3.51 \(\int \frac{1}{(a+b \csc (c+d x))^4} \, dx\)

Optimal. Leaf size=239 \[ \frac{b \left (-8 a^4 b^2+7 a^2 b^4+8 a^6-2 b^6\right ) \tanh ^{-1}\left (\frac{a+b \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{a^4 d \left (a^2-b^2\right )^{7/2}}-\frac{b^2 \left (-17 a^2 b^2+26 a^4+6 b^4\right ) \cot (c+d x)}{6 a^3 d \left (a^2-b^2\right )^3 (a+b \csc (c+d x))}-\frac{b^2 \left (8 a^2-3 b^2\right ) \cot (c+d x)}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \csc (c+d x))^2}-\frac{b^2 \cot (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \csc (c+d x))^3}+\frac{x}{a^4} \]

[Out]

x/a^4 + (b*(8*a^6 - 8*a^4*b^2 + 7*a^2*b^4 - 2*b^6)*ArcTanh[(a + b*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^4*(a^
2 - b^2)^(7/2)*d) - (b^2*Cot[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Csc[c + d*x])^3) - (b^2*(8*a^2 - 3*b^2)*Cot[c
+ d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Csc[c + d*x])^2) - (b^2*(26*a^4 - 17*a^2*b^2 + 6*b^4)*Cot[c + d*x])/(6*
a^3*(a^2 - b^2)^3*d*(a + b*Csc[c + d*x]))

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Rubi [A]  time = 0.50425, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {3785, 4060, 3919, 3831, 2660, 618, 206} \[ \frac{b \left (-8 a^4 b^2+7 a^2 b^4+8 a^6-2 b^6\right ) \tanh ^{-1}\left (\frac{a+b \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{a^4 d \left (a^2-b^2\right )^{7/2}}-\frac{b^2 \left (-17 a^2 b^2+26 a^4+6 b^4\right ) \cot (c+d x)}{6 a^3 d \left (a^2-b^2\right )^3 (a+b \csc (c+d x))}-\frac{b^2 \left (8 a^2-3 b^2\right ) \cot (c+d x)}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \csc (c+d x))^2}-\frac{b^2 \cot (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \csc (c+d x))^3}+\frac{x}{a^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Csc[c + d*x])^(-4),x]

[Out]

x/a^4 + (b*(8*a^6 - 8*a^4*b^2 + 7*a^2*b^4 - 2*b^6)*ArcTanh[(a + b*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^4*(a^
2 - b^2)^(7/2)*d) - (b^2*Cot[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Csc[c + d*x])^3) - (b^2*(8*a^2 - 3*b^2)*Cot[c
+ d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Csc[c + d*x])^2) - (b^2*(26*a^4 - 17*a^2*b^2 + 6*b^4)*Cot[c + d*x])/(6*
a^3*(a^2 - b^2)^3*d*(a + b*Csc[c + d*x]))

Rule 3785

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[(b^2*Cot[c + d*x]*(a + b*Csc[c + d*x])^(n +
1))/(a*d*(n + 1)*(a^2 - b^2)), x] + Dist[1/(a*(n + 1)*(a^2 - b^2)), Int[(a + b*Csc[c + d*x])^(n + 1)*Simp[(a^
2 - b^2)*(n + 1) - a*b*(n + 1)*Csc[c + d*x] + b^2*(n + 2)*Csc[c + d*x]^2, x], x], x] /; FreeQ[{a, b, c, d}, x]
&& NeQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]

Rule 4060

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) +
(a_))^(m_), x_Symbol] :> Simp[((A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1))/(a*f*(m + 1
)*(a^2 - b^2)), x] + Dist[1/(a*(m + 1)*(a^2 - b^2)), Int[(a + b*Csc[e + f*x])^(m + 1)*Simp[A*(a^2 - b^2)*(m +
1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m + 2)*Csc[e + f*x]^2, x], x], x] /;
FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]

Rule 3919

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(c*x)/a,
x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0]

Rule 3831

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[1/(1 + (a*Sin[e
+ f*x])/b), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 2660

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(2*e)/d, Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{(a+b \csc (c+d x))^4} \, dx &=-\frac{b^2 \cot (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \csc (c+d x))^3}-\frac{\int \frac{-3 \left (a^2-b^2\right )+3 a b \csc (c+d x)-2 b^2 \csc ^2(c+d x)}{(a+b \csc (c+d x))^3} \, dx}{3 a \left (a^2-b^2\right )}\\ &=-\frac{b^2 \cot (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \csc (c+d x))^3}-\frac{b^2 \left (8 a^2-3 b^2\right ) \cot (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \csc (c+d x))^2}+\frac{\int \frac{6 \left (a^2-b^2\right )^2-2 a b \left (6 a^2-b^2\right ) \csc (c+d x)+b^2 \left (8 a^2-3 b^2\right ) \csc ^2(c+d x)}{(a+b \csc (c+d x))^2} \, dx}{6 a^2 \left (a^2-b^2\right )^2}\\ &=-\frac{b^2 \cot (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \csc (c+d x))^3}-\frac{b^2 \left (8 a^2-3 b^2\right ) \cot (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \csc (c+d x))^2}-\frac{b^2 \left (26 a^4-17 a^2 b^2+6 b^4\right ) \cot (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \csc (c+d x))}-\frac{\int \frac{-6 \left (a^2-b^2\right )^3+3 a b \left (6 a^4-2 a^2 b^2+b^4\right ) \csc (c+d x)}{a+b \csc (c+d x)} \, dx}{6 a^3 \left (a^2-b^2\right )^3}\\ &=\frac{x}{a^4}-\frac{b^2 \cot (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \csc (c+d x))^3}-\frac{b^2 \left (8 a^2-3 b^2\right ) \cot (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \csc (c+d x))^2}-\frac{b^2 \left (26 a^4-17 a^2 b^2+6 b^4\right ) \cot (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \csc (c+d x))}-\frac{\left (b \left (8 a^6-8 a^4 b^2+7 a^2 b^4-2 b^6\right )\right ) \int \frac{\csc (c+d x)}{a+b \csc (c+d x)} \, dx}{2 a^4 \left (a^2-b^2\right )^3}\\ &=\frac{x}{a^4}-\frac{b^2 \cot (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \csc (c+d x))^3}-\frac{b^2 \left (8 a^2-3 b^2\right ) \cot (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \csc (c+d x))^2}-\frac{b^2 \left (26 a^4-17 a^2 b^2+6 b^4\right ) \cot (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \csc (c+d x))}-\frac{\left (8 a^6-8 a^4 b^2+7 a^2 b^4-2 b^6\right ) \int \frac{1}{1+\frac{a \sin (c+d x)}{b}} \, dx}{2 a^4 \left (a^2-b^2\right )^3}\\ &=\frac{x}{a^4}-\frac{b^2 \cot (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \csc (c+d x))^3}-\frac{b^2 \left (8 a^2-3 b^2\right ) \cot (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \csc (c+d x))^2}-\frac{b^2 \left (26 a^4-17 a^2 b^2+6 b^4\right ) \cot (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \csc (c+d x))}-\frac{\left (8 a^6-8 a^4 b^2+7 a^2 b^4-2 b^6\right ) \operatorname{Subst}\left (\int \frac{1}{1+\frac{2 a x}{b}+x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^4 \left (a^2-b^2\right )^3 d}\\ &=\frac{x}{a^4}-\frac{b^2 \cot (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \csc (c+d x))^3}-\frac{b^2 \left (8 a^2-3 b^2\right ) \cot (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \csc (c+d x))^2}-\frac{b^2 \left (26 a^4-17 a^2 b^2+6 b^4\right ) \cot (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \csc (c+d x))}+\frac{\left (2 \left (8 a^6-8 a^4 b^2+7 a^2 b^4-2 b^6\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left (1-\frac{a^2}{b^2}\right )-x^2} \, dx,x,\frac{2 a}{b}+2 \tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^4 \left (a^2-b^2\right )^3 d}\\ &=\frac{x}{a^4}+\frac{b \left (8 a^6-8 a^4 b^2+7 a^2 b^4-2 b^6\right ) \tanh ^{-1}\left (\frac{b \left (\frac{a}{b}+\tan \left (\frac{1}{2} (c+d x)\right )\right )}{\sqrt{a^2-b^2}}\right )}{a^4 \left (a^2-b^2\right )^{7/2} d}-\frac{b^2 \cot (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \csc (c+d x))^3}-\frac{b^2 \left (8 a^2-3 b^2\right ) \cot (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \csc (c+d x))^2}-\frac{b^2 \left (26 a^4-17 a^2 b^2+6 b^4\right ) \cot (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \csc (c+d x))}\\ \end{align*}

Mathematica [A]  time = 2.09468, size = 279, normalized size = 1.17 \[ \frac{\csc ^3(c+d x) (a \sin (c+d x)+b) \left (\frac{a b^3 \left (12 a^2-7 b^2\right ) \cot (c+d x) (a \sin (c+d x)+b)}{(a-b)^2 (a+b)^2}-\frac{a b^2 \left (-32 a^2 b^2+36 a^4+11 b^4\right ) \cot (c+d x) (a \sin (c+d x)+b)^2}{(a-b)^3 (a+b)^3}-\frac{6 b \left (8 a^4 b^2-7 a^2 b^4-8 a^6+2 b^6\right ) \csc (c+d x) (a \sin (c+d x)+b)^3 \tan ^{-1}\left (\frac{a+b \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b^2-a^2}}\right )}{\left (b^2-a^2\right )^{7/2}}+\frac{2 a b^4 \cot (c+d x)}{(b-a) (a+b)}+6 (c+d x) \csc (c+d x) (a \sin (c+d x)+b)^3\right )}{6 a^4 d (a+b \csc (c+d x))^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Csc[c + d*x]^3*(b + a*Sin[c + d*x])*((2*a*b^4*Cot[c + d*x])/((-a + b)*(a + b)) + (a*b^3*(12*a^2 - 7*b^2)*Cot[
c + d*x]*(b + a*Sin[c + d*x]))/((a - b)^2*(a + b)^2) - (a*b^2*(36*a^4 - 32*a^2*b^2 + 11*b^4)*Cot[c + d*x]*(b +
a*Sin[c + d*x])^2)/((a - b)^3*(a + b)^3) + 6*(c + d*x)*Csc[c + d*x]*(b + a*Sin[c + d*x])^3 - (6*b*(-8*a^6 + 8
*a^4*b^2 - 7*a^2*b^4 + 2*b^6)*ArcTan[(a + b*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]]*Csc[c + d*x]*(b + a*Sin[c + d*
x])^3)/(-a^2 + b^2)^(7/2)))/(6*a^4*d*(a + b*Csc[c + d*x])^4)

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Maple [B]  time = 0.147, size = 1912, normalized size = 8. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*csc(d*x+c))^4,x)

[Out]

-46/d*a^2*b^3/(tan(1/2*d*x+1/2*c)^2*b+2*a*tan(1/2*d*x+1/2*c)+b)^3/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)*tan(1/2*d*x+1/
2*c)-2/d/a^3*b^8/(tan(1/2*d*x+1/2*c)^2*b+2*a*tan(1/2*d*x+1/2*c)+b)^3/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)*tan(1/2*d*x
+1/2*c)^4-12/d/a^2*b^7/(tan(1/2*d*x+1/2*c)^2*b+2*a*tan(1/2*d*x+1/2*c)+b)^3/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)*tan(1
/2*d*x+1/2*c)^3+2/d/a^4*b^7/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/(-a^2+b^2)^(1/2)*arctan(1/2*(2*b*tan(1/2*d*x+1/2*c)+
2*a)/(-a^2+b^2)^(1/2))-7/d/a^2*b^5/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/(-a^2+b^2)^(1/2)*arctan(1/2*(2*b*tan(1/2*d*x+
1/2*c)+2*a)/(-a^2+b^2)^(1/2))-104/3/d*a^4*b/(tan(1/2*d*x+1/2*c)^2*b+2*a*tan(1/2*d*x+1/2*c)+b)^3/(a^6-3*a^4*b^2
+3*a^2*b^4-b^6)*tan(1/2*d*x+1/2*c)^3+17/3/d/a*b^6/(tan(1/2*d*x+1/2*c)^2*b+2*a*tan(1/2*d*x+1/2*c)+b)^3/(a^6-3*a
^4*b^2+3*a^2*b^4-b^6)-2/d/a^3*b^8/(tan(1/2*d*x+1/2*c)^2*b+2*a*tan(1/2*d*x+1/2*c)+b)^3/(a^6-3*a^4*b^2+3*a^2*b^4
-b^6)-8/d/a*b^6/(tan(1/2*d*x+1/2*c)^2*b+2*a*tan(1/2*d*x+1/2*c)+b)^3/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)*tan(1/2*d*x+
1/2*c)^2-4/d/a^3*b^8/(tan(1/2*d*x+1/2*c)^2*b+2*a*tan(1/2*d*x+1/2*c)+b)^3/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)*tan(1/2
*d*x+1/2*c)^2-28/d*a^3*b^2/(tan(1/2*d*x+1/2*c)^2*b+2*a*tan(1/2*d*x+1/2*c)+b)^3/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)*t
an(1/2*d*x+1/2*c)^4-11/d/a^2*b^7/(tan(1/2*d*x+1/2*c)^2*b+2*a*tan(1/2*d*x+1/2*c)+b)^3/(a^6-3*a^4*b^2+3*a^2*b^4-
b^6)*tan(1/2*d*x+1/2*c)+4/d*a*b^4/(tan(1/2*d*x+1/2*c)^2*b+2*a*tan(1/2*d*x+1/2*c)+b)^3/(a^6-3*a^4*b^2+3*a^2*b^4
-b^6)*tan(1/2*d*x+1/2*c)^4+1/d/a*b^6/(tan(1/2*d*x+1/2*c)^2*b+2*a*tan(1/2*d*x+1/2*c)+b)^3/(a^6-3*a^4*b^2+3*a^2*
b^4-b^6)*tan(1/2*d*x+1/2*c)^4-1/d/a^2*b^7/(tan(1/2*d*x+1/2*c)^2*b+2*a*tan(1/2*d*x+1/2*c)+b)^3/(a^6-3*a^4*b^2+3
*a^2*b^4-b^6)*tan(1/2*d*x+1/2*c)^5-76/d*a^3*b^2/(tan(1/2*d*x+1/2*c)^2*b+2*a*tan(1/2*d*x+1/2*c)+b)^3/(a^6-3*a^4
*b^2+3*a^2*b^4-b^6)*tan(1/2*d*x+1/2*c)^2+2/d/a^4*arctan(tan(1/2*d*x+1/2*c))+38/d*a*b^4/(tan(1/2*d*x+1/2*c)^2*b
+2*a*tan(1/2*d*x+1/2*c)+b)^3/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)*tan(1/2*d*x+1/2*c)^2-8/d*a^2*b/(a^6-3*a^4*b^2+3*a^2
*b^4-b^6)/(-a^2+b^2)^(1/2)*arctan(1/2*(2*b*tan(1/2*d*x+1/2*c)+2*a)/(-a^2+b^2)^(1/2))+8/d*b^3/(a^6-3*a^4*b^2+3*
a^2*b^4-b^6)/(-a^2+b^2)^(1/2)*arctan(1/2*(2*b*tan(1/2*d*x+1/2*c)+2*a)/(-a^2+b^2)^(1/2))+26/d*b^5/(tan(1/2*d*x+
1/2*c)^2*b+2*a*tan(1/2*d*x+1/2*c)+b)^3/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)*tan(1/2*d*x+1/2*c)^3-26/3/d*a*b^4/(tan(1/
2*d*x+1/2*c)^2*b+2*a*tan(1/2*d*x+1/2*c)+b)^3/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)+2/d*b^5/(tan(1/2*d*x+1/2*c)^2*b+2*a
*tan(1/2*d*x+1/2*c)+b)^3/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)*tan(1/2*d*x+1/2*c)^5+32/d*b^5/(tan(1/2*d*x+1/2*c)^2*b+2
*a*tan(1/2*d*x+1/2*c)+b)^3/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)*tan(1/2*d*x+1/2*c)-6/d*a^2*b^3/(tan(1/2*d*x+1/2*c)^2*
b+2*a*tan(1/2*d*x+1/2*c)+b)^3/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)*tan(1/2*d*x+1/2*c)^5-88/3/d*a^2*b^3/(tan(1/2*d*x+1
/2*c)^2*b+2*a*tan(1/2*d*x+1/2*c)+b)^3/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)*tan(1/2*d*x+1/2*c)^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*csc(d*x+c))^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 0.777519, size = 3389, normalized size = 14.18 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*csc(d*x+c))^4,x, algorithm="fricas")

[Out]

[1/12*(36*(a^10*b - 4*a^8*b^3 + 6*a^6*b^5 - 4*a^4*b^7 + a^2*b^9)*d*x*cos(d*x + c)^2 - 2*(36*a^9*b^2 - 68*a^7*b
^4 + 43*a^5*b^6 - 11*a^3*b^8)*cos(d*x + c)^3 - 12*(3*a^10*b - 11*a^8*b^3 + 14*a^6*b^5 - 6*a^4*b^7 - a^2*b^9 +
b^11)*d*x - 3*(24*a^8*b^2 - 16*a^6*b^4 + 13*a^4*b^6 + a^2*b^8 - 2*b^10 - 3*(8*a^8*b^2 - 8*a^6*b^4 + 7*a^4*b^6
- 2*a^2*b^8)*cos(d*x + c)^2 + (8*a^9*b + 16*a^7*b^3 - 17*a^5*b^5 + 19*a^3*b^7 - 6*a*b^9 - (8*a^9*b - 8*a^7*b^3
+ 7*a^5*b^5 - 2*a^3*b^7)*cos(d*x + c)^2)*sin(d*x + c))*sqrt(a^2 - b^2)*log(((a^2 - 2*b^2)*cos(d*x + c)^2 + 2*
a*b*sin(d*x + c) + a^2 + b^2 + 2*(b*cos(d*x + c)*sin(d*x + c) + a*cos(d*x + c))*sqrt(a^2 - b^2))/(a^2*cos(d*x
+ c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2)) + 12*(6*a^9*b^2 - 7*a^7*b^4 + 2*a^3*b^8 - a*b^10)*cos(d*x + c) + 6*(
2*(a^11 - 4*a^9*b^2 + 6*a^7*b^4 - 4*a^5*b^6 + a^3*b^8)*d*x*cos(d*x + c)^2 - 2*(a^11 - a^9*b^2 - 6*a^7*b^4 + 14
*a^5*b^6 - 11*a^3*b^8 + 3*a*b^10)*d*x + 5*(4*a^8*b^3 - 7*a^6*b^5 + 4*a^4*b^7 - a^2*b^9)*cos(d*x + c))*sin(d*x
+ c))/(3*(a^14*b - 4*a^12*b^3 + 6*a^10*b^5 - 4*a^8*b^7 + a^6*b^9)*d*cos(d*x + c)^2 - (3*a^14*b - 11*a^12*b^3 +
14*a^10*b^5 - 6*a^8*b^7 - a^6*b^9 + a^4*b^11)*d + ((a^15 - 4*a^13*b^2 + 6*a^11*b^4 - 4*a^9*b^6 + a^7*b^8)*d*c
os(d*x + c)^2 - (a^15 - a^13*b^2 - 6*a^11*b^4 + 14*a^9*b^6 - 11*a^7*b^8 + 3*a^5*b^10)*d)*sin(d*x + c)), 1/6*(1
8*(a^10*b - 4*a^8*b^3 + 6*a^6*b^5 - 4*a^4*b^7 + a^2*b^9)*d*x*cos(d*x + c)^2 - (36*a^9*b^2 - 68*a^7*b^4 + 43*a^
5*b^6 - 11*a^3*b^8)*cos(d*x + c)^3 - 6*(3*a^10*b - 11*a^8*b^3 + 14*a^6*b^5 - 6*a^4*b^7 - a^2*b^9 + b^11)*d*x -
3*(24*a^8*b^2 - 16*a^6*b^4 + 13*a^4*b^6 + a^2*b^8 - 2*b^10 - 3*(8*a^8*b^2 - 8*a^6*b^4 + 7*a^4*b^6 - 2*a^2*b^8
)*cos(d*x + c)^2 + (8*a^9*b + 16*a^7*b^3 - 17*a^5*b^5 + 19*a^3*b^7 - 6*a*b^9 - (8*a^9*b - 8*a^7*b^3 + 7*a^5*b^
5 - 2*a^3*b^7)*cos(d*x + c)^2)*sin(d*x + c))*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*sin(d*x + c) + a)/((
a^2 - b^2)*cos(d*x + c))) + 6*(6*a^9*b^2 - 7*a^7*b^4 + 2*a^3*b^8 - a*b^10)*cos(d*x + c) + 3*(2*(a^11 - 4*a^9*b
^2 + 6*a^7*b^4 - 4*a^5*b^6 + a^3*b^8)*d*x*cos(d*x + c)^2 - 2*(a^11 - a^9*b^2 - 6*a^7*b^4 + 14*a^5*b^6 - 11*a^3
*b^8 + 3*a*b^10)*d*x + 5*(4*a^8*b^3 - 7*a^6*b^5 + 4*a^4*b^7 - a^2*b^9)*cos(d*x + c))*sin(d*x + c))/(3*(a^14*b
- 4*a^12*b^3 + 6*a^10*b^5 - 4*a^8*b^7 + a^6*b^9)*d*cos(d*x + c)^2 - (3*a^14*b - 11*a^12*b^3 + 14*a^10*b^5 - 6*
a^8*b^7 - a^6*b^9 + a^4*b^11)*d + ((a^15 - 4*a^13*b^2 + 6*a^11*b^4 - 4*a^9*b^6 + a^7*b^8)*d*cos(d*x + c)^2 - (
a^15 - a^13*b^2 - 6*a^11*b^4 + 14*a^9*b^6 - 11*a^7*b^8 + 3*a^5*b^10)*d)*sin(d*x + c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b \csc{\left (c + d x \right )}\right )^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*csc(d*x+c))**4,x)

[Out]

Integral((a + b*csc(c + d*x))**(-4), x)

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Giac [B]  time = 1.39036, size = 722, normalized size = 3.02 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*csc(d*x+c))^4,x, algorithm="giac")

[Out]

-1/3*(3*(8*a^6*b - 8*a^4*b^3 + 7*a^2*b^5 - 2*b^7)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(b) + arctan((b*tan(1/2
*d*x + 1/2*c) + a)/sqrt(-a^2 + b^2)))/((a^10 - 3*a^8*b^2 + 3*a^6*b^4 - a^4*b^6)*sqrt(-a^2 + b^2)) + (18*a^5*b^
3*tan(1/2*d*x + 1/2*c)^5 - 6*a^3*b^5*tan(1/2*d*x + 1/2*c)^5 + 3*a*b^7*tan(1/2*d*x + 1/2*c)^5 + 84*a^6*b^2*tan(
1/2*d*x + 1/2*c)^4 - 12*a^4*b^4*tan(1/2*d*x + 1/2*c)^4 - 3*a^2*b^6*tan(1/2*d*x + 1/2*c)^4 + 6*b^8*tan(1/2*d*x
+ 1/2*c)^4 + 104*a^7*b*tan(1/2*d*x + 1/2*c)^3 + 88*a^5*b^3*tan(1/2*d*x + 1/2*c)^3 - 78*a^3*b^5*tan(1/2*d*x + 1
/2*c)^3 + 36*a*b^7*tan(1/2*d*x + 1/2*c)^3 + 228*a^6*b^2*tan(1/2*d*x + 1/2*c)^2 - 114*a^4*b^4*tan(1/2*d*x + 1/2
*c)^2 + 24*a^2*b^6*tan(1/2*d*x + 1/2*c)^2 + 12*b^8*tan(1/2*d*x + 1/2*c)^2 + 138*a^5*b^3*tan(1/2*d*x + 1/2*c) -
96*a^3*b^5*tan(1/2*d*x + 1/2*c) + 33*a*b^7*tan(1/2*d*x + 1/2*c) + 26*a^4*b^4 - 17*a^2*b^6 + 6*b^8)/((a^9 - 3*
a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*(b*tan(1/2*d*x + 1/2*c)^2 + 2*a*tan(1/2*d*x + 1/2*c) + b)^3) - 3*(d*x + c)/a^4)
/d