3.1142 \(\int \frac{\cot ^4(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx\)

Optimal. Leaf size=340 \[ -\frac{3 b \left (-11 a^2 b^2+2 a^4+10 b^4\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^7 d \sqrt{a^2-b^2}}-\frac{b \left (13 a^2-30 b^2\right ) \cot (c+d x)}{2 a^6 d}-\frac{3 \left (-24 a^2 b^2+a^4+40 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^7 d}-\frac{\left (3 a^2-10 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{2 a^4 b d}+\frac{3 \left (7 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^5 d}+\frac{\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^3 b d (a+b \sin (c+d x))}+\frac{\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2} \]

[Out]

(-3*b*(2*a^4 - 11*a^2*b^2 + 10*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^7*Sqrt[a^2 - b^2]*d)
- (3*(a^4 - 24*a^2*b^2 + 40*b^4)*ArcTanh[Cos[c + d*x]])/(8*a^7*d) - (b*(13*a^2 - 30*b^2)*Cot[c + d*x])/(2*a^6*
d) + (3*(7*a^2 - 20*b^2)*Cot[c + d*x]*Csc[c + d*x])/(8*a^5*d) - ((3*a^2 - 10*b^2)*Cot[c + d*x]*Csc[c + d*x]^2)
/(2*a^4*b*d) + ((2*a^2 - 3*b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/(4*a^2*b*d*(a + b*Sin[c + d*x])^2) - (Cot[c + d*x
]*Csc[c + d*x]^3)/(4*a*d*(a + b*Sin[c + d*x])^2) + ((4*a^2 - 15*b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/(4*a^3*b*d*(
a + b*Sin[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 1.45867, antiderivative size = 340, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {2890, 3055, 3001, 3770, 2660, 618, 204} \[ -\frac{3 b \left (-11 a^2 b^2+2 a^4+10 b^4\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^7 d \sqrt{a^2-b^2}}-\frac{b \left (13 a^2-30 b^2\right ) \cot (c+d x)}{2 a^6 d}-\frac{3 \left (-24 a^2 b^2+a^4+40 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^7 d}-\frac{\left (3 a^2-10 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{2 a^4 b d}+\frac{3 \left (7 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^5 d}+\frac{\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^3 b d (a+b \sin (c+d x))}+\frac{\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2} \]

Antiderivative was successfully verified.

[In]

Int[(Cot[c + d*x]^4*Csc[c + d*x])/(a + b*Sin[c + d*x])^3,x]

[Out]

(-3*b*(2*a^4 - 11*a^2*b^2 + 10*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^7*Sqrt[a^2 - b^2]*d)
- (3*(a^4 - 24*a^2*b^2 + 40*b^4)*ArcTanh[Cos[c + d*x]])/(8*a^7*d) - (b*(13*a^2 - 30*b^2)*Cot[c + d*x])/(2*a^6*
d) + (3*(7*a^2 - 20*b^2)*Cot[c + d*x]*Csc[c + d*x])/(8*a^5*d) - ((3*a^2 - 10*b^2)*Cot[c + d*x]*Csc[c + d*x]^2)
/(2*a^4*b*d) + ((2*a^2 - 3*b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/(4*a^2*b*d*(a + b*Sin[c + d*x])^2) - (Cot[c + d*x
]*Csc[c + d*x]^3)/(4*a*d*(a + b*Sin[c + d*x])^2) + ((4*a^2 - 15*b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/(4*a^3*b*d*(
a + b*Sin[c + d*x]))

Rule 2890

Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Simp[(Cos[e + f*x]*(d*Sin[e + f*x])^(n + 1)*(a + b*Sin[e + f*x])^(m + 1))/(a*d*f*(n + 1)), x] +
(Dist[1/(a^2*b*d*(n + 1)*(m + 1)), Int[(d*Sin[e + f*x])^(n + 1)*(a + b*Sin[e + f*x])^(m + 1)*Simp[a^2*(n + 1)
*(n + 2) - b^2*(m + n + 2)*(m + n + 3) + a*b*(m + 1)*Sin[e + f*x] - (a^2*(n + 1)*(n + 3) - b^2*(m + n + 2)*(m
+ n + 4))*Sin[e + f*x]^2, x], x], x] - Simp[((a^2*(n + 1) - b^2*(m + n + 2))*Cos[e + f*x]*(d*Sin[e + f*x])^(n
+ 2)*(a + b*Sin[e + f*x])^(m + 1))/(a^2*b*d^2*f*(n + 1)*(m + 1)), x]) /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2
- b^2, 0] && IntegersQ[2*m, 2*n] && LtQ[m, -1] && LtQ[n, -1]

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 ^4(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx &=\frac{\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}+\frac{\int \frac{\csc ^4(c+d x) \left (6 \left (2 a^2-5 b^2\right )-2 a b \sin (c+d x)-8 \left (a^2-3 b^2\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{8 a^2 b}\\ &=\frac{\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}+\frac{\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^3 b d (a+b \sin (c+d x))}+\frac{\int \frac{\csc ^4(c+d x) \left (12 \left (3 a^4-13 a^2 b^2+10 b^4\right )-6 a b \left (a^2-b^2\right ) \sin (c+d x)-6 \left (4 a^4-19 a^2 b^2+15 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{8 a^3 b \left (a^2-b^2\right )}\\ &=-\frac{\left (3 a^2-10 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{2 a^4 b d}+\frac{\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}+\frac{\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^3 b d (a+b \sin (c+d x))}+\frac{\int \frac{\csc ^3(c+d x) \left (-18 b \left (7 a^4-27 a^2 b^2+20 b^4\right )+30 a b^2 \left (a^2-b^2\right ) \sin (c+d x)+24 b \left (3 a^4-13 a^2 b^2+10 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{24 a^4 b \left (a^2-b^2\right )}\\ &=\frac{3 \left (7 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^5 d}-\frac{\left (3 a^2-10 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{2 a^4 b d}+\frac{\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}+\frac{\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^3 b d (a+b \sin (c+d x))}+\frac{\int \frac{\csc ^2(c+d x) \left (24 b^2 \left (13 a^4-43 a^2 b^2+30 b^4\right )+6 a b \left (3 a^4-23 a^2 b^2+20 b^4\right ) \sin (c+d x)-18 b^2 \left (7 a^4-27 a^2 b^2+20 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{48 a^5 b \left (a^2-b^2\right )}\\ &=-\frac{b \left (13 a^2-30 b^2\right ) \cot (c+d x)}{2 a^6 d}+\frac{3 \left (7 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^5 d}-\frac{\left (3 a^2-10 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{2 a^4 b d}+\frac{\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}+\frac{\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^3 b d (a+b \sin (c+d x))}+\frac{\int \frac{\csc (c+d x) \left (18 b \left (a^6-25 a^4 b^2+64 a^2 b^4-40 b^6\right )-18 a b^2 \left (7 a^4-27 a^2 b^2+20 b^4\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{48 a^6 b \left (a^2-b^2\right )}\\ &=-\frac{b \left (13 a^2-30 b^2\right ) \cot (c+d x)}{2 a^6 d}+\frac{3 \left (7 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^5 d}-\frac{\left (3 a^2-10 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{2 a^4 b d}+\frac{\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}+\frac{\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^3 b d (a+b \sin (c+d x))}-\frac{\left (3 b \left (2 a^4-11 a^2 b^2+10 b^4\right )\right ) \int \frac{1}{a+b \sin (c+d x)} \, dx}{2 a^7}+\frac{\left (3 \left (a^4-24 a^2 b^2+40 b^4\right )\right ) \int \csc (c+d x) \, dx}{8 a^7}\\ &=-\frac{3 \left (a^4-24 a^2 b^2+40 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^7 d}-\frac{b \left (13 a^2-30 b^2\right ) \cot (c+d x)}{2 a^6 d}+\frac{3 \left (7 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^5 d}-\frac{\left (3 a^2-10 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{2 a^4 b d}+\frac{\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}+\frac{\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^3 b d (a+b \sin (c+d x))}-\frac{\left (3 b \left (2 a^4-11 a^2 b^2+10 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^7 d}\\ &=-\frac{3 \left (a^4-24 a^2 b^2+40 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^7 d}-\frac{b \left (13 a^2-30 b^2\right ) \cot (c+d x)}{2 a^6 d}+\frac{3 \left (7 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^5 d}-\frac{\left (3 a^2-10 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{2 a^4 b d}+\frac{\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}+\frac{\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^3 b d (a+b \sin (c+d x))}+\frac{\left (6 b \left (2 a^4-11 a^2 b^2+10 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^7 d}\\ &=-\frac{3 b \left (2 a^4-11 a^2 b^2+10 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^7 \sqrt{a^2-b^2} d}-\frac{3 \left (a^4-24 a^2 b^2+40 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^7 d}-\frac{b \left (13 a^2-30 b^2\right ) \cot (c+d x)}{2 a^6 d}+\frac{3 \left (7 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^5 d}-\frac{\left (3 a^2-10 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{2 a^4 b d}+\frac{\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}+\frac{\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^3 b d (a+b \sin (c+d x))}\\ \end{align*}

Mathematica [A]  time = 4.50849, size = 347, normalized size = 1.02 \[ -\frac{\frac{384 b \left (-11 a^2 b^2+2 a^4+10 b^4\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{\sqrt{a^2-b^2}}-48 \left (-24 a^2 b^2+a^4+40 b^4\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+48 \left (-24 a^2 b^2+a^4+40 b^4\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+\frac{2 a \cot (c+d x) \csc ^5(c+d x) \left (20 a^2 b^3 \sin (c+d x)-50 a^2 b^3 \sin (3 (c+d x))+26 a^2 b^3 \sin (5 (c+d x))+4 \left (-93 a^3 b^2+5 a^5+180 a b^4\right ) \cos (2 (c+d x))+\left (83 a^3 b^2-180 a b^4\right ) \cos (4 (c+d x))+289 a^3 b^2+100 a^4 b \sin (c+d x)-44 a^4 b \sin (3 (c+d x))-4 a^5-540 a b^4-600 b^5 \sin (c+d x)+300 b^5 \sin (3 (c+d x))-60 b^5 \sin (5 (c+d x))\right )}{(a \csc (c+d x)+b)^2}}{128 a^7 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cot[c + d*x]^4*Csc[c + d*x])/(a + b*Sin[c + d*x])^3,x]

[Out]

-((384*b*(2*a^4 - 11*a^2*b^2 + 10*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + 48*
(a^4 - 24*a^2*b^2 + 40*b^4)*Log[Cos[(c + d*x)/2]] - 48*(a^4 - 24*a^2*b^2 + 40*b^4)*Log[Sin[(c + d*x)/2]] + (2*
a*Cot[c + d*x]*Csc[c + d*x]^5*(-4*a^5 + 289*a^3*b^2 - 540*a*b^4 + 4*(5*a^5 - 93*a^3*b^2 + 180*a*b^4)*Cos[2*(c
+ d*x)] + (83*a^3*b^2 - 180*a*b^4)*Cos[4*(c + d*x)] + 100*a^4*b*Sin[c + d*x] + 20*a^2*b^3*Sin[c + d*x] - 600*b
^5*Sin[c + d*x] - 44*a^4*b*Sin[3*(c + d*x)] - 50*a^2*b^3*Sin[3*(c + d*x)] + 300*b^5*Sin[3*(c + d*x)] + 26*a^2*
b^3*Sin[5*(c + d*x)] - 60*b^5*Sin[5*(c + d*x)]))/(b + a*Csc[c + d*x])^2)/(128*a^7*d)

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Maple [B]  time = 0.223, size = 889, normalized size = 2.6 \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)^4*csc(d*x+c)^5/(a+b*sin(d*x+c))^3,x)

[Out]

-1/d/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*b^4-17/d/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)*b^3+33/d/a^5*b^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))-5/d/a^6*b^3*tan(1/2*d*x+1/2*c)+15/d/a^7*ln(tan(1/2*d*x+1/2*c))*b^4+1/8/d/a
^4*b/tan(1/2*d*x+1/2*c)^3+11/d*b^4/a^5/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2-3/4/d/a^5/tan(1/2*d
*x+1/2*c)^2*b^2+3/8/d/a^3*ln(tan(1/2*d*x+1/2*c))+5/d*b^3/a^6/tan(1/2*d*x+1/2*c)-1/8/d/a^4*tan(1/2*d*x+1/2*c)^3
*b+3/4/d/a^5*b^2*tan(1/2*d*x+1/2*c)^2-6/d/a^3*b^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)^2-6/d/a^3*b/(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))-1/8/d/a^3*tan(
1/2*d*x+1/2*c)^2-7/d/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*b^3-9/d/a^5*
ln(tan(1/2*d*x+1/2*c))*b^2-15/8/d*b/a^4/tan(1/2*d*x+1/2*c)+32/d*b^5/a^6/(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)-30/d*b^5/a^7/(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^5/a^6/(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+22/d*b^6/a^7/(t
an(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)^2-1/64/d/a^3/tan(1/2*d*x+1/2*c)^4+1/64/d/
a^3*tan(1/2*d*x+1/2*c)^4+15/8/d/a^4*tan(1/2*d*x+1/2*c)*b-6/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*b^2+1/8/d/a^3/tan(1/2*d*x+1/2*c)^2

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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)^4*csc(d*x+c)^5/(a+b*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 7.43483, size = 5862, normalized size = 17.24 \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)^4*csc(d*x+c)^5/(a+b*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

[1/16*(2*(83*a^6*b^2 - 263*a^4*b^4 + 180*a^2*b^6)*cos(d*x + c)^5 + 2*(5*a^8 - 181*a^6*b^2 + 536*a^4*b^4 - 360*
a^2*b^6)*cos(d*x + c)^3 + 12*(2*a^6*b - 9*a^4*b^3 - a^2*b^5 + 10*b^7 - (2*a^4*b^3 - 11*a^2*b^5 + 10*b^7)*cos(d
*x + c)^6 + (2*a^6*b - 5*a^4*b^3 - 23*a^2*b^5 + 30*b^7)*cos(d*x + c)^4 - (4*a^6*b - 16*a^4*b^3 - 13*a^2*b^5 +
30*b^7)*cos(d*x + c)^2 + 2*(2*a^5*b^2 - 11*a^3*b^4 + 10*a*b^6 + (2*a^5*b^2 - 11*a^3*b^4 + 10*a*b^6)*cos(d*x +
c)^4 - 2*(2*a^5*b^2 - 11*a^3*b^4 + 10*a*b^6)*cos(d*x + c)^2)*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*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)) - 6*(a^8 - 32*a^6*b^2 + 91*a^4*b^4 - 60*a^2*b^
6)*cos(d*x + c) + 3*(a^8 - 24*a^6*b^2 + 39*a^4*b^4 + 24*a^2*b^6 - 40*b^8 - (a^6*b^2 - 25*a^4*b^4 + 64*a^2*b^6
- 40*b^8)*cos(d*x + c)^6 + (a^8 - 22*a^6*b^2 - 11*a^4*b^4 + 152*a^2*b^6 - 120*b^8)*cos(d*x + c)^4 - (2*a^8 - 4
7*a^6*b^2 + 53*a^4*b^4 + 112*a^2*b^6 - 120*b^8)*cos(d*x + c)^2 + 2*(a^7*b - 25*a^5*b^3 + 64*a^3*b^5 - 40*a*b^7
+ (a^7*b - 25*a^5*b^3 + 64*a^3*b^5 - 40*a*b^7)*cos(d*x + c)^4 - 2*(a^7*b - 25*a^5*b^3 + 64*a^3*b^5 - 40*a*b^7
)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) - 3*(a^8 - 24*a^6*b^2 + 39*a^4*b^4 + 24*a^2*b^6 -
40*b^8 - (a^6*b^2 - 25*a^4*b^4 + 64*a^2*b^6 - 40*b^8)*cos(d*x + c)^6 + (a^8 - 22*a^6*b^2 - 11*a^4*b^4 + 152*a^
2*b^6 - 120*b^8)*cos(d*x + c)^4 - (2*a^8 - 47*a^6*b^2 + 53*a^4*b^4 + 112*a^2*b^6 - 120*b^8)*cos(d*x + c)^2 + 2
*(a^7*b - 25*a^5*b^3 + 64*a^3*b^5 - 40*a*b^7 + (a^7*b - 25*a^5*b^3 + 64*a^3*b^5 - 40*a*b^7)*cos(d*x + c)^4 - 2
*(a^7*b - 25*a^5*b^3 + 64*a^3*b^5 - 40*a*b^7)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2) + 4*(
2*(13*a^5*b^3 - 43*a^3*b^5 + 30*a*b^7)*cos(d*x + c)^5 - (11*a^7*b + 21*a^5*b^3 - 152*a^3*b^5 + 120*a*b^7)*cos(
d*x + c)^3 + 3*(3*a^7*b - a^5*b^3 - 22*a^3*b^5 + 20*a*b^7)*cos(d*x + c))*sin(d*x + c))/((a^9*b^2 - a^7*b^4)*d*
cos(d*x + c)^6 - (a^11 + 2*a^9*b^2 - 3*a^7*b^4)*d*cos(d*x + c)^4 + (2*a^11 + a^9*b^2 - 3*a^7*b^4)*d*cos(d*x +
c)^2 - (a^11 - a^7*b^4)*d - 2*((a^10*b - a^8*b^3)*d*cos(d*x + c)^4 - 2*(a^10*b - a^8*b^3)*d*cos(d*x + c)^2 + (
a^10*b - a^8*b^3)*d)*sin(d*x + c)), 1/16*(2*(83*a^6*b^2 - 263*a^4*b^4 + 180*a^2*b^6)*cos(d*x + c)^5 + 2*(5*a^8
- 181*a^6*b^2 + 536*a^4*b^4 - 360*a^2*b^6)*cos(d*x + c)^3 - 24*(2*a^6*b - 9*a^4*b^3 - a^2*b^5 + 10*b^7 - (2*a
^4*b^3 - 11*a^2*b^5 + 10*b^7)*cos(d*x + c)^6 + (2*a^6*b - 5*a^4*b^3 - 23*a^2*b^5 + 30*b^7)*cos(d*x + c)^4 - (4
*a^6*b - 16*a^4*b^3 - 13*a^2*b^5 + 30*b^7)*cos(d*x + c)^2 + 2*(2*a^5*b^2 - 11*a^3*b^4 + 10*a*b^6 + (2*a^5*b^2
- 11*a^3*b^4 + 10*a*b^6)*cos(d*x + c)^4 - 2*(2*a^5*b^2 - 11*a^3*b^4 + 10*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))) - 6*(a^8 - 32*a^6*b^2 + 91*a^4*b^
4 - 60*a^2*b^6)*cos(d*x + c) + 3*(a^8 - 24*a^6*b^2 + 39*a^4*b^4 + 24*a^2*b^6 - 40*b^8 - (a^6*b^2 - 25*a^4*b^4
+ 64*a^2*b^6 - 40*b^8)*cos(d*x + c)^6 + (a^8 - 22*a^6*b^2 - 11*a^4*b^4 + 152*a^2*b^6 - 120*b^8)*cos(d*x + c)^4
- (2*a^8 - 47*a^6*b^2 + 53*a^4*b^4 + 112*a^2*b^6 - 120*b^8)*cos(d*x + c)^2 + 2*(a^7*b - 25*a^5*b^3 + 64*a^3*b
^5 - 40*a*b^7 + (a^7*b - 25*a^5*b^3 + 64*a^3*b^5 - 40*a*b^7)*cos(d*x + c)^4 - 2*(a^7*b - 25*a^5*b^3 + 64*a^3*b
^5 - 40*a*b^7)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) - 3*(a^8 - 24*a^6*b^2 + 39*a^4*b^4 +
24*a^2*b^6 - 40*b^8 - (a^6*b^2 - 25*a^4*b^4 + 64*a^2*b^6 - 40*b^8)*cos(d*x + c)^6 + (a^8 - 22*a^6*b^2 - 11*a^4
*b^4 + 152*a^2*b^6 - 120*b^8)*cos(d*x + c)^4 - (2*a^8 - 47*a^6*b^2 + 53*a^4*b^4 + 112*a^2*b^6 - 120*b^8)*cos(d
*x + c)^2 + 2*(a^7*b - 25*a^5*b^3 + 64*a^3*b^5 - 40*a*b^7 + (a^7*b - 25*a^5*b^3 + 64*a^3*b^5 - 40*a*b^7)*cos(d
*x + c)^4 - 2*(a^7*b - 25*a^5*b^3 + 64*a^3*b^5 - 40*a*b^7)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*cos(d*x + c)
+ 1/2) + 4*(2*(13*a^5*b^3 - 43*a^3*b^5 + 30*a*b^7)*cos(d*x + c)^5 - (11*a^7*b + 21*a^5*b^3 - 152*a^3*b^5 + 12
0*a*b^7)*cos(d*x + c)^3 + 3*(3*a^7*b - a^5*b^3 - 22*a^3*b^5 + 20*a*b^7)*cos(d*x + c))*sin(d*x + c))/((a^9*b^2
- a^7*b^4)*d*cos(d*x + c)^6 - (a^11 + 2*a^9*b^2 - 3*a^7*b^4)*d*cos(d*x + c)^4 + (2*a^11 + a^9*b^2 - 3*a^7*b^4)
*d*cos(d*x + c)^2 - (a^11 - a^7*b^4)*d - 2*((a^10*b - a^8*b^3)*d*cos(d*x + c)^4 - 2*(a^10*b - a^8*b^3)*d*cos(d
*x + c)^2 + (a^10*b - a^8*b^3)*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)**4*csc(d*x+c)**5/(a+b*sin(d*x+c))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.38712, size = 743, normalized size = 2.19 \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)^4*csc(d*x+c)^5/(a+b*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/64*(24*(a^4 - 24*a^2*b^2 + 40*b^4)*log(abs(tan(1/2*d*x + 1/2*c)))/a^7 - 192*(2*a^4*b - 11*a^2*b^3 + 10*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)))/(sqrt(a^2 - b
^2)*a^7) - 64*(7*a^3*b^3*tan(1/2*d*x + 1/2*c)^3 - 12*a*b^5*tan(1/2*d*x + 1/2*c)^3 + 6*a^4*b^2*tan(1/2*d*x + 1/
2*c)^2 + a^2*b^4*tan(1/2*d*x + 1/2*c)^2 - 22*b^6*tan(1/2*d*x + 1/2*c)^2 + 17*a^3*b^3*tan(1/2*d*x + 1/2*c) - 32
*a*b^5*tan(1/2*d*x + 1/2*c) + 6*a^4*b^2 - 11*a^2*b^4)/((a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) +
a)^2*a^7) - (50*a^4*tan(1/2*d*x + 1/2*c)^4 - 1200*a^2*b^2*tan(1/2*d*x + 1/2*c)^4 + 2000*b^4*tan(1/2*d*x + 1/2*
c)^4 + 120*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 320*a*b^3*tan(1/2*d*x + 1/2*c)^3 - 8*a^4*tan(1/2*d*x + 1/2*c)^2 + 48
*a^2*b^2*tan(1/2*d*x + 1/2*c)^2 - 8*a^3*b*tan(1/2*d*x + 1/2*c) + a^4)/(a^7*tan(1/2*d*x + 1/2*c)^4) + (a^9*tan(
1/2*d*x + 1/2*c)^4 - 8*a^8*b*tan(1/2*d*x + 1/2*c)^3 - 8*a^9*tan(1/2*d*x + 1/2*c)^2 + 48*a^7*b^2*tan(1/2*d*x +
1/2*c)^2 + 120*a^8*b*tan(1/2*d*x + 1/2*c) - 320*a^6*b^3*tan(1/2*d*x + 1/2*c))/a^12)/d