\(\int \frac {\csc ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx\) [211]
Optimal result
Integrand size = 24, antiderivative size = 197 \[
\int \frac {\csc ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {b^2 \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{11/4} \sqrt {\sqrt {a}-\sqrt {b}} d}+\frac {b^2 \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{11/4} \sqrt {\sqrt {a}+\sqrt {b}} d}-\frac {(a+b) \cot (c+d x)}{a^2 d}-\frac {(3 a+b) \cot ^3(c+d x)}{3 a^2 d}-\frac {3 \cot ^5(c+d x)}{5 a d}-\frac {\cot ^7(c+d x)}{7 a d}
\]
[Out]
-(a+b)*cot(d*x+c)/a^2/d-1/3*(3*a+b)*cot(d*x+c)^3/a^2/d-3/5*cot(d*x+c)^5/a/d-1/7*cot(d*x+c)^7/a/d+1/2*b^2*arcta
n((a^(1/2)-b^(1/2))^(1/2)*tan(d*x+c)/a^(1/4))/a^(11/4)/d/(a^(1/2)-b^(1/2))^(1/2)+1/2*b^2*arctan((a^(1/2)+b^(1/
2))^(1/2)*tan(d*x+c)/a^(1/4))/a^(11/4)/d/(a^(1/2)+b^(1/2))^(1/2)
Rubi [A] (verified)
Time = 0.15 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.00, number of
steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3296, 1301, 1180, 211}
\[
\int \frac {\csc ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {b^2 \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{11/4} d \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {b^2 \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{11/4} d \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {(3 a+b) \cot ^3(c+d x)}{3 a^2 d}-\frac {(a+b) \cot (c+d x)}{a^2 d}-\frac {\cot ^7(c+d x)}{7 a d}-\frac {3 \cot ^5(c+d x)}{5 a d}
\]
[In]
Int[Csc[c + d*x]^8/(a - b*Sin[c + d*x]^4),x]
[Out]
(b^2*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(11/4)*Sqrt[Sqrt[a] - Sqrt[b]]*d) + (b^2*Arc
Tan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(11/4)*Sqrt[Sqrt[a] + Sqrt[b]]*d) - ((a + b)*Cot[c +
d*x])/(a^2*d) - ((3*a + b)*Cot[c + d*x]^3)/(3*a^2*d) - (3*Cot[c + d*x]^5)/(5*a*d) - Cot[c + d*x]^7/(7*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 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 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*}
\text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (1+x^2\right )^5}{x^8 \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^8}+\frac {3}{a x^6}+\frac {3 a+b}{a^2 x^4}+\frac {a+b}{a^2 x^2}+\frac {b^2 \left (1+x^2\right )}{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 {(3 a+b) \cot ^3(c+d x)}{3 a^2 d}-\frac {3 \cot ^5(c+d x)}{5 a d}-\frac {\cot ^7(c+d x)}{7 a d}+\frac {b^2 \text {Subst}\left (\int \frac {1+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 {(3 a+b) \cot ^3(c+d x)}{3 a^2 d}-\frac {3 \cot ^5(c+d x)}{5 a d}-\frac {\cot ^7(c+d x)}{7 a d}+\frac {\left (\left (\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^{5/2} d}+\frac {\left (\left (1-\frac {\sqrt {b}}{\sqrt {a}}\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^2 \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{11/4} \sqrt {\sqrt {a}-\sqrt {b}} d}+\frac {b^2 \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{11/4} \sqrt {\sqrt {a}+\sqrt {b}} d}-\frac {(a+b) \cot (c+d x)}{a^2 d}-\frac {(3 a+b) \cot ^3(c+d x)}{3 a^2 d}-\frac {3 \cot ^5(c+d x)}{5 a d}-\frac {\cot ^7(c+d x)}{7 a d} \\
\end{align*}
Mathematica [A] (verified)
Time = 6.33 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.98
\[
\int \frac {\csc ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\frac {105 b^2 \arctan \left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a+\sqrt {a} \sqrt {b}}}-\frac {105 b^2 \text {arctanh}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tan (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {-a+\sqrt {a} \sqrt {b}}}-2 \sqrt {a} \cot (c+d x) \left (48 a+70 b+(24 a+35 b) \csc ^2(c+d x)+18 a \csc ^4(c+d x)+15 a \csc ^6(c+d x)\right )}{210 a^{5/2} d}
\]
[In]
Integrate[Csc[c + d*x]^8/(a - b*Sin[c + d*x]^4),x]
[Out]
((105*b^2*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/Sqrt[a + Sqrt[a]*Sqrt[b]] - (1
05*b^2*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/Sqrt[-a + Sqrt[a]*Sqrt[b]] - 2*
Sqrt[a]*Cot[c + d*x]*(48*a + 70*b + (24*a + 35*b)*Csc[c + d*x]^2 + 18*a*Csc[c + d*x]^4 + 15*a*Csc[c + d*x]^6))
/(210*a^(5/2)*d)
Maple [A] (verified)
Time = 2.95 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.08
| | |
| method | result | size |
| | |
| derivativedivides |
\(\frac {-\frac {1}{7 a \tan \left (d x +c \right )^{7}}-\frac {a +b}{a^{2} \tan \left (d x +c \right )}-\frac {3 a +b}{3 a^{2} \tan \left (d x +c \right )^{3}}-\frac {3}{5 a \tan \left (d x +c \right )^{5}}+\frac {b^{2} \left (a -b \right ) \left (\frac {\left (\sqrt {a b}+b \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {\left (\sqrt {a b}-b \right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{a^{2}}}{d}\) |
\(213\) |
| default |
\(\frac {-\frac {1}{7 a \tan \left (d x +c \right )^{7}}-\frac {a +b}{a^{2} \tan \left (d x +c \right )}-\frac {3 a +b}{3 a^{2} \tan \left (d x +c \right )^{3}}-\frac {3}{5 a \tan \left (d x +c \right )^{5}}+\frac {b^{2} \left (a -b \right ) \left (\frac {\left (\sqrt {a b}+b \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {\left (\sqrt {a b}-b \right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{a^{2}}}{d}\) |
\(213\) |
| risch |
\(\frac {4 i \left (105 b \,{\mathrm e}^{10 i \left (d x +c \right )}-455 \,{\mathrm e}^{8 i \left (d x +c \right )} b +840 a \,{\mathrm e}^{6 i \left (d x +c \right )}+770 b \,{\mathrm e}^{6 i \left (d x +c \right )}-504 a \,{\mathrm e}^{4 i \left (d x +c \right )}-630 b \,{\mathrm e}^{4 i \left (d x +c \right )}+168 a \,{\mathrm e}^{2 i \left (d x +c \right )}+245 b \,{\mathrm e}^{2 i \left (d x +c \right )}-24 a -35 b \right )}{105 d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{7}}+256 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (1099511627776 a^{12} d^{4}-1099511627776 a^{11} b \,d^{4}\right ) \textit {\_Z}^{4}+2097152 a^{6} b^{4} d^{2} \textit {\_Z}^{2}+b^{8}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\frac {2147483648 i d^{3} a^{10}}{b^{7}}-\frac {2147483648 i d^{3} a^{9}}{b^{6}}\right ) \textit {\_R}^{3}+\left (-\frac {2097152 d^{2} a^{7}}{b^{5}}+\frac {2097152 d^{2} a^{6}}{b^{4}}\right ) \textit {\_R}^{2}+\left (\frac {2048 i d \,a^{4}}{b^{3}}+\frac {2048 i d \,a^{3}}{b^{2}}\right ) \textit {\_R} -\frac {2 a}{b}-1\right )\right )\) |
\(272\) |
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[In]
int(csc(d*x+c)^8/(a-b*sin(d*x+c)^4),x,method=_RETURNVERBOSE)
[Out]
1/d*(-1/7/a/tan(d*x+c)^7-(a+b)/a^2/tan(d*x+c)-1/3*(3*a+b)/a^2/tan(d*x+c)^3-3/5/a/tan(d*x+c)^5+b^2/a^2*(a-b)*(1
/2*((a*b)^(1/2)+b)/(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))+1/2*((a*b)^(1/2)-b)/(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))))
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 1585 vs. \(2 (155) = 310\).
Time = 0.44 (sec) , antiderivative size = 1585, normalized size of antiderivative = 8.05
\[
\int \frac {\csc ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display}
\]
[In]
integrate(csc(d*x+c)^8/(a-b*sin(d*x+c)^4),x, algorithm="fricas")
[Out]
-1/840*(16*(24*a + 35*b)*cos(d*x + c)^7 - 56*(24*a + 35*b)*cos(d*x + c)^5 + 560*(3*a + 4*b)*cos(d*x + c)^3 + 1
05*(a^2*d*cos(d*x + c)^6 - 3*a^2*d*cos(d*x + c)^4 + 3*a^2*d*cos(d*x + c)^2 - a^2*d)*sqrt(-(b^4 + (a^6 - a^5*b)
*sqrt(b^9/((a^13 - 2*a^12*b + a^11*b^2)*d^4))*d^2)/((a^6 - a^5*b)*d^2))*log(1/4*b^7*cos(d*x + c)^2 - 1/4*b^7 -
1/4*(2*(a^7*b^2 - a^6*b^3)*d^2*cos(d*x + c)^2 - (a^7*b^2 - a^6*b^3)*d^2)*sqrt(b^9/((a^13 - 2*a^12*b + a^11*b^
2)*d^4)) + 1/2*(a^3*b^5*d*cos(d*x + c)*sin(d*x + c) - (a^10 - a^9*b)*sqrt(b^9/((a^13 - 2*a^12*b + a^11*b^2)*d^
4))*d^3*cos(d*x + c)*sin(d*x + c))*sqrt(-(b^4 + (a^6 - a^5*b)*sqrt(b^9/((a^13 - 2*a^12*b + a^11*b^2)*d^4))*d^2
)/((a^6 - a^5*b)*d^2)))*sin(d*x + c) - 105*(a^2*d*cos(d*x + c)^6 - 3*a^2*d*cos(d*x + c)^4 + 3*a^2*d*cos(d*x +
c)^2 - a^2*d)*sqrt(-(b^4 + (a^6 - a^5*b)*sqrt(b^9/((a^13 - 2*a^12*b + a^11*b^2)*d^4))*d^2)/((a^6 - a^5*b)*d^2)
)*log(1/4*b^7*cos(d*x + c)^2 - 1/4*b^7 - 1/4*(2*(a^7*b^2 - a^6*b^3)*d^2*cos(d*x + c)^2 - (a^7*b^2 - a^6*b^3)*d
^2)*sqrt(b^9/((a^13 - 2*a^12*b + a^11*b^2)*d^4)) - 1/2*(a^3*b^5*d*cos(d*x + c)*sin(d*x + c) - (a^10 - a^9*b)*s
qrt(b^9/((a^13 - 2*a^12*b + a^11*b^2)*d^4))*d^3*cos(d*x + c)*sin(d*x + c))*sqrt(-(b^4 + (a^6 - a^5*b)*sqrt(b^9
/((a^13 - 2*a^12*b + a^11*b^2)*d^4))*d^2)/((a^6 - a^5*b)*d^2)))*sin(d*x + c) - 105*(a^2*d*cos(d*x + c)^6 - 3*a
^2*d*cos(d*x + c)^4 + 3*a^2*d*cos(d*x + c)^2 - a^2*d)*sqrt(-(b^4 - (a^6 - a^5*b)*sqrt(b^9/((a^13 - 2*a^12*b +
a^11*b^2)*d^4))*d^2)/((a^6 - a^5*b)*d^2))*log(-1/4*b^7*cos(d*x + c)^2 + 1/4*b^7 - 1/4*(2*(a^7*b^2 - a^6*b^3)*d
^2*cos(d*x + c)^2 - (a^7*b^2 - a^6*b^3)*d^2)*sqrt(b^9/((a^13 - 2*a^12*b + a^11*b^2)*d^4)) + 1/2*(a^3*b^5*d*cos
(d*x + c)*sin(d*x + c) + (a^10 - a^9*b)*sqrt(b^9/((a^13 - 2*a^12*b + a^11*b^2)*d^4))*d^3*cos(d*x + c)*sin(d*x
+ c))*sqrt(-(b^4 - (a^6 - a^5*b)*sqrt(b^9/((a^13 - 2*a^12*b + a^11*b^2)*d^4))*d^2)/((a^6 - a^5*b)*d^2)))*sin(d
*x + c) + 105*(a^2*d*cos(d*x + c)^6 - 3*a^2*d*cos(d*x + c)^4 + 3*a^2*d*cos(d*x + c)^2 - a^2*d)*sqrt(-(b^4 - (a
^6 - a^5*b)*sqrt(b^9/((a^13 - 2*a^12*b + a^11*b^2)*d^4))*d^2)/((a^6 - a^5*b)*d^2))*log(-1/4*b^7*cos(d*x + c)^2
+ 1/4*b^7 - 1/4*(2*(a^7*b^2 - a^6*b^3)*d^2*cos(d*x + c)^2 - (a^7*b^2 - a^6*b^3)*d^2)*sqrt(b^9/((a^13 - 2*a^12
*b + a^11*b^2)*d^4)) - 1/2*(a^3*b^5*d*cos(d*x + c)*sin(d*x + c) + (a^10 - a^9*b)*sqrt(b^9/((a^13 - 2*a^12*b +
a^11*b^2)*d^4))*d^3*cos(d*x + c)*sin(d*x + c))*sqrt(-(b^4 - (a^6 - a^5*b)*sqrt(b^9/((a^13 - 2*a^12*b + a^11*b^
2)*d^4))*d^2)/((a^6 - a^5*b)*d^2)))*sin(d*x + c) - 840*(a + b)*cos(d*x + c))/((a^2*d*cos(d*x + c)^6 - 3*a^2*d*
cos(d*x + c)^4 + 3*a^2*d*cos(d*x + c)^2 - a^2*d)*sin(d*x + c))
Sympy [F(-1)]
Timed out. \[
\int \frac {\csc ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Timed out}
\]
[In]
integrate(csc(d*x+c)**8/(a-b*sin(d*x+c)**4),x)
[Out]
Timed out
Maxima [F]
\[
\int \frac {\csc ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int { -\frac {\csc \left (d x + c\right )^{8}}{b \sin \left (d x + c\right )^{4} - a} \,d x }
\]
[In]
integrate(csc(d*x+c)^8/(a-b*sin(d*x+c)^4),x, algorithm="maxima")
[Out]
4/105*(735*b*cos(4*d*x + 4*c)*sin(2*d*x + 2*c) - 7*(15*b*sin(10*d*x + 10*c) - 65*b*sin(8*d*x + 8*c) + 10*(12*a
+ 11*b)*sin(6*d*x + 6*c) - 18*(4*a + 5*b)*sin(4*d*x + 4*c) + (24*a + 35*b)*sin(2*d*x + 2*c))*cos(14*d*x + 14*
c) + 49*(15*b*sin(10*d*x + 10*c) - 65*b*sin(8*d*x + 8*c) + 10*(12*a + 11*b)*sin(6*d*x + 6*c) - 18*(4*a + 5*b)*
sin(4*d*x + 4*c) + (24*a + 35*b)*sin(2*d*x + 2*c))*cos(12*d*x + 12*c) + 147*(40*b*sin(8*d*x + 8*c) - 5*(24*a +
17*b)*sin(6*d*x + 6*c) + 3*(24*a + 25*b)*sin(4*d*x + 4*c) - 6*(4*a + 5*b)*sin(2*d*x + 2*c))*cos(10*d*x + 10*c
) + 245*(15*(8*a + 3*b)*sin(6*d*x + 6*c) - 3*(24*a + 17*b)*sin(4*d*x + 4*c) + 2*(12*a + 11*b)*sin(2*d*x + 2*c)
)*cos(8*d*x + 8*c) + 245*(24*b*sin(4*d*x + 4*c) - 13*b*sin(2*d*x + 2*c))*cos(6*d*x + 6*c) - 420*(a^2*b^2*d*cos
(14*d*x + 14*c)^2 + 49*a^2*b^2*d*cos(12*d*x + 12*c)^2 + 441*a^2*b^2*d*cos(10*d*x + 10*c)^2 + 1225*a^2*b^2*d*co
s(8*d*x + 8*c)^2 + 1225*a^2*b^2*d*cos(6*d*x + 6*c)^2 + 441*a^2*b^2*d*cos(4*d*x + 4*c)^2 + 49*a^2*b^2*d*cos(2*d
*x + 2*c)^2 + a^2*b^2*d*sin(14*d*x + 14*c)^2 + 49*a^2*b^2*d*sin(12*d*x + 12*c)^2 + 441*a^2*b^2*d*sin(10*d*x +
10*c)^2 + 1225*a^2*b^2*d*sin(8*d*x + 8*c)^2 + 1225*a^2*b^2*d*sin(6*d*x + 6*c)^2 + 441*a^2*b^2*d*sin(4*d*x + 4*
c)^2 - 294*a^2*b^2*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 49*a^2*b^2*d*sin(2*d*x + 2*c)^2 - 14*a^2*b^2*d*cos(2*
d*x + 2*c) + a^2*b^2*d - 2*(7*a^2*b^2*d*cos(12*d*x + 12*c) - 21*a^2*b^2*d*cos(10*d*x + 10*c) + 35*a^2*b^2*d*co
s(8*d*x + 8*c) - 35*a^2*b^2*d*cos(6*d*x + 6*c) + 21*a^2*b^2*d*cos(4*d*x + 4*c) - 7*a^2*b^2*d*cos(2*d*x + 2*c)
+ a^2*b^2*d)*cos(14*d*x + 14*c) - 14*(21*a^2*b^2*d*cos(10*d*x + 10*c) - 35*a^2*b^2*d*cos(8*d*x + 8*c) + 35*a^2
*b^2*d*cos(6*d*x + 6*c) - 21*a^2*b^2*d*cos(4*d*x + 4*c) + 7*a^2*b^2*d*cos(2*d*x + 2*c) - a^2*b^2*d)*cos(12*d*x
+ 12*c) - 42*(35*a^2*b^2*d*cos(8*d*x + 8*c) - 35*a^2*b^2*d*cos(6*d*x + 6*c) + 21*a^2*b^2*d*cos(4*d*x + 4*c) -
7*a^2*b^2*d*cos(2*d*x + 2*c) + a^2*b^2*d)*cos(10*d*x + 10*c) - 70*(35*a^2*b^2*d*cos(6*d*x + 6*c) - 21*a^2*b^2
*d*cos(4*d*x + 4*c) + 7*a^2*b^2*d*cos(2*d*x + 2*c) - a^2*b^2*d)*cos(8*d*x + 8*c) - 70*(21*a^2*b^2*d*cos(4*d*x
+ 4*c) - 7*a^2*b^2*d*cos(2*d*x + 2*c) + a^2*b^2*d)*cos(6*d*x + 6*c) - 42*(7*a^2*b^2*d*cos(2*d*x + 2*c) - a^2*b
^2*d)*cos(4*d*x + 4*c) - 14*(a^2*b^2*d*sin(12*d*x + 12*c) - 3*a^2*b^2*d*sin(10*d*x + 10*c) + 5*a^2*b^2*d*sin(8
*d*x + 8*c) - 5*a^2*b^2*d*sin(6*d*x + 6*c) + 3*a^2*b^2*d*sin(4*d*x + 4*c) - a^2*b^2*d*sin(2*d*x + 2*c))*sin(14
*d*x + 14*c) - 98*(3*a^2*b^2*d*sin(10*d*x + 10*c) - 5*a^2*b^2*d*sin(8*d*x + 8*c) + 5*a^2*b^2*d*sin(6*d*x + 6*c
) - 3*a^2*b^2*d*sin(4*d*x + 4*c) + a^2*b^2*d*sin(2*d*x + 2*c))*sin(12*d*x + 12*c) - 294*(5*a^2*b^2*d*sin(8*d*x
+ 8*c) - 5*a^2*b^2*d*sin(6*d*x + 6*c) + 3*a^2*b^2*d*sin(4*d*x + 4*c) - a^2*b^2*d*sin(2*d*x + 2*c))*sin(10*d*x
+ 10*c) - 490*(5*a^2*b^2*d*sin(6*d*x + 6*c) - 3*a^2*b^2*d*sin(4*d*x + 4*c) + a^2*b^2*d*sin(2*d*x + 2*c))*sin(
8*d*x + 8*c) - 490*(3*a^2*b^2*d*sin(4*d*x + 4*c) - a^2*b^2*d*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*integrate((b*
cos(8*d*x + 8*c)*cos(4*d*x + 4*c) - 4*b*cos(6*d*x + 6*c)*cos(4*d*x + 4*c) - 2*(8*a - 3*b)*cos(4*d*x + 4*c)^2 +
b*sin(8*d*x + 8*c)*sin(4*d*x + 4*c) - 4*b*sin(6*d*x + 6*c)*sin(4*d*x + 4*c) - 2*(8*a - 3*b)*sin(4*d*x + 4*c)^
2 - 4*b*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) - (4*b*cos(2*d*x + 2*c) - b)*cos(4*d*x + 4*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) + (105*b*cos(10*d*x + 10*c) - 455*b*cos(8*d*x + 8*c) + 70*(12*a + 11*b)*cos(6*
d*x + 6*c) - 126*(4*a + 5*b)*cos(4*d*x + 4*c) + 7*(24*a + 35*b)*cos(2*d*x + 2*c) - 24*a - 35*b)*sin(14*d*x + 1
4*c) - 7*(105*b*cos(10*d*x + 10*c) - 455*b*cos(8*d*x + 8*c) + 70*(12*a + 11*b)*cos(6*d*x + 6*c) - 126*(4*a + 5
*b)*cos(4*d*x + 4*c) + 7*(24*a + 35*b)*cos(2*d*x + 2*c) - 24*a - 35*b)*sin(12*d*x + 12*c) - 21*(280*b*cos(8*d*
x + 8*c) - 35*(24*a + 17*b)*cos(6*d*x + 6*c) + 21*(24*a + 25*b)*cos(4*d*x + 4*c) - 42*(4*a + 5*b)*cos(2*d*x +
2*c) + 24*a + 30*b)*sin(10*d*x + 10*c) - 35*(105*(8*a + 3*b)*cos(6*d*x + 6*c) - 21*(24*a + 17*b)*cos(4*d*x + 4
*c) + 14*(12*a + 11*b)*cos(2*d*x + 2*c) - 24*a - 22*b)*sin(8*d*x + 8*c) - 35*(168*b*cos(4*d*x + 4*c) - 91*b*co
s(2*d*x + 2*c) + 13*b)*sin(6*d*x + 6*c) - 105*(7*b*cos(2*d*x + 2*c) - b)*sin(4*d*x + 4*c))/(a^2*d*cos(14*d*x +
14*c)^2 + 49*a^2*d*cos(12*d*x + 12*c)^2 + 441*a^2*d*cos(10*d*x + 10*c)^2 + 1225*a^2*d*cos(8*d*x + 8*c)^2 + 12
25*a^2*d*cos(6*d*x + 6*c)^2 + 441*a^2*d*cos(4*d*x + 4*c)^2 + 49*a^2*d*cos(2*d*x + 2*c)^2 + a^2*d*sin(14*d*x +
14*c)^2 + 49*a^2*d*sin(12*d*x + 12*c)^2 + 441*a^2*d*sin(10*d*x + 10*c)^2 + 1225*a^2*d*sin(8*d*x + 8*c)^2 + 122
5*a^2*d*sin(6*d*x + 6*c)^2 + 441*a^2*d*sin(4*d*x + 4*c)^2 - 294*a^2*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 49*a
^2*d*sin(2*d*x + 2*c)^2 - 14*a^2*d*cos(2*d*x + 2*c) + a^2*d - 2*(7*a^2*d*cos(12*d*x + 12*c) - 21*a^2*d*cos(10*
d*x + 10*c) + 35*a^2*d*cos(8*d*x + 8*c) - 35*a^2*d*cos(6*d*x + 6*c) + 21*a^2*d*cos(4*d*x + 4*c) - 7*a^2*d*cos(
2*d*x + 2*c) + a^2*d)*cos(14*d*x + 14*c) - 14*(21*a^2*d*cos(10*d*x + 10*c) - 35*a^2*d*cos(8*d*x + 8*c) + 35*a^
2*d*cos(6*d*x + 6*c) - 21*a^2*d*cos(4*d*x + 4*c) + 7*a^2*d*cos(2*d*x + 2*c) - a^2*d)*cos(12*d*x + 12*c) - 42*(
35*a^2*d*cos(8*d*x + 8*c) - 35*a^2*d*cos(6*d*x + 6*c) + 21*a^2*d*cos(4*d*x + 4*c) - 7*a^2*d*cos(2*d*x + 2*c) +
a^2*d)*cos(10*d*x + 10*c) - 70*(35*a^2*d*cos(6*d*x + 6*c) - 21*a^2*d*cos(4*d*x + 4*c) + 7*a^2*d*cos(2*d*x + 2
*c) - a^2*d)*cos(8*d*x + 8*c) - 70*(21*a^2*d*cos(4*d*x + 4*c) - 7*a^2*d*cos(2*d*x + 2*c) + a^2*d)*cos(6*d*x +
6*c) - 42*(7*a^2*d*cos(2*d*x + 2*c) - a^2*d)*cos(4*d*x + 4*c) - 14*(a^2*d*sin(12*d*x + 12*c) - 3*a^2*d*sin(10*
d*x + 10*c) + 5*a^2*d*sin(8*d*x + 8*c) - 5*a^2*d*sin(6*d*x + 6*c) + 3*a^2*d*sin(4*d*x + 4*c) - a^2*d*sin(2*d*x
+ 2*c))*sin(14*d*x + 14*c) - 98*(3*a^2*d*sin(10*d*x + 10*c) - 5*a^2*d*sin(8*d*x + 8*c) + 5*a^2*d*sin(6*d*x +
6*c) - 3*a^2*d*sin(4*d*x + 4*c) + a^2*d*sin(2*d*x + 2*c))*sin(12*d*x + 12*c) - 294*(5*a^2*d*sin(8*d*x + 8*c) -
5*a^2*d*sin(6*d*x + 6*c) + 3*a^2*d*sin(4*d*x + 4*c) - a^2*d*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) - 490*(5*a^2
*d*sin(6*d*x + 6*c) - 3*a^2*d*sin(4*d*x + 4*c) + a^2*d*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) - 490*(3*a^2*d*sin(4
*d*x + 4*c) - a^2*d*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 467 vs. \(2 (155) = 310\).
Time = 0.81 (sec) , antiderivative size = 467, normalized size of antiderivative = 2.37
\[
\int \frac {\csc ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\frac {105 \, {\left (3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{2} b^{2} - 6 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a b^{3} - \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\tan \left (d x + c\right )}{\sqrt {\frac {a^{3} + \sqrt {a^{6} - {\left (a^{3} - a^{2} b\right )} a^{3}}}{a^{3} - a^{2} b}}}\right )\right )} {\left | a - b \right |}}{3 \, a^{7} - 12 \, a^{6} b + 14 \, a^{5} b^{2} - 4 \, a^{4} b^{3} - a^{3} b^{4}} + \frac {105 \, {\left (3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{2} b^{2} - 6 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a b^{3} - \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\tan \left (d x + c\right )}{\sqrt {\frac {a^{3} - \sqrt {a^{6} - {\left (a^{3} - a^{2} b\right )} a^{3}}}{a^{3} - a^{2} b}}}\right )\right )} {\left | a - b \right |}}{3 \, a^{7} - 12 \, a^{6} b + 14 \, a^{5} b^{2} - 4 \, a^{4} b^{3} - a^{3} b^{4}} - \frac {2 \, {\left (105 \, a \tan \left (d x + c\right )^{6} + 105 \, b \tan \left (d x + c\right )^{6} + 105 \, a \tan \left (d x + c\right )^{4} + 35 \, b \tan \left (d x + c\right )^{4} + 63 \, a \tan \left (d x + c\right )^{2} + 15 \, a\right )}}{a^{2} \tan \left (d x + c\right )^{7}}}{210 \, d}
\]
[In]
integrate(csc(d*x+c)^8/(a-b*sin(d*x+c)^4),x, algorithm="giac")
[Out]
1/210*(105*(3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^2*b^2 - 6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a*b^3 - sqrt
(a^2 - a*b + sqrt(a*b)*(a - b))*b^4)*(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) + 105*
(3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b^2 - 6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a*b^3 - sqrt(a^2 - a*b
- sqrt(a*b)*(a - b))*b^4)*(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*(105*a*tan(d*
x + c)^6 + 105*b*tan(d*x + c)^6 + 105*a*tan(d*x + c)^4 + 35*b*tan(d*x + c)^4 + 63*a*tan(d*x + c)^2 + 15*a)/(a^
2*tan(d*x + c)^7))/d
Mupad [B] (verification not implemented)
Time = 16.82 (sec) , antiderivative size = 1704, normalized size of antiderivative = 8.65
\[
\int \frac {\csc ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display}
\]
[In]
int(1/(sin(c + d*x)^8*(a - b*sin(c + d*x)^4)),x)
[Out]
(atan((((((a^11*b^9)^(1/2) + a^6*b^4)/(16*(a^11*b - a^12)))^(1/2)*(16*a^9*b^5 - 32*a^10*b^4 + 16*a^11*b^3 + ta
n(c + d*x)*(((a^11*b^9)^(1/2) + a^6*b^4)/(16*(a^11*b - a^12)))^(1/2)*(64*a^14*b + 64*a^12*b^3 - 128*a^13*b^2))
- tan(c + d*x)*(4*a^6*b^7 - 4*a^8*b^5))*(((a^11*b^9)^(1/2) + a^6*b^4)/(16*(a^11*b - a^12)))^(1/2)*1i - ((((a^
11*b^9)^(1/2) + a^6*b^4)/(16*(a^11*b - a^12)))^(1/2)*(16*a^9*b^5 - 32*a^10*b^4 + 16*a^11*b^3 - tan(c + d*x)*((
(a^11*b^9)^(1/2) + a^6*b^4)/(16*(a^11*b - a^12)))^(1/2)*(64*a^14*b + 64*a^12*b^3 - 128*a^13*b^2)) + tan(c + d*
x)*(4*a^6*b^7 - 4*a^8*b^5))*(((a^11*b^9)^(1/2) + a^6*b^4)/(16*(a^11*b - a^12)))^(1/2)*1i)/(((((a^11*b^9)^(1/2)
+ a^6*b^4)/(16*(a^11*b - a^12)))^(1/2)*(16*a^9*b^5 - 32*a^10*b^4 + 16*a^11*b^3 + tan(c + d*x)*(((a^11*b^9)^(1
/2) + a^6*b^4)/(16*(a^11*b - a^12)))^(1/2)*(64*a^14*b + 64*a^12*b^3 - 128*a^13*b^2)) - tan(c + d*x)*(4*a^6*b^7
- 4*a^8*b^5))*(((a^11*b^9)^(1/2) + a^6*b^4)/(16*(a^11*b - a^12)))^(1/2) + ((((a^11*b^9)^(1/2) + a^6*b^4)/(16*
(a^11*b - a^12)))^(1/2)*(16*a^9*b^5 - 32*a^10*b^4 + 16*a^11*b^3 - tan(c + d*x)*(((a^11*b^9)^(1/2) + a^6*b^4)/(
16*(a^11*b - a^12)))^(1/2)*(64*a^14*b + 64*a^12*b^3 - 128*a^13*b^2)) + tan(c + d*x)*(4*a^6*b^7 - 4*a^8*b^5))*(
((a^11*b^9)^(1/2) + a^6*b^4)/(16*(a^11*b - a^12)))^(1/2) - 2*a^4*b^8 + 2*a^5*b^7))*(((a^11*b^9)^(1/2) + a^6*b^
4)/(16*(a^11*b - a^12)))^(1/2)*2i)/d + (atan((((-((a^11*b^9)^(1/2) - a^6*b^4)/(16*(a^11*b - a^12)))^(1/2)*(16*
a^9*b^5 - 32*a^10*b^4 + 16*a^11*b^3 + tan(c + d*x)*(-((a^11*b^9)^(1/2) - a^6*b^4)/(16*(a^11*b - a^12)))^(1/2)*
(64*a^14*b + 64*a^12*b^3 - 128*a^13*b^2)) - tan(c + d*x)*(4*a^6*b^7 - 4*a^8*b^5))*(-((a^11*b^9)^(1/2) - a^6*b^
4)/(16*(a^11*b - a^12)))^(1/2)*1i - ((-((a^11*b^9)^(1/2) - a^6*b^4)/(16*(a^11*b - a^12)))^(1/2)*(16*a^9*b^5 -
32*a^10*b^4 + 16*a^11*b^3 - tan(c + d*x)*(-((a^11*b^9)^(1/2) - a^6*b^4)/(16*(a^11*b - a^12)))^(1/2)*(64*a^14*b
+ 64*a^12*b^3 - 128*a^13*b^2)) + tan(c + d*x)*(4*a^6*b^7 - 4*a^8*b^5))*(-((a^11*b^9)^(1/2) - a^6*b^4)/(16*(a^
11*b - a^12)))^(1/2)*1i)/(((-((a^11*b^9)^(1/2) - a^6*b^4)/(16*(a^11*b - a^12)))^(1/2)*(16*a^9*b^5 - 32*a^10*b^
4 + 16*a^11*b^3 + tan(c + d*x)*(-((a^11*b^9)^(1/2) - a^6*b^4)/(16*(a^11*b - a^12)))^(1/2)*(64*a^14*b + 64*a^12
*b^3 - 128*a^13*b^2)) - tan(c + d*x)*(4*a^6*b^7 - 4*a^8*b^5))*(-((a^11*b^9)^(1/2) - a^6*b^4)/(16*(a^11*b - a^1
2)))^(1/2) + ((-((a^11*b^9)^(1/2) - a^6*b^4)/(16*(a^11*b - a^12)))^(1/2)*(16*a^9*b^5 - 32*a^10*b^4 + 16*a^11*b
^3 - tan(c + d*x)*(-((a^11*b^9)^(1/2) - a^6*b^4)/(16*(a^11*b - a^12)))^(1/2)*(64*a^14*b + 64*a^12*b^3 - 128*a^
13*b^2)) + tan(c + d*x)*(4*a^6*b^7 - 4*a^8*b^5))*(-((a^11*b^9)^(1/2) - a^6*b^4)/(16*(a^11*b - a^12)))^(1/2) -
2*a^4*b^8 + 2*a^5*b^7))*(-((a^11*b^9)^(1/2) - a^6*b^4)/(16*(a^11*b - a^12)))^(1/2)*2i)/d - (1/(7*a) + (3*tan(c
+ d*x)^2)/(5*a) + (tan(c + d*x)^6*(a + b))/a^2 + (tan(c + d*x)^4*(3*a + b))/(3*a^2))/(d*tan(c + d*x)^7)