3.3.17 \(\int \frac {\csc (c+d x)}{(a-b \sin ^4(c+d x))^2} \, dx\) [217]

Optimal. Leaf size=325 \[ -\frac {\sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 a^{3/2} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} d}-\frac {\sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^2 \sqrt {\sqrt {a}-\sqrt {b}} d}-\frac {\tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac {\sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 a^{3/2} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} d}+\frac {\sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^2 \sqrt {\sqrt {a}+\sqrt {b}} d}-\frac {b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{4 a (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )} \]

[Out]

-arctanh(cos(d*x+c))/a^2/d-1/4*b*cos(d*x+c)*(2-cos(d*x+c)^2)/a/(a-b)/d/(a-b+2*b*cos(d*x+c)^2-b*cos(d*x+c)^4)-1
/8*b^(1/4)*arctan(b^(1/4)*cos(d*x+c)/(a^(1/2)-b^(1/2))^(1/2))/a^(3/2)/d/(a^(1/2)-b^(1/2))^(3/2)+1/8*b^(1/4)*ar
ctanh(b^(1/4)*cos(d*x+c)/(a^(1/2)+b^(1/2))^(1/2))/a^(3/2)/d/(a^(1/2)+b^(1/2))^(3/2)-1/2*b^(1/4)*arctan(b^(1/4)
*cos(d*x+c)/(a^(1/2)-b^(1/2))^(1/2))/a^2/d/(a^(1/2)-b^(1/2))^(1/2)+1/2*b^(1/4)*arctanh(b^(1/4)*cos(d*x+c)/(a^(
1/2)+b^(1/2))^(1/2))/a^2/d/(a^(1/2)+b^(1/2))^(1/2)

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Rubi [A]
time = 0.26, antiderivative size = 325, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {3294, 1252, 213, 1192, 1180, 211, 214} \begin {gather*} -\frac {\sqrt [4]{b} \text {ArcTan}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 a^{3/2} d \left (\sqrt {a}-\sqrt {b}\right )^{3/2}}+\frac {\sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 a^{3/2} d \left (\sqrt {a}+\sqrt {b}\right )^{3/2}}-\frac {\sqrt [4]{b} \text {ArcTan}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^2 d \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^2 d \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {\tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac {b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{4 a d (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*(b^(1/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(a^(3/2)*(Sqrt[a] - Sqrt[b])^(3/2)*d) -
(b^(1/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a^2*Sqrt[Sqrt[a] - Sqrt[b]]*d) - ArcTanh[C
os[c + d*x]]/(a^2*d) + (b^(1/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(8*a^(3/2)*(Sqrt[a] +
Sqrt[b])^(3/2)*d) + (b^(1/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*a^2*Sqrt[Sqrt[a] + S
qrt[b]]*d) - (b*Cos[c + d*x]*(2 - Cos[c + d*x]^2))/(4*a*(a - b)*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]
^4))

Rule 211

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

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 214

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

Rule 1180

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

Rule 1192

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[x*(a*b*e - d*(b^2 - 2*a
*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Dist[1/(2*a*(p + 1)
*(b^2 - 4*a*c)), Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a +
b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e
^2, 0] && LtQ[p, -1] && IntegerQ[2*p]

Rule 1252

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d
+ e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b^2 - 4*a*c, 0] && ((Intege
rQ[p] && IntegerQ[q]) || IGtQ[p, 0] || IGtQ[q, 0])

Rule 3294

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, Dist[-ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4
)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps

\begin {align*} \int \frac {\csc (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx &=-\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \left (a-b+2 b x^2-b x^4\right )^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {\text {Subst}\left (\int \left (-\frac {1}{a^2 \left (-1+x^2\right )}+\frac {b-b x^2}{a \left (a-b+2 b x^2-b x^4\right )^2}+\frac {b-b x^2}{a^2 \left (a-b+2 b x^2-b x^4\right )}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\cos (c+d x)\right )}{a^2 d}-\frac {\text {Subst}\left (\int \frac {b-b x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cos (c+d x)\right )}{a^2 d}-\frac {\text {Subst}\left (\int \frac {b-b x^2}{\left (a-b+2 b x^2-b x^4\right )^2} \, dx,x,\cos (c+d x)\right )}{a d}\\ &=-\frac {\tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac {b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{4 a (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {-4 a b^2+2 a b^2 x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cos (c+d x)\right )}{8 a^2 (a-b) b d}+\frac {b \text {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{2 a^2 d}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{2 a^2 d}\\ &=-\frac {\sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^2 \sqrt {\sqrt {a}-\sqrt {b}} d}-\frac {\tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac {\sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^2 \sqrt {\sqrt {a}+\sqrt {b}} d}-\frac {b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{4 a (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}+\frac {b \text {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{8 a^{3/2} \left (\sqrt {a}-\sqrt {b}\right ) d}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{8 a^{3/2} \left (\sqrt {a}+\sqrt {b}\right ) d}\\ &=-\frac {\sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 a^{3/2} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} d}-\frac {\sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^2 \sqrt {\sqrt {a}-\sqrt {b}} d}-\frac {\tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac {\sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 a^{3/2} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} d}+\frac {\sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^2 \sqrt {\sqrt {a}+\sqrt {b}} d}-\frac {b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{4 a (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.59, size = 600, normalized size = 1.85 \begin {gather*} \frac {\frac {16 a b (-5 \cos (c+d x)+\cos (3 (c+d x)))}{(a-b) (8 a-3 b+4 b \cos (2 (c+d x))-b \cos (4 (c+d x)))}-32 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+32 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-\frac {i b \text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {-10 a \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+8 b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+5 i a \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )-4 i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+38 a \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-24 b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-19 i a \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2+12 i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-38 a \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4+24 b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4+19 i a \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4-12 i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4+10 a \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^6-8 b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^6-5 i a \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^6+4 i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^6}{-b \text {$\#$1}-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-3 b \text {$\#$1}^5+b \text {$\#$1}^7}\&\right ]}{a-b}}{32 a^2 d} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((16*a*b*(-5*Cos[c + d*x] + Cos[3*(c + d*x)]))/((a - b)*(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)]
)) - 32*Log[Cos[(c + d*x)/2]] + 32*Log[Sin[(c + d*x)/2]] - (I*b*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 -
4*b*#1^6 + b*#1^8 & , (-10*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + 8*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x]
- #1)] + (5*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - (4*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + 38*a*ArcTan[S
in[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - 24*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - (19*I)*a*Log[1 -
2*Cos[c + d*x]*#1 + #1^2]*#1^2 + (12*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - 38*a*ArcTan[Sin[c + d*x]/(C
os[c + d*x] - #1)]*#1^4 + 24*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + (19*I)*a*Log[1 - 2*Cos[c + d*x]
*#1 + #1^2]*#1^4 - (12*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + 10*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] -
#1)]*#1^6 - 8*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 - (5*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6
+ (4*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(a
- b))/(32*a^2*d)

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Maple [A]
time = 1.14, size = 243, normalized size = 0.75

method	result	size
derivativedivides	\(\frac {\frac {b \left (\frac {\frac {a \left (\cos ^{3}\left (d x +c \right )\right )}{4 a -4 b}-\frac {a \cos \left (d x +c \right )}{2 \left (a -b \right )}}{a -b +2 b \left (\cos ^{2}\left (d x +c \right )\right )-b \left (\cos ^{4}\left (d x +c \right )\right )}+\frac {b \left (\frac {\left (-5 a \sqrt {a b}+4 \sqrt {a b}\, b -a b \right ) \arctan \left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 \sqrt {a b}\, b \sqrt {\left (\sqrt {a b}-b \right ) b}}-\frac {\left (-5 a \sqrt {a b}+4 \sqrt {a b}\, b +a b \right ) \arctanh \left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 \sqrt {a b}\, b \sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{4 a -4 b}\right )}{a^{2}}-\frac {\ln \left (1+\cos \left (d x +c \right )\right )}{2 a^{2}}+\frac {\ln \left (\cos \left (d x +c \right )-1\right )}{2 a^{2}}}{d}\)	\(243\)
default	\(\frac {\frac {b \left (\frac {\frac {a \left (\cos ^{3}\left (d x +c \right )\right )}{4 a -4 b}-\frac {a \cos \left (d x +c \right )}{2 \left (a -b \right )}}{a -b +2 b \left (\cos ^{2}\left (d x +c \right )\right )-b \left (\cos ^{4}\left (d x +c \right )\right )}+\frac {b \left (\frac {\left (-5 a \sqrt {a b}+4 \sqrt {a b}\, b -a b \right ) \arctan \left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 \sqrt {a b}\, b \sqrt {\left (\sqrt {a b}-b \right ) b}}-\frac {\left (-5 a \sqrt {a b}+4 \sqrt {a b}\, b +a b \right ) \arctanh \left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 \sqrt {a b}\, b \sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{4 a -4 b}\right )}{a^{2}}-\frac {\ln \left (1+\cos \left (d x +c \right )\right )}{2 a^{2}}+\frac {\ln \left (\cos \left (d x +c \right )-1\right )}{2 a^{2}}}{d}\)	\(243\)
risch	\(\frac {b \left ({\mathrm e}^{7 i \left (d x +c \right )}-5 \,{\mathrm e}^{5 i \left (d x +c \right )}-5 \,{\mathrm e}^{3 i \left (d x +c \right )}+{\mathrm e}^{i \left (d x +c \right )}\right )}{2 a \left (-a +b \right ) d \left (b \,{\mathrm e}^{8 i \left (d x +c \right )}-4 b \,{\mathrm e}^{6 i \left (d x +c \right )}-16 a \,{\mathrm e}^{4 i \left (d x +c \right )}+6 b \,{\mathrm e}^{4 i \left (d x +c \right )}-4 b \,{\mathrm e}^{2 i \left (d x +c \right )}+b \right )}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{a^{2} d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{a^{2} d}+2 i \left (\munderset {\textit {\_R} =\RootOf \left (\left (1048576 a^{11} d^{4}-3145728 a^{10} b \,d^{4}+3145728 a^{9} b^{2} d^{4}-1048576 a^{8} b^{3} d^{4}\right ) \textit {\_Z}^{4}+\left (-71680 a^{6} b \,d^{2}+96256 a^{5} b^{2} d^{2}-32768 a^{4} b^{3} d^{2}\right ) \textit {\_Z}^{2}-625 a^{2} b +800 a \,b^{2}-256 b^{3}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\left (-\frac {327680 i a^{10} d^{3}}{625 a^{3} b -1125 a^{2} b^{2}+664 a \,b^{3}-128 b^{4}}+\frac {1179648 i a^{9} d^{3} b}{625 a^{3} b -1125 a^{2} b^{2}+664 a \,b^{3}-128 b^{4}}-\frac {1572864 i a^{8} d^{3} b^{2}}{625 a^{3} b -1125 a^{2} b^{2}+664 a \,b^{3}-128 b^{4}}+\frac {917504 i a^{7} d^{3} b^{3}}{625 a^{3} b -1125 a^{2} b^{2}+664 a \,b^{3}-128 b^{4}}-\frac {196608 i a^{6} b^{4} d^{3}}{625 a^{3} b -1125 a^{2} b^{2}+664 a \,b^{3}-128 b^{4}}\right ) \textit {\_R}^{3}+\left (\frac {20800 i a^{5} d b}{625 a^{3} b -1125 a^{2} b^{2}+664 a \,b^{3}-128 b^{4}}-\frac {39296 i a^{4} d \,b^{2}}{625 a^{3} b -1125 a^{2} b^{2}+664 a \,b^{3}-128 b^{4}}+\frac {24640 i a^{3} d \,b^{3}}{625 a^{3} b -1125 a^{2} b^{2}+664 a \,b^{3}-128 b^{4}}-\frac {5120 i a^{2} b^{4} d}{625 a^{3} b -1125 a^{2} b^{2}+664 a \,b^{3}-128 b^{4}}\right ) \textit {\_R} \right ) {\mathrm e}^{i \left (d x +c \right )}+1\right )\right )\)	\(644\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(b/a^2*((1/4*a/(a-b)*cos(d*x+c)^3-1/2*a/(a-b)*cos(d*x+c))/(a-b+2*b*cos(d*x+c)^2-b*cos(d*x+c)^4)+1/4/(a-b)*
b*(1/2*(-5*a*(a*b)^(1/2)+4*(a*b)^(1/2)*b-a*b)/(a*b)^(1/2)/b/(((a*b)^(1/2)-b)*b)^(1/2)*arctan(b*cos(d*x+c)/(((a
*b)^(1/2)-b)*b)^(1/2))-1/2*(-5*a*(a*b)^(1/2)+4*(a*b)^(1/2)*b+a*b)/(a*b)^(1/2)/b/(((a*b)^(1/2)+b)*b)^(1/2)*arct
anh(b*cos(d*x+c)/(((a*b)^(1/2)+b)*b)^(1/2))))-1/2/a^2*ln(1+cos(d*x+c))+1/2/a^2*ln(cos(d*x+c)-1))

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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(csc(d*x+c)/(a-b*sin(d*x+c)^4)^2,x, algorithm="maxima")

[Out]

1/2*(4*a*b^2*cos(2*d*x + 2*c)*cos(d*x + c) - 20*a*b^2*sin(3*d*x + 3*c)*sin(2*d*x + 2*c) + 4*a*b^2*sin(2*d*x +
2*c)*sin(d*x + c) - a*b^2*cos(d*x + c) - (a*b^2*cos(7*d*x + 7*c) - 5*a*b^2*cos(5*d*x + 5*c) - 5*a*b^2*cos(3*d*
x + 3*c) + a*b^2*cos(d*x + c))*cos(8*d*x + 8*c) + (4*a*b^2*cos(6*d*x + 6*c) + 4*a*b^2*cos(2*d*x + 2*c) - a*b^2
+ 2*(8*a^2*b - 3*a*b^2)*cos(4*d*x + 4*c))*cos(7*d*x + 7*c) - 4*(5*a*b^2*cos(5*d*x + 5*c) + 5*a*b^2*cos(3*d*x
+ 3*c) - a*b^2*cos(d*x + c))*cos(6*d*x + 6*c) - 5*(4*a*b^2*cos(2*d*x + 2*c) - a*b^2 + 2*(8*a^2*b - 3*a*b^2)*co
s(4*d*x + 4*c))*cos(5*d*x + 5*c) - 2*(5*(8*a^2*b - 3*a*b^2)*cos(3*d*x + 3*c) - (8*a^2*b - 3*a*b^2)*cos(d*x + c
))*cos(4*d*x + 4*c) - 5*(4*a*b^2*cos(2*d*x + 2*c) - a*b^2)*cos(3*d*x + 3*c) - 2*((a^3*b^2 - a^2*b^3)*d*cos(8*d
*x + 8*c)^2 + 16*(a^3*b^2 - a^2*b^3)*d*cos(6*d*x + 6*c)^2 + 4*(64*a^5 - 112*a^4*b + 57*a^3*b^2 - 9*a^2*b^3)*d*
cos(4*d*x + 4*c)^2 + 16*(a^3*b^2 - a^2*b^3)*d*cos(2*d*x + 2*c)^2 + (a^3*b^2 - a^2*b^3)*d*sin(8*d*x + 8*c)^2 +
16*(a^3*b^2 - a^2*b^3)*d*sin(6*d*x + 6*c)^2 + 4*(64*a^5 - 112*a^4*b + 57*a^3*b^2 - 9*a^2*b^3)*d*sin(4*d*x + 4*
c)^2 + 16*(8*a^4*b - 11*a^3*b^2 + 3*a^2*b^3)*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*(a^3*b^2 - a^2*b^3)*d*si
n(2*d*x + 2*c)^2 - 8*(a^3*b^2 - a^2*b^3)*d*cos(2*d*x + 2*c) + (a^3*b^2 - a^2*b^3)*d - 2*(4*(a^3*b^2 - a^2*b^3)
*d*cos(6*d*x + 6*c) + 2*(8*a^4*b - 11*a^3*b^2 + 3*a^2*b^3)*d*cos(4*d*x + 4*c) + 4*(a^3*b^2 - a^2*b^3)*d*cos(2*
d*x + 2*c) - (a^3*b^2 - a^2*b^3)*d)*cos(8*d*x + 8*c) + 8*(2*(8*a^4*b - 11*a^3*b^2 + 3*a^2*b^3)*d*cos(4*d*x + 4
*c) + 4*(a^3*b^2 - a^2*b^3)*d*cos(2*d*x + 2*c) - (a^3*b^2 - a^2*b^3)*d)*cos(6*d*x + 6*c) + 4*(4*(8*a^4*b - 11*
a^3*b^2 + 3*a^2*b^3)*d*cos(2*d*x + 2*c) - (8*a^4*b - 11*a^3*b^2 + 3*a^2*b^3)*d)*cos(4*d*x + 4*c) - 4*(2*(a^3*b
^2 - a^2*b^3)*d*sin(6*d*x + 6*c) + (8*a^4*b - 11*a^3*b^2 + 3*a^2*b^3)*d*sin(4*d*x + 4*c) + 2*(a^3*b^2 - a^2*b^
3)*d*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 16*((8*a^4*b - 11*a^3*b^2 + 3*a^2*b^3)*d*sin(4*d*x + 4*c) + 2*(a^3*b
^2 - a^2*b^3)*d*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*integrate(-1/2*(4*(19*a*b^2 - 12*b^3)*cos(3*d*x + 3*c)*sin
(2*d*x + 2*c) - 4*(5*a*b^2 - 4*b^3)*cos(d*x + c)*sin(2*d*x + 2*c) + 4*(5*a*b^2 - 4*b^3)*cos(2*d*x + 2*c)*sin(d
*x + c) + ((5*a*b^2 - 4*b^3)*sin(7*d*x + 7*c) - (19*a*b^2 - 12*b^3)*sin(5*d*x + 5*c) + (19*a*b^2 - 12*b^3)*sin
(3*d*x + 3*c) - (5*a*b^2 - 4*b^3)*sin(d*x + c))*cos(8*d*x + 8*c) + 2*(2*(5*a*b^2 - 4*b^3)*sin(6*d*x + 6*c) + (
40*a^2*b - 47*a*b^2 + 12*b^3)*sin(4*d*x + 4*c) + 2*(5*a*b^2 - 4*b^3)*sin(2*d*x + 2*c))*cos(7*d*x + 7*c) + 4*((
19*a*b^2 - 12*b^3)*sin(5*d*x + 5*c) - (19*a*b^2 - 12*b^3)*sin(3*d*x + 3*c) + (5*a*b^2 - 4*b^3)*sin(d*x + c))*c
os(6*d*x + 6*c) - 2*((152*a^2*b - 153*a*b^2 + 36*b^3)*sin(4*d*x + 4*c) + 2*(19*a*b^2 - 12*b^3)*sin(2*d*x + 2*c
))*cos(5*d*x + 5*c) - 2*((152*a^2*b - 153*a*b^2 + 36*b^3)*sin(3*d*x + 3*c) - (40*a^2*b - 47*a*b^2 + 12*b^3)*si
n(d*x + c))*cos(4*d*x + 4*c) - ((5*a*b^2 - 4*b^3)*cos(7*d*x + 7*c) - (19*a*b^2 - 12*b^3)*cos(5*d*x + 5*c) + (1
9*a*b^2 - 12*b^3)*cos(3*d*x + 3*c) - (5*a*b^2 - 4*b^3)*cos(d*x + c))*sin(8*d*x + 8*c) + (5*a*b^2 - 4*b^3 - 4*(
5*a*b^2 - 4*b^3)*cos(6*d*x + 6*c) - 2*(40*a^2*b - 47*a*b^2 + 12*b^3)*cos(4*d*x + 4*c) - 4*(5*a*b^2 - 4*b^3)*co
s(2*d*x + 2*c))*sin(7*d*x + 7*c) - 4*((19*a*b^2 - 12*b^3)*cos(5*d*x + 5*c) - (19*a*b^2 - 12*b^3)*cos(3*d*x + 3
*c) + (5*a*b^2 - 4*b^3)*cos(d*x + c))*sin(6*d*x + 6*c) - (19*a*b^2 - 12*b^3 - 2*(152*a^2*b - 153*a*b^2 + 36*b^
3)*cos(4*d*x + 4*c) - 4*(19*a*b^2 - 12*b^3)*cos(2*d*x + 2*c))*sin(5*d*x + 5*c) + 2*((152*a^2*b - 153*a*b^2 + 3
6*b^3)*cos(3*d*x + 3*c) - (40*a^2*b - 47*a*b^2 + 12*b^3)*cos(d*x + c))*sin(4*d*x + 4*c) + (19*a*b^2 - 12*b^3 -
4*(19*a*b^2 - 12*b^3)*cos(2*d*x + 2*c))*sin(3*d*x + 3*c) - (5*a*b^2 - 4*b^3)*sin(d*x + c))/(a^3*b^2 - a^2*b^3
+ (a^3*b^2 - a^2*b^3)*cos(8*d*x + 8*c)^2 + 16*(a^3*b^2 - a^2*b^3)*cos(6*d*x + 6*c)^2 + 4*(64*a^5 - 112*a^4*b
+ 57*a^3*b^2 - 9*a^2*b^3)*cos(4*d*x + 4*c)^2 + 16*(a^3*b^2 - a^2*b^3)*cos(2*d*x + 2*c)^2 + (a^3*b^2 - a^2*b^3)
*sin(8*d*x + 8*c)^2 + 16*(a^3*b^2 - a^2*b^3)*sin(6*d*x + 6*c)^2 + 4*(64*a^5 - 112*a^4*b + 57*a^3*b^2 - 9*a^2*b
^3)*sin(4*d*x + 4*c)^2 + 16*(8*a^4*b - 11*a^3*b^2 + 3*a^2*b^3)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*(a^3*b^2
- a^2*b^3)*sin(2*d*x + 2*c)^2 + 2*(a^3*b^2 - a^2*b^3 - 4*(a^3*b^2 - a^2*b^3)*cos(6*d*x + 6*c) - 2*(8*a^4*b -
11*a^3*b^2 + 3*a^2*b^3)*cos(4*d*x + 4*c) - 4*(a^3*b^2 - a^2*b^3)*cos(2*d*x + 2*c))*cos(8*d*x + 8*c) - 8*(a^3*b
^2 - a^2*b^3 - 2*(8*a^4*b - 11*a^3*b^2 + 3*a^2*b^3)*cos(4*d*x + 4*c) - 4*(a^3*b^2 - a^2*b^3)*cos(2*d*x + 2*c))
*cos(6*d*x + 6*c) - 4*(8*a^4*b - 11*a^3*b^2 + 3*a^2*b^3 - 4*(8*a^4*b - 11*a^3*b^2 + 3*a^2*b^3)*cos(2*d*x + 2*c
))*cos(4*d*x + 4*c) - 8*(a^3*b^2 - a^2*b^3)*cos(2*d*x + 2*c) - 4*(2*(a^3*b^2 - a^2*b^3)*sin(6*d*x + 6*c) + (8*
a^4*b - 11*a^3*b^2 + 3*a^2*b^3)*sin(4*d*x + 4*c) + 2*(a^3*b^2 - a^2*b^3)*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) +
16*((8*a^4*b - 11*a^3*b^2 + 3*a^2*b^3)*sin(4*d*x + 4*c) + 2*(a^3*b^2 - a^2*b^3)*sin(2*d*x + 2*c))*sin(6*d*x +
6*c)), x) - (a*b^2 - b^3 + (a*b^2 - b^3)*cos(8*...

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2711 vs. \(2 (244) = 488\).
time = 0.96, size = 2711, normalized size = 8.34 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(4*a*b*cos(d*x + c)^3 - 8*a*b*cos(d*x + c) + ((a^3*b - a^2*b^2)*d*cos(d*x + c)^4 - 2*(a^3*b - a^2*b^2)*d
*cos(d*x + c)^2 - (a^4 - 2*a^3*b + a^2*b^2)*d)*sqrt(-((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((625*a^4*
b - 1450*a^3*b^2 + 1241*a^2*b^3 - 464*a*b^4 + 64*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)) + 35*a^2*b - 47*a*b^2 + 16*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2))*l
og((625*a^3*b - 1125*a^2*b^2 + 664*a*b^3 - 128*b^4)*cos(d*x + c) + ((5*a^10 - 18*a^9*b + 24*a^8*b^2 - 14*a^7*b
^3 + 3*a^6*b^4)*d^3*sqrt((625*a^4*b - 1450*a^3*b^2 + 1241*a^2*b^3 - 464*a*b^4 + 64*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)) - 2*(75*a^5*b - 137*a^4*b^2 + 82*a^3*b^3 - 1
6*a^2*b^4)*d)*sqrt(-((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((625*a^4*b - 1450*a^3*b^2 + 1241*a^2*b^3 -
464*a*b^4 + 64*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)) +
35*a^2*b - 47*a*b^2 + 16*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2))) - ((a^3*b - a^2*b^2)*d*cos(d*x +
c)^4 - 2*(a^3*b - a^2*b^2)*d*cos(d*x + c)^2 - (a^4 - 2*a^3*b + a^2*b^2)*d)*sqrt(((a^7 - 3*a^6*b + 3*a^5*b^2 -
a^4*b^3)*d^2*sqrt((625*a^4*b - 1450*a^3*b^2 + 1241*a^2*b^3 - 464*a*b^4 + 64*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)) - 35*a^2*b + 47*a*b^2 - 16*b^3)/((a^7 - 3*a^6*b + 3
*a^5*b^2 - a^4*b^3)*d^2))*log((625*a^3*b - 1125*a^2*b^2 + 664*a*b^3 - 128*b^4)*cos(d*x + c) + ((5*a^10 - 18*a^
9*b + 24*a^8*b^2 - 14*a^7*b^3 + 3*a^6*b^4)*d^3*sqrt((625*a^4*b - 1450*a^3*b^2 + 1241*a^2*b^3 - 464*a*b^4 + 64*
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)) + 2*(75*a^5*b - 1
37*a^4*b^2 + 82*a^3*b^3 - 16*a^2*b^4)*d)*sqrt(((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((625*a^4*b - 145
0*a^3*b^2 + 1241*a^2*b^3 - 464*a*b^4 + 64*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)) - 35*a^2*b + 47*a*b^2 - 16*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2))) - ((a^3
*b - a^2*b^2)*d*cos(d*x + c)^4 - 2*(a^3*b - a^2*b^2)*d*cos(d*x + c)^2 - (a^4 - 2*a^3*b + a^2*b^2)*d)*sqrt(-((a
^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((625*a^4*b - 1450*a^3*b^2 + 1241*a^2*b^3 - 464*a*b^4 + 64*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)) + 35*a^2*b - 47*a*b^2 +
16*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2))*log(-(625*a^3*b - 1125*a^2*b^2 + 664*a*b^3 - 128*b^4)*cos
(d*x + c) + ((5*a^10 - 18*a^9*b + 24*a^8*b^2 - 14*a^7*b^3 + 3*a^6*b^4)*d^3*sqrt((625*a^4*b - 1450*a^3*b^2 + 12
41*a^2*b^3 - 464*a*b^4 + 64*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)) - 2*(75*a^5*b - 137*a^4*b^2 + 82*a^3*b^3 - 16*a^2*b^4)*d)*sqrt(-((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b
^3)*d^2*sqrt((625*a^4*b - 1450*a^3*b^2 + 1241*a^2*b^3 - 464*a*b^4 + 64*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)) + 35*a^2*b - 47*a*b^2 + 16*b^3)/((a^7 - 3*a^6*b + 3*a^5*
b^2 - a^4*b^3)*d^2))) + ((a^3*b - a^2*b^2)*d*cos(d*x + c)^4 - 2*(a^3*b - a^2*b^2)*d*cos(d*x + c)^2 - (a^4 - 2*
a^3*b + a^2*b^2)*d)*sqrt(((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((625*a^4*b - 1450*a^3*b^2 + 1241*a^2*
b^3 - 464*a*b^4 + 64*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)) - 35*a^2*b + 47*a*b^2 - 16*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2))*log(-(625*a^3*b - 1125*a^2*b^
2 + 664*a*b^3 - 128*b^4)*cos(d*x + c) + ((5*a^10 - 18*a^9*b + 24*a^8*b^2 - 14*a^7*b^3 + 3*a^6*b^4)*d^3*sqrt((6
25*a^4*b - 1450*a^3*b^2 + 1241*a^2*b^3 - 464*a*b^4 + 64*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 1
5*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) + 2*(75*a^5*b - 137*a^4*b^2 + 82*a^3*b^3 - 16*a^2*b^4)*d)*sqrt(((a^7 -
3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((625*a^4*b - 1450*a^3*b^2 + 1241*a^2*b^3 - 464*a*b^4 + 64*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)) - 35*a^2*b + 47*a*b^2 - 16*b^
3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2))) + 8*((a*b - b^2)*cos(d*x + c)^4 - 2*(a*b - b^2)*cos(d*x + c)^
2 - a^2 + 2*a*b - b^2)*log(1/2*cos(d*x + c) + 1/2) - 8*((a*b - b^2)*cos(d*x + c)^4 - 2*(a*b - b^2)*cos(d*x + c
)^2 - a^2 + 2*a*b - b^2)*log(-1/2*cos(d*x + c) + 1/2))/((a^3*b - a^2*b^2)*d*cos(d*x + c)^4 - 2*(a^3*b - a^2*b^
2)*d*cos(d*x + c)^2 - (a^4 - 2*a^3*b + a^2*b^2)*d)

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Warning, need to choose a branch for the
root of a polynomial with parameters. This might be wrong.The choice was done assuming [sageVARa,sageVARb]=[-
89,-82]Warning, need

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Mupad [B]
time = 17.55, size = 2500, normalized size = 7.69 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b*cos(c + d*x)^3)/(4*a*(a - b)) - (b*cos(c + d*x))/(2*a*(a - b)))/(d*(a - b + 2*b*cos(c + d*x)^2 - b*cos(c +
d*x)^4)) - (atan(((((3072*a^3*b^7 - 10944*a^4*b^6 + 9776*a^5*b^5)/(256*(a^7 - 2*a^6*b + a^5*b^2)) - (((49152*
a^7*b^7 - 155648*a^8*b^6 + 172032*a^9*b^5 - 65536*a^10*b^4)/(256*(a^7 - 2*a^6*b + a^5*b^2)) - (cos(c + d*x)*((
25*a^2*(a^9*b)^(1/2) + 8*b^2*(a^9*b)^(1/2) + 35*a^6*b + 16*a^4*b^3 - 47*a^5*b^2 - 29*a*b*(a^9*b)^(1/2))/(256*(
3*a^10*b - a^11 + a^8*b^3 - 3*a^9*b^2)))^(1/2)*(98304*a^8*b^7 - 262144*a^9*b^6 + 229376*a^10*b^5 - 65536*a^11*
b^4))/(128*(a^6 - 2*a^5*b + a^4*b^2)))*((25*a^2*(a^9*b)^(1/2) + 8*b^2*(a^9*b)^(1/2) + 35*a^6*b + 16*a^4*b^3 -
47*a^5*b^2 - 29*a*b*(a^9*b)^(1/2))/(256*(3*a^10*b - a^11 + a^8*b^3 - 3*a^9*b^2)))^(1/2) + (cos(c + d*x)*(18432
*a^4*b^7 - 45440*a^5*b^6 + 29312*a^6*b^5))/(128*(a^6 - 2*a^5*b + a^4*b^2)))*((25*a^2*(a^9*b)^(1/2) + 8*b^2*(a^
9*b)^(1/2) + 35*a^6*b + 16*a^4*b^3 - 47*a^5*b^2 - 29*a*b*(a^9*b)^(1/2))/(256*(3*a^10*b - a^11 + a^8*b^3 - 3*a^
9*b^2)))^(1/2))*((25*a^2*(a^9*b)^(1/2) + 8*b^2*(a^9*b)^(1/2) + 35*a^6*b + 16*a^4*b^3 - 47*a^5*b^2 - 29*a*b*(a^
9*b)^(1/2))/(256*(3*a^10*b - a^11 + a^8*b^3 - 3*a^9*b^2)))^(1/2) + (cos(c + d*x)*(768*b^7 - 2048*a*b^6 + 1425*
a^2*b^5))/(128*(a^6 - 2*a^5*b + a^4*b^2)))*((25*a^2*(a^9*b)^(1/2) + 8*b^2*(a^9*b)^(1/2) + 35*a^6*b + 16*a^4*b^
3 - 47*a^5*b^2 - 29*a*b*(a^9*b)^(1/2))/(256*(3*a^10*b - a^11 + a^8*b^3 - 3*a^9*b^2)))^(1/2)*1i - (((3072*a^3*b
^7 - 10944*a^4*b^6 + 9776*a^5*b^5)/(256*(a^7 - 2*a^6*b + a^5*b^2)) - (((49152*a^7*b^7 - 155648*a^8*b^6 + 17203
2*a^9*b^5 - 65536*a^10*b^4)/(256*(a^7 - 2*a^6*b + a^5*b^2)) + (cos(c + d*x)*((25*a^2*(a^9*b)^(1/2) + 8*b^2*(a^
9*b)^(1/2) + 35*a^6*b + 16*a^4*b^3 - 47*a^5*b^2 - 29*a*b*(a^9*b)^(1/2))/(256*(3*a^10*b - a^11 + a^8*b^3 - 3*a^
9*b^2)))^(1/2)*(98304*a^8*b^7 - 262144*a^9*b^6 + 229376*a^10*b^5 - 65536*a^11*b^4))/(128*(a^6 - 2*a^5*b + a^4*
b^2)))*((25*a^2*(a^9*b)^(1/2) + 8*b^2*(a^9*b)^(1/2) + 35*a^6*b + 16*a^4*b^3 - 47*a^5*b^2 - 29*a*b*(a^9*b)^(1/2
))/(256*(3*a^10*b - a^11 + a^8*b^3 - 3*a^9*b^2)))^(1/2) - (cos(c + d*x)*(18432*a^4*b^7 - 45440*a^5*b^6 + 29312
*a^6*b^5))/(128*(a^6 - 2*a^5*b + a^4*b^2)))*((25*a^2*(a^9*b)^(1/2) + 8*b^2*(a^9*b)^(1/2) + 35*a^6*b + 16*a^4*b
^3 - 47*a^5*b^2 - 29*a*b*(a^9*b)^(1/2))/(256*(3*a^10*b - a^11 + a^8*b^3 - 3*a^9*b^2)))^(1/2))*((25*a^2*(a^9*b)
^(1/2) + 8*b^2*(a^9*b)^(1/2) + 35*a^6*b + 16*a^4*b^3 - 47*a^5*b^2 - 29*a*b*(a^9*b)^(1/2))/(256*(3*a^10*b - a^1
1 + a^8*b^3 - 3*a^9*b^2)))^(1/2) - (cos(c + d*x)*(768*b^7 - 2048*a*b^6 + 1425*a^2*b^5))/(128*(a^6 - 2*a^5*b +
a^4*b^2)))*((25*a^2*(a^9*b)^(1/2) + 8*b^2*(a^9*b)^(1/2) + 35*a^6*b + 16*a^4*b^3 - 47*a^5*b^2 - 29*a*b*(a^9*b)^
(1/2))/(256*(3*a^10*b - a^11 + a^8*b^3 - 3*a^9*b^2)))^(1/2)*1i)/((((3072*a^3*b^7 - 10944*a^4*b^6 + 9776*a^5*b^
5)/(256*(a^7 - 2*a^6*b + a^5*b^2)) - (((49152*a^7*b^7 - 155648*a^8*b^6 + 172032*a^9*b^5 - 65536*a^10*b^4)/(256
*(a^7 - 2*a^6*b + a^5*b^2)) - (cos(c + d*x)*((25*a^2*(a^9*b)^(1/2) + 8*b^2*(a^9*b)^(1/2) + 35*a^6*b + 16*a^4*b
^3 - 47*a^5*b^2 - 29*a*b*(a^9*b)^(1/2))/(256*(3*a^10*b - a^11 + a^8*b^3 - 3*a^9*b^2)))^(1/2)*(98304*a^8*b^7 -
262144*a^9*b^6 + 229376*a^10*b^5 - 65536*a^11*b^4))/(128*(a^6 - 2*a^5*b + a^4*b^2)))*((25*a^2*(a^9*b)^(1/2) +
8*b^2*(a^9*b)^(1/2) + 35*a^6*b + 16*a^4*b^3 - 47*a^5*b^2 - 29*a*b*(a^9*b)^(1/2))/(256*(3*a^10*b - a^11 + a^8*b
^3 - 3*a^9*b^2)))^(1/2) + (cos(c + d*x)*(18432*a^4*b^7 - 45440*a^5*b^6 + 29312*a^6*b^5))/(128*(a^6 - 2*a^5*b +
a^4*b^2)))*((25*a^2*(a^9*b)^(1/2) + 8*b^2*(a^9*b)^(1/2) + 35*a^6*b + 16*a^4*b^3 - 47*a^5*b^2 - 29*a*b*(a^9*b)
^(1/2))/(256*(3*a^10*b - a^11 + a^8*b^3 - 3*a^9*b^2)))^(1/2))*((25*a^2*(a^9*b)^(1/2) + 8*b^2*(a^9*b)^(1/2) + 3
5*a^6*b + 16*a^4*b^3 - 47*a^5*b^2 - 29*a*b*(a^9*b)^(1/2))/(256*(3*a^10*b - a^11 + a^8*b^3 - 3*a^9*b^2)))^(1/2)
+ (cos(c + d*x)*(768*b^7 - 2048*a*b^6 + 1425*a^2*b^5))/(128*(a^6 - 2*a^5*b + a^4*b^2)))*((25*a^2*(a^9*b)^(1/2
) + 8*b^2*(a^9*b)^(1/2) + 35*a^6*b + 16*a^4*b^3 - 47*a^5*b^2 - 29*a*b*(a^9*b)^(1/2))/(256*(3*a^10*b - a^11 + a
^8*b^3 - 3*a^9*b^2)))^(1/2) - (125*a*b^5 - 80*b^6)/(128*(a^7 - 2*a^6*b + a^5*b^2)) + (((3072*a^3*b^7 - 10944*a
^4*b^6 + 9776*a^5*b^5)/(256*(a^7 - 2*a^6*b + a^5*b^2)) - (((49152*a^7*b^7 - 155648*a^8*b^6 + 172032*a^9*b^5 -
65536*a^10*b^4)/(256*(a^7 - 2*a^6*b + a^5*b^2)) + (cos(c + d*x)*((25*a^2*(a^9*b)^(1/2) + 8*b^2*(a^9*b)^(1/2) +
35*a^6*b + 16*a^4*b^3 - 47*a^5*b^2 - 29*a*b*(a^9*b)^(1/2))/(256*(3*a^10*b - a^11 + a^8*b^3 - 3*a^9*b^2)))^(1/
2)*(98304*a^8*b^7 - 262144*a^9*b^6 + 229376*a^10*b^5 - 65536*a^11*b^4))/(128*(a^6 - 2*a^5*b + a^4*b^2)))*((25*
a^2*(a^9*b)^(1/2) + 8*b^2*(a^9*b)^(1/2) + 35*a^6*b + 16*a^4*b^3 - 47*a^5*b^2 - 29*a*b*(a^9*b)^(1/2))/(256*(3*a
^10*b - a^11 + a^8*b^3 - 3*a^9*b^2)))^(1/2) - (cos(c + d*x)*(18432*a^4*b^7 - 45440*a^5*b^6 + 29312*a^6*b^5))/(
128*(a^6 - 2*a^5*b + a^4*b^2)))*((25*a^2*(a^9*b)^(1/2) + 8*b^2*(a^9*b)^(1/2) + 35*a^6*b + 16*a^4*b^3 - 47*a^5*
b^2 - 29*a*b*(a^9*b)^(1/2))/(256*(3*a^10*b - a^11 + a^8*b^3 - 3*a^9*b^2)))^(1/2))*((25*a^2*(a^9*b)^(1/2) + 8*b
^2*(a^9*b)^(1/2) + 35*a^6*b + 16*a^4*b^3 - 47*a...

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