3.1090 \(\int \frac{\cot ^2(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx\)

Optimal. Leaf size=269 \[ \frac{b \left (-19 a^2 b^2+6 a^4+12 b^4\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^5 d \left (a^2-b^2\right )^{3/2}}+\frac{b \left (11 a^2-12 b^2\right ) \cot (c+d x)}{2 a^4 d \left (a^2-b^2\right )}+\frac{\left (a^2-12 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^5 d}-\frac{\left (5 a^2-6 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^3 d \left (a^2-b^2\right )}+\frac{\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))}+\frac{\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2} \]

[Out]

(b*(6*a^4 - 19*a^2*b^2 + 12*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^5*(a^2 - b^2)^(3/2)*d) +
((a^2 - 12*b^2)*ArcTanh[Cos[c + d*x]])/(2*a^5*d) + (b*(11*a^2 - 12*b^2)*Cot[c + d*x])/(2*a^4*(a^2 - b^2)*d) -
((5*a^2 - 6*b^2)*Cot[c + d*x]*Csc[c + d*x])/(2*a^3*(a^2 - b^2)*d) + (Cot[c + d*x]*Csc[c + d*x])/(2*a*d*(a + b
*Sin[c + d*x])^2) + ((3*a^2 - 4*b^2)*Cot[c + d*x]*Csc[c + d*x])/(2*a^2*(a^2 - b^2)*d*(a + b*Sin[c + d*x]))

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Rubi [A]  time = 1.15081, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {2889, 3056, 3055, 3001, 3770, 2660, 618, 204} \[ \frac{b \left (-19 a^2 b^2+6 a^4+12 b^4\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^5 d \left (a^2-b^2\right )^{3/2}}+\frac{b \left (11 a^2-12 b^2\right ) \cot (c+d x)}{2 a^4 d \left (a^2-b^2\right )}+\frac{\left (a^2-12 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^5 d}-\frac{\left (5 a^2-6 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^3 d \left (a^2-b^2\right )}+\frac{\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))}+\frac{\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(6*a^4 - 19*a^2*b^2 + 12*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^5*(a^2 - b^2)^(3/2)*d) +
((a^2 - 12*b^2)*ArcTanh[Cos[c + d*x]])/(2*a^5*d) + (b*(11*a^2 - 12*b^2)*Cot[c + d*x])/(2*a^4*(a^2 - b^2)*d) -
((5*a^2 - 6*b^2)*Cot[c + d*x]*Csc[c + d*x])/(2*a^3*(a^2 - b^2)*d) + (Cot[c + d*x]*Csc[c + d*x])/(2*a*d*(a + b
*Sin[c + d*x])^2) + ((3*a^2 - 4*b^2)*Cot[c + d*x]*Csc[c + d*x])/(2*a^2*(a^2 - b^2)*d*(a + b*Sin[c + d*x]))

Rule 2889

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

Rule 3056

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*s
in[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((A*b^2 + 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] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), I
nt[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[a*(m + 1)*(b*c - a*d)*(A + C) + d*(A*b^2 + a^2*C)*
(m + n + 2) - (c*(A*b^2 + a^2*C) + b*(m + 1)*(b*c - a*d)*(A + C))*Sin[e + f*x] - d*(A*b^2 + a^2*C)*(m + n + 3)
*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
&& NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ
[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) || EqQ[a, 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 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 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, 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{\cot ^2(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx &=\int \frac{\csc ^3(c+d x) \left (1-\sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^3} \, dx\\ &=\frac{\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac{\int \frac{\csc ^3(c+d x) \left (4 \left (a^2-b^2\right )-3 \left (a^2-b^2\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{2 a \left (a^2-b^2\right )}\\ &=\frac{\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac{\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac{\int \frac{\csc ^3(c+d x) \left (2 \left (5 a^4-11 a^2 b^2+6 b^4\right )-a b \left (a^2-b^2\right ) \sin (c+d x)-2 \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (5 a^2-6 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}+\frac{\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac{\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac{\int \frac{\csc ^2(c+d x) \left (-2 b \left (11 a^4-23 a^2 b^2+12 b^4\right )-2 a \left (a^4-3 a^2 b^2+2 b^4\right ) \sin (c+d x)+2 b \left (5 a^4-11 a^2 b^2+6 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{4 a^3 \left (a^2-b^2\right )^2}\\ &=\frac{b \left (11 a^2-12 b^2\right ) \cot (c+d x)}{2 a^4 \left (a^2-b^2\right ) d}-\frac{\left (5 a^2-6 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}+\frac{\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac{\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac{\int \frac{\csc (c+d x) \left (-2 \left (a^2-12 b^2\right ) \left (a^2-b^2\right )^2+2 a b \left (5 a^4-11 a^2 b^2+6 b^4\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{4 a^4 \left (a^2-b^2\right )^2}\\ &=\frac{b \left (11 a^2-12 b^2\right ) \cot (c+d x)}{2 a^4 \left (a^2-b^2\right ) d}-\frac{\left (5 a^2-6 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}+\frac{\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac{\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}-\frac{\left (a^2-12 b^2\right ) \int \csc (c+d x) \, dx}{2 a^5}+\frac{\left (b \left (6 a^4-19 a^2 b^2+12 b^4\right )\right ) \int \frac{1}{a+b \sin (c+d x)} \, dx}{2 a^5 \left (a^2-b^2\right )}\\ &=\frac{\left (a^2-12 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^5 d}+\frac{b \left (11 a^2-12 b^2\right ) \cot (c+d x)}{2 a^4 \left (a^2-b^2\right ) d}-\frac{\left (5 a^2-6 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}+\frac{\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac{\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac{\left (b \left (6 a^4-19 a^2 b^2+12 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 ) d}\\ &=\frac{\left (a^2-12 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^5 d}+\frac{b \left (11 a^2-12 b^2\right ) \cot (c+d x)}{2 a^4 \left (a^2-b^2\right ) d}-\frac{\left (5 a^2-6 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}+\frac{\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac{\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}-\frac{\left (2 b \left (6 a^4-19 a^2 b^2+12 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 ) d}\\ &=\frac{b \left (6 a^4-19 a^2 b^2+12 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 )^{3/2} d}+\frac{\left (a^2-12 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^5 d}+\frac{b \left (11 a^2-12 b^2\right ) \cot (c+d x)}{2 a^4 \left (a^2-b^2\right ) d}-\frac{\left (5 a^2-6 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}+\frac{\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac{\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}\\ \end{align*}

Mathematica [A]  time = 6.31739, size = 330, normalized size = 1.23 \[ \frac{\left (12 b^2-a^2\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{2 a^5 d}+\frac{\left (a^2-12 b^2\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{2 a^5 d}+\frac{b^2 \cos (c+d x)}{2 a^3 d (a+b \sin (c+d x))^2}+\frac{5 a^2 b^2 \cos (c+d x)-6 b^4 \cos (c+d x)}{2 a^4 d (a-b) (a+b) (a+b \sin (c+d x))}+\frac{b \left (-19 a^2 b^2+6 a^4+12 b^4\right ) \tan ^{-1}\left (\frac{\sec \left (\frac{1}{2} (c+d x)\right ) \left (a \sin \left (\frac{1}{2} (c+d x)\right )+b \cos \left (\frac{1}{2} (c+d x)\right )\right )}{\sqrt{a^2-b^2}}\right )}{a^5 d \left (a^2-b^2\right )^{3/2}}-\frac{3 b \tan \left (\frac{1}{2} (c+d x)\right )}{2 a^4 d}+\frac{3 b \cot \left (\frac{1}{2} (c+d x)\right )}{2 a^4 d}-\frac{\csc ^2\left (\frac{1}{2} (c+d x)\right )}{8 a^3 d}+\frac{\sec ^2\left (\frac{1}{2} (c+d x)\right )}{8 a^3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(6*a^4 - 19*a^2*b^2 + 12*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)^(3/2)*d) + (3*b*Cot[(c + d*x)/2])/(2*a^4*d) - Csc[(c + d*x)/2]^2/(8*a^3*d) + ((a^2 -
12*b^2)*Log[Cos[(c + d*x)/2]])/(2*a^5*d) + ((-a^2 + 12*b^2)*Log[Sin[(c + d*x)/2]])/(2*a^5*d) + Sec[(c + d*x)/
2]^2/(8*a^3*d) + (b^2*Cos[c + d*x])/(2*a^3*d*(a + b*Sin[c + d*x])^2) + (5*a^2*b^2*Cos[c + d*x] - 6*b^4*Cos[c +
d*x])/(2*a^4*(a - b)*(a + b)*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 = 0.207, size = 803, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^2*csc(d*x+c)^3/(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+7/d/a^2/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/
2*c)*b+a)^2*b^3/(a^2-b^2)*tan(1/2*d*x+1/2*c)^3-8/d*b^5/a^4/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2
/(a^2-b^2)*tan(1/2*d*x+1/2*c)^3+6/d*b^2/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2/a/(a^2-b^2)*tan(1/
2*d*x+1/2*c)^2+5/d/a^3/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2/(a^2-b^2)*tan(1/2*d*x+1/2*c)^2*b^4-
14/d*b^6/a^5/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2/(a^2-b^2)*tan(1/2*d*x+1/2*c)^2+17/d/a^2/(tan(
1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*b^3/(a^2-b^2)*tan(1/2*d*x+1/2*c)-20/d*b^5/a^4/(tan(1/2*d*x+1/2*
c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2/(a^2-b^2)*tan(1/2*d*x+1/2*c)+6/d/a/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/
2*c)*b+a)^2*b^2/(a^2-b^2)-7/d*b^4/a^3/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2/(a^2-b^2)+6/d/a*b/(a
^2-b^2)^(3/2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))-19/d/a^3*b^3/(a^2-b^2)^(3/2)*arctan(1/2
*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))+12/d*b^5/a^5/(a^2-b^2)^(3/2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)
+2*b)/(a^2-b^2)^(1/2))-1/8/d/a^3/tan(1/2*d*x+1/2*c)^2-1/2/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(cos(d*x+c)^2*csc(d*x+c)^3/(a+b*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 6.44917, size = 4301, normalized size = 15.99 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(2*(17*a^6*b^2 - 35*a^4*b^4 + 18*a^2*b^6)*cos(d*x + c)^3 - (6*a^6*b - 13*a^4*b^3 - 7*a^2*b^5 + 12*b^7 +
(6*a^4*b^3 - 19*a^2*b^5 + 12*b^7)*cos(d*x + c)^4 - (6*a^6*b - 7*a^4*b^3 - 26*a^2*b^5 + 24*b^7)*cos(d*x + c)^2
+ 2*(6*a^5*b^2 - 19*a^3*b^4 + 12*a*b^6 - (6*a^5*b^2 - 19*a^3*b^4 + 12*a*b^6)*cos(d*x + c)^2)*sin(d*x + c))*sqr
t(-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*cos(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)) + 2*(a^8 - 19*
a^6*b^2 + 36*a^4*b^4 - 18*a^2*b^6)*cos(d*x + c) - (a^8 - 13*a^6*b^2 + 11*a^4*b^4 + 13*a^2*b^6 - 12*b^8 + (a^6*
b^2 - 14*a^4*b^4 + 25*a^2*b^6 - 12*b^8)*cos(d*x + c)^4 - (a^8 - 12*a^6*b^2 - 3*a^4*b^4 + 38*a^2*b^6 - 24*b^8)*
cos(d*x + c)^2 + 2*(a^7*b - 14*a^5*b^3 + 25*a^3*b^5 - 12*a*b^7 - (a^7*b - 14*a^5*b^3 + 25*a^3*b^5 - 12*a*b^7)*
cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) + (a^8 - 13*a^6*b^2 + 11*a^4*b^4 + 13*a^2*b^6 - 12*b
^8 + (a^6*b^2 - 14*a^4*b^4 + 25*a^2*b^6 - 12*b^8)*cos(d*x + c)^4 - (a^8 - 12*a^6*b^2 - 3*a^4*b^4 + 38*a^2*b^6
- 24*b^8)*cos(d*x + c)^2 + 2*(a^7*b - 14*a^5*b^3 + 25*a^3*b^5 - 12*a*b^7 - (a^7*b - 14*a^5*b^3 + 25*a^3*b^5 -
12*a*b^7)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2) + 2*((11*a^5*b^3 - 23*a^3*b^5 + 12*a*b^7)
*cos(d*x + c)^3 - (4*a^7*b + 3*a^5*b^3 - 19*a^3*b^5 + 12*a*b^7)*cos(d*x + c))*sin(d*x + c))/((a^9*b^2 - 2*a^7*
b^4 + a^5*b^6)*d*cos(d*x + c)^4 - (a^11 - 3*a^7*b^4 + 2*a^5*b^6)*d*cos(d*x + c)^2 + (a^11 - a^9*b^2 - a^7*b^4
+ a^5*b^6)*d - 2*((a^10*b - 2*a^8*b^3 + a^6*b^5)*d*cos(d*x + c)^2 - (a^10*b - 2*a^8*b^3 + a^6*b^5)*d)*sin(d*x
+ c)), -1/4*(2*(17*a^6*b^2 - 35*a^4*b^4 + 18*a^2*b^6)*cos(d*x + c)^3 + 2*(6*a^6*b - 13*a^4*b^3 - 7*a^2*b^5 + 1
2*b^7 + (6*a^4*b^3 - 19*a^2*b^5 + 12*b^7)*cos(d*x + c)^4 - (6*a^6*b - 7*a^4*b^3 - 26*a^2*b^5 + 24*b^7)*cos(d*x
+ c)^2 + 2*(6*a^5*b^2 - 19*a^3*b^4 + 12*a*b^6 - (6*a^5*b^2 - 19*a^3*b^4 + 12*a*b^6)*cos(d*x + c)^2)*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))) + 2*(a^8 - 19*a^6*b^2 + 36*a
^4*b^4 - 18*a^2*b^6)*cos(d*x + c) - (a^8 - 13*a^6*b^2 + 11*a^4*b^4 + 13*a^2*b^6 - 12*b^8 + (a^6*b^2 - 14*a^4*b
^4 + 25*a^2*b^6 - 12*b^8)*cos(d*x + c)^4 - (a^8 - 12*a^6*b^2 - 3*a^4*b^4 + 38*a^2*b^6 - 24*b^8)*cos(d*x + c)^2
+ 2*(a^7*b - 14*a^5*b^3 + 25*a^3*b^5 - 12*a*b^7 - (a^7*b - 14*a^5*b^3 + 25*a^3*b^5 - 12*a*b^7)*cos(d*x + c)^2
)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) + (a^8 - 13*a^6*b^2 + 11*a^4*b^4 + 13*a^2*b^6 - 12*b^8 + (a^6*b^2
- 14*a^4*b^4 + 25*a^2*b^6 - 12*b^8)*cos(d*x + c)^4 - (a^8 - 12*a^6*b^2 - 3*a^4*b^4 + 38*a^2*b^6 - 24*b^8)*cos(
d*x + c)^2 + 2*(a^7*b - 14*a^5*b^3 + 25*a^3*b^5 - 12*a*b^7 - (a^7*b - 14*a^5*b^3 + 25*a^3*b^5 - 12*a*b^7)*cos(
d*x + c)^2)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2) + 2*((11*a^5*b^3 - 23*a^3*b^5 + 12*a*b^7)*cos(d*x + c)^
3 - (4*a^7*b + 3*a^5*b^3 - 19*a^3*b^5 + 12*a*b^7)*cos(d*x + c))*sin(d*x + c))/((a^9*b^2 - 2*a^7*b^4 + a^5*b^6)
*d*cos(d*x + c)^4 - (a^11 - 3*a^7*b^4 + 2*a^5*b^6)*d*cos(d*x + c)^2 + (a^11 - a^9*b^2 - a^7*b^4 + a^5*b^6)*d -
2*((a^10*b - 2*a^8*b^3 + a^6*b^5)*d*cos(d*x + c)^2 - (a^10*b - 2*a^8*b^3 + a^6*b^5)*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(cos(d*x+c)**2*csc(d*x+c)**3/(a+b*sin(d*x+c))**3,x)

[Out]

Timed out

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Giac [B]  time = 1.33237, size = 710, normalized size = 2.64 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(8*(6*a^4*b - 19*a^2*b^3 + 12*b^5)*(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^7 - a^5*b^2)*sqrt(a^2 - b^2)) + (2*a^6*tan(1/2*d*x + 1/2*c)^6 - 26*a^4*b^2*tan(1
/2*d*x + 1/2*c)^6 + 24*a^2*b^4*tan(1/2*d*x + 1/2*c)^6 + 20*a^5*b*tan(1/2*d*x + 1/2*c)^5 - 60*a^3*b^3*tan(1/2*d
*x + 1/2*c)^5 + 32*a*b^5*tan(1/2*d*x + 1/2*c)^5 + 3*a^6*tan(1/2*d*x + 1/2*c)^4 + 53*a^4*b^2*tan(1/2*d*x + 1/2*
c)^4 - 64*a^2*b^4*tan(1/2*d*x + 1/2*c)^4 - 16*b^6*tan(1/2*d*x + 1/2*c)^4 + 28*a^5*b*tan(1/2*d*x + 1/2*c)^3 + 6
0*a^3*b^3*tan(1/2*d*x + 1/2*c)^3 - 112*a*b^5*tan(1/2*d*x + 1/2*c)^3 + 68*a^4*b^2*tan(1/2*d*x + 1/2*c)^2 - 76*a
^2*b^4*tan(1/2*d*x + 1/2*c)^2 + 8*a^5*b*tan(1/2*d*x + 1/2*c) - 8*a^3*b^3*tan(1/2*d*x + 1/2*c) - a^6 + a^4*b^2)
/((a^7 - a^5*b^2)*(a*tan(1/2*d*x + 1/2*c)^3 + 2*b*tan(1/2*d*x + 1/2*c)^2 + a*tan(1/2*d*x + 1/2*c))^2) - 4*(a^2
- 12*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