3.1477 \(\int \frac{\csc ^3(c+d x) \sec ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx\)

Optimal. Leaf size=470 \[ \frac{b^5 \left (2 a^2+b^2\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^3 d \left (a^2-b^2\right )^{7/2}}+\frac{2 b^5 \left (-17 a^2 b^2+15 a^4+6 b^4\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^5 d \left (a^2-b^2\right )^{7/2}}+\frac{2 b^5 \left (5 a^2-3 b^2\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^3 d \left (a^2-b^2\right )^{7/2}}+\frac{b^6 \left (5 a^2-3 b^2\right ) \cos (c+d x)}{a^4 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))}+\frac{3 b^6 \cos (c+d x)}{2 a^2 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))}+\frac{b^6 \cos (c+d x)}{2 a^3 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^2}-\frac{\left (a^2+6 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac{3 b \cot (c+d x)}{a^4 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac{\cos (c+d x)}{2 d (a+b)^3 (1-\sin (c+d x))}+\frac{\cos (c+d x)}{2 d (a-b)^3 (\sin (c+d x)+1)} \]

[Out]

(2*b^5*(5*a^2 - 3*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^3*(a^2 - b^2)^(7/2)*d) + (b^5*(2*a
^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^3*(a^2 - b^2)^(7/2)*d) + (2*b^5*(15*a^4 - 17*a^
2*b^2 + 6*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^5*(a^2 - b^2)^(7/2)*d) - ArcTanh[Cos[c + d
*x]]/(2*a^3*d) - ((a^2 + 6*b^2)*ArcTanh[Cos[c + d*x]])/(a^5*d) + (3*b*Cot[c + d*x])/(a^4*d) - (Cot[c + d*x]*Cs
c[c + d*x])/(2*a^3*d) + Cos[c + d*x]/(2*(a + b)^3*d*(1 - Sin[c + d*x])) + Cos[c + d*x]/(2*(a - b)^3*d*(1 + Sin
[c + d*x])) + (b^6*Cos[c + d*x])/(2*a^3*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^2) + (3*b^6*Cos[c + d*x])/(2*a^2*
(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])) + (b^6*(5*a^2 - 3*b^2)*Cos[c + d*x])/(a^4*(a^2 - b^2)^3*d*(a + b*Sin[c +
d*x]))

________________________________________________________________________________________

Rubi [A]  time = 0.608277, antiderivative size = 470, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 12, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.414, Rules used = {2897, 3770, 3767, 8, 3768, 2648, 2664, 2754, 12, 2660, 618, 204} \[ \frac{b^5 \left (2 a^2+b^2\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^3 d \left (a^2-b^2\right )^{7/2}}+\frac{2 b^5 \left (-17 a^2 b^2+15 a^4+6 b^4\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^5 d \left (a^2-b^2\right )^{7/2}}+\frac{2 b^5 \left (5 a^2-3 b^2\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^3 d \left (a^2-b^2\right )^{7/2}}+\frac{b^6 \left (5 a^2-3 b^2\right ) \cos (c+d x)}{a^4 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))}+\frac{3 b^6 \cos (c+d x)}{2 a^2 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))}+\frac{b^6 \cos (c+d x)}{2 a^3 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^2}-\frac{\left (a^2+6 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac{3 b \cot (c+d x)}{a^4 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac{\cos (c+d x)}{2 d (a+b)^3 (1-\sin (c+d x))}+\frac{\cos (c+d x)}{2 d (a-b)^3 (\sin (c+d x)+1)} \]

Antiderivative was successfully verified.

[In]

Int[(Csc[c + d*x]^3*Sec[c + d*x]^2)/(a + b*Sin[c + d*x])^3,x]

[Out]

(2*b^5*(5*a^2 - 3*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^3*(a^2 - b^2)^(7/2)*d) + (b^5*(2*a
^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^3*(a^2 - b^2)^(7/2)*d) + (2*b^5*(15*a^4 - 17*a^
2*b^2 + 6*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^5*(a^2 - b^2)^(7/2)*d) - ArcTanh[Cos[c + d
*x]]/(2*a^3*d) - ((a^2 + 6*b^2)*ArcTanh[Cos[c + d*x]])/(a^5*d) + (3*b*Cot[c + d*x])/(a^4*d) - (Cot[c + d*x]*Cs
c[c + d*x])/(2*a^3*d) + Cos[c + d*x]/(2*(a + b)^3*d*(1 - Sin[c + d*x])) + Cos[c + d*x]/(2*(a - b)^3*d*(1 + Sin
[c + d*x])) + (b^6*Cos[c + d*x])/(2*a^3*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^2) + (3*b^6*Cos[c + d*x])/(2*a^2*
(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])) + (b^6*(5*a^2 - 3*b^2)*Cos[c + d*x])/(a^4*(a^2 - b^2)^3*d*(a + b*Sin[c +
d*x]))

Rule 2897

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(
m_), x_Symbol] :> Int[ExpandTrig[(d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m*(1 - sin[e + f*x]^2)^(p/2), x], x]
/; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[m, 2*n, p/2] && (LtQ[m, -1] || (EqQ[m, -1] && G
tQ[p, 0]))

Rule 3770

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

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3768

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

Rule 2648

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> -Simp[Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 2664

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

Rule 2754

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 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])

Rubi steps

\begin{align*} \int \frac{\csc ^3(c+d x) \sec ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx &=\int \left (\frac{\left (a^2+6 b^2\right ) \csc (c+d x)}{a^5}-\frac{3 b \csc ^2(c+d x)}{a^4}+\frac{\csc ^3(c+d x)}{a^3}-\frac{1}{2 (a+b)^3 (-1+\sin (c+d x))}-\frac{1}{2 (a-b)^3 (1+\sin (c+d x))}+\frac{b^5}{a^3 \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}+\frac{b^5 \left (5 a^2-3 b^2\right )}{a^4 \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^2}+\frac{b^5 \left (15 a^4-17 a^2 b^2+6 b^4\right )}{a^5 \left (a^2-b^2\right )^3 (a+b \sin (c+d x))}\right ) \, dx\\ &=\frac{\int \csc ^3(c+d x) \, dx}{a^3}-\frac{\int \frac{1}{1+\sin (c+d x)} \, dx}{2 (a-b)^3}-\frac{(3 b) \int \csc ^2(c+d x) \, dx}{a^4}-\frac{\int \frac{1}{-1+\sin (c+d x)} \, dx}{2 (a+b)^3}+\frac{\left (b^5 \left (5 a^2-3 b^2\right )\right ) \int \frac{1}{(a+b \sin (c+d x))^2} \, dx}{a^4 \left (a^2-b^2\right )^2}+\frac{b^5 \int \frac{1}{(a+b \sin (c+d x))^3} \, dx}{a^3 \left (a^2-b^2\right )}+\frac{\left (a^2+6 b^2\right ) \int \csc (c+d x) \, dx}{a^5}+\frac{\left (b^5 \left (15 a^4-17 a^2 b^2+6 b^4\right )\right ) \int \frac{1}{a+b \sin (c+d x)} \, dx}{a^5 \left (a^2-b^2\right )^3}\\ &=-\frac{\left (a^2+6 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac{\cos (c+d x)}{2 (a+b)^3 d (1-\sin (c+d x))}+\frac{\cos (c+d x)}{2 (a-b)^3 d (1+\sin (c+d x))}+\frac{b^6 \cos (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^2}+\frac{b^6 \left (5 a^2-3 b^2\right ) \cos (c+d x)}{a^4 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}+\frac{\int \csc (c+d x) \, dx}{2 a^3}+\frac{\left (b^5 \left (5 a^2-3 b^2\right )\right ) \int \frac{a}{a+b \sin (c+d x)} \, dx}{a^4 \left (a^2-b^2\right )^3}-\frac{b^5 \int \frac{-2 a+b \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx}{2 a^3 \left (a^2-b^2\right )^2}+\frac{(3 b) \operatorname{Subst}(\int 1 \, dx,x,\cot (c+d x))}{a^4 d}+\frac{\left (2 b^5 \left (15 a^4-17 a^2 b^2+6 b^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^5 \left (a^2-b^2\right )^3 d}\\ &=-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac{\left (a^2+6 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac{3 b \cot (c+d x)}{a^4 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac{\cos (c+d x)}{2 (a+b)^3 d (1-\sin (c+d x))}+\frac{\cos (c+d x)}{2 (a-b)^3 d (1+\sin (c+d x))}+\frac{b^6 \cos (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^2}+\frac{3 b^6 \cos (c+d x)}{2 a^2 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}+\frac{b^6 \left (5 a^2-3 b^2\right ) \cos (c+d x)}{a^4 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}+\frac{b^5 \int \frac{2 a^2+b^2}{a+b \sin (c+d x)} \, dx}{2 a^3 \left (a^2-b^2\right )^3}+\frac{\left (b^5 \left (5 a^2-3 b^2\right )\right ) \int \frac{1}{a+b \sin (c+d x)} \, dx}{a^3 \left (a^2-b^2\right )^3}-\frac{\left (4 b^5 \left (15 a^4-17 a^2 b^2+6 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{1}{2} (c+d x)\right )\right )}{a^5 \left (a^2-b^2\right )^3 d}\\ &=\frac{2 b^5 \left (15 a^4-17 a^2 b^2+6 b^4\right ) \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{a^5 \left (a^2-b^2\right )^{7/2} d}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac{\left (a^2+6 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac{3 b \cot (c+d x)}{a^4 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac{\cos (c+d x)}{2 (a+b)^3 d (1-\sin (c+d x))}+\frac{\cos (c+d x)}{2 (a-b)^3 d (1+\sin (c+d x))}+\frac{b^6 \cos (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^2}+\frac{3 b^6 \cos (c+d x)}{2 a^2 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}+\frac{b^6 \left (5 a^2-3 b^2\right ) \cos (c+d x)}{a^4 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}+\frac{\left (b^5 \left (2 a^2+b^2\right )\right ) \int \frac{1}{a+b \sin (c+d x)} \, dx}{2 a^3 \left (a^2-b^2\right )^3}+\frac{\left (2 b^5 \left (5 a^2-3 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^3 \left (a^2-b^2\right )^3 d}\\ &=\frac{2 b^5 \left (15 a^4-17 a^2 b^2+6 b^4\right ) \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{a^5 \left (a^2-b^2\right )^{7/2} d}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac{\left (a^2+6 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac{3 b \cot (c+d x)}{a^4 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac{\cos (c+d x)}{2 (a+b)^3 d (1-\sin (c+d x))}+\frac{\cos (c+d x)}{2 (a-b)^3 d (1+\sin (c+d x))}+\frac{b^6 \cos (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^2}+\frac{3 b^6 \cos (c+d x)}{2 a^2 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}+\frac{b^6 \left (5 a^2-3 b^2\right ) \cos (c+d x)}{a^4 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}-\frac{\left (4 b^5 \left (5 a^2-3 b^2\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{1}{2} (c+d x)\right )\right )}{a^3 \left (a^2-b^2\right )^3 d}+\frac{\left (b^5 \left (2 a^2+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^3 \left (a^2-b^2\right )^3 d}\\ &=\frac{2 b^5 \left (5 a^2-3 b^2\right ) \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{a^3 \left (a^2-b^2\right )^{7/2} d}+\frac{2 b^5 \left (15 a^4-17 a^2 b^2+6 b^4\right ) \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{a^5 \left (a^2-b^2\right )^{7/2} d}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac{\left (a^2+6 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac{3 b \cot (c+d x)}{a^4 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac{\cos (c+d x)}{2 (a+b)^3 d (1-\sin (c+d x))}+\frac{\cos (c+d x)}{2 (a-b)^3 d (1+\sin (c+d x))}+\frac{b^6 \cos (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^2}+\frac{3 b^6 \cos (c+d x)}{2 a^2 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}+\frac{b^6 \left (5 a^2-3 b^2\right ) \cos (c+d x)}{a^4 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}-\frac{\left (2 b^5 \left (2 a^2+b^2\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{1}{2} (c+d x)\right )\right )}{a^3 \left (a^2-b^2\right )^3 d}\\ &=\frac{2 b^5 \left (5 a^2-3 b^2\right ) \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{a^3 \left (a^2-b^2\right )^{7/2} d}+\frac{b^5 \left (2 a^2+b^2\right ) \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{a^3 \left (a^2-b^2\right )^{7/2} d}+\frac{2 b^5 \left (15 a^4-17 a^2 b^2+6 b^4\right ) \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{a^5 \left (a^2-b^2\right )^{7/2} d}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac{\left (a^2+6 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac{3 b \cot (c+d x)}{a^4 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac{\cos (c+d x)}{2 (a+b)^3 d (1-\sin (c+d x))}+\frac{\cos (c+d x)}{2 (a-b)^3 d (1+\sin (c+d x))}+\frac{b^6 \cos (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^2}+\frac{3 b^6 \cos (c+d x)}{2 a^2 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}+\frac{b^6 \left (5 a^2-3 b^2\right ) \cos (c+d x)}{a^4 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}\\ \end{align*}

Mathematica [A]  time = 6.77968, size = 432, normalized size = 0.92 \[ \frac{3 \left (a^2+4 b^2\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{2 a^5 d}-\frac{3 \left (a^2+4 b^2\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{2 a^5 d}+\frac{b^6 \cos (c+d x)}{2 a^3 d (a-b)^2 (a+b)^2 (a+b \sin (c+d x))^2}+\frac{13 a^2 b^6 \cos (c+d x)-6 b^8 \cos (c+d x)}{2 a^4 d (a-b)^3 (a+b)^3 (a+b \sin (c+d x))}+\frac{3 b^5 \left (-13 a^2 b^2+14 a^4+4 b^4\right ) \tan ^{-1}\left (\frac{\sec \left (\frac{1}{2} (c+d x)\right ) \left (a \sin \left (\frac{1}{2} (c+d x)\right )+b \cos \left (\frac{1}{2} (c+d x)\right )\right )}{\sqrt{a^2-b^2}}\right )}{a^5 d \left (a^2-b^2\right )^{7/2}}-\frac{3 b \tan \left (\frac{1}{2} (c+d x)\right )}{2 a^4 d}+\frac{3 b \cot \left (\frac{1}{2} (c+d x)\right )}{2 a^4 d}-\frac{\csc ^2\left (\frac{1}{2} (c+d x)\right )}{8 a^3 d}+\frac{\sec ^2\left (\frac{1}{2} (c+d x)\right )}{8 a^3 d}+\frac{\sin \left (\frac{1}{2} (c+d x)\right )}{d (a+b)^3 \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}-\frac{\sin \left (\frac{1}{2} (c+d x)\right )}{d (a-b)^3 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*b^5*(14*a^4 - 13*a^2*b^2 + 4*b^4)*ArcTan[(Sec[(c + d*x)/2]*(b*Cos[(c + d*x)/2] + a*Sin[(c + d*x)/2]))/Sqrt[
a^2 - b^2]])/(a^5*(a^2 - b^2)^(7/2)*d) + (3*b*Cot[(c + d*x)/2])/(2*a^4*d) - Csc[(c + d*x)/2]^2/(8*a^3*d) - (3*
(a^2 + 4*b^2)*Log[Cos[(c + d*x)/2]])/(2*a^5*d) + (3*(a^2 + 4*b^2)*Log[Sin[(c + d*x)/2]])/(2*a^5*d) + Sec[(c +
d*x)/2]^2/(8*a^3*d) + Sin[(c + d*x)/2]/((a + b)^3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - Sin[(c + d*x)/2]/
((a - b)^3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (b^6*Cos[c + d*x])/(2*a^3*(a - b)^2*(a + b)^2*d*(a + b*S
in[c + d*x])^2) + (13*a^2*b^6*Cos[c + d*x] - 6*b^8*Cos[c + d*x])/(2*a^4*(a - b)^3*(a + b)^3*d*(a + b*Sin[c + d
*x])) - (3*b*Tan[(c + d*x)/2])/(2*a^4*d)

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Maple [B]  time = 0.198, size = 897, normalized size = 1.9 \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)^3*sec(d*x+c)^2/(a+b*sin(d*x+c))^3,x)

[Out]

1/8/d/a^3*tan(1/2*d*x+1/2*c)^2-3/2/d/a^4*tan(1/2*d*x+1/2*c)*b-1/d/(a+b)^3/(tan(1/2*d*x+1/2*c)-1)+15/d*b^7/(a-b
)^3/(a+b)^3/a^2/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)^3-8/d*b^9/(a-b)^3/(a+b)
^3/a^4/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)^3+14/d*b^6/(a-b)^3/(a+b)^3/(tan(
1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2/a*tan(1/2*d*x+1/2*c)^2+21/d*b^8/(a-b)^3/(a+b)^3/a^3/(tan(1/2*d*
x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)^2-14/d*b^10/(a-b)^3/(a+b)^3/a^5/(tan(1/2*d*x+1/2*c
)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)^2+41/d*b^7/(a-b)^3/(a+b)^3/a^2/(tan(1/2*d*x+1/2*c)^2*a+2*
tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)-20/d*b^9/(a-b)^3/(a+b)^3/a^4/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*
x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)+14/d*b^6/(a-b)^3/(a+b)^3/a/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a
)^2-7/d*b^8/(a-b)^3/(a+b)^3/a^3/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2+42/d*b^5/(a-b)^3/(a+b)^3/a
/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))-39/d*b^7/(a-b)^3/(a+b)^3/a^3/(a^2-b^
2)^(1/2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))+12/d*b^9/(a-b)^3/(a+b)^3/a^5/(a^2-b^2)^(1/2)
*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))+1/d/(a-b)^3/(tan(1/2*d*x+1/2*c)+1)-1/8/d/a^3/tan(1/2
*d*x+1/2*c)^2+3/2/d/a^3*ln(tan(1/2*d*x+1/2*c))+6/d/a^5*ln(tan(1/2*d*x+1/2*c))*b^2+3/2/d*b/a^4/tan(1/2*d*x+1/2*
c)

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 34.9756, size = 6075, normalized size = 12.93 \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)^3*sec(d*x+c)^2/(a+b*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

[1/4*(4*a^12 - 12*a^10*b^2 + 12*a^8*b^4 - 4*a^6*b^6 - 2*(21*a^10*b^2 - 56*a^8*b^4 + 82*a^6*b^6 - 65*a^4*b^8 +
18*a^2*b^10)*cos(d*x + c)^4 - 2*(3*a^12 - 31*a^10*b^2 + 68*a^8*b^4 - 88*a^6*b^6 + 66*a^4*b^8 - 18*a^2*b^10)*co
s(d*x + c)^2 + 3*((14*a^4*b^7 - 13*a^2*b^9 + 4*b^11)*cos(d*x + c)^5 - (14*a^6*b^5 + 15*a^4*b^7 - 22*a^2*b^9 +
8*b^11)*cos(d*x + c)^3 + (14*a^6*b^5 + a^4*b^7 - 9*a^2*b^9 + 4*b^11)*cos(d*x + c) - 2*((14*a^5*b^6 - 13*a^3*b^
8 + 4*a*b^10)*cos(d*x + c)^3 - (14*a^5*b^6 - 13*a^3*b^8 + 4*a*b^10)*cos(d*x + c))*sin(d*x + c))*sqrt(-a^2 + b^
2)*log(-((2*a^2 - b^2)*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2 - 2*(a*cos(d*x + c)*sin(d*x + c) + b*co
s(d*x + c))*sqrt(-a^2 + b^2))/(b^2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2)) - 3*((a^10*b^2 - 10*a^6*b
^6 + 20*a^4*b^8 - 15*a^2*b^10 + 4*b^12)*cos(d*x + c)^5 - (a^12 + 2*a^10*b^2 - 10*a^8*b^4 + 25*a^4*b^8 - 26*a^2
*b^10 + 8*b^12)*cos(d*x + c)^3 + (a^12 + a^10*b^2 - 10*a^8*b^4 + 10*a^6*b^6 + 5*a^4*b^8 - 11*a^2*b^10 + 4*b^12
)*cos(d*x + c) - 2*((a^11*b - 10*a^7*b^5 + 20*a^5*b^7 - 15*a^3*b^9 + 4*a*b^11)*cos(d*x + c)^3 - (a^11*b - 10*a
^7*b^5 + 20*a^5*b^7 - 15*a^3*b^9 + 4*a*b^11)*cos(d*x + c))*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) + 3*((a^1
0*b^2 - 10*a^6*b^6 + 20*a^4*b^8 - 15*a^2*b^10 + 4*b^12)*cos(d*x + c)^5 - (a^12 + 2*a^10*b^2 - 10*a^8*b^4 + 25*
a^4*b^8 - 26*a^2*b^10 + 8*b^12)*cos(d*x + c)^3 + (a^12 + a^10*b^2 - 10*a^8*b^4 + 10*a^6*b^6 + 5*a^4*b^8 - 11*a
^2*b^10 + 4*b^12)*cos(d*x + c) - 2*((a^11*b - 10*a^7*b^5 + 20*a^5*b^7 - 15*a^3*b^9 + 4*a*b^11)*cos(d*x + c)^3
- (a^11*b - 10*a^7*b^5 + 20*a^5*b^7 - 15*a^3*b^9 + 4*a*b^11)*cos(d*x + c))*sin(d*x + c))*log(-1/2*cos(d*x + c)
+ 1/2) - 2*(2*a^11*b - 6*a^9*b^3 + 6*a^7*b^5 - 2*a^5*b^7 + (12*a^9*b^3 - 28*a^7*b^5 + 47*a^5*b^7 - 43*a^3*b^9
+ 12*a*b^11)*cos(d*x + c)^4 - (6*a^11*b - 10*a^9*b^3 + 2*a^7*b^5 + 29*a^5*b^7 - 39*a^3*b^9 + 12*a*b^11)*cos(d
*x + c)^2)*sin(d*x + c))/((a^13*b^2 - 4*a^11*b^4 + 6*a^9*b^6 - 4*a^7*b^8 + a^5*b^10)*d*cos(d*x + c)^5 - (a^15
- 2*a^13*b^2 - 2*a^11*b^4 + 8*a^9*b^6 - 7*a^7*b^8 + 2*a^5*b^10)*d*cos(d*x + c)^3 + (a^15 - 3*a^13*b^2 + 2*a^11
*b^4 + 2*a^9*b^6 - 3*a^7*b^8 + a^5*b^10)*d*cos(d*x + c) - 2*((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)^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))*sin(d*x +
c)), 1/4*(4*a^12 - 12*a^10*b^2 + 12*a^8*b^4 - 4*a^6*b^6 - 2*(21*a^10*b^2 - 56*a^8*b^4 + 82*a^6*b^6 - 65*a^4*b^
8 + 18*a^2*b^10)*cos(d*x + c)^4 - 2*(3*a^12 - 31*a^10*b^2 + 68*a^8*b^4 - 88*a^6*b^6 + 66*a^4*b^8 - 18*a^2*b^10
)*cos(d*x + c)^2 - 6*((14*a^4*b^7 - 13*a^2*b^9 + 4*b^11)*cos(d*x + c)^5 - (14*a^6*b^5 + 15*a^4*b^7 - 22*a^2*b^
9 + 8*b^11)*cos(d*x + c)^3 + (14*a^6*b^5 + a^4*b^7 - 9*a^2*b^9 + 4*b^11)*cos(d*x + c) - 2*((14*a^5*b^6 - 13*a^
3*b^8 + 4*a*b^10)*cos(d*x + c)^3 - (14*a^5*b^6 - 13*a^3*b^8 + 4*a*b^10)*cos(d*x + c))*sin(d*x + c))*sqrt(a^2 -
b^2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) - 3*((a^10*b^2 - 10*a^6*b^6 + 20*a^4*b^8 -
15*a^2*b^10 + 4*b^12)*cos(d*x + c)^5 - (a^12 + 2*a^10*b^2 - 10*a^8*b^4 + 25*a^4*b^8 - 26*a^2*b^10 + 8*b^12)*co
s(d*x + c)^3 + (a^12 + a^10*b^2 - 10*a^8*b^4 + 10*a^6*b^6 + 5*a^4*b^8 - 11*a^2*b^10 + 4*b^12)*cos(d*x + c) - 2
*((a^11*b - 10*a^7*b^5 + 20*a^5*b^7 - 15*a^3*b^9 + 4*a*b^11)*cos(d*x + c)^3 - (a^11*b - 10*a^7*b^5 + 20*a^5*b^
7 - 15*a^3*b^9 + 4*a*b^11)*cos(d*x + c))*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) + 3*((a^10*b^2 - 10*a^6*b^6
+ 20*a^4*b^8 - 15*a^2*b^10 + 4*b^12)*cos(d*x + c)^5 - (a^12 + 2*a^10*b^2 - 10*a^8*b^4 + 25*a^4*b^8 - 26*a^2*b
^10 + 8*b^12)*cos(d*x + c)^3 + (a^12 + a^10*b^2 - 10*a^8*b^4 + 10*a^6*b^6 + 5*a^4*b^8 - 11*a^2*b^10 + 4*b^12)*
cos(d*x + c) - 2*((a^11*b - 10*a^7*b^5 + 20*a^5*b^7 - 15*a^3*b^9 + 4*a*b^11)*cos(d*x + c)^3 - (a^11*b - 10*a^7
*b^5 + 20*a^5*b^7 - 15*a^3*b^9 + 4*a*b^11)*cos(d*x + c))*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2) - 2*(2*a^1
1*b - 6*a^9*b^3 + 6*a^7*b^5 - 2*a^5*b^7 + (12*a^9*b^3 - 28*a^7*b^5 + 47*a^5*b^7 - 43*a^3*b^9 + 12*a*b^11)*cos(
d*x + c)^4 - (6*a^11*b - 10*a^9*b^3 + 2*a^7*b^5 + 29*a^5*b^7 - 39*a^3*b^9 + 12*a*b^11)*cos(d*x + c)^2)*sin(d*x
+ c))/((a^13*b^2 - 4*a^11*b^4 + 6*a^9*b^6 - 4*a^7*b^8 + a^5*b^10)*d*cos(d*x + c)^5 - (a^15 - 2*a^13*b^2 - 2*a
^11*b^4 + 8*a^9*b^6 - 7*a^7*b^8 + 2*a^5*b^10)*d*cos(d*x + c)^3 + (a^15 - 3*a^13*b^2 + 2*a^11*b^4 + 2*a^9*b^6 -
3*a^7*b^8 + a^5*b^10)*d*cos(d*x + c) - 2*((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)^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))*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)**3*sec(d*x+c)**2/(a+b*sin(d*x+c))**3,x)

[Out]

Timed out

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Giac [B]  time = 1.33171, size = 1215, normalized size = 2.59 \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)^3*sec(d*x+c)^2/(a+b*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/8*(24*(14*a^4*b^5 - 13*a^2*b^7 + 4*b^9)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*d*x + 1
/2*c) + b)/sqrt(a^2 - b^2)))/((a^11 - 3*a^9*b^2 + 3*a^7*b^4 - a^5*b^6)*sqrt(a^2 - b^2)) + 16*(3*a^2*b*tan(1/2*
d*x + 1/2*c) + b^3*tan(1/2*d*x + 1/2*c) - a^3 - 3*a*b^2)/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*(tan(1/2*d*x + 1
/2*c)^2 - 1)) - (6*a^10*tan(1/2*d*x + 1/2*c)^6 + 6*a^8*b^2*tan(1/2*d*x + 1/2*c)^6 - 54*a^6*b^4*tan(1/2*d*x + 1
/2*c)^6 + 66*a^4*b^6*tan(1/2*d*x + 1/2*c)^6 - 24*a^2*b^8*tan(1/2*d*x + 1/2*c)^6 + 12*a^9*b*tan(1/2*d*x + 1/2*c
)^5 + 60*a^7*b^3*tan(1/2*d*x + 1/2*c)^5 - 252*a^5*b^5*tan(1/2*d*x + 1/2*c)^5 + 156*a^3*b^7*tan(1/2*d*x + 1/2*c
)^5 - 32*a*b^9*tan(1/2*d*x + 1/2*c)^5 + 13*a^10*tan(1/2*d*x + 1/2*c)^4 - 15*a^8*b^2*tan(1/2*d*x + 1/2*c)^4 + 6
3*a^6*b^4*tan(1/2*d*x + 1/2*c)^4 - 341*a^4*b^6*tan(1/2*d*x + 1/2*c)^4 + 96*a^2*b^8*tan(1/2*d*x + 1/2*c)^4 + 16
*b^10*tan(1/2*d*x + 1/2*c)^4 + 4*a^9*b*tan(1/2*d*x + 1/2*c)^3 + 36*a^7*b^3*tan(1/2*d*x + 1/2*c)^3 - 132*a^5*b^
5*tan(1/2*d*x + 1/2*c)^3 - 188*a^3*b^7*tan(1/2*d*x + 1/2*c)^3 + 112*a*b^9*tan(1/2*d*x + 1/2*c)^3 + 8*a^10*tan(
1/2*d*x + 1/2*c)^2 - 44*a^8*b^2*tan(1/2*d*x + 1/2*c)^2 + 84*a^6*b^4*tan(1/2*d*x + 1/2*c)^2 - 180*a^4*b^6*tan(1
/2*d*x + 1/2*c)^2 + 76*a^2*b^8*tan(1/2*d*x + 1/2*c)^2 - 8*a^9*b*tan(1/2*d*x + 1/2*c) + 24*a^7*b^3*tan(1/2*d*x
+ 1/2*c) - 24*a^5*b^5*tan(1/2*d*x + 1/2*c) + 8*a^3*b^7*tan(1/2*d*x + 1/2*c) + a^10 - 3*a^8*b^2 + 3*a^6*b^4 - a
^4*b^6)/((a^11 - 3*a^9*b^2 + 3*a^7*b^4 - a^5*b^6)*(a*tan(1/2*d*x + 1/2*c)^3 + 2*b*tan(1/2*d*x + 1/2*c)^2 + a*t
an(1/2*d*x + 1/2*c))^2) + 12*(a^2 + 4*b^2)*log(abs(tan(1/2*d*x + 1/2*c)))/a^5 + (a^3*tan(1/2*d*x + 1/2*c)^2 -
12*a^2*b*tan(1/2*d*x + 1/2*c))/a^6)/d