\(\int \frac {\csc ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx\) [209]
Optimal result
Integrand size = 24, antiderivative size = 149 \[
\int \frac {\csc ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {b \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{7/4} \sqrt {\sqrt {a}-\sqrt {b}} d}+\frac {b \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{7/4} \sqrt {\sqrt {a}+\sqrt {b}} d}-\frac {\cot (c+d x)}{a d}-\frac {\cot ^3(c+d x)}{3 a d}
\]
[Out]
-cot(d*x+c)/a/d-1/3*cot(d*x+c)^3/a/d+1/2*b*arctan((a^(1/2)-b^(1/2))^(1/2)*tan(d*x+c)/a^(1/4))/a^(7/4)/d/(a^(1/
2)-b^(1/2))^(1/2)+1/2*b*arctan((a^(1/2)+b^(1/2))^(1/2)*tan(d*x+c)/a^(1/4))/a^(7/4)/d/(a^(1/2)+b^(1/2))^(1/2)
Rubi [A] (verified)
Time = 0.13 (sec) , antiderivative size = 149, 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}
\[
\int \frac {\csc ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {b \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{7/4} d \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {b \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{7/4} d \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {\cot ^3(c+d x)}{3 a d}-\frac {\cot (c+d x)}{a d}
\]
[In]
Int[Csc[c + d*x]^4/(a - b*Sin[c + d*x]^4),x]
[Out]
(b*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(7/4)*Sqrt[Sqrt[a] - Sqrt[b]]*d) + (b*ArcTan[(
Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(7/4)*Sqrt[Sqrt[a] + Sqrt[b]]*d) - Cot[c + d*x]/(a*d) - C
ot[c + d*x]^3/(3*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 )^3}{x^4 \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^4}+\frac {1}{a x^2}+\frac {b \left (1+x^2\right )}{a \left (a+2 a x^2+(a-b) x^4\right )}\right ) \, dx,x,\tan (c+d x)\right )}{d} \\ & = -\frac {\cot (c+d x)}{a d}-\frac {\cot ^3(c+d x)}{3 a d}+\frac {b \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 d} \\ & = -\frac {\cot (c+d x)}{a d}-\frac {\cot ^3(c+d x)}{3 a d}+\frac {\left (\left (\sqrt {a}+\sqrt {b}\right ) b\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^{3/2} d}+\frac {\left (\left (1-\frac {\sqrt {b}}{\sqrt {a}}\right ) b\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 d} \\ & = \frac {b \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{7/4} \sqrt {\sqrt {a}-\sqrt {b}} d}+\frac {b \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{7/4} \sqrt {\sqrt {a}+\sqrt {b}} d}-\frac {\cot (c+d x)}{a d}-\frac {\cot ^3(c+d x)}{3 a d} \\
\end{align*}
Mathematica [A] (verified)
Time = 1.64 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.11
\[
\int \frac {\csc ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\frac {3 b \arctan \left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a+\sqrt {a} \sqrt {b}}}-\frac {3 b \text {arctanh}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tan (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {-a+\sqrt {a} \sqrt {b}}}-4 \sqrt {a} \cot (c+d x)-2 \sqrt {a} \cot (c+d x) \csc ^2(c+d x)}{6 a^{3/2} d}
\]
[In]
Integrate[Csc[c + d*x]^4/(a - b*Sin[c + d*x]^4),x]
[Out]
((3*b*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/Sqrt[a + Sqrt[a]*Sqrt[b]] - (3*b*A
rcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/Sqrt[-a + Sqrt[a]*Sqrt[b]] - 4*Sqrt[a]*
Cot[c + d*x] - 2*Sqrt[a]*Cot[c + d*x]*Csc[c + d*x]^2)/(6*a^(3/2)*d)
Maple [A] (verified)
Time = 1.44 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.19
| | |
| method | result | size |
| | |
| derivativedivides |
\(\frac {\frac {b \left (a -b \right ) \left (\frac {\left (\sqrt {a b}+b \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {\left (\sqrt {a b}-b \right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{a}-\frac {1}{3 a \tan \left (d x +c \right )^{3}}-\frac {1}{a \tan \left (d x +c \right )}}{d}\) |
\(177\) |
| default |
\(\frac {\frac {b \left (a -b \right ) \left (\frac {\left (\sqrt {a b}+b \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {\left (\sqrt {a b}-b \right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{a}-\frac {1}{3 a \tan \left (d x +c \right )^{3}}-\frac {1}{a \tan \left (d x +c \right )}}{d}\) |
\(177\) |
| risch |
\(\frac {4 i \left (3 \,{\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{3 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}+16 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (16777216 a^{8} d^{4}-16777216 a^{7} b \,d^{4}\right ) \textit {\_Z}^{4}+8192 a^{4} b^{2} d^{2} \textit {\_Z}^{2}+b^{4}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\frac {524288 i d^{3} a^{7}}{b^{4}}-\frac {524288 i a^{6} d^{3}}{b^{3}}\right ) \textit {\_R}^{3}+\left (-\frac {8192 d^{2} a^{5}}{b^{3}}+\frac {8192 d^{2} a^{4}}{b^{2}}\right ) \textit {\_R}^{2}+\left (\frac {128 i d \,a^{3}}{b^{2}}+\frac {128 i a^{2} d}{b}\right ) \textit {\_R} -\frac {2 a}{b}-1\right )\right )\) |
\(182\) |
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[In]
int(csc(d*x+c)^4/(a-b*sin(d*x+c)^4),x,method=_RETURNVERBOSE)
[Out]
1/d*(1/a*b*(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)))-1/3/a/tan(d*x+c)^3-1/a/tan(d*x+c))
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 1365 vs. \(2 (111) = 222\).
Time = 0.41 (sec) , antiderivative size = 1365, normalized size of antiderivative = 9.16
\[
\int \frac {\csc ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display}
\]
[In]
integrate(csc(d*x+c)^4/(a-b*sin(d*x+c)^4),x, algorithm="fricas")
[Out]
-1/24*(3*(a*d*cos(d*x + c)^2 - a*d)*sqrt(-((a^4 - a^3*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) + b^2)/
((a^4 - a^3*b)*d^2))*log(1/4*b^4*cos(d*x + c)^2 - 1/4*b^4 - 1/4*(2*(a^5*b - a^4*b^2)*d^2*cos(d*x + c)^2 - (a^5
*b - a^4*b^2)*d^2)*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) + 1/2*(a^2*b^3*d*cos(d*x + c)*sin(d*x + c) - (a^7
- a^6*b)*d^3*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4))*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^4 - a^3*b)*d^2*sq
rt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) + b^2)/((a^4 - a^3*b)*d^2)))*sin(d*x + c) - 3*(a*d*cos(d*x + c)^2 - a*
d)*sqrt(-((a^4 - a^3*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) + b^2)/((a^4 - a^3*b)*d^2))*log(1/4*b^4*
cos(d*x + c)^2 - 1/4*b^4 - 1/4*(2*(a^5*b - a^4*b^2)*d^2*cos(d*x + c)^2 - (a^5*b - a^4*b^2)*d^2)*sqrt(b^5/((a^9
- 2*a^8*b + a^7*b^2)*d^4)) - 1/2*(a^2*b^3*d*cos(d*x + c)*sin(d*x + c) - (a^7 - a^6*b)*d^3*sqrt(b^5/((a^9 - 2*
a^8*b + a^7*b^2)*d^4))*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^4 - a^3*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)
*d^4)) + b^2)/((a^4 - a^3*b)*d^2)))*sin(d*x + c) - 3*(a*d*cos(d*x + c)^2 - a*d)*sqrt(((a^4 - a^3*b)*d^2*sqrt(b
^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) - b^2)/((a^4 - a^3*b)*d^2))*log(-1/4*b^4*cos(d*x + c)^2 + 1/4*b^4 - 1/4*(2
*(a^5*b - a^4*b^2)*d^2*cos(d*x + c)^2 - (a^5*b - a^4*b^2)*d^2)*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) + 1/2
*(a^2*b^3*d*cos(d*x + c)*sin(d*x + c) + (a^7 - a^6*b)*d^3*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4))*cos(d*x +
c)*sin(d*x + c))*sqrt(((a^4 - a^3*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) - b^2)/((a^4 - a^3*b)*d^2))
)*sin(d*x + c) + 3*(a*d*cos(d*x + c)^2 - a*d)*sqrt(((a^4 - a^3*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)
) - b^2)/((a^4 - a^3*b)*d^2))*log(-1/4*b^4*cos(d*x + c)^2 + 1/4*b^4 - 1/4*(2*(a^5*b - a^4*b^2)*d^2*cos(d*x + c
)^2 - (a^5*b - a^4*b^2)*d^2)*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) - 1/2*(a^2*b^3*d*cos(d*x + c)*sin(d*x +
c) + (a^7 - a^6*b)*d^3*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4))*cos(d*x + c)*sin(d*x + c))*sqrt(((a^4 - a^3*
b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) - b^2)/((a^4 - a^3*b)*d^2)))*sin(d*x + c) + 16*cos(d*x + c)^3
- 24*cos(d*x + c))/((a*d*cos(d*x + c)^2 - a*d)*sin(d*x + c))
Sympy [F]
\[
\int \frac {\csc ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int \frac {\csc ^{4}{\left (c + d x \right )}}{a - b \sin ^{4}{\left (c + d x \right )}}\, dx
\]
[In]
integrate(csc(d*x+c)**4/(a-b*sin(d*x+c)**4),x)
[Out]
Integral(csc(c + d*x)**4/(a - b*sin(c + d*x)**4), x)
Maxima [F]
\[
\int \frac {\csc ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int { -\frac {\csc \left (d x + c\right )^{4}}{b \sin \left (d x + c\right )^{4} - a} \,d x }
\]
[In]
integrate(csc(d*x+c)^4/(a-b*sin(d*x+c)^4),x, algorithm="maxima")
[Out]
-4/3*(12*(a*b*d*cos(6*d*x + 6*c)^2 + 9*a*b*d*cos(4*d*x + 4*c)^2 + 9*a*b*d*cos(2*d*x + 2*c)^2 + a*b*d*sin(6*d*x
+ 6*c)^2 + 9*a*b*d*sin(4*d*x + 4*c)^2 - 18*a*b*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 9*a*b*d*sin(2*d*x + 2*c)
^2 - 6*a*b*d*cos(2*d*x + 2*c) + a*b*d - 2*(3*a*b*d*cos(4*d*x + 4*c) - 3*a*b*d*cos(2*d*x + 2*c) + a*b*d)*cos(6*
d*x + 6*c) - 6*(3*a*b*d*cos(2*d*x + 2*c) - a*b*d)*cos(4*d*x + 4*c) - 6*(a*b*d*sin(4*d*x + 4*c) - a*b*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*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*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(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) - (3*cos(2*d*x + 2*c) - 1)*sin(6*d*x + 6*c) + 3*(3*cos(2*d*x + 2*c) - 1)*sin(4*d*x + 4
*c) + 3*cos(6*d*x + 6*c)*sin(2*d*x + 2*c) - 9*cos(4*d*x + 4*c)*sin(2*d*x + 2*c))/(a*d*cos(6*d*x + 6*c)^2 + 9*a
*d*cos(4*d*x + 4*c)^2 + 9*a*d*cos(2*d*x + 2*c)^2 + a*d*sin(6*d*x + 6*c)^2 + 9*a*d*sin(4*d*x + 4*c)^2 - 18*a*d*
sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 9*a*d*sin(2*d*x + 2*c)^2 - 6*a*d*cos(2*d*x + 2*c) + a*d - 2*(3*a*d*cos(4*d
*x + 4*c) - 3*a*d*cos(2*d*x + 2*c) + a*d)*cos(6*d*x + 6*c) - 6*(3*a*d*cos(2*d*x + 2*c) - a*d)*cos(4*d*x + 4*c)
- 6*(a*d*sin(4*d*x + 4*c) - a*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. 937 vs. \(2 (111) = 222\).
Time = 0.70 (sec) , antiderivative size = 937, normalized size of antiderivative = 6.29
\[
\int \frac {\csc ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\frac {3 \, {\left ({\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 )} a^{2} {\left | a - b \right |} + {\left (3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{4} b - 9 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{3} b^{2} + 5 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{2} b^{3} + \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a b^{4}\right )} {\left | a - b \right |} {\left | a \right |} - {\left (3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{4} b - 6 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{3} b^{2} - \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} b^{3}\right )} {\left | a - b \right |}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\tan \left (d x + c\right )}{\sqrt {\frac {a^{2} + \sqrt {a^{4} - {\left (a^{2} - a b\right )} a^{2}}}{a^{2} - a b}}}\right )\right )}}{{\left (3 \, a^{8} - 15 \, a^{7} b + 26 \, a^{6} b^{2} - 18 \, a^{5} b^{3} + 3 \, a^{4} b^{4} + a^{3} b^{5}\right )} {\left | a \right |}} + \frac {3 \, {\left ({\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 )} a^{2} {\left | a - b \right |} + {\left (3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{4} b - 9 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{3} b^{2} + 5 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{2} b^{3} + \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a b^{4}\right )} {\left | a - b \right |} {\left | a \right |} - {\left (3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{4} b - 6 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{3} b^{2} - \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} b^{3}\right )} {\left | a - b \right |}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\tan \left (d x + c\right )}{\sqrt {\frac {a^{2} - \sqrt {a^{4} - {\left (a^{2} - a b\right )} a^{2}}}{a^{2} - a b}}}\right )\right )}}{{\left (3 \, a^{8} - 15 \, a^{7} b + 26 \, a^{6} b^{2} - 18 \, a^{5} b^{3} + 3 \, a^{4} b^{4} + a^{3} b^{5}\right )} {\left | a \right |}} - \frac {2 \, {\left (3 \, \tan \left (d x + c\right )^{2} + 1\right )}}{a \tan \left (d x + c\right )^{3}}}{6 \, d}
\]
[In]
integrate(csc(d*x+c)^4/(a-b*sin(d*x+c)^4),x, algorithm="giac")
[Out]
1/6*(3*((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)*a^2*abs(a - b) + (3*sqrt(a^2 - a*b + sqrt(a*b)*(
a - b))*a^4*b - 9*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^3*b^2 + 5*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^2*b^3
+ sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a*b^4)*abs(a - b)*abs(a) - (3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a
*b)*a^4*b - 6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^2 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt
(a*b)*a^2*b^3)*abs(a - b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^2 + sqrt(a^4 - (a^2 - a
*b)*a^2))/(a^2 - a*b))))/((3*a^8 - 15*a^7*b + 26*a^6*b^2 - 18*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*abs(a)) + 3*((3*s
qrt(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 - s
qrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^3)*a^2*abs(a - b) + (3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^4*
b - 9*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^3*b^2 + 5*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b^3 + sqrt(a^2 -
a*b - sqrt(a*b)*(a - b))*a*b^4)*abs(a - b)*abs(a) - (3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b -
6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^2 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^
3)*abs(a - b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^2 - sqrt(a^4 - (a^2 - a*b)*a^2))/(a
^2 - a*b))))/((3*a^8 - 15*a^7*b + 26*a^6*b^2 - 18*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*abs(a)) - 2*(3*tan(d*x + c)^2
+ 1)/(a*tan(d*x + c)^3))/d
Mupad [B] (verification not implemented)
Time = 15.91 (sec) , antiderivative size = 1670, normalized size of antiderivative = 11.21
\[
\int \frac {\csc ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display}
\]
[In]
int(1/(sin(c + d*x)^4*(a - b*sin(c + d*x)^4)),x)
[Out]
(atan((((((a^7*b^5)^(1/2) + a^4*b^2)/(16*(a^7*b - a^8)))^(1/2)*(16*a^5*b^4 - 32*a^6*b^3 + 16*a^7*b^2 + tan(c +
d*x)*(((a^7*b^5)^(1/2) + a^4*b^2)/(16*(a^7*b - a^8)))^(1/2)*(64*a^9*b + 64*a^7*b^3 - 128*a^8*b^2)) - tan(c +
d*x)*(4*a^3*b^5 - 4*a^5*b^3))*(((a^7*b^5)^(1/2) + a^4*b^2)/(16*(a^7*b - a^8)))^(1/2)*1i - ((((a^7*b^5)^(1/2) +
a^4*b^2)/(16*(a^7*b - a^8)))^(1/2)*(16*a^5*b^4 - 32*a^6*b^3 + 16*a^7*b^2 - tan(c + d*x)*(((a^7*b^5)^(1/2) + a
^4*b^2)/(16*(a^7*b - a^8)))^(1/2)*(64*a^9*b + 64*a^7*b^3 - 128*a^8*b^2)) + tan(c + d*x)*(4*a^3*b^5 - 4*a^5*b^3
))*(((a^7*b^5)^(1/2) + a^4*b^2)/(16*(a^7*b - a^8)))^(1/2)*1i)/(((((a^7*b^5)^(1/2) + a^4*b^2)/(16*(a^7*b - a^8)
))^(1/2)*(16*a^5*b^4 - 32*a^6*b^3 + 16*a^7*b^2 + tan(c + d*x)*(((a^7*b^5)^(1/2) + a^4*b^2)/(16*(a^7*b - a^8)))
^(1/2)*(64*a^9*b + 64*a^7*b^3 - 128*a^8*b^2)) - tan(c + d*x)*(4*a^3*b^5 - 4*a^5*b^3))*(((a^7*b^5)^(1/2) + a^4*
b^2)/(16*(a^7*b - a^8)))^(1/2) + ((((a^7*b^5)^(1/2) + a^4*b^2)/(16*(a^7*b - a^8)))^(1/2)*(16*a^5*b^4 - 32*a^6*
b^3 + 16*a^7*b^2 - tan(c + d*x)*(((a^7*b^5)^(1/2) + a^4*b^2)/(16*(a^7*b - a^8)))^(1/2)*(64*a^9*b + 64*a^7*b^3
- 128*a^8*b^2)) + tan(c + d*x)*(4*a^3*b^5 - 4*a^5*b^3))*(((a^7*b^5)^(1/2) + a^4*b^2)/(16*(a^7*b - a^8)))^(1/2)
- 2*a^2*b^5 + 2*a^3*b^4))*(((a^7*b^5)^(1/2) + a^4*b^2)/(16*(a^7*b - a^8)))^(1/2)*2i)/d + (atan((((-((a^7*b^5)
^(1/2) - a^4*b^2)/(16*(a^7*b - a^8)))^(1/2)*(16*a^5*b^4 - 32*a^6*b^3 + 16*a^7*b^2 + tan(c + d*x)*(-((a^7*b^5)^
(1/2) - a^4*b^2)/(16*(a^7*b - a^8)))^(1/2)*(64*a^9*b + 64*a^7*b^3 - 128*a^8*b^2)) - tan(c + d*x)*(4*a^3*b^5 -
4*a^5*b^3))*(-((a^7*b^5)^(1/2) - a^4*b^2)/(16*(a^7*b - a^8)))^(1/2)*1i - ((-((a^7*b^5)^(1/2) - a^4*b^2)/(16*(a
^7*b - a^8)))^(1/2)*(16*a^5*b^4 - 32*a^6*b^3 + 16*a^7*b^2 - tan(c + d*x)*(-((a^7*b^5)^(1/2) - a^4*b^2)/(16*(a^
7*b - a^8)))^(1/2)*(64*a^9*b + 64*a^7*b^3 - 128*a^8*b^2)) + tan(c + d*x)*(4*a^3*b^5 - 4*a^5*b^3))*(-((a^7*b^5)
^(1/2) - a^4*b^2)/(16*(a^7*b - a^8)))^(1/2)*1i)/(((-((a^7*b^5)^(1/2) - a^4*b^2)/(16*(a^7*b - a^8)))^(1/2)*(16*
a^5*b^4 - 32*a^6*b^3 + 16*a^7*b^2 + tan(c + d*x)*(-((a^7*b^5)^(1/2) - a^4*b^2)/(16*(a^7*b - a^8)))^(1/2)*(64*a
^9*b + 64*a^7*b^3 - 128*a^8*b^2)) - tan(c + d*x)*(4*a^3*b^5 - 4*a^5*b^3))*(-((a^7*b^5)^(1/2) - a^4*b^2)/(16*(a
^7*b - a^8)))^(1/2) + ((-((a^7*b^5)^(1/2) - a^4*b^2)/(16*(a^7*b - a^8)))^(1/2)*(16*a^5*b^4 - 32*a^6*b^3 + 16*a
^7*b^2 - tan(c + d*x)*(-((a^7*b^5)^(1/2) - a^4*b^2)/(16*(a^7*b - a^8)))^(1/2)*(64*a^9*b + 64*a^7*b^3 - 128*a^8
*b^2)) + tan(c + d*x)*(4*a^3*b^5 - 4*a^5*b^3))*(-((a^7*b^5)^(1/2) - a^4*b^2)/(16*(a^7*b - a^8)))^(1/2) - 2*a^2
*b^5 + 2*a^3*b^4))*(-((a^7*b^5)^(1/2) - a^4*b^2)/(16*(a^7*b - a^8)))^(1/2)*2i)/d - (1/(3*a) + tan(c + d*x)^2/a
)/(d*tan(c + d*x)^3)