3.1272 \(\int \frac {\cos ^2(c+d x) \cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx\)

Optimal. Leaf size=329 \[ \frac {5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}+\frac {5 \left (a^2-4 b^2\right ) \sqrt {a^2-b^2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^6 d}-\frac {5 b \left (3 a^2-4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^6 d}-\frac {5 \left (a^2-2 b^2\right ) \cot (c+d x)}{2 a^4 d (a+b \sin (c+d x))}-\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x)}{3 a^3 d (a+b \sin (c+d x))^2}+\frac {\left (3 a^4+35 a^2 b^2-60 b^4\right ) \cot (c+d x)}{6 a^5 b^2 d}-\frac {a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2} \]

[Out]

-5/2*b*(3*a^2-4*b^2)*arctanh(cos(d*x+c))/a^6/d+1/6*(3*a^4+35*a^2*b^2-60*b^4)*cot(d*x+c)/a^5/b^2/d-cos(d*x+c)/b
/d/(a+b*sin(d*x+c))^2-1/2*a*cot(d*x+c)/b^2/d/(a+b*sin(d*x+c))^2-1/3*(3*a^2-5*b^2)*cot(d*x+c)/a^3/d/(a+b*sin(d*
x+c))^2+5/6*b*cot(d*x+c)*csc(d*x+c)/a^2/d/(a+b*sin(d*x+c))^2-1/3*cot(d*x+c)*csc(d*x+c)^2/a/d/(a+b*sin(d*x+c))^
2-5/2*(a^2-2*b^2)*cot(d*x+c)/a^4/d/(a+b*sin(d*x+c))+5*(a^2-4*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^6/d

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Rubi [A]  time = 1.33, antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2896, 3055, 3001, 3770, 2660, 618, 204} \[ \frac {5 \left (a^2-4 b^2\right ) \sqrt {a^2-b^2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^6 d}+\frac {\left (35 a^2 b^2+3 a^4-60 b^4\right ) \cot (c+d x)}{6 a^5 b^2 d}-\frac {5 b \left (3 a^2-4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^6 d}-\frac {5 \left (a^2-2 b^2\right ) \cot (c+d x)}{2 a^4 d (a+b \sin (c+d x))}-\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x)}{3 a^3 d (a+b \sin (c+d x))^2}+\frac {5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}-\frac {a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^2*Cot[c + d*x]^4)/(a + b*Sin[c + d*x])^3,x]

[Out]

(5*(a^2 - 4*b^2)*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^6*d) - (5*b*(3*a^2 - 4*b
^2)*ArcTanh[Cos[c + d*x]])/(2*a^6*d) + ((3*a^4 + 35*a^2*b^2 - 60*b^4)*Cot[c + d*x])/(6*a^5*b^2*d) - Cos[c + d*
x]/(b*d*(a + b*Sin[c + d*x])^2) - (a*Cot[c + d*x])/(2*b^2*d*(a + b*Sin[c + d*x])^2) - ((3*a^2 - 5*b^2)*Cot[c +
d*x])/(3*a^3*d*(a + b*Sin[c + d*x])^2) + (5*b*Cot[c + d*x]*Csc[c + d*x])/(6*a^2*d*(a + b*Sin[c + d*x])^2) - (
Cot[c + d*x]*Csc[c + d*x]^2)/(3*a*d*(a + b*Sin[c + d*x])^2) - (5*(a^2 - 2*b^2)*Cot[c + d*x])/(2*a^4*d*(a + b*S
in[c + d*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 2896

Int[cos[(e_.) + (f_.)*(x_)]^6*((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^2*d^2*(n + 1)*(n + 2)*(m + n + 5)*(m + n + 6)), Int[(d*Sin[e + f*x])^(n + 2)*(a + b*Sin[e + f*
x])^m*Simp[a^4*(n + 1)*(n + 2)*(n + 3)*(n + 5) - a^2*b^2*(n + 2)*(2*n + 1)*(m + n + 5)*(m + n + 6) + b^4*(m +
n + 2)*(m + n + 3)*(m + n + 5)*(m + n + 6) + a*b*m*(a^2*(n + 1)*(n + 2) - b^2*(m + n + 5)*(m + n + 6))*Sin[e +
f*x] - (a^4*(n + 1)*(n + 2)*(4 + n)*(n + 5) + b^4*(m + n + 2)*(m + n + 4)*(m + n + 5)*(m + n + 6) - a^2*b^2*(
n + 1)*(n + 2)*(m + n + 5)*(2*n + 2*m + 13))*Sin[e + f*x]^2, x], x], x] - Simp[(b*(m + n + 2)*Cos[e + f*x]*(d*
Sin[e + f*x])^(n + 2)*(a + b*Sin[e + f*x])^(m + 1))/(a^2*d^2*f*(n + 1)*(n + 2)), x] - Simp[(a*(n + 5)*Cos[e +
f*x]*(d*Sin[e + f*x])^(n + 3)*(a + b*Sin[e + f*x])^(m + 1))/(b^2*d^3*f*(m + n + 5)*(m + n + 6)), x] + Simp[(Co
s[e + f*x]*(d*Sin[e + f*x])^(n + 4)*(a + b*Sin[e + f*x])^(m + 1))/(b*d^4*f*(m + n + 6)), x]) /; FreeQ[{a, b, d
, e, f, m, n}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n] && NeQ[n, -1] && NeQ[n, -2] && NeQ[m + n + 5, 0]
&& NeQ[m + n + 6, 0] &&  !IGtQ[m, 0]

Rule 3001

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

Rule 3055

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

Rule 3770

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

Rubi steps

\begin {align*} \int \frac {\cos ^2(c+d x) \cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx &=-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac {a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}+\frac {5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac {\int \frac {\csc ^2(c+d x) \left (-2 \left (3 a^4+14 a^2 b^2-20 b^4\right )-6 a b \left (3 a^2-b^2\right ) \sin (c+d x)+6 b^2 \left (2 a^2-5 b^2\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^3} \, dx}{12 a^2 b^2}\\ &=-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac {a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}-\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x)}{3 a^3 d (a+b \sin (c+d x))^2}+\frac {5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac {\int \frac {\csc ^2(c+d x) \left (-4 \left (3 a^6+17 a^4 b^2-50 a^2 b^4+30 b^6\right )-4 a b \left (6 a^4-11 a^2 b^2+5 b^4\right ) \sin (c+d x)+16 b^2 \left (3 a^4-8 a^2 b^2+5 b^4\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{24 a^3 b^2 \left (a^2-b^2\right )}\\ &=-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac {a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}-\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x)}{3 a^3 d (a+b \sin (c+d x))^2}+\frac {5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}-\frac {5 \left (a^2-2 b^2\right ) \cot (c+d x)}{2 a^4 d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^2(c+d x) \left (-4 \left (a^2-b^2\right )^2 \left (3 a^4+35 a^2 b^2-60 b^4\right )-4 a b \left (3 a^2-10 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x)+60 b^2 \left (a^2-2 b^2\right ) \left (a^2-b^2\right )^2 \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{24 a^4 b^2 \left (a^2-b^2\right )^2}\\ &=\frac {\left (3 a^4+35 a^2 b^2-60 b^4\right ) \cot (c+d x)}{6 a^5 b^2 d}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac {a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}-\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x)}{3 a^3 d (a+b \sin (c+d x))^2}+\frac {5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}-\frac {5 \left (a^2-2 b^2\right ) \cot (c+d x)}{2 a^4 d (a+b \sin (c+d x))}+\frac {\int \frac {\csc (c+d x) \left (60 b^3 \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right )^2+60 a b^2 \left (a^2-2 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{24 a^5 b^2 \left (a^2-b^2\right )^2}\\ &=\frac {\left (3 a^4+35 a^2 b^2-60 b^4\right ) \cot (c+d x)}{6 a^5 b^2 d}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac {a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}-\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x)}{3 a^3 d (a+b \sin (c+d x))^2}+\frac {5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}-\frac {5 \left (a^2-2 b^2\right ) \cot (c+d x)}{2 a^4 d (a+b \sin (c+d x))}+\frac {\left (5 b \left (3 a^2-4 b^2\right )\right ) \int \csc (c+d x) \, dx}{2 a^6}+\frac {\left (5 \left (a^2-4 b^2\right ) \left (a^2-b^2\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{2 a^6}\\ &=-\frac {5 b \left (3 a^2-4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^6 d}+\frac {\left (3 a^4+35 a^2 b^2-60 b^4\right ) \cot (c+d x)}{6 a^5 b^2 d}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac {a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}-\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x)}{3 a^3 d (a+b \sin (c+d x))^2}+\frac {5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}-\frac {5 \left (a^2-2 b^2\right ) \cot (c+d x)}{2 a^4 d (a+b \sin (c+d x))}+\frac {\left (5 \left (a^2-4 b^2\right ) \left (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^6 d}\\ &=-\frac {5 b \left (3 a^2-4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^6 d}+\frac {\left (3 a^4+35 a^2 b^2-60 b^4\right ) \cot (c+d x)}{6 a^5 b^2 d}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac {a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}-\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x)}{3 a^3 d (a+b \sin (c+d x))^2}+\frac {5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}-\frac {5 \left (a^2-2 b^2\right ) \cot (c+d x)}{2 a^4 d (a+b \sin (c+d x))}-\frac {\left (10 \left (a^2-4 b^2\right ) \left (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^6 d}\\ &=\frac {5 \left (a^2-4 b^2\right ) \sqrt {a^2-b^2} \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^6 d}-\frac {5 b \left (3 a^2-4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^6 d}+\frac {\left (3 a^4+35 a^2 b^2-60 b^4\right ) \cot (c+d x)}{6 a^5 b^2 d}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac {a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}-\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x)}{3 a^3 d (a+b \sin (c+d x))^2}+\frac {5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}-\frac {5 \left (a^2-2 b^2\right ) \cot (c+d x)}{2 a^4 d (a+b \sin (c+d x))}\\ \end {align*}

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Mathematica [A]  time = 6.27, size = 490, normalized size = 1.49 \[ \frac {3 b \csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 a^4 d}-\frac {3 b \sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 a^4 d}-\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{24 a^3 d}+\frac {\tan \left (\frac {1}{2} (c+d x)\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{24 a^3 d}+\frac {5 \left (3 a^2 b-4 b^3\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^6 d}-\frac {5 \left (3 a^2 b-4 b^3\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^6 d}+\frac {\csc \left (\frac {1}{2} (c+d x)\right ) \left (7 a^2 \cos \left (\frac {1}{2} (c+d x)\right )-18 b^2 \cos \left (\frac {1}{2} (c+d x)\right )\right )}{6 a^5 d}+\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (18 b^2 \sin \left (\frac {1}{2} (c+d x)\right )-7 a^2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{6 a^5 d}+\frac {a^4 (-\cos (c+d x))+2 a^2 b^2 \cos (c+d x)-b^4 \cos (c+d x)}{2 a^4 b d (a+b \sin (c+d x))^2}+\frac {5 \left (a^4-5 a^2 b^2+4 b^4\right ) \tan ^{-1}\left (\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (a \sin \left (\frac {1}{2} (c+d x)\right )+b \cos \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^6 d \sqrt {a^2-b^2}}+\frac {a^4 \cos (c+d x)+7 a^2 b^2 \cos (c+d x)-8 b^4 \cos (c+d x)}{2 a^5 b d (a+b \sin (c+d x))} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(Cos[c + d*x]^2*Cot[c + d*x]^4)/(a + b*Sin[c + d*x])^3,x]

[Out]

(5*(a^4 - 5*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^6*Sqrt[a^2 - b^2]*d) + ((7*a^2*Cos[(c + d*x)/2] - 18*b^2*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(6*a^5*d)
+ (3*b*Csc[(c + d*x)/2]^2)/(8*a^4*d) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(24*a^3*d) - (5*(3*a^2*b - 4*b^3
)*Log[Cos[(c + d*x)/2]])/(2*a^6*d) + (5*(3*a^2*b - 4*b^3)*Log[Sin[(c + d*x)/2]])/(2*a^6*d) - (3*b*Sec[(c + d*x
)/2]^2)/(8*a^4*d) + (Sec[(c + d*x)/2]*(-7*a^2*Sin[(c + d*x)/2] + 18*b^2*Sin[(c + d*x)/2]))/(6*a^5*d) + (-(a^4*
Cos[c + d*x]) + 2*a^2*b^2*Cos[c + d*x] - b^4*Cos[c + d*x])/(2*a^4*b*d*(a + b*Sin[c + d*x])^2) + (a^4*Cos[c + d
*x] + 7*a^2*b^2*Cos[c + d*x] - 8*b^4*Cos[c + d*x])/(2*a^5*b*d*(a + b*Sin[c + d*x])) + (Sec[(c + d*x)/2]^2*Tan[
(c + d*x)/2])/(24*a^3*d)

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fricas [B]  time = 1.46, size = 1553, normalized size = 4.72 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^4/(a+b*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

[1/12*(2*(3*a^5 + 35*a^3*b^2 - 60*a*b^4)*cos(d*x + c)^5 - 20*(2*a^5 + 5*a^3*b^2 - 12*a*b^4)*cos(d*x + c)^3 - 1
5*(2*(a^3*b - 4*a*b^3)*cos(d*x + c)^4 + 2*a^3*b - 8*a*b^3 - 4*(a^3*b - 4*a*b^3)*cos(d*x + c)^2 + ((a^2*b^2 - 4
*b^4)*cos(d*x + c)^4 + a^4 - 3*a^2*b^2 - 4*b^4 - (a^4 - 2*a^2*b^2 - 8*b^4)*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)) + 30*(a^5 + a^3*b
^2 - 4*a*b^4)*cos(d*x + c) - 15*(6*a^3*b^2 - 8*a*b^4 + 2*(3*a^3*b^2 - 4*a*b^4)*cos(d*x + c)^4 - 4*(3*a^3*b^2 -
4*a*b^4)*cos(d*x + c)^2 + (3*a^4*b - a^2*b^3 - 4*b^5 + (3*a^2*b^3 - 4*b^5)*cos(d*x + c)^4 - (3*a^4*b + 2*a^2*
b^3 - 8*b^5)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) + 15*(6*a^3*b^2 - 8*a*b^4 + 2*(3*a^3*b^
2 - 4*a*b^4)*cos(d*x + c)^4 - 4*(3*a^3*b^2 - 4*a*b^4)*cos(d*x + c)^2 + (3*a^4*b - a^2*b^3 - 4*b^5 + (3*a^2*b^3
- 4*b^5)*cos(d*x + c)^4 - (3*a^4*b + 2*a^2*b^3 - 8*b^5)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*cos(d*x + c) +
1/2) - 10*((11*a^4*b - 18*a^2*b^3)*cos(d*x + c)^3 - 6*(2*a^4*b - 3*a^2*b^3)*cos(d*x + c))*sin(d*x + c))/(2*a^
7*b*d*cos(d*x + c)^4 - 4*a^7*b*d*cos(d*x + c)^2 + 2*a^7*b*d + (a^6*b^2*d*cos(d*x + c)^4 - (a^8 + 2*a^6*b^2)*d*
cos(d*x + c)^2 + (a^8 + a^6*b^2)*d)*sin(d*x + c)), 1/12*(2*(3*a^5 + 35*a^3*b^2 - 60*a*b^4)*cos(d*x + c)^5 - 20
*(2*a^5 + 5*a^3*b^2 - 12*a*b^4)*cos(d*x + c)^3 - 30*(2*(a^3*b - 4*a*b^3)*cos(d*x + c)^4 + 2*a^3*b - 8*a*b^3 -
4*(a^3*b - 4*a*b^3)*cos(d*x + c)^2 + ((a^2*b^2 - 4*b^4)*cos(d*x + c)^4 + a^4 - 3*a^2*b^2 - 4*b^4 - (a^4 - 2*a^
2*b^2 - 8*b^4)*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))) + 30*(a^5 + a^3*b^2 - 4*a*b^4)*cos(d*x + c) - 15*(6*a^3*b^2 - 8*a*b^4 + 2*(3*a^3*b^2 - 4*a*b^4)*co
s(d*x + c)^4 - 4*(3*a^3*b^2 - 4*a*b^4)*cos(d*x + c)^2 + (3*a^4*b - a^2*b^3 - 4*b^5 + (3*a^2*b^3 - 4*b^5)*cos(d
*x + c)^4 - (3*a^4*b + 2*a^2*b^3 - 8*b^5)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) + 15*(6*a^
3*b^2 - 8*a*b^4 + 2*(3*a^3*b^2 - 4*a*b^4)*cos(d*x + c)^4 - 4*(3*a^3*b^2 - 4*a*b^4)*cos(d*x + c)^2 + (3*a^4*b -
a^2*b^3 - 4*b^5 + (3*a^2*b^3 - 4*b^5)*cos(d*x + c)^4 - (3*a^4*b + 2*a^2*b^3 - 8*b^5)*cos(d*x + c)^2)*sin(d*x
+ c))*log(-1/2*cos(d*x + c) + 1/2) - 10*((11*a^4*b - 18*a^2*b^3)*cos(d*x + c)^3 - 6*(2*a^4*b - 3*a^2*b^3)*cos(
d*x + c))*sin(d*x + c))/(2*a^7*b*d*cos(d*x + c)^4 - 4*a^7*b*d*cos(d*x + c)^2 + 2*a^7*b*d + (a^6*b^2*d*cos(d*x
+ c)^4 - (a^8 + 2*a^6*b^2)*d*cos(d*x + c)^2 + (a^8 + a^6*b^2)*d)*sin(d*x + c))]

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giac [A]  time = 0.29, size = 478, normalized size = 1.45 \[ \frac {\frac {60 \, {\left (3 \, a^{2} b - 4 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{6}} + \frac {120 \, {\left (a^{4} - 5 \, a^{2} b^{2} + 4 \, b^{4}\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^{6}} - \frac {24 \, {\left (a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 11 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 10 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 18 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 25 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 26 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 9 \, a^{4} b + 9 \, a^{2} b^{3}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )}^{2} a^{6}} + \frac {a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 27 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 72 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{9}} - \frac {330 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 440 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 27 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 72 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3}}{a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^4/(a+b*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/24*(60*(3*a^2*b - 4*b^3)*log(abs(tan(1/2*d*x + 1/2*c)))/a^6 + 120*(a^4 - 5*a^2*b^2 + 4*b^4)*(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^6) - 24*(a
^5*tan(1/2*d*x + 1/2*c)^3 - 11*a^3*b^2*tan(1/2*d*x + 1/2*c)^3 + 10*a*b^4*tan(1/2*d*x + 1/2*c)^3 - 9*a^4*b*tan(
1/2*d*x + 1/2*c)^2 - 9*a^2*b^3*tan(1/2*d*x + 1/2*c)^2 + 18*b^5*tan(1/2*d*x + 1/2*c)^2 - a^5*tan(1/2*d*x + 1/2*
c) - 25*a^3*b^2*tan(1/2*d*x + 1/2*c) + 26*a*b^4*tan(1/2*d*x + 1/2*c) - 9*a^4*b + 9*a^2*b^3)/((a*tan(1/2*d*x +
1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)^2*a^6) + (a^6*tan(1/2*d*x + 1/2*c)^3 - 9*a^5*b*tan(1/2*d*x + 1/2*c)^2
- 27*a^6*tan(1/2*d*x + 1/2*c) + 72*a^4*b^2*tan(1/2*d*x + 1/2*c))/a^9 - (330*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 44
0*b^3*tan(1/2*d*x + 1/2*c)^3 - 27*a^3*tan(1/2*d*x + 1/2*c)^2 + 72*a*b^2*tan(1/2*d*x + 1/2*c)^2 - 9*a^2*b*tan(1
/2*d*x + 1/2*c) + a^3)/(a^6*tan(1/2*d*x + 1/2*c)^3))/d

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maple [B]  time = 0.86, size = 873, normalized size = 2.65 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*csc(d*x+c)^4/(a+b*sin(d*x+c))^3,x)

[Out]

1/24/d/a^3*tan(1/2*d*x+1/2*c)^3-3/8/d/a^4*tan(1/2*d*x+1/2*c)^2*b-9/8/d/a^3*tan(1/2*d*x+1/2*c)+3/d/a^5*b^2*tan(
1/2*d*x+1/2*c)-1/24/d/a^3/tan(1/2*d*x+1/2*c)^3+9/8/d/a^3/tan(1/2*d*x+1/2*c)-3/d/a^5/tan(1/2*d*x+1/2*c)*b^2+3/8
/d/a^4*b/tan(1/2*d*x+1/2*c)^2+15/2/d/a^4*b*ln(tan(1/2*d*x+1/2*c))-10/d/a^6*b^3*ln(tan(1/2*d*x+1/2*c))-1/d/a/(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)^3+11/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*tan(1/2*d*x+1/2*c)^3*b^2-10/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*ta
n(1/2*d*x+1/2*c)^3*b^4+9/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*tan(1/2*d*x+1/2*c)^2*b+9/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)^2*b^3-18/d/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)^2*b^5+1/d/a/(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)+25/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*tan(1/2*d*x+1/2*c)*b^2-26
/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)*b^4+9/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-9/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*b^3+5/d/a^2/(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))-25/d/a^4/(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))*b^2+20/d/a^6/(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))*b^4

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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)^4/(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 = 12.41, size = 1082, normalized size = 3.29 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^6/(sin(c + d*x)^4*(a + b*sin(c + d*x))^3),x)

[Out]

tan(c/2 + (d*x)/2)^3/(24*a^3*d) + (tan(c/2 + (d*x)/2)^4*((77*a^4)/3 - 304*b^4 + 200*a^2*b^2) + tan(c/2 + (d*x)
/2)^6*(a^4 - 80*b^4 + 64*a^2*b^2) - a^4/3 + tan(c/2 + (d*x)/2)^2*((25*a^4)/3 - (40*a^2*b^2)/3) - tan(c/2 + (d*
x)/2)^3*(156*a*b^3 - (338*a^3*b)/3) - (3*tan(c/2 + (d*x)/2)^5*(48*b^5 - 37*a^4*b + 8*a^2*b^3))/a + (5*a^3*b*ta
n(c/2 + (d*x)/2))/3)/(d*(8*a^7*tan(c/2 + (d*x)/2)^3 + 8*a^7*tan(c/2 + (d*x)/2)^7 + tan(c/2 + (d*x)/2)^5*(16*a^
7 + 32*a^5*b^2) + 32*a^6*b*tan(c/2 + (d*x)/2)^4 + 32*a^6*b*tan(c/2 + (d*x)/2)^6)) - (tan(c/2 + (d*x)/2)*((3*(a
^2 + 4*b^2))/(8*a^5) + 3/(4*a^3) - (9*b^2)/(2*a^5)))/d + (log(tan(c/2 + (d*x)/2))*(15*a^2*b - 20*b^3))/(2*a^6*
d) - (3*b*tan(c/2 + (d*x)/2)^2)/(8*a^4*d) + (atan((((b^2 - a^2)^(1/2)*(a^2 - 4*b^2)*((5*a^10 + 40*a^6*b^4 - 40
*a^8*b^2)/a^10 + (tan(c/2 + (d*x)/2)*(25*a^8*b + 80*a^4*b^5 - 100*a^6*b^3))/a^9 - (5*(2*a^2*b - (tan(c/2 + (d*
x)/2)*(6*a^12 - 8*a^10*b^2))/a^9)*(b^2 - a^2)^(1/2)*(a^2 - 4*b^2))/(2*a^6))*5i)/(2*a^6) + ((b^2 - a^2)^(1/2)*(
a^2 - 4*b^2)*((5*a^10 + 40*a^6*b^4 - 40*a^8*b^2)/a^10 + (tan(c/2 + (d*x)/2)*(25*a^8*b + 80*a^4*b^5 - 100*a^6*b
^3))/a^9 + (5*(2*a^2*b - (tan(c/2 + (d*x)/2)*(6*a^12 - 8*a^10*b^2))/a^9)*(b^2 - a^2)^(1/2)*(a^2 - 4*b^2))/(2*a
^6))*5i)/(2*a^6))/((75*a^6*b - 400*b^7 + 800*a^2*b^5 - 475*a^4*b^3)/a^10 + (2*tan(c/2 + (d*x)/2)*(25*a^6 - 200
*b^6 + 350*a^2*b^4 - 175*a^4*b^2))/a^9 + (5*(b^2 - a^2)^(1/2)*(a^2 - 4*b^2)*((5*a^10 + 40*a^6*b^4 - 40*a^8*b^2
)/a^10 + (tan(c/2 + (d*x)/2)*(25*a^8*b + 80*a^4*b^5 - 100*a^6*b^3))/a^9 - (5*(2*a^2*b - (tan(c/2 + (d*x)/2)*(6
*a^12 - 8*a^10*b^2))/a^9)*(b^2 - a^2)^(1/2)*(a^2 - 4*b^2))/(2*a^6)))/(2*a^6) - (5*(b^2 - a^2)^(1/2)*(a^2 - 4*b
^2)*((5*a^10 + 40*a^6*b^4 - 40*a^8*b^2)/a^10 + (tan(c/2 + (d*x)/2)*(25*a^8*b + 80*a^4*b^5 - 100*a^6*b^3))/a^9
+ (5*(2*a^2*b - (tan(c/2 + (d*x)/2)*(6*a^12 - 8*a^10*b^2))/a^9)*(b^2 - a^2)^(1/2)*(a^2 - 4*b^2))/(2*a^6)))/(2*
a^6)))*(b^2 - a^2)^(1/2)*(a^2 - 4*b^2)*5i)/(a^6*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)**4/(a+b*sin(d*x+c))**3,x)

[Out]

Timed out

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