3.3.10 \(\int \frac {\csc ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx\) [210]

Optimal. Leaf size=178 \[ \frac {b^{3/2} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{9/4} \sqrt {\sqrt {a}-\sqrt {b}} d}-\frac {b^{3/2} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{9/4} \sqrt {\sqrt {a}+\sqrt {b}} d}-\frac {(a+b) \cot (c+d x)}{a^2 d}-\frac {2 \cot ^3(c+d x)}{3 a d}-\frac {\cot ^5(c+d x)}{5 a d} \]

[Out]

-(a+b)*cot(d*x+c)/a^2/d-2/3*cot(d*x+c)^3/a/d-1/5*cot(d*x+c)^5/a/d+1/2*b^(3/2)*arctan((a^(1/2)-b^(1/2))^(1/2)*t
an(d*x+c)/a^(1/4))/a^(9/4)/d/(a^(1/2)-b^(1/2))^(1/2)-1/2*b^(3/2)*arctan((a^(1/2)+b^(1/2))^(1/2)*tan(d*x+c)/a^(
1/4))/a^(9/4)/d/(a^(1/2)+b^(1/2))^(1/2)

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Rubi [A]
time = 0.14, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3296, 1301, 1144, 211} \begin {gather*} \frac {b^{3/2} \text {ArcTan}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{9/4} d \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {b^{3/2} \text {ArcTan}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{9/4} d \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {(a+b) \cot (c+d x)}{a^2 d}-\frac {\cot ^5(c+d x)}{5 a d}-\frac {2 \cot ^3(c+d x)}{3 a d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^6/(a - b*Sin[c + d*x]^4),x]

[Out]

(b^(3/2)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(9/4)*Sqrt[Sqrt[a] - Sqrt[b]]*d) - (b^(3
/2)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(9/4)*Sqrt[Sqrt[a] + Sqrt[b]]*d) - ((a + b)*C
ot[c + d*x])/(a^2*d) - (2*Cot[c + d*x]^3)/(3*a*d) - Cot[c + d*x]^5/(5*a*d)

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 1144

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

Rule 1301

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[Ex
pandIntegrand[(f*x)^m*((d + e*x^2)^q/(a + b*x^2 + c*x^4)), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^
2 - 4*a*c, 0] && IntegerQ[q] && 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 {\csc ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1+x^2\right )^4}{x^6 \left (a+2 a x^2+(a-b) x^4\right )} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {1}{a x^6}+\frac {2}{a x^4}+\frac {a+b}{a^2 x^2}+\frac {b^2 x^2}{a^2 \left (a+2 a x^2+(a-b) x^4\right )}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac {(a+b) \cot (c+d x)}{a^2 d}-\frac {2 \cot ^3(c+d x)}{3 a d}-\frac {\cot ^5(c+d x)}{5 a d}+\frac {b^2 \text {Subst}\left (\int \frac {x^2}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{a^2 d}\\ &=-\frac {(a+b) \cot (c+d x)}{a^2 d}-\frac {2 \cot ^3(c+d x)}{3 a d}-\frac {\cot ^5(c+d x)}{5 a d}+\frac {\left (\left (\sqrt {a}+\sqrt {b}\right ) b^{3/2}\right ) \text {Subst}\left (\int \frac {1}{a+\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{2 a^2 d}+\frac {\left (\left (1-\frac {\sqrt {a}}{\sqrt {b}}\right ) b^2\right ) \text {Subst}\left (\int \frac {1}{a-\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{2 a^2 d}\\ &=\frac {b^{3/2} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{9/4} \sqrt {\sqrt {a}-\sqrt {b}} d}-\frac {b^{3/2} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{9/4} \sqrt {\sqrt {a}+\sqrt {b}} d}-\frac {(a+b) \cot (c+d x)}{a^2 d}-\frac {2 \cot ^3(c+d x)}{3 a d}-\frac {\cot ^5(c+d x)}{5 a d}\\ \end {align*}

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Mathematica [A]
time = 3.14, size = 174, normalized size = 0.98 \begin {gather*} -\frac {\frac {15 b^{3/2} \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 {15 b^{3/2} \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}}}+2 \cot (c+d x) \left (8 a+15 b+4 a \csc ^2(c+d x)+3 a \csc ^4(c+d x)\right )}{30 a^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]^6/(a - b*Sin[c + d*x]^4),x]

[Out]

-1/30*((15*b^(3/2)*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/Sqrt[a + Sqrt[a]*Sqrt
[b]] + (15*b^(3/2)*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/Sqrt[-a + Sqrt[a]*S
qrt[b]] + 2*Cot[c + d*x]*(8*a + 15*b + 4*a*Csc[c + d*x]^2 + 3*a*Csc[c + d*x]^4))/(a^2*d)

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Maple [A]
time = 0.53, size = 195, normalized size = 1.10

method	result	size
derivativedivides	\(\frac {-\frac {1}{5 a \tan \left (d x +c \right )^{5}}-\frac {a +b}{a^{2} \tan \left (d x +c \right )}-\frac {2}{3 a \tan \left (d x +c \right )^{3}}+\frac {b^{2} \left (a -b \right ) \left (\frac {\left (\sqrt {a b}+a \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \left (a -b \right ) \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {\left (\sqrt {a b}-a \right ) \arctanh \left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{a^{2}}}{d}\)	\(195\)
default	\(\frac {-\frac {1}{5 a \tan \left (d x +c \right )^{5}}-\frac {a +b}{a^{2} \tan \left (d x +c \right )}-\frac {2}{3 a \tan \left (d x +c \right )^{3}}+\frac {b^{2} \left (a -b \right ) \left (\frac {\left (\sqrt {a b}+a \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \left (a -b \right ) \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {\left (\sqrt {a b}-a \right ) \arctanh \left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{a^{2}}}{d}\)	\(195\)
risch	\(-\frac {2 i \left (15 b \,{\mathrm e}^{8 i \left (d x +c \right )}-60 b \,{\mathrm e}^{6 i \left (d x +c \right )}+80 a \,{\mathrm e}^{4 i \left (d x +c \right )}+90 b \,{\mathrm e}^{4 i \left (d x +c \right )}-40 a \,{\mathrm e}^{2 i \left (d x +c \right )}-60 b \,{\mathrm e}^{2 i \left (d x +c \right )}+8 a +15 b \right )}{15 d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}-64 \left (\munderset {\textit {\_R} =\RootOf \left (\left (4294967296 a^{10} d^{4}-4294967296 a^{9} b \,d^{4}\right ) \textit {\_Z}^{4}+131072 a^{5} b^{3} d^{2} \textit {\_Z}^{2}+b^{6}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\frac {33554432 i a^{8} d^{3}}{b^{5}}-\frac {33554432 i d^{3} a^{7}}{b^{4}}\right ) \textit {\_R}^{3}+\left (-\frac {131072 d^{2} a^{6}}{b^{4}}+\frac {131072 d^{2} a^{5}}{b^{3}}\right ) \textit {\_R}^{2}+\frac {1024 i d \,a^{3} \textit {\_R}}{b^{2}}-\frac {2 a}{b}-1\right )\right )\)	\(236\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)^6/(a-b*sin(d*x+c)^4),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^6/(a-b*sin(d*x+c)^4),x, algorithm="maxima")

[Out]

1/15*(300*b*cos(4*d*x + 4*c)*sin(2*d*x + 2*c) + 10*(3*b*sin(8*d*x + 8*c) - 12*b*sin(6*d*x + 6*c) + 2*(8*a + 9*
b)*sin(4*d*x + 4*c) - 4*(2*a + 3*b)*sin(2*d*x + 2*c))*cos(10*d*x + 10*c) + 50*(6*b*sin(6*d*x + 6*c) - 4*(4*a +
3*b)*sin(4*d*x + 4*c) + (8*a + 9*b)*sin(2*d*x + 2*c))*cos(8*d*x + 8*c) + 200*((8*a + 3*b)*sin(4*d*x + 4*c) -
(4*a + 3*b)*sin(2*d*x + 2*c))*cos(6*d*x + 6*c) + 15*(a^2*d*cos(10*d*x + 10*c)^2 + 25*a^2*d*cos(8*d*x + 8*c)^2
+ 100*a^2*d*cos(6*d*x + 6*c)^2 + 100*a^2*d*cos(4*d*x + 4*c)^2 + 25*a^2*d*cos(2*d*x + 2*c)^2 + a^2*d*sin(10*d*x
+ 10*c)^2 + 25*a^2*d*sin(8*d*x + 8*c)^2 + 100*a^2*d*sin(6*d*x + 6*c)^2 + 100*a^2*d*sin(4*d*x + 4*c)^2 - 100*a
^2*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 25*a^2*d*sin(2*d*x + 2*c)^2 - 10*a^2*d*cos(2*d*x + 2*c) + a^2*d - 2*(
5*a^2*d*cos(8*d*x + 8*c) - 10*a^2*d*cos(6*d*x + 6*c) + 10*a^2*d*cos(4*d*x + 4*c) - 5*a^2*d*cos(2*d*x + 2*c) +
a^2*d)*cos(10*d*x + 10*c) - 10*(10*a^2*d*cos(6*d*x + 6*c) - 10*a^2*d*cos(4*d*x + 4*c) + 5*a^2*d*cos(2*d*x + 2*
c) - a^2*d)*cos(8*d*x + 8*c) - 20*(10*a^2*d*cos(4*d*x + 4*c) - 5*a^2*d*cos(2*d*x + 2*c) + a^2*d)*cos(6*d*x + 6
*c) - 20*(5*a^2*d*cos(2*d*x + 2*c) - a^2*d)*cos(4*d*x + 4*c) - 10*(a^2*d*sin(8*d*x + 8*c) - 2*a^2*d*sin(6*d*x
+ 6*c) + 2*a^2*d*sin(4*d*x + 4*c) - a^2*d*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) - 50*(2*a^2*d*sin(6*d*x + 6*c)
- 2*a^2*d*sin(4*d*x + 4*c) + a^2*d*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) - 100*(2*a^2*d*sin(4*d*x + 4*c) - a^2*d*
sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*integrate(-4*(4*b^3*cos(6*d*x + 6*c)^2 + 4*b^3*cos(2*d*x + 2*c)^2 + 4*b^3*
sin(6*d*x + 6*c)^2 + 4*b^3*sin(2*d*x + 2*c)^2 - b^3*cos(2*d*x + 2*c) - 4*(8*a*b^2 - 3*b^3)*cos(4*d*x + 4*c)^2
- 4*(8*a*b^2 - 3*b^3)*sin(4*d*x + 4*c)^2 + 2*(8*a*b^2 - 7*b^3)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) - (b^3*cos(6*
d*x + 6*c) - 2*b^3*cos(4*d*x + 4*c) + b^3*cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + (8*b^3*cos(2*d*x + 2*c) - b^3 +
2*(8*a*b^2 - 7*b^3)*cos(4*d*x + 4*c))*cos(6*d*x + 6*c) + 2*(b^3 + (8*a*b^2 - 7*b^3)*cos(2*d*x + 2*c))*cos(4*d
*x + 4*c) - (b^3*sin(6*d*x + 6*c) - 2*b^3*sin(4*d*x + 4*c) + b^3*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 2*(4*b^3
*sin(2*d*x + 2*c) + (8*a*b^2 - 7*b^3)*sin(4*d*x + 4*c))*sin(6*d*x + 6*c))/(a^2*b^2*cos(8*d*x + 8*c)^2 + 16*a^2
*b^2*cos(6*d*x + 6*c)^2 + 16*a^2*b^2*cos(2*d*x + 2*c)^2 + a^2*b^2*sin(8*d*x + 8*c)^2 + 16*a^2*b^2*sin(6*d*x +
6*c)^2 + 16*a^2*b^2*sin(2*d*x + 2*c)^2 - 8*a^2*b^2*cos(2*d*x + 2*c) + a^2*b^2 + 4*(64*a^4 - 48*a^3*b + 9*a^2*b
^2)*cos(4*d*x + 4*c)^2 + 4*(64*a^4 - 48*a^3*b + 9*a^2*b^2)*sin(4*d*x + 4*c)^2 + 16*(8*a^3*b - 3*a^2*b^2)*sin(4
*d*x + 4*c)*sin(2*d*x + 2*c) - 2*(4*a^2*b^2*cos(6*d*x + 6*c) + 4*a^2*b^2*cos(2*d*x + 2*c) - a^2*b^2 + 2*(8*a^3
*b - 3*a^2*b^2)*cos(4*d*x + 4*c))*cos(8*d*x + 8*c) + 8*(4*a^2*b^2*cos(2*d*x + 2*c) - a^2*b^2 + 2*(8*a^3*b - 3*
a^2*b^2)*cos(4*d*x + 4*c))*cos(6*d*x + 6*c) - 4*(8*a^3*b - 3*a^2*b^2 - 4*(8*a^3*b - 3*a^2*b^2)*cos(2*d*x + 2*c
))*cos(4*d*x + 4*c) - 4*(2*a^2*b^2*sin(6*d*x + 6*c) + 2*a^2*b^2*sin(2*d*x + 2*c) + (8*a^3*b - 3*a^2*b^2)*sin(4
*d*x + 4*c))*sin(8*d*x + 8*c) + 16*(2*a^2*b^2*sin(2*d*x + 2*c) + (8*a^3*b - 3*a^2*b^2)*sin(4*d*x + 4*c))*sin(6
*d*x + 6*c)), x) - 2*(15*b*cos(8*d*x + 8*c) - 60*b*cos(6*d*x + 6*c) + 10*(8*a + 9*b)*cos(4*d*x + 4*c) - 20*(2*
a + 3*b)*cos(2*d*x + 2*c) + 8*a + 15*b)*sin(10*d*x + 10*c) - 10*(30*b*cos(6*d*x + 6*c) - 20*(4*a + 3*b)*cos(4*
d*x + 4*c) + 5*(8*a + 9*b)*cos(2*d*x + 2*c) - 8*a - 12*b)*sin(8*d*x + 8*c) - 20*(10*(8*a + 3*b)*cos(4*d*x + 4*
c) - 10*(4*a + 3*b)*cos(2*d*x + 2*c) + 8*a + 9*b)*sin(6*d*x + 6*c) - 60*(5*b*cos(2*d*x + 2*c) - 2*b)*sin(4*d*x
+ 4*c) - 30*b*sin(2*d*x + 2*c))/(a^2*d*cos(10*d*x + 10*c)^2 + 25*a^2*d*cos(8*d*x + 8*c)^2 + 100*a^2*d*cos(6*d
*x + 6*c)^2 + 100*a^2*d*cos(4*d*x + 4*c)^2 + 25*a^2*d*cos(2*d*x + 2*c)^2 + a^2*d*sin(10*d*x + 10*c)^2 + 25*a^2
*d*sin(8*d*x + 8*c)^2 + 100*a^2*d*sin(6*d*x + 6*c)^2 + 100*a^2*d*sin(4*d*x + 4*c)^2 - 100*a^2*d*sin(4*d*x + 4*
c)*sin(2*d*x + 2*c) + 25*a^2*d*sin(2*d*x + 2*c)^2 - 10*a^2*d*cos(2*d*x + 2*c) + a^2*d - 2*(5*a^2*d*cos(8*d*x +
8*c) - 10*a^2*d*cos(6*d*x + 6*c) + 10*a^2*d*cos(4*d*x + 4*c) - 5*a^2*d*cos(2*d*x + 2*c) + a^2*d)*cos(10*d*x +
10*c) - 10*(10*a^2*d*cos(6*d*x + 6*c) - 10*a^2*d*cos(4*d*x + 4*c) + 5*a^2*d*cos(2*d*x + 2*c) - a^2*d)*cos(8*d
*x + 8*c) - 20*(10*a^2*d*cos(4*d*x + 4*c) - 5*a^2*d*cos(2*d*x + 2*c) + a^2*d)*cos(6*d*x + 6*c) - 20*(5*a^2*d*c
os(2*d*x + 2*c) - a^2*d)*cos(4*d*x + 4*c) - 10*(a^2*d*sin(8*d*x + 8*c) - 2*a^2*d*sin(6*d*x + 6*c) + 2*a^2*d*si
n(4*d*x + 4*c) - a^2*d*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) - 50*(2*a^2*d*sin(6*d*x + 6*c) - 2*a^2*d*sin(4*d*x
+ 4*c) + a^2*d*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) - 100*(2*a^2*d*sin(4*d*x + 4*c) - a^2*d*sin(2*d*x + 2*c))*s
in(6*d*x + 6*c))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1477 vs. \(2 (134) = 268\).
time = 0.58, size = 1477, normalized size = 8.30 \begin {gather*} -\frac {8 \, {\left (8 \, a + 15 \, b\right )} \cos \left (d x + c\right )^{5} - 80 \, {\left (2 \, a + 3 \, b\right )} \cos \left (d x + c\right )^{3} - 15 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} - 2 \, a^{2} d \cos \left (d x + c\right )^{2} + a^{2} d\right )} \sqrt {-\frac {{\left (a^{5} - a^{4} b\right )} \sqrt {\frac {b^{7}}{{\left (a^{11} - 2 \, a^{10} b + a^{9} b^{2}\right )} d^{4}}} d^{2} + b^{3}}{{\left (a^{5} - a^{4} b\right )} d^{2}}} \log \left (\frac {1}{4} \, b^{5} \cos \left (d x + c\right )^{2} - \frac {1}{4} \, b^{5} - \frac {1}{4} \, {\left (2 \, {\left (a^{6} b - a^{5} b^{2}\right )} d^{2} \cos \left (d x + c\right )^{2} - {\left (a^{6} b - a^{5} b^{2}\right )} d^{2}\right )} \sqrt {\frac {b^{7}}{{\left (a^{11} - 2 \, a^{10} b + a^{9} b^{2}\right )} d^{4}}} + \frac {1}{2} \, {\left (a^{3} b^{3} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - {\left (a^{8} - a^{7} b\right )} \sqrt {\frac {b^{7}}{{\left (a^{11} - 2 \, a^{10} b + a^{9} b^{2}\right )} d^{4}}} d^{3} \cos \left (d x + c\right ) \sin \left (d x + c\right )\right )} \sqrt {-\frac {{\left (a^{5} - a^{4} b\right )} \sqrt {\frac {b^{7}}{{\left (a^{11} - 2 \, a^{10} b + a^{9} b^{2}\right )} d^{4}}} d^{2} + b^{3}}{{\left (a^{5} - a^{4} b\right )} d^{2}}}\right ) \sin \left (d x + c\right ) + 15 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} - 2 \, a^{2} d \cos \left (d x + c\right )^{2} + a^{2} d\right )} \sqrt {-\frac {{\left (a^{5} - a^{4} b\right )} \sqrt {\frac {b^{7}}{{\left (a^{11} - 2 \, a^{10} b + a^{9} b^{2}\right )} d^{4}}} d^{2} + b^{3}}{{\left (a^{5} - a^{4} b\right )} d^{2}}} \log \left (\frac {1}{4} \, b^{5} \cos \left (d x + c\right )^{2} - \frac {1}{4} \, b^{5} - \frac {1}{4} \, {\left (2 \, {\left (a^{6} b - a^{5} b^{2}\right )} d^{2} \cos \left (d x + c\right )^{2} - {\left (a^{6} b - a^{5} b^{2}\right )} d^{2}\right )} \sqrt {\frac {b^{7}}{{\left (a^{11} - 2 \, a^{10} b + a^{9} b^{2}\right )} d^{4}}} - \frac {1}{2} \, {\left (a^{3} b^{3} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - {\left (a^{8} - a^{7} b\right )} \sqrt {\frac {b^{7}}{{\left (a^{11} - 2 \, a^{10} b + a^{9} b^{2}\right )} d^{4}}} d^{3} \cos \left (d x + c\right ) \sin \left (d x + c\right )\right )} \sqrt {-\frac {{\left (a^{5} - a^{4} b\right )} \sqrt {\frac {b^{7}}{{\left (a^{11} - 2 \, a^{10} b + a^{9} b^{2}\right )} d^{4}}} d^{2} + b^{3}}{{\left (a^{5} - a^{4} b\right )} d^{2}}}\right ) \sin \left (d x + c\right ) + 15 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} - 2 \, a^{2} d \cos \left (d x + c\right )^{2} + a^{2} d\right )} \sqrt {\frac {{\left (a^{5} - a^{4} b\right )} \sqrt {\frac {b^{7}}{{\left (a^{11} - 2 \, a^{10} b + a^{9} b^{2}\right )} d^{4}}} d^{2} - b^{3}}{{\left (a^{5} - a^{4} b\right )} d^{2}}} \log \left (-\frac {1}{4} \, b^{5} \cos \left (d x + c\right )^{2} + \frac {1}{4} \, b^{5} - \frac {1}{4} \, {\left (2 \, {\left (a^{6} b - a^{5} b^{2}\right )} d^{2} \cos \left (d x + c\right )^{2} - {\left (a^{6} b - a^{5} b^{2}\right )} d^{2}\right )} \sqrt {\frac {b^{7}}{{\left (a^{11} - 2 \, a^{10} b + a^{9} b^{2}\right )} d^{4}}} + \frac {1}{2} \, {\left (a^{3} b^{3} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{8} - a^{7} b\right )} \sqrt {\frac {b^{7}}{{\left (a^{11} - 2 \, a^{10} b + a^{9} b^{2}\right )} d^{4}}} d^{3} \cos \left (d x + c\right ) \sin \left (d x + c\right )\right )} \sqrt {\frac {{\left (a^{5} - a^{4} b\right )} \sqrt {\frac {b^{7}}{{\left (a^{11} - 2 \, a^{10} b + a^{9} b^{2}\right )} d^{4}}} d^{2} - b^{3}}{{\left (a^{5} - a^{4} b\right )} d^{2}}}\right ) \sin \left (d x + c\right ) - 15 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} - 2 \, a^{2} d \cos \left (d x + c\right )^{2} + a^{2} d\right )} \sqrt {\frac {{\left (a^{5} - a^{4} b\right )} \sqrt {\frac {b^{7}}{{\left (a^{11} - 2 \, a^{10} b + a^{9} b^{2}\right )} d^{4}}} d^{2} - b^{3}}{{\left (a^{5} - a^{4} b\right )} d^{2}}} \log \left (-\frac {1}{4} \, b^{5} \cos \left (d x + c\right )^{2} + \frac {1}{4} \, b^{5} - \frac {1}{4} \, {\left (2 \, {\left (a^{6} b - a^{5} b^{2}\right )} d^{2} \cos \left (d x + c\right )^{2} - {\left (a^{6} b - a^{5} b^{2}\right )} d^{2}\right )} \sqrt {\frac {b^{7}}{{\left (a^{11} - 2 \, a^{10} b + a^{9} b^{2}\right )} d^{4}}} - \frac {1}{2} \, {\left (a^{3} b^{3} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{8} - a^{7} b\right )} \sqrt {\frac {b^{7}}{{\left (a^{11} - 2 \, a^{10} b + a^{9} b^{2}\right )} d^{4}}} d^{3} \cos \left (d x + c\right ) \sin \left (d x + c\right )\right )} \sqrt {\frac {{\left (a^{5} - a^{4} b\right )} \sqrt {\frac {b^{7}}{{\left (a^{11} - 2 \, a^{10} b + a^{9} b^{2}\right )} d^{4}}} d^{2} - b^{3}}{{\left (a^{5} - a^{4} b\right )} d^{2}}}\right ) \sin \left (d x + c\right ) + 120 \, {\left (a + b\right )} \cos \left (d x + c\right )}{120 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} - 2 \, a^{2} d \cos \left (d x + c\right )^{2} + a^{2} d\right )} \sin \left (d x + c\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^6/(a-b*sin(d*x+c)^4),x, algorithm="fricas")

[Out]

-1/120*(8*(8*a + 15*b)*cos(d*x + c)^5 - 80*(2*a + 3*b)*cos(d*x + c)^3 - 15*(a^2*d*cos(d*x + c)^4 - 2*a^2*d*cos
(d*x + c)^2 + a^2*d)*sqrt(-((a^5 - a^4*b)*sqrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4))*d^2 + b^3)/((a^5 - a^4*b
)*d^2))*log(1/4*b^5*cos(d*x + c)^2 - 1/4*b^5 - 1/4*(2*(a^6*b - a^5*b^2)*d^2*cos(d*x + c)^2 - (a^6*b - a^5*b^2)
*d^2)*sqrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4)) + 1/2*(a^3*b^3*d*cos(d*x + c)*sin(d*x + c) - (a^8 - a^7*b)*s
qrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4))*d^3*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^5 - a^4*b)*sqrt(b^7/((a^11
- 2*a^10*b + a^9*b^2)*d^4))*d^2 + b^3)/((a^5 - a^4*b)*d^2)))*sin(d*x + c) + 15*(a^2*d*cos(d*x + c)^4 - 2*a^2*
d*cos(d*x + c)^2 + a^2*d)*sqrt(-((a^5 - a^4*b)*sqrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4))*d^2 + b^3)/((a^5 -
a^4*b)*d^2))*log(1/4*b^5*cos(d*x + c)^2 - 1/4*b^5 - 1/4*(2*(a^6*b - a^5*b^2)*d^2*cos(d*x + c)^2 - (a^6*b - a^5
*b^2)*d^2)*sqrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4)) - 1/2*(a^3*b^3*d*cos(d*x + c)*sin(d*x + c) - (a^8 - a^7
*b)*sqrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4))*d^3*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^5 - a^4*b)*sqrt(b^7/(
(a^11 - 2*a^10*b + a^9*b^2)*d^4))*d^2 + b^3)/((a^5 - a^4*b)*d^2)))*sin(d*x + c) + 15*(a^2*d*cos(d*x + c)^4 - 2
*a^2*d*cos(d*x + c)^2 + a^2*d)*sqrt(((a^5 - a^4*b)*sqrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4))*d^2 - b^3)/((a^
5 - a^4*b)*d^2))*log(-1/4*b^5*cos(d*x + c)^2 + 1/4*b^5 - 1/4*(2*(a^6*b - a^5*b^2)*d^2*cos(d*x + c)^2 - (a^6*b
- a^5*b^2)*d^2)*sqrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4)) + 1/2*(a^3*b^3*d*cos(d*x + c)*sin(d*x + c) + (a^8
- a^7*b)*sqrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4))*d^3*cos(d*x + c)*sin(d*x + c))*sqrt(((a^5 - a^4*b)*sqrt(b
^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4))*d^2 - b^3)/((a^5 - a^4*b)*d^2)))*sin(d*x + c) - 15*(a^2*d*cos(d*x + c)^4
- 2*a^2*d*cos(d*x + c)^2 + a^2*d)*sqrt(((a^5 - a^4*b)*sqrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4))*d^2 - b^3)/
((a^5 - a^4*b)*d^2))*log(-1/4*b^5*cos(d*x + c)^2 + 1/4*b^5 - 1/4*(2*(a^6*b - a^5*b^2)*d^2*cos(d*x + c)^2 - (a^
6*b - a^5*b^2)*d^2)*sqrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4)) - 1/2*(a^3*b^3*d*cos(d*x + c)*sin(d*x + c) + (
a^8 - a^7*b)*sqrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4))*d^3*cos(d*x + c)*sin(d*x + c))*sqrt(((a^5 - a^4*b)*sq
rt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4))*d^2 - b^3)/((a^5 - a^4*b)*d^2)))*sin(d*x + c) + 120*(a + b)*cos(d*x
+ c))/((a^2*d*cos(d*x + c)^4 - 2*a^2*d*cos(d*x + c)^2 + a^2*d)*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(csc(d*x+c)**6/(a-b*sin(d*x+c)**4),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 471 vs. \(2 (134) = 268\).
time = 0.78, size = 471, normalized size = 2.65 \begin {gather*} -\frac {\frac {15 \, {\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 (\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} + \sqrt {a^{6} - {\left (a^{3} - a^{2} b\right )} a^{3}}}{a^{3} - a^{2} b}}}\right )\right )} {\left | a - b \right |}}{3 \, a^{7} - 12 \, a^{6} b + 14 \, a^{5} b^{2} - 4 \, a^{4} b^{3} - a^{3} b^{4}} + \frac {15 \, {\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 (\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} - \sqrt {a^{6} - {\left (a^{3} - a^{2} b\right )} a^{3}}}{a^{3} - a^{2} b}}}\right )\right )} {\left | a - b \right |}}{3 \, a^{7} - 12 \, a^{6} b + 14 \, a^{5} b^{2} - 4 \, a^{4} b^{3} - a^{3} b^{4}} + \frac {2 \, {\left (15 \, a \tan \left (d x + c\right )^{4} + 15 \, b \tan \left (d x + c\right )^{4} + 10 \, a \tan \left (d x + c\right )^{2} + 3 \, a\right )}}{a^{2} \tan \left (d x + c\right )^{5}}}{30 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^6/(a-b*sin(d*x+c)^4),x, algorithm="giac")

[Out]

-1/30*(15*(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)*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x
+ c)/sqrt((a^3 + sqrt(a^6 - (a^3 - a^2*b)*a^3))/(a^3 - a^2*b))))*abs(a - b)/(3*a^7 - 12*a^6*b + 14*a^5*b^2 -
4*a^4*b^3 - a^3*b^4) + 15*(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)*(pi*floor((d*x + c)/pi + 1/2)
+ arctan(tan(d*x + c)/sqrt((a^3 - sqrt(a^6 - (a^3 - a^2*b)*a^3))/(a^3 - a^2*b))))*abs(a - b)/(3*a^7 - 12*a^6*b
+ 14*a^5*b^2 - 4*a^4*b^3 - a^3*b^4) + 2*(15*a*tan(d*x + c)^4 + 15*b*tan(d*x + c)^4 + 10*a*tan(d*x + c)^2 + 3*
a)/(a^2*tan(d*x + c)^5))/d

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Mupad [B]
time = 15.26, size = 416, normalized size = 2.34 \begin {gather*} \frac {2\,\mathrm {atanh}\left (\frac {2\,\left (\mathrm {tan}\left (c+d\,x\right )\,\left (4\,a^7\,b^6-4\,a^9\,b^4\right )-\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (\sqrt {a^9\,b^7}+a^5\,b^3\right )\,\left (64\,a^{14}\,b-128\,a^{13}\,b^2+64\,a^{12}\,b^3\right )}{16\,\left (a^9\,b-a^{10}\right )}\right )\,\sqrt {\frac {\sqrt {a^9\,b^7}+a^5\,b^3}{16\,\left (a^9\,b-a^{10}\right )}}}{2\,a^5\,b^7-2\,a^6\,b^6}\right )\,\sqrt {\frac {\sqrt {a^9\,b^7}+a^5\,b^3}{16\,\left (a^9\,b-a^{10}\right )}}}{d}+\frac {2\,\mathrm {atanh}\left (\frac {2\,\left (\mathrm {tan}\left (c+d\,x\right )\,\left (4\,a^7\,b^6-4\,a^9\,b^4\right )+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (\sqrt {a^9\,b^7}-a^5\,b^3\right )\,\left (64\,a^{14}\,b-128\,a^{13}\,b^2+64\,a^{12}\,b^3\right )}{16\,\left (a^9\,b-a^{10}\right )}\right )\,\sqrt {-\frac {\sqrt {a^9\,b^7}-a^5\,b^3}{16\,\left (a^9\,b-a^{10}\right )}}}{2\,a^5\,b^7-2\,a^6\,b^6}\right )\,\sqrt {-\frac {\sqrt {a^9\,b^7}-a^5\,b^3}{16\,\left (a^9\,b-a^{10}\right )}}}{d}-\frac {\frac {1}{5\,a}+\frac {2\,{\mathrm {tan}\left (c+d\,x\right )}^2}{3\,a}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (a+b\right )}{a^2}}{d\,{\mathrm {tan}\left (c+d\,x\right )}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(c + d*x)^6*(a - b*sin(c + d*x)^4)),x)

[Out]

(2*atanh((2*(tan(c + d*x)*(4*a^7*b^6 - 4*a^9*b^4) - (tan(c + d*x)*((a^9*b^7)^(1/2) + a^5*b^3)*(64*a^14*b + 64*
a^12*b^3 - 128*a^13*b^2))/(16*(a^9*b - a^10)))*(((a^9*b^7)^(1/2) + a^5*b^3)/(16*(a^9*b - a^10)))^(1/2))/(2*a^5
*b^7 - 2*a^6*b^6))*(((a^9*b^7)^(1/2) + a^5*b^3)/(16*(a^9*b - a^10)))^(1/2))/d + (2*atanh((2*(tan(c + d*x)*(4*a
^7*b^6 - 4*a^9*b^4) + (tan(c + d*x)*((a^9*b^7)^(1/2) - a^5*b^3)*(64*a^14*b + 64*a^12*b^3 - 128*a^13*b^2))/(16*
(a^9*b - a^10)))*(-((a^9*b^7)^(1/2) - a^5*b^3)/(16*(a^9*b - a^10)))^(1/2))/(2*a^5*b^7 - 2*a^6*b^6))*(-((a^9*b^
7)^(1/2) - a^5*b^3)/(16*(a^9*b - a^10)))^(1/2))/d - (1/(5*a) + (2*tan(c + d*x)^2)/(3*a) + (tan(c + d*x)^4*(a +
b))/a^2)/(d*tan(c + d*x)^5)

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