3.200 \(\int \frac {\csc ^2(x)}{(a+b \sin (x))^3} \, dx\)

Optimal. Leaf size=187 \[ \frac {3 b \tanh ^{-1}(\cos (x))}{a^4}-\frac {3 b^2 \left (2 a^2-b^2\right ) \cot (x)}{2 a^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}-\frac {b^2 \cot (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}+\frac {3 b^2 \left (4 a^4-5 a^2 b^2+2 b^4\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2}}\right )}{a^4 \left (a^2-b^2\right )^{5/2}}-\frac {\left (2 a^4-11 a^2 b^2+6 b^4\right ) \cot (x)}{2 a^3 \left (a^2-b^2\right )^2} \]

[Out]

3*b^2*(4*a^4-5*a^2*b^2+2*b^4)*arctan((b+a*tan(1/2*x))/(a^2-b^2)^(1/2))/a^4/(a^2-b^2)^(5/2)+3*b*arctanh(cos(x))
/a^4-1/2*(2*a^4-11*a^2*b^2+6*b^4)*cot(x)/a^3/(a^2-b^2)^2-1/2*b^2*cot(x)/a/(a^2-b^2)/(a+b*sin(x))^2-3/2*b^2*(2*
a^2-b^2)*cot(x)/a^2/(a^2-b^2)^2/(a+b*sin(x))

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Rubi [A]  time = 0.64, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {2802, 3055, 3001, 3770, 2660, 618, 204} \[ \frac {3 b^2 \left (-5 a^2 b^2+4 a^4+2 b^4\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2}}\right )}{a^4 \left (a^2-b^2\right )^{5/2}}-\frac {\left (-11 a^2 b^2+2 a^4+6 b^4\right ) \cot (x)}{2 a^3 \left (a^2-b^2\right )^2}-\frac {3 b^2 \left (2 a^2-b^2\right ) \cot (x)}{2 a^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}-\frac {b^2 \cot (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}+\frac {3 b \tanh ^{-1}(\cos (x))}{a^4} \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]^2/(a + b*Sin[x])^3,x]

[Out]

(3*b^2*(4*a^4 - 5*a^2*b^2 + 2*b^4)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^4*(a^2 - b^2)^(5/2)) + (3*b*Ar
cTanh[Cos[x]])/a^4 - ((2*a^4 - 11*a^2*b^2 + 6*b^4)*Cot[x])/(2*a^3*(a^2 - b^2)^2) - (b^2*Cot[x])/(2*a*(a^2 - b^
2)*(a + b*Sin[x])^2) - (3*b^2*(2*a^2 - b^2)*Cot[x])/(2*a^2*(a^2 - b^2)^2*(a + b*Sin[x]))

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 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 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 2802

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -S
imp[(b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 -
b^2)), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n
*Simp[a*(b*c - a*d)*(m + 1) + b^2*d*(m + n + 2) - (b^2*c + b*(b*c - a*d)*(m + 1))*Sin[e + f*x] - b^2*d*(m + n
+ 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] &
& NeQ[c^2 - d^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*n] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !
(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) || EqQ[a, 0])))

Rule 3001

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)])), x_Symbol] :> Dist[(A*b - a*B)/(b*c - a*d), Int[1/(a + b*Sin[e + f*x]), x], x] + Dist[(B*c - A
*d)/(b*c - a*d), Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0]
&& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 3055

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((A*b^2 - a*b*B + a^2*C)*Cos[e +
f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)), x] + Dis
t[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b
*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*
c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /;
FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && Lt
Q[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&
!IntegerQ[m]) || EqQ[a, 0])))

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin {align*} \int \frac {\csc ^2(x)}{(a+b \sin (x))^3} \, dx &=-\frac {b^2 \cot (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}+\frac {\int \frac {\csc ^2(x) \left (2 a^2-3 b^2-2 a b \sin (x)+2 b^2 \sin ^2(x)\right )}{(a+b \sin (x))^2} \, dx}{2 a \left (a^2-b^2\right )}\\ &=-\frac {b^2 \cot (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {3 b^2 \left (2 a^2-b^2\right ) \cot (x)}{2 a^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}+\frac {\int \frac {\csc ^2(x) \left (2 a^4-11 a^2 b^2+6 b^4-a b \left (4 a^2-b^2\right ) \sin (x)+3 b^2 \left (2 a^2-b^2\right ) \sin ^2(x)\right )}{a+b \sin (x)} \, dx}{2 a^2 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (2 a^4-11 a^2 b^2+6 b^4\right ) \cot (x)}{2 a^3 \left (a^2-b^2\right )^2}-\frac {b^2 \cot (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {3 b^2 \left (2 a^2-b^2\right ) \cot (x)}{2 a^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}+\frac {\int \frac {\csc (x) \left (-6 b \left (a^2-b^2\right )^2+3 a b^2 \left (2 a^2-b^2\right ) \sin (x)\right )}{a+b \sin (x)} \, dx}{2 a^3 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (2 a^4-11 a^2 b^2+6 b^4\right ) \cot (x)}{2 a^3 \left (a^2-b^2\right )^2}-\frac {b^2 \cot (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {3 b^2 \left (2 a^2-b^2\right ) \cot (x)}{2 a^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}-\frac {(3 b) \int \csc (x) \, dx}{a^4}+\frac {\left (3 b^2 \left (4 a^4-5 a^2 b^2+2 b^4\right )\right ) \int \frac {1}{a+b \sin (x)} \, dx}{2 a^4 \left (a^2-b^2\right )^2}\\ &=\frac {3 b \tanh ^{-1}(\cos (x))}{a^4}-\frac {\left (2 a^4-11 a^2 b^2+6 b^4\right ) \cot (x)}{2 a^3 \left (a^2-b^2\right )^2}-\frac {b^2 \cot (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {3 b^2 \left (2 a^2-b^2\right ) \cot (x)}{2 a^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}+\frac {\left (3 b^2 \left (4 a^4-5 a^2 b^2+2 b^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{a^4 \left (a^2-b^2\right )^2}\\ &=\frac {3 b \tanh ^{-1}(\cos (x))}{a^4}-\frac {\left (2 a^4-11 a^2 b^2+6 b^4\right ) \cot (x)}{2 a^3 \left (a^2-b^2\right )^2}-\frac {b^2 \cot (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {3 b^2 \left (2 a^2-b^2\right ) \cot (x)}{2 a^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}-\frac {\left (6 b^2 \left (4 a^4-5 a^2 b^2+2 b^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {x}{2}\right )\right )}{a^4 \left (a^2-b^2\right )^2}\\ &=\frac {3 b^2 \left (4 a^4-5 a^2 b^2+2 b^4\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{a^4 \left (a^2-b^2\right )^{5/2}}+\frac {3 b \tanh ^{-1}(\cos (x))}{a^4}-\frac {\left (2 a^4-11 a^2 b^2+6 b^4\right ) \cot (x)}{2 a^3 \left (a^2-b^2\right )^2}-\frac {b^2 \cot (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {3 b^2 \left (2 a^2-b^2\right ) \cot (x)}{2 a^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}\\ \end {align*}

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Mathematica [A]  time = 1.38, size = 174, normalized size = 0.93 \[ \frac {\frac {a^2 b^3 \cos (x)}{(a-b) (a+b) (a+b \sin (x))^2}+\frac {a b^3 \left (7 a^2-4 b^2\right ) \cos (x)}{(a-b)^2 (a+b)^2 (a+b \sin (x))}+\frac {6 b^2 \left (4 a^4-5 a^2 b^2+2 b^4\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+a \tan \left (\frac {x}{2}\right )-a \cot \left (\frac {x}{2}\right )-6 b \log \left (\sin \left (\frac {x}{2}\right )\right )+6 b \log \left (\cos \left (\frac {x}{2}\right )\right )}{2 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]^2/(a + b*Sin[x])^3,x]

[Out]

((6*b^2*(4*a^4 - 5*a^2*b^2 + 2*b^4)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) - a*Cot[x/2] +
6*b*Log[Cos[x/2]] - 6*b*Log[Sin[x/2]] + (a^2*b^3*Cos[x])/((a - b)*(a + b)*(a + b*Sin[x])^2) + (a*b^3*(7*a^2 -
4*b^2)*Cos[x])/((a - b)^2*(a + b)^2*(a + b*Sin[x])) + a*Tan[x/2])/(2*a^4)

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fricas [B]  time = 1.42, size = 1436, normalized size = 7.68 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^2/(a+b*sin(x))^3,x, algorithm="fricas")

[Out]

[1/4*(2*(2*a^7*b^2 - 13*a^5*b^4 + 17*a^3*b^6 - 6*a*b^8)*cos(x)^3 - 2*(4*a^8*b - 20*a^6*b^3 + 25*a^4*b^5 - 9*a^
2*b^7)*cos(x)*sin(x) - 3*(8*a^5*b^3 - 10*a^3*b^5 + 4*a*b^7 - 2*(4*a^5*b^3 - 5*a^3*b^5 + 2*a*b^7)*cos(x)^2 + (4
*a^6*b^2 - a^4*b^4 - 3*a^2*b^6 + 2*b^8 - (4*a^4*b^4 - 5*a^2*b^6 + 2*b^8)*cos(x)^2)*sin(x))*sqrt(-a^2 + b^2)*lo
g(((2*a^2 - b^2)*cos(x)^2 - 2*a*b*sin(x) - a^2 - b^2 + 2*(a*cos(x)*sin(x) + b*cos(x))*sqrt(-a^2 + b^2))/(b^2*c
os(x)^2 - 2*a*b*sin(x) - a^2 - b^2)) - 2*(2*a^9 - 4*a^7*b^2 - 7*a^5*b^4 + 15*a^3*b^6 - 6*a*b^8)*cos(x) + 6*(2*
a^7*b^2 - 6*a^5*b^4 + 6*a^3*b^6 - 2*a*b^8 - 2*(a^7*b^2 - 3*a^5*b^4 + 3*a^3*b^6 - a*b^8)*cos(x)^2 + (a^8*b - 2*
a^6*b^3 + 2*a^2*b^7 - b^9 - (a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^7 - b^9)*cos(x)^2)*sin(x))*log(1/2*cos(x) + 1/2) -
6*(2*a^7*b^2 - 6*a^5*b^4 + 6*a^3*b^6 - 2*a*b^8 - 2*(a^7*b^2 - 3*a^5*b^4 + 3*a^3*b^6 - a*b^8)*cos(x)^2 + (a^8*b
- 2*a^6*b^3 + 2*a^2*b^7 - b^9 - (a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^7 - b^9)*cos(x)^2)*sin(x))*log(-1/2*cos(x) + 1
/2))/(2*a^11*b - 6*a^9*b^3 + 6*a^7*b^5 - 2*a^5*b^7 - 2*(a^11*b - 3*a^9*b^3 + 3*a^7*b^5 - a^5*b^7)*cos(x)^2 + (
a^12 - 2*a^10*b^2 + 2*a^6*b^6 - a^4*b^8 - (a^10*b^2 - 3*a^8*b^4 + 3*a^6*b^6 - a^4*b^8)*cos(x)^2)*sin(x)), 1/2*
((2*a^7*b^2 - 13*a^5*b^4 + 17*a^3*b^6 - 6*a*b^8)*cos(x)^3 - (4*a^8*b - 20*a^6*b^3 + 25*a^4*b^5 - 9*a^2*b^7)*co
s(x)*sin(x) - 3*(8*a^5*b^3 - 10*a^3*b^5 + 4*a*b^7 - 2*(4*a^5*b^3 - 5*a^3*b^5 + 2*a*b^7)*cos(x)^2 + (4*a^6*b^2
- a^4*b^4 - 3*a^2*b^6 + 2*b^8 - (4*a^4*b^4 - 5*a^2*b^6 + 2*b^8)*cos(x)^2)*sin(x))*sqrt(a^2 - b^2)*arctan(-(a*s
in(x) + b)/(sqrt(a^2 - b^2)*cos(x))) - (2*a^9 - 4*a^7*b^2 - 7*a^5*b^4 + 15*a^3*b^6 - 6*a*b^8)*cos(x) + 3*(2*a^
7*b^2 - 6*a^5*b^4 + 6*a^3*b^6 - 2*a*b^8 - 2*(a^7*b^2 - 3*a^5*b^4 + 3*a^3*b^6 - a*b^8)*cos(x)^2 + (a^8*b - 2*a^
6*b^3 + 2*a^2*b^7 - b^9 - (a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^7 - b^9)*cos(x)^2)*sin(x))*log(1/2*cos(x) + 1/2) - 3*
(2*a^7*b^2 - 6*a^5*b^4 + 6*a^3*b^6 - 2*a*b^8 - 2*(a^7*b^2 - 3*a^5*b^4 + 3*a^3*b^6 - a*b^8)*cos(x)^2 + (a^8*b -
2*a^6*b^3 + 2*a^2*b^7 - b^9 - (a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^7 - b^9)*cos(x)^2)*sin(x))*log(-1/2*cos(x) + 1/2
))/(2*a^11*b - 6*a^9*b^3 + 6*a^7*b^5 - 2*a^5*b^7 - 2*(a^11*b - 3*a^9*b^3 + 3*a^7*b^5 - a^5*b^7)*cos(x)^2 + (a^
12 - 2*a^10*b^2 + 2*a^6*b^6 - a^4*b^8 - (a^10*b^2 - 3*a^8*b^4 + 3*a^6*b^6 - a^4*b^8)*cos(x)^2)*sin(x))]

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giac [A]  time = 0.23, size = 280, normalized size = 1.50 \[ \frac {3 \, {\left (4 \, a^{4} b^{2} - 5 \, a^{2} b^{4} + 2 \, b^{6}\right )} {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (a) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, x\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{8} - 2 \, a^{6} b^{2} + a^{4} b^{4}\right )} \sqrt {a^{2} - b^{2}}} + \frac {9 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, x\right )^{3} - 6 \, a b^{6} \tan \left (\frac {1}{2} \, x\right )^{3} + 8 \, a^{4} b^{3} \tan \left (\frac {1}{2} \, x\right )^{2} + 11 \, a^{2} b^{5} \tan \left (\frac {1}{2} \, x\right )^{2} - 10 \, b^{7} \tan \left (\frac {1}{2} \, x\right )^{2} + 23 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, x\right ) - 14 \, a b^{6} \tan \left (\frac {1}{2} \, x\right ) + 8 \, a^{4} b^{3} - 5 \, a^{2} b^{5}}{{\left (a^{8} - 2 \, a^{6} b^{2} + a^{4} b^{4}\right )} {\left (a \tan \left (\frac {1}{2} \, x\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, x\right ) + a\right )}^{2}} - \frac {3 \, b \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{a^{4}} + \frac {\tan \left (\frac {1}{2} \, x\right )}{2 \, a^{3}} + \frac {6 \, b \tan \left (\frac {1}{2} \, x\right ) - a}{2 \, a^{4} \tan \left (\frac {1}{2} \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^2/(a+b*sin(x))^3,x, algorithm="giac")

[Out]

3*(4*a^4*b^2 - 5*a^2*b^4 + 2*b^6)*(pi*floor(1/2*x/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*x) + b)/sqrt(a^2 - b^2)
))/((a^8 - 2*a^6*b^2 + a^4*b^4)*sqrt(a^2 - b^2)) + (9*a^3*b^4*tan(1/2*x)^3 - 6*a*b^6*tan(1/2*x)^3 + 8*a^4*b^3*
tan(1/2*x)^2 + 11*a^2*b^5*tan(1/2*x)^2 - 10*b^7*tan(1/2*x)^2 + 23*a^3*b^4*tan(1/2*x) - 14*a*b^6*tan(1/2*x) + 8
*a^4*b^3 - 5*a^2*b^5)/((a^8 - 2*a^6*b^2 + a^4*b^4)*(a*tan(1/2*x)^2 + 2*b*tan(1/2*x) + a)^2) - 3*b*log(abs(tan(
1/2*x)))/a^4 + 1/2*tan(1/2*x)/a^3 + 1/2*(6*b*tan(1/2*x) - a)/(a^4*tan(1/2*x))

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maple [B]  time = 0.16, size = 641, normalized size = 3.43 \[ \frac {\tan \left (\frac {x}{2}\right )}{2 a^{3}}-\frac {1}{2 a^{3} \tan \left (\frac {x}{2}\right )}-\frac {3 b \ln \left (\tan \left (\frac {x}{2}\right )\right )}{a^{4}}+\frac {9 b^{4} \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{a \left (\left (\tan ^{2}\left (\frac {x}{2}\right )\right ) a +2 \tan \left (\frac {x}{2}\right ) b +a \right )^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {6 b^{6} \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{a^{3} \left (\left (\tan ^{2}\left (\frac {x}{2}\right )\right ) a +2 \tan \left (\frac {x}{2}\right ) b +a \right )^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {8 b^{3} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{\left (\left (\tan ^{2}\left (\frac {x}{2}\right )\right ) a +2 \tan \left (\frac {x}{2}\right ) b +a \right )^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {11 b^{5} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{a^{2} \left (\left (\tan ^{2}\left (\frac {x}{2}\right )\right ) a +2 \tan \left (\frac {x}{2}\right ) b +a \right )^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {10 b^{7} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{a^{4} \left (\left (\tan ^{2}\left (\frac {x}{2}\right )\right ) a +2 \tan \left (\frac {x}{2}\right ) b +a \right )^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {23 b^{4} \tan \left (\frac {x}{2}\right )}{a \left (\left (\tan ^{2}\left (\frac {x}{2}\right )\right ) a +2 \tan \left (\frac {x}{2}\right ) b +a \right )^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {14 b^{6} \tan \left (\frac {x}{2}\right )}{a^{3} \left (\left (\tan ^{2}\left (\frac {x}{2}\right )\right ) a +2 \tan \left (\frac {x}{2}\right ) b +a \right )^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {8 b^{3}}{\left (\left (\tan ^{2}\left (\frac {x}{2}\right )\right ) a +2 \tan \left (\frac {x}{2}\right ) b +a \right )^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {5 b^{5}}{a^{2} \left (\left (\tan ^{2}\left (\frac {x}{2}\right )\right ) a +2 \tan \left (\frac {x}{2}\right ) b +a \right )^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {12 \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right ) b^{2}}{\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {a^{2}-b^{2}}}-\frac {15 b^{4} \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {a^{2}-b^{2}}}+\frac {6 b^{6} \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{4} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {a^{2}-b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(x)^2/(a+b*sin(x))^3,x)

[Out]

1/2/a^3*tan(1/2*x)-1/2/a^3/tan(1/2*x)-3/a^4*b*ln(tan(1/2*x))+9/a*b^4/(tan(1/2*x)^2*a+2*tan(1/2*x)*b+a)^2/(a^4-
2*a^2*b^2+b^4)*tan(1/2*x)^3-6/a^3*b^6/(tan(1/2*x)^2*a+2*tan(1/2*x)*b+a)^2/(a^4-2*a^2*b^2+b^4)*tan(1/2*x)^3+8*b
^3/(tan(1/2*x)^2*a+2*tan(1/2*x)*b+a)^2/(a^4-2*a^2*b^2+b^4)*tan(1/2*x)^2+11/a^2*b^5/(tan(1/2*x)^2*a+2*tan(1/2*x
)*b+a)^2/(a^4-2*a^2*b^2+b^4)*tan(1/2*x)^2-10/a^4*b^7/(tan(1/2*x)^2*a+2*tan(1/2*x)*b+a)^2/(a^4-2*a^2*b^2+b^4)*t
an(1/2*x)^2+23/a*b^4/(tan(1/2*x)^2*a+2*tan(1/2*x)*b+a)^2/(a^4-2*a^2*b^2+b^4)*tan(1/2*x)-14/a^3*b^6/(tan(1/2*x)
^2*a+2*tan(1/2*x)*b+a)^2/(a^4-2*a^2*b^2+b^4)*tan(1/2*x)+8*b^3/(tan(1/2*x)^2*a+2*tan(1/2*x)*b+a)^2/(a^4-2*a^2*b
^2+b^4)-5/a^2*b^5/(tan(1/2*x)^2*a+2*tan(1/2*x)*b+a)^2/(a^4-2*a^2*b^2+b^4)+12/(a^4-2*a^2*b^2+b^4)/(a^2-b^2)^(1/
2)*arctan(1/2*(2*a*tan(1/2*x)+2*b)/(a^2-b^2)^(1/2))*b^2-15/a^2*b^4/(a^4-2*a^2*b^2+b^4)/(a^2-b^2)^(1/2)*arctan(
1/2*(2*a*tan(1/2*x)+2*b)/(a^2-b^2)^(1/2))+6/a^4*b^6/(a^4-2*a^2*b^2+b^4)/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/
2*x)+2*b)/(a^2-b^2)^(1/2))

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^2/(a+b*sin(x))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`
for more details)Is 4*b^2-4*a^2 positive or negative?

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mupad [B]  time = 9.02, size = 2295, normalized size = 12.27 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(x)^2*(a + b*sin(x))^3),x)

[Out]

tan(x/2)/(2*a^3) - (a^2 + (2*tan(x/2)*(7*a*b^5 + 2*a^5*b - 12*a^3*b^3))/(a^4 + b^4 - 2*a^2*b^2) + (tan(x/2)^4*
(a^6 + 12*b^6 - 17*a^2*b^4 - 2*a^4*b^2))/(a^4 + b^4 - 2*a^2*b^2) + (2*tan(x/2)^2*(a^6 + 16*b^6 - 26*a^2*b^4))/
(a^4 + b^4 - 2*a^2*b^2) + (2*tan(x/2)^3*(2*a^6*b + 10*b^7 - 9*a^2*b^5 - 12*a^4*b^3))/(a*(a^4 + b^4 - 2*a^2*b^2
)))/(tan(x/2)^3*(4*a^5 + 8*a^3*b^2) + 2*a^5*tan(x/2) + 2*a^5*tan(x/2)^5 + 8*a^4*b*tan(x/2)^2 + 8*a^4*b*tan(x/2
)^4) - (3*b*log(tan(x/2)))/a^4 - (b^2*atan(((b^2*(-(a + b)^5*(a - b)^5)^(1/2)*(4*a^4 + 2*b^4 - 5*a^2*b^2)*((12
*a^4*b^6 - 27*a^6*b^4 + 18*a^8*b^2)/(a^10 + a^6*b^4 - 2*a^8*b^2) - (tan(x/2)*(6*a^12*b - 24*a^2*b^11 + 108*a^4
*b^9 - 192*a^6*b^7 + 162*a^8*b^5 - 60*a^10*b^3))/(a^13 + a^5*b^8 - 4*a^7*b^6 + 6*a^9*b^4 - 4*a^11*b^2) + (3*b^
2*((2*a^12*b + 2*a^8*b^5 - 4*a^10*b^3)/(a^10 + a^6*b^4 - 2*a^8*b^2) - (tan(x/2)*(6*a^16 - 8*a^6*b^10 + 38*a^8*
b^8 - 72*a^10*b^6 + 68*a^12*b^4 - 32*a^14*b^2))/(a^13 + a^5*b^8 - 4*a^7*b^6 + 6*a^9*b^4 - 4*a^11*b^2))*(-(a +
b)^5*(a - b)^5)^(1/2)*(4*a^4 + 2*b^4 - 5*a^2*b^2))/(2*(a^14 - a^4*b^10 + 5*a^6*b^8 - 10*a^8*b^6 + 10*a^10*b^4
- 5*a^12*b^2)))*3i)/(2*(a^14 - a^4*b^10 + 5*a^6*b^8 - 10*a^8*b^6 + 10*a^10*b^4 - 5*a^12*b^2)) - (b^2*(-(a + b)
^5*(a - b)^5)^(1/2)*(4*a^4 + 2*b^4 - 5*a^2*b^2)*((tan(x/2)*(6*a^12*b - 24*a^2*b^11 + 108*a^4*b^9 - 192*a^6*b^7
+ 162*a^8*b^5 - 60*a^10*b^3))/(a^13 + a^5*b^8 - 4*a^7*b^6 + 6*a^9*b^4 - 4*a^11*b^2) - (12*a^4*b^6 - 27*a^6*b^
4 + 18*a^8*b^2)/(a^10 + a^6*b^4 - 2*a^8*b^2) + (3*b^2*((2*a^12*b + 2*a^8*b^5 - 4*a^10*b^3)/(a^10 + a^6*b^4 - 2
*a^8*b^2) - (tan(x/2)*(6*a^16 - 8*a^6*b^10 + 38*a^8*b^8 - 72*a^10*b^6 + 68*a^12*b^4 - 32*a^14*b^2))/(a^13 + a^
5*b^8 - 4*a^7*b^6 + 6*a^9*b^4 - 4*a^11*b^2))*(-(a + b)^5*(a - b)^5)^(1/2)*(4*a^4 + 2*b^4 - 5*a^2*b^2))/(2*(a^1
4 - a^4*b^10 + 5*a^6*b^8 - 10*a^8*b^6 + 10*a^10*b^4 - 5*a^12*b^2)))*3i)/(2*(a^14 - a^4*b^10 + 5*a^6*b^8 - 10*a
^8*b^6 + 10*a^10*b^4 - 5*a^12*b^2)))/((2*(18*b^7 - 45*a^2*b^5 + 36*a^4*b^3))/(a^10 + a^6*b^4 - 2*a^8*b^2) + (2
*tan(x/2)*(18*b^10 - 81*a^2*b^8 + 126*a^4*b^6 - 72*a^6*b^4))/(a^13 + a^5*b^8 - 4*a^7*b^6 + 6*a^9*b^4 - 4*a^11*
b^2) + (3*b^2*(-(a + b)^5*(a - b)^5)^(1/2)*(4*a^4 + 2*b^4 - 5*a^2*b^2)*((12*a^4*b^6 - 27*a^6*b^4 + 18*a^8*b^2)
/(a^10 + a^6*b^4 - 2*a^8*b^2) - (tan(x/2)*(6*a^12*b - 24*a^2*b^11 + 108*a^4*b^9 - 192*a^6*b^7 + 162*a^8*b^5 -
60*a^10*b^3))/(a^13 + a^5*b^8 - 4*a^7*b^6 + 6*a^9*b^4 - 4*a^11*b^2) + (3*b^2*((2*a^12*b + 2*a^8*b^5 - 4*a^10*b
^3)/(a^10 + a^6*b^4 - 2*a^8*b^2) - (tan(x/2)*(6*a^16 - 8*a^6*b^10 + 38*a^8*b^8 - 72*a^10*b^6 + 68*a^12*b^4 - 3
2*a^14*b^2))/(a^13 + a^5*b^8 - 4*a^7*b^6 + 6*a^9*b^4 - 4*a^11*b^2))*(-(a + b)^5*(a - b)^5)^(1/2)*(4*a^4 + 2*b^
4 - 5*a^2*b^2))/(2*(a^14 - a^4*b^10 + 5*a^6*b^8 - 10*a^8*b^6 + 10*a^10*b^4 - 5*a^12*b^2))))/(2*(a^14 - a^4*b^1
0 + 5*a^6*b^8 - 10*a^8*b^6 + 10*a^10*b^4 - 5*a^12*b^2)) + (3*b^2*(-(a + b)^5*(a - b)^5)^(1/2)*(4*a^4 + 2*b^4 -
5*a^2*b^2)*((tan(x/2)*(6*a^12*b - 24*a^2*b^11 + 108*a^4*b^9 - 192*a^6*b^7 + 162*a^8*b^5 - 60*a^10*b^3))/(a^13
+ a^5*b^8 - 4*a^7*b^6 + 6*a^9*b^4 - 4*a^11*b^2) - (12*a^4*b^6 - 27*a^6*b^4 + 18*a^8*b^2)/(a^10 + a^6*b^4 - 2*
a^8*b^2) + (3*b^2*((2*a^12*b + 2*a^8*b^5 - 4*a^10*b^3)/(a^10 + a^6*b^4 - 2*a^8*b^2) - (tan(x/2)*(6*a^16 - 8*a^
6*b^10 + 38*a^8*b^8 - 72*a^10*b^6 + 68*a^12*b^4 - 32*a^14*b^2))/(a^13 + a^5*b^8 - 4*a^7*b^6 + 6*a^9*b^4 - 4*a^
11*b^2))*(-(a + b)^5*(a - b)^5)^(1/2)*(4*a^4 + 2*b^4 - 5*a^2*b^2))/(2*(a^14 - a^4*b^10 + 5*a^6*b^8 - 10*a^8*b^
6 + 10*a^10*b^4 - 5*a^12*b^2))))/(2*(a^14 - a^4*b^10 + 5*a^6*b^8 - 10*a^8*b^6 + 10*a^10*b^4 - 5*a^12*b^2))))*(
-(a + b)^5*(a - b)^5)^(1/2)*(4*a^4 + 2*b^4 - 5*a^2*b^2)*3i)/(a^14 - a^4*b^10 + 5*a^6*b^8 - 10*a^8*b^6 + 10*a^1
0*b^4 - 5*a^12*b^2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc ^{2}{\relax (x )}}{\left (a + b \sin {\relax (x )}\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)**2/(a+b*sin(x))**3,x)

[Out]

Integral(csc(x)**2/(a + b*sin(x))**3, x)

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