256*a^4*b^4 + 768*a^5*b^3 - 768*a^6*b^2))/(4*(a^2*b - a^3)))*((24*a^2*(a^7*b)^(1/2) + 9
*b^2*(a^7*b)^(1/2) - 15*a^5*b + 16*a^6 + 3*a^4*b^2 - 29*a*b*(a^7*b)^(1/2))/(256*(3*a^9*b - a^10 + a^7*b^3 - 3*
a^8*b^2)))^(1/2) + (tan(c + d*x)*(16*a^3*b - 26*a*b^3 + 9*b^4 + 9*a^2*b^2))/(4*(a^2*b - a^3)))*((24*a^2*(a^7*b
)^(1/2) + 9*b^2*(a^7*b)^(1/2) - 15*a^5*b + 16*a^6 + 3*a^4*b^2 - 29*a*b*(a^7*b)^(1/2))/(256*(3*a^9*b - a^10 + a
^7*b^3 - 3*a^8*b^2)))^(1/2)))*((24*a^2*(a^7*b)^(1/2) + 9*b^2*(a^7*b)^(1/2) - 15*a^5*b + 16*a^6 + 3*a^4*b^2 - 2
9*a*b*(a^7*b)^(1/2))/(256*(3*a^9*b - a^10 + a^7*b^3 - 3*a^8*b^2)))^(1/2)*2i)/d - (atan(((((512*a^6*b - 384*a^3
*b^4 + 1280*a^4*b^3 - 1408*a^5*b^2)/(32*(a^3*b - a^4)) - (tan(c + d*x)*(-(24*a^2*(a^7*b)^(1/2) + 9*b^2*(a^7*b)
^(1/2) + 15*a^5*b - 16*a^6 - 3*a^4*b^2 - 29*a*b*(a^7*b)^(1/2))/(256*(3*a^9*b - a^10 + a^7*b^3 - 3*a^8*b^2)))^(
1/2)*(256*a^7*b - 256*a^4*b^4 + 768*a^5*b^3 - 768*a^6*b^2))/(4*(a^2*b - a^3)))*(-(24*a^2*(a^7*b)^(1/2) + 9*b^2
*(a^7*b)^(1/2) + 15*a^5*b - 16*a^6 - 3*a^4*b^2 - 29*a*b*(a^7*b)^(1/2))/(256*(3*a^9*b - a^10 + a^7*b^3 - 3*a^8*
b^2)))^(1/2) - (tan(c + d*x)*(16*a^3*b - 26*a*b^3 + 9*b^4 + 9*a^2*b^2))/(4*(a^2*b - a^3)))*(-(24*a^2*(a^7*b)^(
1/2) + 9*b^2*(a^7*b)^(1/2) + 15*a^5*b - 16*a^6 - 3*a^4*b^2 - 29*a*b*(a^7*b)^(1/2))/(256*(3*a^9*b - a^10 + a^7*
b^3 - 3*a^8*b^2)))^(1/2)*1i - (((512*a^6*b - 384*a^3*b^4 + 1280*a^4*b^3 - 1408*a^5*b^2)/(32*(a^3*b - a^4)) + (
tan(c + d*x)*(-(24*a^2*(a^7*b)^(1/2) + 9*b^2*(a^7*b)^(1/2) + 15*a^5*b - 16*a^6 - 3*a^4*b^2 - 29*a*b*(a^7*b)^(1
/2))/(256*(3*a^9*b - a^10 + a^7*b^3 - 3*a^8*b^2)))^(1/2)*(256*a^7*b - 256*a^4*b^4 + 768*a^5*b^3 - 768*a^6*b^2)
)/(4*(a^2*b - a^3)))*(-(24*a^2*(a^7*b)^(1/2) + 9*b^2*(a^7*b)^(1/2) + 15*a^5*b - 16*a^6 - 3*a^4*b^2 - 29*a*b*(a
^7*b)^(1/2))/(256*(3*a^9*b - a^10 + a^7*b^3 - 3*a^8*b^2)))^(1/2) + (tan(c + d*x)*(16*a^3*b - 26*a*b^3 + 9*b^4
+ 9*a^2*b^2))/(4*(a^2*b - a^3)))*(-(24*a^2*(a^7*b)^(1/2) + 9*b^2*(a^7*b)^(1/2) + 15*a^5*b - 16*a^6 - 3*a^4*b^2
- 29*a*b*(a^7*b)^(1/2))/(256*(3*a^9*b - a^10 + a^7*b^3 - 3*a^8*b^2)))^(1/2)*1i)/((32*a^2*b - 34*a*b^2 + 9*b^3
)/(16*(a^3*b - a^4)) + (((512*a^6*b - 384*a^3*b^4 + 1280*a^4*b^3 - 1408*a^5*b^2)/(32*(a^3*b - a^4)) - (tan(c +
d*x)*(-(24*a^2*(a^7*b)^(1/2) + 9*b^2*(a^7*b)^(1/2) + 15*a^5*b - 16*a^6 - 3*a^4*b^2 - 29*a*b*(a^7*b)^(1/2))/(2
56*(3*a^9*b - a^10 + a^7*b^3 - 3*a^8*b^2)))^(1/2)*(256*a^7*b - 256*a^4*b^4 + 768*a^5*b^3 - 768*a^6*b^2))/(4*(a
^2*b - a^3)))*(-(24*a^2*(a^7*b)^(1/2) + 9*b^2*(a^7*b)^(1/2) + 15*a^5*b - 16*a^6 - 3*a^4*b^2 - 29*a*b*(a^7*b)^(
1/2))/(256*(3*a^9*b - a^10 + a^7*b^3 - 3*a^8*b^...

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