3.3.1 \(\int \frac {\csc ^3(c+d x)}{a-b \sin ^4(c+d x)} \, dx\) [201]

Optimal. Leaf size=184 \[ -\frac {b^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^{3/2} \sqrt {\sqrt {a}-\sqrt {b}} d}-\frac {\tanh ^{-1}(\cos (c+d x))}{2 a d}-\frac {b^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^{3/2} \sqrt {\sqrt {a}+\sqrt {b}} d}-\frac {1}{4 a d (1-\cos (c+d x))}+\frac {1}{4 a d (1+\cos (c+d x))} \]

[Out]

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

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Rubi [A]
time = 0.14, antiderivative size = 184, 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 = {3294, 1184, 213, 1107, 211, 214} \begin {gather*} -\frac {b^{3/4} \text {ArcTan}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^{3/2} d \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {b^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^{3/2} d \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {1}{4 a d (1-\cos (c+d x))}+\frac {1}{4 a d (\cos (c+d x)+1)}-\frac {\tanh ^{-1}(\cos (c+d x))}{2 a d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^3/(a - b*Sin[c + d*x]^4),x]

[Out]

-1/2*(b^(3/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(a^(3/2)*Sqrt[Sqrt[a] - Sqrt[b]]*d) - Ar
cTanh[Cos[c + d*x]]/(2*a*d) - (b^(3/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*a^(3/2)*Sqr
t[Sqrt[a] + Sqrt[b]]*d) - 1/(4*a*d*(1 - Cos[c + d*x])) + 1/(4*a*d*(1 + Cos[c + d*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 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 1107

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

Rule 1184

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(d + e*x
^2)^q/(a + b*x^2 + c*x^4), 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] && IntegerQ[q]

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

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.24, size = 242, normalized size = 1.32 \begin {gather*} \frac {-\csc ^2\left (\frac {1}{2} (c+d x)\right )-4 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+4 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+4 i b \text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {-2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}+i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}+2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^3-i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3}{-b-8 a \text {$\#$1}^2+3 b \text {$\#$1}^2-3 b \text {$\#$1}^4+b \text {$\#$1}^6}\&\right ]+\sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 a d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]^3/(a - b*Sin[c + d*x]^4),x]

[Out]

(-Csc[(c + d*x)/2]^2 - 4*Log[Cos[(c + d*x)/2]] + 4*Log[Sin[(c + d*x)/2]] + (4*I)*b*RootSum[b - 4*b*#1^2 - 16*a
*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 + I*Log[1 - 2*Cos[c +
d*x]*#1 + #1^2]*#1 + 2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1
^3)/(-b - 8*a*#1^2 + 3*b*#1^2 - 3*b*#1^4 + b*#1^6) & ] + Sec[(c + d*x)/2]^2)/(8*a*d)

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Maple [A]
time = 0.64, size = 152, normalized size = 0.83

method	result	size
derivativedivides	\(\frac {\frac {1}{4 a \left (\cos \left (d x +c \right )-1\right )}+\frac {\ln \left (\cos \left (d x +c \right )-1\right )}{4 a}+\frac {1}{4 a \left (1+\cos \left (d x +c \right )\right )}-\frac {\ln \left (1+\cos \left (d x +c \right )\right )}{4 a}+\frac {b^{2} \left (-\frac {\arctanh \left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+b \right ) b}}-\frac {\arctan \left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{a}}{d}\)	\(152\)
default	\(\frac {\frac {1}{4 a \left (\cos \left (d x +c \right )-1\right )}+\frac {\ln \left (\cos \left (d x +c \right )-1\right )}{4 a}+\frac {1}{4 a \left (1+\cos \left (d x +c \right )\right )}-\frac {\ln \left (1+\cos \left (d x +c \right )\right )}{4 a}+\frac {b^{2} \left (-\frac {\arctanh \left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+b \right ) b}}-\frac {\arctan \left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{a}}{d}\)	\(152\)
risch	\(\frac {{\mathrm e}^{3 i \left (d x +c \right )}+{\mathrm e}^{i \left (d x +c \right )}}{d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 a d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 a d}-8 i \left (\munderset {\textit {\_R} =\RootOf \left (\left (1048576 a^{7} d^{4}-1048576 a^{6} b \,d^{4}\right ) \textit {\_Z}^{4}-2048 a^{3} b^{2} d^{2} \textit {\_Z}^{2}-b^{3}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\left (-\frac {65536 i d^{3} a^{5}}{b^{2}}+\frac {65536 i d^{3} a^{4}}{b}\right ) \textit {\_R}^{3}+\left (\frac {64 i a^{2} d}{b}+64 i a d \right ) \textit {\_R} \right ) {\mathrm e}^{i \left (d x +c \right )}+1\right )\right )\)	\(202\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)^3/(a-b*sin(d*x+c)^4),x,method=_RETURNVERBOSE)

[Out]

1/d*(1/4/a/(cos(d*x+c)-1)+1/4/a*ln(cos(d*x+c)-1)+1/4/a/(1+cos(d*x+c))-1/4/a*ln(1+cos(d*x+c))+1/a*b^2*(-1/2/(a*
b)^(1/2)/(((a*b)^(1/2)+b)*b)^(1/2)*arctanh(b*cos(d*x+c)/(((a*b)^(1/2)+b)*b)^(1/2))-1/2/(a*b)^(1/2)/(((a*b)^(1/
2)-b)*b)^(1/2)*arctan(b*cos(d*x+c)/(((a*b)^(1/2)-b)*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(csc(d*x+c)^3/(a-b*sin(d*x+c)^4),x, algorithm="maxima")

[Out]

1/4*(4*(cos(3*d*x + 3*c) + cos(d*x + c))*cos(4*d*x + 4*c) - 4*(2*cos(2*d*x + 2*c) - 1)*cos(3*d*x + 3*c) - 8*co
s(2*d*x + 2*c)*cos(d*x + c) + 4*(a*d*cos(4*d*x + 4*c)^2 + 4*a*d*cos(2*d*x + 2*c)^2 + a*d*sin(4*d*x + 4*c)^2 -
4*a*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 4*a*d*sin(2*d*x + 2*c)^2 - 4*a*d*cos(2*d*x + 2*c) + a*d - 2*(2*a*d*c
os(2*d*x + 2*c) - a*d)*cos(4*d*x + 4*c))*integrate(8*(4*b^2*cos(3*d*x + 3*c)*sin(2*d*x + 2*c) + 2*(8*a*b - 3*b
^2)*cos(3*d*x + 3*c)*sin(4*d*x + 4*c) - 2*(8*a*b - 3*b^2)*cos(4*d*x + 4*c)*sin(3*d*x + 3*c) - (b^2*sin(5*d*x +
5*c) - b^2*sin(3*d*x + 3*c))*cos(8*d*x + 8*c) + 4*(b^2*sin(5*d*x + 5*c) - b^2*sin(3*d*x + 3*c))*cos(6*d*x + 6
*c) - 2*(2*b^2*sin(2*d*x + 2*c) + (8*a*b - 3*b^2)*sin(4*d*x + 4*c))*cos(5*d*x + 5*c) + (b^2*cos(5*d*x + 5*c) -
b^2*cos(3*d*x + 3*c))*sin(8*d*x + 8*c) - 4*(b^2*cos(5*d*x + 5*c) - b^2*cos(3*d*x + 3*c))*sin(6*d*x + 6*c) + (
4*b^2*cos(2*d*x + 2*c) - b^2 + 2*(8*a*b - 3*b^2)*cos(4*d*x + 4*c))*sin(5*d*x + 5*c) - (4*b^2*cos(2*d*x + 2*c)
- b^2)*sin(3*d*x + 3*c))/(a*b^2*cos(8*d*x + 8*c)^2 + 16*a*b^2*cos(6*d*x + 6*c)^2 + 16*a*b^2*cos(2*d*x + 2*c)^2
+ a*b^2*sin(8*d*x + 8*c)^2 + 16*a*b^2*sin(6*d*x + 6*c)^2 + 16*a*b^2*sin(2*d*x + 2*c)^2 - 8*a*b^2*cos(2*d*x +
2*c) + a*b^2 + 4*(64*a^3 - 48*a^2*b + 9*a*b^2)*cos(4*d*x + 4*c)^2 + 4*(64*a^3 - 48*a^2*b + 9*a*b^2)*sin(4*d*x
+ 4*c)^2 + 16*(8*a^2*b - 3*a*b^2)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) - 2*(4*a*b^2*cos(6*d*x + 6*c) + 4*a*b^2*co
s(2*d*x + 2*c) - a*b^2 + 2*(8*a^2*b - 3*a*b^2)*cos(4*d*x + 4*c))*cos(8*d*x + 8*c) + 8*(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(6*d*x + 6*c) - 4*(8*a^2*b - 3*a*b^2 - 4*(8*a^2*b - 3*a
*b^2)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - 4*(2*a*b^2*sin(6*d*x + 6*c) + 2*a*b^2*sin(2*d*x + 2*c) + (8*a^2*b -
3*a*b^2)*sin(4*d*x + 4*c))*sin(8*d*x + 8*c) + 16*(2*a*b^2*sin(2*d*x + 2*c) + (8*a^2*b - 3*a*b^2)*sin(4*d*x +
4*c))*sin(6*d*x + 6*c)), x) + (2*(2*cos(2*d*x + 2*c) - 1)*cos(4*d*x + 4*c) - cos(4*d*x + 4*c)^2 - 4*cos(2*d*x
+ 2*c)^2 - sin(4*d*x + 4*c)^2 + 4*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) - 4*sin(2*d*x + 2*c)^2 + 4*cos(2*d*x + 2*c
) - 1)*log(cos(d*x)^2 + 2*cos(d*x)*cos(c) + cos(c)^2 + sin(d*x)^2 - 2*sin(d*x)*sin(c) + sin(c)^2) - (2*(2*cos(
2*d*x + 2*c) - 1)*cos(4*d*x + 4*c) - cos(4*d*x + 4*c)^2 - 4*cos(2*d*x + 2*c)^2 - sin(4*d*x + 4*c)^2 + 4*sin(4*
d*x + 4*c)*sin(2*d*x + 2*c) - 4*sin(2*d*x + 2*c)^2 + 4*cos(2*d*x + 2*c) - 1)*log(cos(d*x)^2 - 2*cos(d*x)*cos(c
) + cos(c)^2 + sin(d*x)^2 + 2*sin(d*x)*sin(c) + sin(c)^2) + 4*(sin(3*d*x + 3*c) + sin(d*x + c))*sin(4*d*x + 4*
c) - 8*sin(3*d*x + 3*c)*sin(2*d*x + 2*c) - 8*sin(2*d*x + 2*c)*sin(d*x + c) + 4*cos(d*x + c))/(a*d*cos(4*d*x +
4*c)^2 + 4*a*d*cos(2*d*x + 2*c)^2 + a*d*sin(4*d*x + 4*c)^2 - 4*a*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 4*a*d*s
in(2*d*x + 2*c)^2 - 4*a*d*cos(2*d*x + 2*c) + a*d - 2*(2*a*d*cos(2*d*x + 2*c) - a*d)*cos(4*d*x + 4*c))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 924 vs. \(2 (136) = 272\).
time = 0.55, size = 924, normalized size = 5.02 \begin {gather*} -\frac {{\left (a d \cos \left (d x + c\right )^{2} - a d\right )} \sqrt {-\frac {{\left (a^{4} - a^{3} b\right )} d^{2} \sqrt {\frac {b^{3}}{{\left (a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right )} d^{4}}} + b^{2}}{{\left (a^{4} - a^{3} b\right )} d^{2}}} \log \left (b^{2} \cos \left (d x + c\right ) - {\left ({\left (a^{5} - a^{4} b\right )} d^{3} \sqrt {\frac {b^{3}}{{\left (a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right )} d^{4}}} - a^{2} b d\right )} \sqrt {-\frac {{\left (a^{4} - a^{3} b\right )} d^{2} \sqrt {\frac {b^{3}}{{\left (a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right )} d^{4}}} + b^{2}}{{\left (a^{4} - a^{3} b\right )} d^{2}}}\right ) - {\left (a d \cos \left (d x + c\right )^{2} - a d\right )} \sqrt {\frac {{\left (a^{4} - a^{3} b\right )} d^{2} \sqrt {\frac {b^{3}}{{\left (a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right )} d^{4}}} - b^{2}}{{\left (a^{4} - a^{3} b\right )} d^{2}}} \log \left (b^{2} \cos \left (d x + c\right ) - {\left ({\left (a^{5} - a^{4} b\right )} d^{3} \sqrt {\frac {b^{3}}{{\left (a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right )} d^{4}}} + a^{2} b d\right )} \sqrt {\frac {{\left (a^{4} - a^{3} b\right )} d^{2} \sqrt {\frac {b^{3}}{{\left (a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right )} d^{4}}} - b^{2}}{{\left (a^{4} - a^{3} b\right )} d^{2}}}\right ) - {\left (a d \cos \left (d x + c\right )^{2} - a d\right )} \sqrt {-\frac {{\left (a^{4} - a^{3} b\right )} d^{2} \sqrt {\frac {b^{3}}{{\left (a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right )} d^{4}}} + b^{2}}{{\left (a^{4} - a^{3} b\right )} d^{2}}} \log \left (-b^{2} \cos \left (d x + c\right ) - {\left ({\left (a^{5} - a^{4} b\right )} d^{3} \sqrt {\frac {b^{3}}{{\left (a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right )} d^{4}}} - a^{2} b d\right )} \sqrt {-\frac {{\left (a^{4} - a^{3} b\right )} d^{2} \sqrt {\frac {b^{3}}{{\left (a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right )} d^{4}}} + b^{2}}{{\left (a^{4} - a^{3} b\right )} d^{2}}}\right ) + {\left (a d \cos \left (d x + c\right )^{2} - a d\right )} \sqrt {\frac {{\left (a^{4} - a^{3} b\right )} d^{2} \sqrt {\frac {b^{3}}{{\left (a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right )} d^{4}}} - b^{2}}{{\left (a^{4} - a^{3} b\right )} d^{2}}} \log \left (-b^{2} \cos \left (d x + c\right ) - {\left ({\left (a^{5} - a^{4} b\right )} d^{3} \sqrt {\frac {b^{3}}{{\left (a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right )} d^{4}}} + a^{2} b d\right )} \sqrt {\frac {{\left (a^{4} - a^{3} b\right )} d^{2} \sqrt {\frac {b^{3}}{{\left (a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right )} d^{4}}} - b^{2}}{{\left (a^{4} - a^{3} b\right )} d^{2}}}\right ) + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 2 \, \cos \left (d x + c\right )}{4 \, {\left (a d \cos \left (d x + c\right )^{2} - a d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^3/(a-b*sin(d*x+c)^4),x, algorithm="fricas")

[Out]

-1/4*((a*d*cos(d*x + c)^2 - a*d)*sqrt(-((a^4 - a^3*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) + b^2)/((a
^4 - a^3*b)*d^2))*log(b^2*cos(d*x + c) - ((a^5 - a^4*b)*d^3*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) - a^2*b*
d)*sqrt(-((a^4 - a^3*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) + b^2)/((a^4 - a^3*b)*d^2))) - (a*d*cos(
d*x + c)^2 - a*d)*sqrt(((a^4 - a^3*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) - b^2)/((a^4 - a^3*b)*d^2)
)*log(b^2*cos(d*x + c) - ((a^5 - a^4*b)*d^3*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) + a^2*b*d)*sqrt(((a^4 -
a^3*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) - b^2)/((a^4 - a^3*b)*d^2))) - (a*d*cos(d*x + c)^2 - a*d)
*sqrt(-((a^4 - a^3*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) + b^2)/((a^4 - a^3*b)*d^2))*log(-b^2*cos(d
*x + c) - ((a^5 - a^4*b)*d^3*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) - a^2*b*d)*sqrt(-((a^4 - a^3*b)*d^2*sqr
t(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) + b^2)/((a^4 - a^3*b)*d^2))) + (a*d*cos(d*x + c)^2 - a*d)*sqrt(((a^4 -
a^3*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) - b^2)/((a^4 - a^3*b)*d^2))*log(-b^2*cos(d*x + c) - ((a^5
- a^4*b)*d^3*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) + a^2*b*d)*sqrt(((a^4 - a^3*b)*d^2*sqrt(b^3/((a^7 - 2*
a^6*b + a^5*b^2)*d^4)) - b^2)/((a^4 - a^3*b)*d^2))) + (cos(d*x + c)^2 - 1)*log(1/2*cos(d*x + c) + 1/2) - (cos(
d*x + c)^2 - 1)*log(-1/2*cos(d*x + c) + 1/2) - 2*cos(d*x + c))/(a*d*cos(d*x + c)^2 - a*d)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\csc ^{3}{\left (c + d x \right )}}{a - b \sin ^{4}{\left (c + d x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)**3/(a-b*sin(d*x+c)**4),x)

[Out]

Integral(csc(c + d*x)**3/(a - b*sin(c + d*x)**4), x)

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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)^3/(a-b*sin(d*x+c)^4),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]=[-
35,-31]Warning, need

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Mupad [B]
time = 15.19, size = 2779, normalized size = 15.10 \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)^3*(a - b*sin(c + d*x)^4)),x)

[Out]

(atan(cos(c + d*x)*1i)*1i)/(d*(2*a - 2*a*cos(c + d*x)^2)) - cos(c + d*x)/(d*(2*a - 2*a*cos(c + d*x)^2)) - (ata
n(cos(c + d*x)*1i)*cos(c + d*x)^2*1i)/(d*(2*a - 2*a*cos(c + d*x)^2)) + (a*atan((a^13*cos(c + d*x)*(((a^7*b^3)^
(1/2) + a^3*b^2)/(16*a^6*b - 16*a^7))^(5/2)*2048i + a^10*b*cos(c + d*x)*(((a^7*b^3)^(1/2) + a^3*b^2)/(16*a^6*b
- 16*a^7))^(3/2)*64i - a^12*b*cos(c + d*x)*(((a^7*b^3)^(1/2) + a^3*b^2)/(16*a^6*b - 16*a^7))^(5/2)*7168i - a^
4*b^5*cos(c + d*x)*(((a^7*b^3)^(1/2) + a^3*b^2)/(16*a^6*b - 16*a^7))^(1/2)*8i + a^5*b^4*cos(c + d*x)*(((a^7*b^
3)^(1/2) + a^3*b^2)/(16*a^6*b - 16*a^7))^(1/2)*12i - a^7*b^2*cos(c + d*x)*(((a^7*b^3)^(1/2) + a^3*b^2)/(16*a^6
*b - 16*a^7))^(1/2)*4i + a^7*b^4*cos(c + d*x)*(((a^7*b^3)^(1/2) + a^3*b^2)/(16*a^6*b - 16*a^7))^(3/2)*320i - a
^8*b^3*cos(c + d*x)*(((a^7*b^3)^(1/2) + a^3*b^2)/(16*a^6*b - 16*a^7))^(3/2)*576i + a^9*b^2*cos(c + d*x)*(((a^7
*b^3)^(1/2) + a^3*b^2)/(16*a^6*b - 16*a^7))^(3/2)*192i - a^10*b^3*cos(c + d*x)*(((a^7*b^3)^(1/2) + a^3*b^2)/(1
6*a^6*b - 16*a^7))^(5/2)*3072i + a^11*b^2*cos(c + d*x)*(((a^7*b^3)^(1/2) + a^3*b^2)/(16*a^6*b - 16*a^7))^(5/2)
*8192i)/(2*b^3*(a^7*b^3)^(1/2) + a^3*b^5 + a^5*b^3 - a*b^2*(a^7*b^3)^(1/2) + a^2*b*(a^7*b^3)^(1/2)))*(((a^7*b^
3)^(1/2) + a^3*b^2)/(16*a^6*b - 16*a^7))^(1/2)*4i)/(d*(2*a - 2*a*cos(c + d*x)^2)) + (a*atan((a^13*cos(c + d*x)
*(-((a^7*b^3)^(1/2) - a^3*b^2)/(16*a^6*b - 16*a^7))^(5/2)*2048i + a^10*b*cos(c + d*x)*(-((a^7*b^3)^(1/2) - a^3
*b^2)/(16*a^6*b - 16*a^7))^(3/2)*64i - a^12*b*cos(c + d*x)*(-((a^7*b^3)^(1/2) - a^3*b^2)/(16*a^6*b - 16*a^7))^
(5/2)*7168i - a^4*b^5*cos(c + d*x)*(-((a^7*b^3)^(1/2) - a^3*b^2)/(16*a^6*b - 16*a^7))^(1/2)*8i + a^5*b^4*cos(c
+ d*x)*(-((a^7*b^3)^(1/2) - a^3*b^2)/(16*a^6*b - 16*a^7))^(1/2)*12i - a^7*b^2*cos(c + d*x)*(-((a^7*b^3)^(1/2)
- a^3*b^2)/(16*a^6*b - 16*a^7))^(1/2)*4i + a^7*b^4*cos(c + d*x)*(-((a^7*b^3)^(1/2) - a^3*b^2)/(16*a^6*b - 16*
a^7))^(3/2)*320i - a^8*b^3*cos(c + d*x)*(-((a^7*b^3)^(1/2) - a^3*b^2)/(16*a^6*b - 16*a^7))^(3/2)*576i + a^9*b^
2*cos(c + d*x)*(-((a^7*b^3)^(1/2) - a^3*b^2)/(16*a^6*b - 16*a^7))^(3/2)*192i - a^10*b^3*cos(c + d*x)*(-((a^7*b
^3)^(1/2) - a^3*b^2)/(16*a^6*b - 16*a^7))^(5/2)*3072i + a^11*b^2*cos(c + d*x)*(-((a^7*b^3)^(1/2) - a^3*b^2)/(1
6*a^6*b - 16*a^7))^(5/2)*8192i)/(a^3*b^5 - 2*b^3*(a^7*b^3)^(1/2) + a^5*b^3 + a*b^2*(a^7*b^3)^(1/2) - a^2*b*(a^
7*b^3)^(1/2)))*(-((a^7*b^3)^(1/2) - a^3*b^2)/(16*a^6*b - 16*a^7))^(1/2)*4i)/(d*(2*a - 2*a*cos(c + d*x)^2)) - (
a*cos(c + d*x)^2*atan((a^13*cos(c + d*x)*(((a^7*b^3)^(1/2) + a^3*b^2)/(16*a^6*b - 16*a^7))^(5/2)*2048i + a^10*
b*cos(c + d*x)*(((a^7*b^3)^(1/2) + a^3*b^2)/(16*a^6*b - 16*a^7))^(3/2)*64i - a^12*b*cos(c + d*x)*(((a^7*b^3)^(
1/2) + a^3*b^2)/(16*a^6*b - 16*a^7))^(5/2)*7168i - a^4*b^5*cos(c + d*x)*(((a^7*b^3)^(1/2) + a^3*b^2)/(16*a^6*b
- 16*a^7))^(1/2)*8i + a^5*b^4*cos(c + d*x)*(((a^7*b^3)^(1/2) + a^3*b^2)/(16*a^6*b - 16*a^7))^(1/2)*12i - a^7*
b^2*cos(c + d*x)*(((a^7*b^3)^(1/2) + a^3*b^2)/(16*a^6*b - 16*a^7))^(1/2)*4i + a^7*b^4*cos(c + d*x)*(((a^7*b^3)
^(1/2) + a^3*b^2)/(16*a^6*b - 16*a^7))^(3/2)*320i - a^8*b^3*cos(c + d*x)*(((a^7*b^3)^(1/2) + a^3*b^2)/(16*a^6*
b - 16*a^7))^(3/2)*576i + a^9*b^2*cos(c + d*x)*(((a^7*b^3)^(1/2) + a^3*b^2)/(16*a^6*b - 16*a^7))^(3/2)*192i -
a^10*b^3*cos(c + d*x)*(((a^7*b^3)^(1/2) + a^3*b^2)/(16*a^6*b - 16*a^7))^(5/2)*3072i + a^11*b^2*cos(c + d*x)*((
(a^7*b^3)^(1/2) + a^3*b^2)/(16*a^6*b - 16*a^7))^(5/2)*8192i)/(2*b^3*(a^7*b^3)^(1/2) + a^3*b^5 + a^5*b^3 - a*b^
2*(a^7*b^3)^(1/2) + a^2*b*(a^7*b^3)^(1/2)))*(((a^7*b^3)^(1/2) + a^3*b^2)/(16*a^6*b - 16*a^7))^(1/2)*4i)/(d*(2*
a - 2*a*cos(c + d*x)^2)) - (a*cos(c + d*x)^2*atan((a^13*cos(c + d*x)*(-((a^7*b^3)^(1/2) - a^3*b^2)/(16*a^6*b -
16*a^7))^(5/2)*2048i + a^10*b*cos(c + d*x)*(-((a^7*b^3)^(1/2) - a^3*b^2)/(16*a^6*b - 16*a^7))^(3/2)*64i - a^1
2*b*cos(c + d*x)*(-((a^7*b^3)^(1/2) - a^3*b^2)/(16*a^6*b - 16*a^7))^(5/2)*7168i - a^4*b^5*cos(c + d*x)*(-((a^7
*b^3)^(1/2) - a^3*b^2)/(16*a^6*b - 16*a^7))^(1/2)*8i + a^5*b^4*cos(c + d*x)*(-((a^7*b^3)^(1/2) - a^3*b^2)/(16*
a^6*b - 16*a^7))^(1/2)*12i - a^7*b^2*cos(c + d*x)*(-((a^7*b^3)^(1/2) - a^3*b^2)/(16*a^6*b - 16*a^7))^(1/2)*4i
+ a^7*b^4*cos(c + d*x)*(-((a^7*b^3)^(1/2) - a^3*b^2)/(16*a^6*b - 16*a^7))^(3/2)*320i - a^8*b^3*cos(c + d*x)*(-
((a^7*b^3)^(1/2) - a^3*b^2)/(16*a^6*b - 16*a^7))^(3/2)*576i + a^9*b^2*cos(c + d*x)*(-((a^7*b^3)^(1/2) - a^3*b^
2)/(16*a^6*b - 16*a^7))^(3/2)*192i - a^10*b^3*cos(c + d*x)*(-((a^7*b^3)^(1/2) - a^3*b^2)/(16*a^6*b - 16*a^7))^
(5/2)*3072i + a^11*b^2*cos(c + d*x)*(-((a^7*b^3)^(1/2) - a^3*b^2)/(16*a^6*b - 16*a^7))^(5/2)*8192i)/(a^3*b^5 -
2*b^3*(a^7*b^3)^(1/2) + a^5*b^3 + a*b^2*(a^7*b^3)^(1/2) - a^2*b*(a^7*b^3)^(1/2)))*(-((a^7*b^3)^(1/2) - a^3*b^
2)/(16*a^6*b - 16*a^7))^(1/2)*4i)/(d*(2*a - 2*a*cos(c + d*x)^2))

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