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]

(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)

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Rubi [A]  time = 0.235356, antiderivative size = 197, normalized size of antiderivative = 1., 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 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]

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 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 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]

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*}

Mathematica [A]  time = 6.32048, 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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Maple [B]  time = 0.141, size = 624, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}

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]

1/2/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/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^2*b^4/(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^2*b^3/(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^2*b^3/(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^2*b^4/(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))-1/7/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

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Fricas [B]  time = 3.47417, size = 3502, normalized size = 17.78 \begin{align*} \text{result too large to display} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

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]

Exception raised: NotImplementedError