3.1271 \(\int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx\)

Optimal. Leaf size=395 \[ -\frac {3 \left (a^4-b^4\right ) \cos (c+d x)}{a^4 b^2 d (a+b \sin (c+d x))}+\frac {3 b \cot (c+d x)}{a^4 d}-\frac {\tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac {3 \left (a^2-b^2\right ) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}+\frac {3 \left (a^2-2 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{2 a^3 b^2 d (a+b \sin (c+d x))^2}-\frac {6 \left (a^2+b^2\right ) \sqrt {a^2-b^2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^3 b^3 d}+\frac {\left (2 a^2+b^2\right ) \sqrt {a^2-b^2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^3 b^3 d}+\frac {6 \left (a^6+a^2 b^4-2 b^6\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^5 b^3 d \sqrt {a^2-b^2}}-\frac {x}{b^3} \]

[Out]

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

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Rubi [A]  time = 0.52, antiderivative size = 395, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 11, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {2897, 3770, 3767, 8, 3768, 2664, 2754, 12, 2660, 618, 204} \[ -\frac {6 \left (a^2+b^2\right ) \sqrt {a^2-b^2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^3 b^3 d}+\frac {\left (2 a^2+b^2\right ) \sqrt {a^2-b^2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^3 b^3 d}+\frac {6 \left (a^2 b^4+a^6-2 b^6\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^5 b^3 d \sqrt {a^2-b^2}}+\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{2 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac {3 \left (a^2-b^2\right ) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {3 \left (a^4-b^4\right ) \cos (c+d x)}{a^4 b^2 d (a+b \sin (c+d x))}+\frac {3 \left (a^2-2 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac {3 b \cot (c+d x)}{a^4 d}-\frac {\tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a^3 d}-\frac {x}{b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 8

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 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 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 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 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 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 {\cos ^3(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx &=\int \left (-\frac {1}{b^3}-\frac {3 \left (a^2-2 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 {\left (a^2-b^2\right )^3}{a^3 b^3 (a+b \sin (c+d x))^3}-\frac {3 \left (a^2-b^2\right )^2 \left (a^2+b^2\right )}{a^4 b^3 (a+b \sin (c+d x))^2}+\frac {3 \left (a^6+a^2 b^4-2 b^6\right )}{a^5 b^3 (a+b \sin (c+d x))}\right ) \, dx\\ &=-\frac {x}{b^3}+\frac {\int \csc ^3(c+d x) \, dx}{a^3}-\frac {(3 b) \int \csc ^2(c+d x) \, dx}{a^4}-\frac {\left (3 \left (a^2-2 b^2\right )\right ) \int \csc (c+d x) \, dx}{a^5}+\frac {\left (a^2-b^2\right )^3 \int \frac {1}{(a+b \sin (c+d x))^3} \, dx}{a^3 b^3}-\frac {\left (3 \left (a^2-b^2\right )^2 \left (a^2+b^2\right )\right ) \int \frac {1}{(a+b \sin (c+d x))^2} \, dx}{a^4 b^3}+\frac {\left (3 \left (a^6+a^2 b^4-2 b^6\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^5 b^3}\\ &=-\frac {x}{b^3}+\frac {3 \left (a^2-2 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 {\left (a^2-b^2\right )^2 \cos (c+d x)}{2 a^3 b^2 d (a+b \sin (c+d x))^2}-\frac {3 \left (a^4-b^4\right ) \cos (c+d x)}{a^4 b^2 d (a+b \sin (c+d x))}+\frac {\int \csc (c+d x) \, dx}{2 a^3}-\frac {\left (a^2-b^2\right )^2 \int \frac {-2 a+b \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx}{2 a^3 b^3}+\frac {\left (3 \left (a^2-b^2\right )^2 \left (a^2+b^2\right )\right ) \int \frac {a}{a+b \sin (c+d x)} \, dx}{a^4 b^3 \left (-a^2+b^2\right )}+\frac {(3 b) \operatorname {Subst}(\int 1 \, dx,x,\cot (c+d x))}{a^4 d}+\frac {\left (6 \left (a^6+a^2 b^4-2 b^6\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 b^3 d}\\ &=-\frac {x}{b^3}-\frac {\tanh ^{-1}(\cos (c+d x))}{2 a^3 d}+\frac {3 \left (a^2-2 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 {\left (a^2-b^2\right )^2 \cos (c+d x)}{2 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac {3 \left (a^2-b^2\right ) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {3 \left (a^4-b^4\right ) \cos (c+d x)}{a^4 b^2 d (a+b \sin (c+d x))}-\left (3 \left (\frac {a}{b^3}-\frac {b}{a^3}\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx+\frac {\left (a^2-b^2\right ) \int \frac {2 a^2+b^2}{a+b \sin (c+d x)} \, dx}{2 a^3 b^3}-\frac {\left (12 \left (a^6+a^2 b^4-2 b^6\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 b^3 d}\\ &=-\frac {x}{b^3}+\frac {6 \left (a^6+a^2 b^4-2 b^6\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^5 b^3 \sqrt {a^2-b^2} d}-\frac {\tanh ^{-1}(\cos (c+d x))}{2 a^3 d}+\frac {3 \left (a^2-2 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 {\left (a^2-b^2\right )^2 \cos (c+d x)}{2 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac {3 \left (a^2-b^2\right ) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {3 \left (a^4-b^4\right ) \cos (c+d x)}{a^4 b^2 d (a+b \sin (c+d x))}+\frac {\left (\left (a^2-b^2\right ) \left (2 a^2+b^2\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{2 a^3 b^3}-\frac {\left (6 \left (\frac {a}{b^3}-\frac {b}{a^3}\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 )}{d}\\ &=-\frac {x}{b^3}+\frac {6 \left (a^6+a^2 b^4-2 b^6\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^5 b^3 \sqrt {a^2-b^2} d}-\frac {\tanh ^{-1}(\cos (c+d x))}{2 a^3 d}+\frac {3 \left (a^2-2 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 {\left (a^2-b^2\right )^2 \cos (c+d x)}{2 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac {3 \left (a^2-b^2\right ) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {3 \left (a^4-b^4\right ) \cos (c+d x)}{a^4 b^2 d (a+b \sin (c+d x))}+\frac {\left (12 \left (\frac {a}{b^3}-\frac {b}{a^3}\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 )}{d}+\frac {\left (\left (a^2-b^2\right ) \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 b^3 d}\\ &=-\frac {x}{b^3}-\frac {6 \left (\frac {a}{b^3}-\frac {b}{a^3}\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} d}+\frac {6 \left (a^6+a^2 b^4-2 b^6\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^5 b^3 \sqrt {a^2-b^2} d}-\frac {\tanh ^{-1}(\cos (c+d x))}{2 a^3 d}+\frac {3 \left (a^2-2 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 {\left (a^2-b^2\right )^2 \cos (c+d x)}{2 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac {3 \left (a^2-b^2\right ) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {3 \left (a^4-b^4\right ) \cos (c+d x)}{a^4 b^2 d (a+b \sin (c+d x))}-\frac {\left (2 \left (a^2-b^2\right ) \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 b^3 d}\\ &=-\frac {x}{b^3}-\frac {6 \left (\frac {a}{b^3}-\frac {b}{a^3}\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} d}+\frac {\sqrt {a^2-b^2} \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 b^3 d}+\frac {6 \left (a^6+a^2 b^4-2 b^6\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^5 b^3 \sqrt {a^2-b^2} d}-\frac {\tanh ^{-1}(\cos (c+d x))}{2 a^3 d}+\frac {3 \left (a^2-2 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 {\left (a^2-b^2\right )^2 \cos (c+d x)}{2 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac {3 \left (a^2-b^2\right ) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {3 \left (a^4-b^4\right ) \cos (c+d x)}{a^4 b^2 d (a+b \sin (c+d x))}\\ \end {align*}

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Mathematica [A]  time = 6.26, size = 384, normalized size = 0.97 \[ -\frac {3 b \tan \left (\frac {1}{2} (c+d x)\right )}{2 a^4 d}+\frac {3 b \cot \left (\frac {1}{2} (c+d x)\right )}{2 a^4 d}-\frac {\csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 a^3 d}+\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 a^3 d}+\frac {\left (12 b^2-5 a^2\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^5 d}+\frac {\left (5 a^2-12 b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^5 d}-\frac {3 \left (a^4 \cos (c+d x)+a^2 b^2 \cos (c+d x)-2 b^4 \cos (c+d x)\right )}{2 a^4 b^2 d (a+b \sin (c+d x))}+\frac {a^4 \cos (c+d x)-2 a^2 b^2 \cos (c+d x)+b^4 \cos (c+d x)}{2 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac {\left (2 a^6-a^4 b^2+11 a^2 b^4-12 b^6\right ) \tan ^{-1}\left (\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (a \sin \left (\frac {1}{2} (c+d x)\right )+b \cos \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^5 b^3 d \sqrt {a^2-b^2}}-\frac {c+d x}{b^3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((c + d*x)/(b^3*d)) + ((2*a^6 - a^4*b^2 + 11*a^2*b^4 - 12*b^6)*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*b^3*Sqrt[a^2 - b^2]*d) + (3*b*Cot[(c + d*x)/2])/(2*a^4*d) - Csc[(
c + d*x)/2]^2/(8*a^3*d) + ((5*a^2 - 12*b^2)*Log[Cos[(c + d*x)/2]])/(2*a^5*d) + ((-5*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) + (a^4*Cos[c + d*x] - 2*a^2*b^2*Cos[c + d*x] + b^4*Cos[c +
d*x])/(2*a^3*b^2*d*(a + b*Sin[c + d*x])^2) - (3*(a^4*Cos[c + d*x] + a^2*b^2*Cos[c + d*x] - 2*b^4*Cos[c + d*x]
))/(2*a^4*b^2*d*(a + b*Sin[c + d*x])) - (3*b*Tan[(c + d*x)/2])/(2*a^4*d)

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fricas [B]  time = 1.72, size = 1658, normalized size = 4.20 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^3/(a+b*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

[-1/4*(4*a^5*b^2*d*x*cos(d*x + c)^4 - 4*(a^7 + 2*a^5*b^2)*d*x*cos(d*x + c)^2 - 2*(2*a^6*b + 5*a^4*b^3 - 18*a^2
*b^5)*cos(d*x + c)^3 + 4*(a^7 + a^5*b^2)*d*x - (2*a^6 + 3*a^4*b^2 + 13*a^2*b^4 + 12*b^6 + (2*a^4*b^2 + a^2*b^4
+ 12*b^6)*cos(d*x + c)^4 - (2*a^6 + 5*a^4*b^2 + 14*a^2*b^4 + 24*b^6)*cos(d*x + c)^2 + 2*(2*a^5*b + a^3*b^3 +
12*a*b^5 - (2*a^5*b + a^3*b^3 + 12*a*b^5)*cos(d*x + c)^2)*sin(d*x + c))*sqrt(-a^2 + b^2)*log(-((2*a^2 - b^2)*c
os(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)) + 4*(a^6*b + 3*a^4*b^3 - 9*a^2*b^5)*cos(d*x + c)
- (5*a^4*b^3 - 7*a^2*b^5 - 12*b^7 + (5*a^2*b^5 - 12*b^7)*cos(d*x + c)^4 - (5*a^4*b^3 - 2*a^2*b^5 - 24*b^7)*cos
(d*x + c)^2 + 2*(5*a^3*b^4 - 12*a*b^6 - (5*a^3*b^4 - 12*a*b^6)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x +
c) + 1/2) + (5*a^4*b^3 - 7*a^2*b^5 - 12*b^7 + (5*a^2*b^5 - 12*b^7)*cos(d*x + c)^4 - (5*a^4*b^3 - 2*a^2*b^5 -
24*b^7)*cos(d*x + c)^2 + 2*(5*a^3*b^4 - 12*a*b^6 - (5*a^3*b^4 - 12*a*b^6)*cos(d*x + c)^2)*sin(d*x + c))*log(-1
/2*cos(d*x + c) + 1/2) - 2*(4*a^6*b*d*x*cos(d*x + c)^2 - 4*a^6*b*d*x + 3*(a^5*b^2 + a^3*b^4 - 4*a*b^6)*cos(d*x
+ c)^3 - (3*a^5*b^2 - a^3*b^4 - 12*a*b^6)*cos(d*x + c))*sin(d*x + c))/(a^5*b^5*d*cos(d*x + c)^4 - (a^7*b^3 +
2*a^5*b^5)*d*cos(d*x + c)^2 + (a^7*b^3 + a^5*b^5)*d - 2*(a^6*b^4*d*cos(d*x + c)^2 - a^6*b^4*d)*sin(d*x + c)),
-1/4*(4*a^5*b^2*d*x*cos(d*x + c)^4 - 4*(a^7 + 2*a^5*b^2)*d*x*cos(d*x + c)^2 - 2*(2*a^6*b + 5*a^4*b^3 - 18*a^2*
b^5)*cos(d*x + c)^3 + 4*(a^7 + a^5*b^2)*d*x + 2*(2*a^6 + 3*a^4*b^2 + 13*a^2*b^4 + 12*b^6 + (2*a^4*b^2 + a^2*b^
4 + 12*b^6)*cos(d*x + c)^4 - (2*a^6 + 5*a^4*b^2 + 14*a^2*b^4 + 24*b^6)*cos(d*x + c)^2 + 2*(2*a^5*b + a^3*b^3 +
12*a*b^5 - (2*a^5*b + a^3*b^3 + 12*a*b^5)*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))) + 4*(a^6*b + 3*a^4*b^3 - 9*a^2*b^5)*cos(d*x + c) - (5*a^4*b^3 - 7*a^2*
b^5 - 12*b^7 + (5*a^2*b^5 - 12*b^7)*cos(d*x + c)^4 - (5*a^4*b^3 - 2*a^2*b^5 - 24*b^7)*cos(d*x + c)^2 + 2*(5*a^
3*b^4 - 12*a*b^6 - (5*a^3*b^4 - 12*a*b^6)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) + (5*a^4*b
^3 - 7*a^2*b^5 - 12*b^7 + (5*a^2*b^5 - 12*b^7)*cos(d*x + c)^4 - (5*a^4*b^3 - 2*a^2*b^5 - 24*b^7)*cos(d*x + c)^
2 + 2*(5*a^3*b^4 - 12*a*b^6 - (5*a^3*b^4 - 12*a*b^6)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2
) - 2*(4*a^6*b*d*x*cos(d*x + c)^2 - 4*a^6*b*d*x + 3*(a^5*b^2 + a^3*b^4 - 4*a*b^6)*cos(d*x + c)^3 - (3*a^5*b^2
- a^3*b^4 - 12*a*b^6)*cos(d*x + c))*sin(d*x + c))/(a^5*b^5*d*cos(d*x + c)^4 - (a^7*b^3 + 2*a^5*b^5)*d*cos(d*x
+ c)^2 + (a^7*b^3 + a^5*b^5)*d - 2*(a^6*b^4*d*cos(d*x + c)^2 - a^6*b^4*d)*sin(d*x + c))]

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giac [A]  time = 0.29, size = 512, normalized size = 1.30 \[ -\frac {\frac {8 \, {\left (d x + c\right )}}{b^{3}} + \frac {4 \, {\left (5 \, a^{2} - 12 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{5}} - \frac {a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}} - \frac {8 \, {\left (2 \, a^{6} - a^{4} b^{2} + 11 \, a^{2} b^{4} - 12 \, b^{6}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (a) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} a^{5} b^{3}} - \frac {10 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 24 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 8 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 4 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 32 \, a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 16 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 53 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 16 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 16 \, b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 56 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 44 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 112 \, a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 16 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 32 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 76 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{4} b^{2}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}^{2} a^{5} b^{2}}}{8 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^3/(a+b*sin(d*x+c))^3,x, algorithm="giac")

[Out]

-1/8*(8*(d*x + c)/b^3 + 4*(5*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 - 8*(2*a^6 - a^4*b^2 + 11*a^2*b^4 - 12*b^6)*(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^5*b^3) - (10*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 - 8*a^5*b*tan(1/2*d*x + 1/2*c)^5 - 4*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 - 16*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 + 16*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 - 56*a^5*b*tan(1/2*d*x + 1/2*c)^3 - 4
4*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 - 16*a^6*tan(1/2*d*x + 1/2*c)^2 - 32*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^3*b^3*tan(1/2*d*x + 1/2*c) - a^4*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*a^5*b^2))/d

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maple [B]  time = 0.87, size = 943, normalized size = 2.39 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*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-2/d/b^3*arctan(tan(1/2*d*x+1/2*c))-1/8/d/a^3/tan
(1/2*d*x+1/2*c)^2-5/2/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)-1/d/b/(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)^3-7/d/a^2*b/(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/a^4*b^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)^3-2/d*a/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)^2-9/d/(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)^2-3/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+14/d/a^5*b^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)^2-7/d/b/(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)-13/d/a^2*b/(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/a^4*b^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/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-5/d/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/a^3*b^2/(tan(1/2*d*x+1/2*c)^2*a+2*ta
n(1/2*d*x+1/2*c)*b+a)^2+2/d*a/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))-1/d
/a/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))+11/d/a^3*b/(a^2-b^2)^(1/2)*arcta
n(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))-12/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))

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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(cos(d*x+c)^6*csc(d*x+c)^3/(a+b*sin(d*x+c))^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 = 13.02, size = 4381, normalized size = 11.09 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^6/(sin(c + d*x)^3*(a + b*sin(c + d*x))^3),x)

[Out]

tan(c/2 + (d*x)/2)^2/(8*a^3*d) + (2*atan((1800*tan(c/2 + (d*x)/2))/((7120*b^2)/a^2 - (12520*b^4)/a^4 + (24480*
b^6)/a^6 - (31104*b^8)/a^8 + (13824*b^10)/a^10 - (1120*a*tan(c/2 + (d*x)/2))/b + (2320*b*tan(c/2 + (d*x)/2))/a
- (4224*b^3*tan(c/2 + (d*x)/2))/a^3 + (720*a^3*tan(c/2 + (d*x)/2))/b^3 + (2304*b^5*tan(c/2 + (d*x)/2))/a^5 -
1800) - (7120*tan(c/2 + (d*x)/2))/((24480*b^4)/a^4 - (1800*a^2)/b^2 - (12520*b^2)/a^2 - (31104*b^6)/a^6 + (138
24*b^8)/a^8 + (2320*a*tan(c/2 + (d*x)/2))/b - (4224*b*tan(c/2 + (d*x)/2))/a + (2304*b^3*tan(c/2 + (d*x)/2))/a^
3 - (1120*a^3*tan(c/2 + (d*x)/2))/b^3 + (720*a^5*tan(c/2 + (d*x)/2))/b^5 + 7120) + (720*a)/(720*a*tan(c/2 + (d
*x)/2) - (1800*b^3)/a^2 + (7120*b^5)/a^4 - (12520*b^7)/a^6 + (24480*b^9)/a^8 - (31104*b^11)/a^10 + (13824*b^13
)/a^12 - (1120*b^2*tan(c/2 + (d*x)/2))/a + (2320*b^4*tan(c/2 + (d*x)/2))/a^3 - (4224*b^6*tan(c/2 + (d*x)/2))/a
^5 + (2304*b^8*tan(c/2 + (d*x)/2))/a^7) - 4224/((7120*a)/b - 4224*tan(c/2 + (d*x)/2) - (12520*b)/a + (24480*b^
3)/a^3 - (1800*a^3)/b^3 - (31104*b^5)/a^5 + (13824*b^7)/a^7 + (2304*b^2*tan(c/2 + (d*x)/2))/a^2 + (2320*a^2*ta
n(c/2 + (d*x)/2))/b^2 - (1120*a^4*tan(c/2 + (d*x)/2))/b^4 + (720*a^6*tan(c/2 + (d*x)/2))/b^6) + 2320/(2320*tan
(c/2 + (d*x)/2) - (1800*a)/b + (7120*b)/a - (12520*b^3)/a^3 + (24480*b^5)/a^5 - (31104*b^7)/a^7 + (13824*b^9)/
a^9 - (4224*b^2*tan(c/2 + (d*x)/2))/a^2 - (1120*a^2*tan(c/2 + (d*x)/2))/b^2 + (2304*b^4*tan(c/2 + (d*x)/2))/a^
4 + (720*a^4*tan(c/2 + (d*x)/2))/b^4) + (2304*b^2)/((24480*b^3)/a - 12520*a*b + (7120*a^3)/b - (31104*b^5)/a^3
- (1800*a^5)/b^3 + (13824*b^7)/a^5 - 4224*a^2*tan(c/2 + (d*x)/2) + 2304*b^2*tan(c/2 + (d*x)/2) + (2320*a^4*ta
n(c/2 + (d*x)/2))/b^2 - (1120*a^6*tan(c/2 + (d*x)/2))/b^4 + (720*a^8*tan(c/2 + (d*x)/2))/b^6) - 1120/((7120*b^
3)/a^3 - (1800*b)/a - 1120*tan(c/2 + (d*x)/2) - (12520*b^5)/a^5 + (24480*b^7)/a^7 - (31104*b^9)/a^9 + (13824*b
^11)/a^11 + (2320*b^2*tan(c/2 + (d*x)/2))/a^2 + (720*a^2*tan(c/2 + (d*x)/2))/b^2 - (4224*b^4*tan(c/2 + (d*x)/2
))/a^4 + (2304*b^6*tan(c/2 + (d*x)/2))/a^6) - (24480*b^3*tan(c/2 + (d*x)/2))/(24480*b^3 - 12520*a^2*b + (7120*
a^4)/b - (31104*b^5)/a^2 - (1800*a^6)/b^3 + (13824*b^7)/a^4 - 4224*a^3*tan(c/2 + (d*x)/2) + 2304*a*b^2*tan(c/2
+ (d*x)/2) + (2320*a^5*tan(c/2 + (d*x)/2))/b^2 - (1120*a^7*tan(c/2 + (d*x)/2))/b^4 + (720*a^9*tan(c/2 + (d*x)
/2))/b^6) + (31104*b^5*tan(c/2 + (d*x)/2))/(24480*a^2*b^3 - 31104*b^5 - 12520*a^4*b + (7120*a^6)/b + (13824*b^
7)/a^2 - (1800*a^8)/b^3 - 4224*a^5*tan(c/2 + (d*x)/2) + 2304*a^3*b^2*tan(c/2 + (d*x)/2) + (2320*a^7*tan(c/2 +
(d*x)/2))/b^2 - (1120*a^9*tan(c/2 + (d*x)/2))/b^4 + (720*a^11*tan(c/2 + (d*x)/2))/b^6) - (13824*b^7*tan(c/2 +
(d*x)/2))/(13824*b^7 - 12520*a^6*b - 31104*a^2*b^5 + 24480*a^4*b^3 + (7120*a^8)/b - (1800*a^10)/b^3 - 4224*a^7
*tan(c/2 + (d*x)/2) + 2304*a^5*b^2*tan(c/2 + (d*x)/2) + (2320*a^9*tan(c/2 + (d*x)/2))/b^2 - (1120*a^11*tan(c/2
+ (d*x)/2))/b^4 + (720*a^13*tan(c/2 + (d*x)/2))/b^6) + (12520*b*tan(c/2 + (d*x)/2))/((7120*a^2)/b - 4224*a*ta
n(c/2 + (d*x)/2) - 12520*b + (24480*b^3)/a^2 - (1800*a^4)/b^3 - (31104*b^5)/a^4 + (13824*b^7)/a^6 + (2304*b^2*
tan(c/2 + (d*x)/2))/a + (2320*a^3*tan(c/2 + (d*x)/2))/b^2 - (1120*a^5*tan(c/2 + (d*x)/2))/b^4 + (720*a^7*tan(c
/2 + (d*x)/2))/b^6)))/(b^3*d) - (a^3/2 + (tan(c/2 + (d*x)/2)^2*(8*a^5 - 50*a*b^4 + 21*a^3*b^2))/b^2 - 4*a^2*b*
tan(c/2 + (d*x)/2) + (2*tan(c/2 + (d*x)/2)^5*(2*a^4 - 16*b^4 + 11*a^2*b^2))/b + (2*tan(c/2 + (d*x)/2)^3*(14*a^
4 - 52*b^4 + 21*a^2*b^2))/b + (tan(c/2 + (d*x)/2)^4*(16*a^6 - 112*b^6 - 24*a^2*b^4 + 73*a^4*b^2))/(2*a*b^2))/(
d*(4*a^6*tan(c/2 + (d*x)/2)^2 + 4*a^6*tan(c/2 + (d*x)/2)^6 + tan(c/2 + (d*x)/2)^4*(8*a^6 + 16*a^4*b^2) + 16*a^
5*b*tan(c/2 + (d*x)/2)^3 + 16*a^5*b*tan(c/2 + (d*x)/2)^5)) - (3*b*tan(c/2 + (d*x)/2))/(2*a^4*d) - (log(tan(c/2
+ (d*x)/2))*(5*a^2 - 12*b^2))/(2*a^5*d) - (atan((((-(a + b)*(a - b))^(1/2)*(a^4 + 6*b^4 + (a^2*b^2)/2)*((4*ta
n(c/2 + (d*x)/2)*(1728*b^18 - 3888*a^2*b^16 + 3060*a^4*b^14 - 1565*a^6*b^12 + 1658*a^8*b^10 - 1361*a^10*b^8 +
484*a^12*b^6 - 120*a^14*b^4 + 32*a^16*b^2))/(a^12*b^8) - (4*(28*a^15 + 288*a^5*b^10 - 528*a^7*b^8 - 94*a^9*b^6
+ 308*a^11*b^4 - 30*a^13*b^2))/(a^12*b^5) + ((-(a + b)*(a - b))^(1/2)*(a^4 + 6*b^4 + (a^2*b^2)/2)*((4*(1152*a
^4*b^14 - 1824*a^6*b^12 + 920*a^8*b^10 - 318*a^10*b^8 + 70*a^12*b^6 + 32*a^14*b^4 - 24*a^16*b^2))/(a^12*b^5) +
(4*tan(c/2 + (d*x)/2)*(288*a^5*b^16 - 240*a^7*b^14 + 410*a^9*b^12 - 468*a^11*b^10 + 202*a^13*b^8 - 200*a^15*b
^6 + 16*a^17*b^4))/(a^12*b^8) + ((-(a + b)*(a - b))^(1/2)*((4*(384*a^9*b^12 - 448*a^11*b^10 + 120*a^13*b^8 - 2
8*a^15*b^6))/(a^12*b^5) + (4*tan(c/2 + (d*x)/2)*(768*a^8*b^16 - 1136*a^10*b^14 + 484*a^12*b^12 - 120*a^14*b^10
+ 32*a^16*b^8))/(a^12*b^8) + ((-(a + b)*(a - b))^(1/2)*((4*(32*a^14*b^10 - 24*a^16*b^8))/(a^12*b^5) + (4*tan(
c/2 + (d*x)/2)*(128*a^13*b^14 - 136*a^15*b^12 + 16*a^17*b^10))/(a^12*b^8))*(a^4 + 6*b^4 + (a^2*b^2)/2))/(a^5*b
^3))*(a^4 + 6*b^4 + (a^2*b^2)/2))/(a^5*b^3)))/(a^5*b^3))*1i)/(a^5*b^3) - ((-(a + b)*(a - b))^(1/2)*(a^4 + 6*b^
4 + (a^2*b^2)/2)*((4*(28*a^15 + 288*a^5*b^10 - 528*a^7*b^8 - 94*a^9*b^6 + 308*a^11*b^4 - 30*a^13*b^2))/(a^12*b
^5) - (4*tan(c/2 + (d*x)/2)*(1728*b^18 - 3888*a^2*b^16 + 3060*a^4*b^14 - 1565*a^6*b^12 + 1658*a^8*b^10 - 1361*
a^10*b^8 + 484*a^12*b^6 - 120*a^14*b^4 + 32*a^16*b^2))/(a^12*b^8) + ((-(a + b)*(a - b))^(1/2)*(a^4 + 6*b^4 + (
a^2*b^2)/2)*((4*(1152*a^4*b^14 - 1824*a^6*b^12 + 920*a^8*b^10 - 318*a^10*b^8 + 70*a^12*b^6 + 32*a^14*b^4 - 24*
a^16*b^2))/(a^12*b^5) + (4*tan(c/2 + (d*x)/2)*(288*a^5*b^16 - 240*a^7*b^14 + 410*a^9*b^12 - 468*a^11*b^10 + 20
2*a^13*b^8 - 200*a^15*b^6 + 16*a^17*b^4))/(a^12*b^8) - ((-(a + b)*(a - b))^(1/2)*((4*(384*a^9*b^12 - 448*a^11*
b^10 + 120*a^13*b^8 - 28*a^15*b^6))/(a^12*b^5) + (4*tan(c/2 + (d*x)/2)*(768*a^8*b^16 - 1136*a^10*b^14 + 484*a^
12*b^12 - 120*a^14*b^10 + 32*a^16*b^8))/(a^12*b^8) - ((-(a + b)*(a - b))^(1/2)*((4*(32*a^14*b^10 - 24*a^16*b^8
))/(a^12*b^5) + (4*tan(c/2 + (d*x)/2)*(128*a^13*b^14 - 136*a^15*b^12 + 16*a^17*b^10))/(a^12*b^8))*(a^4 + 6*b^4
+ (a^2*b^2)/2))/(a^5*b^3))*(a^4 + 6*b^4 + (a^2*b^2)/2))/(a^5*b^3)))/(a^5*b^3))*1i)/(a^5*b^3))/((8*(1728*b^12
- 70*a^12 - 3888*a^2*b^10 + 1908*a^4*b^8 + 259*a^6*b^6 - 30*a^8*b^4 + 93*a^10*b^2))/(a^12*b^5) + (8*tan(c/2 +
(d*x)/2)*(64*a^15 - 288*a^7*b^8 - 120*a^9*b^6 + 328*a^11*b^4 + 16*a^13*b^2))/(a^12*b^8) - ((-(a + b)*(a - b))^
(1/2)*(a^4 + 6*b^4 + (a^2*b^2)/2)*((4*tan(c/2 + (d*x)/2)*(1728*b^18 - 3888*a^2*b^16 + 3060*a^4*b^14 - 1565*a^6
*b^12 + 1658*a^8*b^10 - 1361*a^10*b^8 + 484*a^12*b^6 - 120*a^14*b^4 + 32*a^16*b^2))/(a^12*b^8) - (4*(28*a^15 +
288*a^5*b^10 - 528*a^7*b^8 - 94*a^9*b^6 + 308*a^11*b^4 - 30*a^13*b^2))/(a^12*b^5) + ((-(a + b)*(a - b))^(1/2)
*(a^4 + 6*b^4 + (a^2*b^2)/2)*((4*(1152*a^4*b^14 - 1824*a^6*b^12 + 920*a^8*b^10 - 318*a^10*b^8 + 70*a^12*b^6 +
32*a^14*b^4 - 24*a^16*b^2))/(a^12*b^5) + (4*tan(c/2 + (d*x)/2)*(288*a^5*b^16 - 240*a^7*b^14 + 410*a^9*b^12 - 4
68*a^11*b^10 + 202*a^13*b^8 - 200*a^15*b^6 + 16*a^17*b^4))/(a^12*b^8) + ((-(a + b)*(a - b))^(1/2)*((4*(384*a^9
*b^12 - 448*a^11*b^10 + 120*a^13*b^8 - 28*a^15*b^6))/(a^12*b^5) + (4*tan(c/2 + (d*x)/2)*(768*a^8*b^16 - 1136*a
^10*b^14 + 484*a^12*b^12 - 120*a^14*b^10 + 32*a^16*b^8))/(a^12*b^8) + ((-(a + b)*(a - b))^(1/2)*((4*(32*a^14*b
^10 - 24*a^16*b^8))/(a^12*b^5) + (4*tan(c/2 + (d*x)/2)*(128*a^13*b^14 - 136*a^15*b^12 + 16*a^17*b^10))/(a^12*b
^8))*(a^4 + 6*b^4 + (a^2*b^2)/2))/(a^5*b^3))*(a^4 + 6*b^4 + (a^2*b^2)/2))/(a^5*b^3)))/(a^5*b^3)))/(a^5*b^3) -
((-(a + b)*(a - b))^(1/2)*(a^4 + 6*b^4 + (a^2*b^2)/2)*((4*(28*a^15 + 288*a^5*b^10 - 528*a^7*b^8 - 94*a^9*b^6 +
308*a^11*b^4 - 30*a^13*b^2))/(a^12*b^5) - (4*tan(c/2 + (d*x)/2)*(1728*b^18 - 3888*a^2*b^16 + 3060*a^4*b^14 -
1565*a^6*b^12 + 1658*a^8*b^10 - 1361*a^10*b^8 + 484*a^12*b^6 - 120*a^14*b^4 + 32*a^16*b^2))/(a^12*b^8) + ((-(a
+ b)*(a - b))^(1/2)*(a^4 + 6*b^4 + (a^2*b^2)/2)*((4*(1152*a^4*b^14 - 1824*a^6*b^12 + 920*a^8*b^10 - 318*a^10*
b^8 + 70*a^12*b^6 + 32*a^14*b^4 - 24*a^16*b^2))/(a^12*b^5) + (4*tan(c/2 + (d*x)/2)*(288*a^5*b^16 - 240*a^7*b^1
4 + 410*a^9*b^12 - 468*a^11*b^10 + 202*a^13*b^8 - 200*a^15*b^6 + 16*a^17*b^4))/(a^12*b^8) - ((-(a + b)*(a - b)
)^(1/2)*((4*(384*a^9*b^12 - 448*a^11*b^10 + 120*a^13*b^8 - 28*a^15*b^6))/(a^12*b^5) + (4*tan(c/2 + (d*x)/2)*(7
68*a^8*b^16 - 1136*a^10*b^14 + 484*a^12*b^12 - 120*a^14*b^10 + 32*a^16*b^8))/(a^12*b^8) - ((-(a + b)*(a - b))^
(1/2)*((4*(32*a^14*b^10 - 24*a^16*b^8))/(a^12*b^5) + (4*tan(c/2 + (d*x)/2)*(128*a^13*b^14 - 136*a^15*b^12 + 16
*a^17*b^10))/(a^12*b^8))*(a^4 + 6*b^4 + (a^2*b^2)/2))/(a^5*b^3))*(a^4 + 6*b^4 + (a^2*b^2)/2))/(a^5*b^3)))/(a^5
*b^3)))/(a^5*b^3)))*(-(a + b)*(a - b))^(1/2)*(a^4 + 6*b^4 + (a^2*b^2)/2)*2i)/(a^5*b^3*d)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**6*csc(d*x+c)**3/(a+b*sin(d*x+c))**3,x)

[Out]

Timed out

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