3.211 \(\int \frac {\csc ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx\)

Optimal. Leaf size=197 \[ \frac {b^2 \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{11/4} d \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {b^2 \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{11/4} d \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {(3 a+b) \cot ^3(c+d x)}{3 a^2 d}-\frac {(a+b) \cot (c+d x)}{a^2 d}-\frac {\cot ^7(c+d x)}{7 a d}-\frac {3 \cot ^5(c+d x)}{5 a d} \]

[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)

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Rubi [A]  time = 0.24, antiderivative size = 197, 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 = {3217, 1287, 1166, 205} \[ \frac {b^2 \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{11/4} d \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {b^2 \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{11/4} d \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {(3 a+b) \cot ^3(c+d x)}{3 a^2 d}-\frac {(a+b) \cot (c+d x)}{a^2 d}-\frac {\cot ^7(c+d x)}{7 a d}-\frac {3 \cot ^5(c+d x)}{5 a d} \]

Antiderivative was successfully verified.

[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 205

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

Rule 1166

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 1287

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 3217

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 ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx &=\frac {\operatorname {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 {\operatorname {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 \operatorname {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 ) \operatorname {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 ) \operatorname {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 \tan ^{-1}\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 \tan ^{-1}\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*}

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Mathematica [A]  time = 6.32, size = 277, normalized size = 1.41 \[ \frac {b^2 \tan ^{-1}\left (\frac {\left (\sqrt {a} \sqrt {b}+b\right ) \tan (c+d x)}{\sqrt {b} \sqrt {\sqrt {a} \sqrt {b}+a}}\right )}{2 a^{5/2} d \sqrt {\sqrt {a} \sqrt {b}+a}}-\frac {b^2 \tanh ^{-1}\left (\frac {\left (\sqrt {a} \sqrt {b}-b\right ) \tan (c+d x)}{\sqrt {b} \sqrt {\sqrt {a} \sqrt {b}-a}}\right )}{2 a^{5/2} d \sqrt {\sqrt {a} \sqrt {b}-a}}+\frac {\csc ^3(c+d x) (-24 a \cos (c+d x)-35 b \cos (c+d x))}{105 a^2 d}-\frac {2 \csc (c+d x) (24 a \cos (c+d x)+35 b \cos (c+d x))}{105 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d}-\frac {6 \cot (c+d x) \csc ^4(c+d x)}{35 a d} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]^8/(a - b*Sin[c + d*x]^4),x]

[Out]

(b^2*ArcTan[((Sqrt[a]*Sqrt[b] + b)*Tan[c + d*x])/(Sqrt[a + Sqrt[a]*Sqrt[b]]*Sqrt[b])])/(2*a^(5/2)*Sqrt[a + Sqr
t[a]*Sqrt[b]]*d) - (b^2*ArcTanh[((Sqrt[a]*Sqrt[b] - b)*Tan[c + d*x])/(Sqrt[-a + Sqrt[a]*Sqrt[b]]*Sqrt[b])])/(2
*a^(5/2)*Sqrt[-a + Sqrt[a]*Sqrt[b]]*d) - (2*(24*a*Cos[c + d*x] + 35*b*Cos[c + d*x])*Csc[c + d*x])/(105*a^2*d)
+ ((-24*a*Cos[c + d*x] - 35*b*Cos[c + d*x])*Csc[c + d*x]^3)/(105*a^2*d) - (6*Cot[c + d*x]*Csc[c + d*x]^4)/(35*
a*d) - (Cot[c + d*x]*Csc[c + d*x]^6)/(7*a*d)

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fricas [B]  time = 0.65, size = 1585, normalized size = 8.05 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [B]  time = 1.02, size = 467, normalized size = 2.37 \[ \frac {\frac {105 \, {\left (3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{2} b^{2} - 6 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a b^{3} - \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\tan \left (d x + c\right )}{\sqrt {\frac {a^{3} + \sqrt {a^{6} - {\left (a^{3} - a^{2} b\right )} a^{3}}}{a^{3} - a^{2} b}}}\right )\right )} {\left | a - b \right |}}{3 \, a^{7} - 12 \, a^{6} b + 14 \, a^{5} b^{2} - 4 \, a^{4} b^{3} - a^{3} b^{4}} + \frac {105 \, {\left (3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{2} b^{2} - 6 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a b^{3} - \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\tan \left (d x + c\right )}{\sqrt {\frac {a^{3} - \sqrt {a^{6} - {\left (a^{3} - a^{2} b\right )} a^{3}}}{a^{3} - a^{2} b}}}\right )\right )} {\left | a - b \right |}}{3 \, a^{7} - 12 \, a^{6} b + 14 \, a^{5} b^{2} - 4 \, a^{4} b^{3} - a^{3} b^{4}} - \frac {2 \, {\left (105 \, a \tan \left (d x + c\right )^{6} + 105 \, b \tan \left (d x + c\right )^{6} + 105 \, a \tan \left (d x + c\right )^{4} + 35 \, b \tan \left (d x + c\right )^{4} + 63 \, a \tan \left (d x + c\right )^{2} + 15 \, a\right )}}{a^{2} \tan \left (d x + c\right )^{7}}}{210 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [B]  time = 0.55, size = 624, normalized size = 3.17 \[ -\frac {1}{7 d a \tan \left (d x +c \right )^{7}}-\frac {1}{d a \tan \left (d x +c \right )}-\frac {b}{d \,a^{2} \tan \left (d x +c \right )}-\frac {1}{d a \tan \left (d x +c \right )^{3}}-\frac {b}{3 d \,a^{2} \tan \left (d x +c \right )^{3}}-\frac {3}{5 d a \tan \left (d x +c \right )^{5}}+\frac {b^{2} \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 d a \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}-\frac {b^{3} \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 d a \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}+\frac {b^{3} \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 d a \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {b^{2} \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 d a \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}-\frac {b^{3} \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 d \,a^{2} \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}+\frac {b^{4} \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 d \,a^{2} \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}-\frac {b^{4} \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 d \,a^{2} \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}-\frac {b^{3} \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 d \,a^{2} \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)^8/(a-b*sin(d*x+c)^4),x)

[Out]

-1/7/d/a/tan(d*x+c)^7-1/d/a/tan(d*x+c)-1/d/a^2/tan(d*x+c)*b-1/d/a/tan(d*x+c)^3-1/3/d/a^2/tan(d*x+c)^3*b-3/5/d/
a/tan(d*x+c)^5+1/2/d/a*b^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))-1/2/d/a*b^3/(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))+1/2/d/a*b^3/(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/d/a*b^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/d*b^3/a^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))+1/2/d*b^4/a^2/(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))-1/2/d*b^4/a^2/(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/d*b^3/a^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))

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^8/(a-b*sin(d*x+c)^4),x, algorithm="maxima")

[Out]

Timed out

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mupad [B]  time = 17.08, size = 1704, normalized size = 8.65 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)**8/(a-b*sin(d*x+c)**4),x)

[Out]

Timed out

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