\(\int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx\) [1140]

Optimal result

Integrand size = 27, antiderivative size = 218 \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {3 b \left (3 a^2-4 b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^5 \sqrt {a^2-b^2} d}+\frac {3 \left (a^2-4 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 a^5 d}-\frac {\left (a^2-12 b^2\right ) \cot (c+d x)}{2 a^4 b d}+\frac {\left (a^2-2 b^2\right ) \cot (c+d x)}{2 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}-\frac {3 b \cot (c+d x)}{a^3 d (a+b \sin (c+d x))} \]

[Out]

3/2*(a^2-4*b^2)*arctanh(cos(d*x+c))/a^5/d-1/2*(a^2-12*b^2)*cot(d*x+c)/a^4/b/d+1/2*(a^2-2*b^2)*cot(d*x+c)/a^2/b
/d/(a+b*sin(d*x+c))^2-1/2*cot(d*x+c)*csc(d*x+c)/a/d/(a+b*sin(d*x+c))^2-3*b*cot(d*x+c)/a^3/d/(a+b*sin(d*x+c))+3
*b*(3*a^2-4*b^2)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/a^5/d/(a^2-b^2)^(1/2)

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2969, 3134, 3080, 3855, 2739, 632, 210} \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {3 b \cot (c+d x)}{a^3 d (a+b \sin (c+d x))}+\frac {\left (a^2-2 b^2\right ) \cot (c+d x)}{2 a^2 b d (a+b \sin (c+d x))^2}+\frac {3 b \left (3 a^2-4 b^2\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^5 d \sqrt {a^2-b^2}}+\frac {3 \left (a^2-4 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 a^5 d}-\frac {\left (a^2-12 b^2\right ) \cot (c+d x)}{2 a^4 b d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2} \]

[In]

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

[Out]

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

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

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 2739

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 2969

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 3080

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 3134

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] + D
ist[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] &&
LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n]
&&  !IntegerQ[m]) || EqQ[a, 0])))

Rule 3855

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

Rubi steps \begin{align*} \text {integral}& = \frac {\left (a^2-2 b^2\right ) \cot (c+d x)}{2 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac {\int \frac {\csc ^2(c+d x) \left (2 \left (a^2-6 b^2\right )-2 a b \sin (c+d x)+8 b^2 \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{4 a^2 b} \\ & = \frac {\left (a^2-2 b^2\right ) \cot (c+d x)}{2 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}-\frac {3 b \cot (c+d x)}{a^3 d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^2(c+d x) \left (2 \left (a^4-13 a^2 b^2+12 b^4\right )-4 a b \left (a^2-b^2\right ) \sin (c+d x)+12 b^2 \left (a^2-b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{4 a^3 b \left (a^2-b^2\right )} \\ & = -\frac {\left (a^2-12 b^2\right ) \cot (c+d x)}{2 a^4 b d}+\frac {\left (a^2-2 b^2\right ) \cot (c+d x)}{2 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}-\frac {3 b \cot (c+d x)}{a^3 d (a+b \sin (c+d x))}+\frac {\int \frac {\csc (c+d x) \left (-6 b \left (a^4-5 a^2 b^2+4 b^4\right )+12 a b^2 \left (a^2-b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{4 a^4 b \left (a^2-b^2\right )} \\ & = -\frac {\left (a^2-12 b^2\right ) \cot (c+d x)}{2 a^4 b d}+\frac {\left (a^2-2 b^2\right ) \cot (c+d x)}{2 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}-\frac {3 b \cot (c+d x)}{a^3 d (a+b \sin (c+d x))}-\frac {\left (3 \left (a^2-4 b^2\right )\right ) \int \csc (c+d x) \, dx}{2 a^5}+\frac {\left (12 a^2 b^2 \left (a^2-b^2\right )+6 b^2 \left (a^4-5 a^2 b^2+4 b^4\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{4 a^5 b \left (a^2-b^2\right )} \\ & = \frac {3 \left (a^2-4 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 a^5 d}-\frac {\left (a^2-12 b^2\right ) \cot (c+d x)}{2 a^4 b d}+\frac {\left (a^2-2 b^2\right ) \cot (c+d x)}{2 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}-\frac {3 b \cot (c+d x)}{a^3 d (a+b \sin (c+d x))}+\frac {\left (3 b \left (3 a^2-4 b^2\right )\right ) \text {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 d} \\ & = \frac {3 \left (a^2-4 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 a^5 d}-\frac {\left (a^2-12 b^2\right ) \cot (c+d x)}{2 a^4 b d}+\frac {\left (a^2-2 b^2\right ) \cot (c+d x)}{2 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}-\frac {3 b \cot (c+d x)}{a^3 d (a+b \sin (c+d x))}-\frac {\left (6 b \left (3 a^2-4 b^2\right )\right ) \text {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 d} \\ & = \frac {3 b \left (3 a^2-4 b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^5 \sqrt {a^2-b^2} d}+\frac {3 \left (a^2-4 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 a^5 d}-\frac {\left (a^2-12 b^2\right ) \cot (c+d x)}{2 a^4 b d}+\frac {\left (a^2-2 b^2\right ) \cot (c+d x)}{2 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}-\frac {3 b \cot (c+d x)}{a^3 d (a+b \sin (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.35 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.46 \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {3 b \left (3 a^2-4 b^2\right ) \arctan \left (\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (b \cos \left (\frac {1}{2} (c+d x)\right )+a \sin \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^5 \sqrt {a^2-b^2} d}+\frac {3 b \cot \left (\frac {1}{2} (c+d x)\right )}{2 a^4 d}-\frac {\csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 a^3 d}+\frac {3 \left (a^2-4 b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^5 d}-\frac {3 \left (a^2-4 b^2\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^5 d}+\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 a^3 d}+\frac {-a^2 \cos (c+d x)+b^2 \cos (c+d x)}{2 a^3 d (a+b \sin (c+d x))^2}+\frac {-a^2 \cos (c+d x)+6 b^2 \cos (c+d x)}{2 a^4 d (a+b \sin (c+d x))}-\frac {3 b \tan \left (\frac {1}{2} (c+d x)\right )}{2 a^4 d} \]

[In]

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

[Out]

(3*b*(3*a^2 - 4*b^2)*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*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) + (3*(a^2 - 4*b^2)*Log[
Cos[(c + d*x)/2]])/(2*a^5*d) - (3*(a^2 - 4*b^2)*Log[Sin[(c + d*x)/2]])/(2*a^5*d) + Sec[(c + d*x)/2]^2/(8*a^3*d
) + (-(a^2*Cos[c + d*x]) + b^2*Cos[c + d*x])/(2*a^3*d*(a + b*Sin[c + d*x])^2) + (-(a^2*Cos[c + d*x]) + 6*b^2*C
os[c + d*x])/(2*a^4*d*(a + b*Sin[c + d*x])) - (3*b*Tan[(c + d*x)/2])/(2*a^4*d)

Maple [A] (verified)

Time = 0.80 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.30

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{2}-6 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{4}}+\frac {\frac {4 \left (\left (-\frac {3}{4} a^{3} b +2 a \,b^{3}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {1}{2} a^{4}+\frac {3}{4} a^{2} b^{2}+\frac {7}{2} b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {5 a b \left (a^{2}-4 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}-\frac {a^{2} \left (2 a^{2}-7 b^{2}\right )}{4}\right )}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {3 b \left (3 a^{2}-4 b^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}}{a^{5}}-\frac {1}{8 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-6 a^{2}+24 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{5}}+\frac {3 b}{2 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\)	\(283\)
default	\(\frac {\frac {\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{2}-6 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{4}}+\frac {\frac {4 \left (\left (-\frac {3}{4} a^{3} b +2 a \,b^{3}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {1}{2} a^{4}+\frac {3}{4} a^{2} b^{2}+\frac {7}{2} b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {5 a b \left (a^{2}-4 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}-\frac {a^{2} \left (2 a^{2}-7 b^{2}\right )}{4}\right )}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {3 b \left (3 a^{2}-4 b^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}}{a^{5}}-\frac {1}{8 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-6 a^{2}+24 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{5}}+\frac {3 b}{2 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\)	\(283\)
risch	\(\frac {-2 i a^{4} {\mathrm e}^{2 i \left (d x +c \right )}+15 i b^{2} a^{2} {\mathrm e}^{6 i \left (d x +c \right )}-36 i b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+6 b^{3} a \,{\mathrm e}^{7 i \left (d x +c \right )}+29 i b^{2} a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-45 i b^{2} {\mathrm e}^{4 i \left (d x +c \right )} a^{2}-12 i b^{4}-54 b^{3} a \,{\mathrm e}^{5 i \left (d x +c \right )}+i a^{2} b^{2}+12 i b^{4} {\mathrm e}^{6 i \left (d x +c \right )}-2 i a^{4} {\mathrm e}^{6 i \left (d x +c \right )}-12 b \,a^{3} {\mathrm e}^{3 i \left (d x +c \right )}+90 b^{3} a \,{\mathrm e}^{3 i \left (d x +c \right )}+36 i b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+4 i a^{4} {\mathrm e}^{4 i \left (d x +c \right )}+4 b \,a^{3} {\mathrm e}^{i \left (d x +c \right )}-42 \,{\mathrm e}^{i \left (d x +c \right )} b^{3} a}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}-b +2 i a \,{\mathrm e}^{i \left (d x +c \right )}\right )^{2} b \,a^{4} d}-\frac {9 i b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, d \,a^{3}}+\frac {6 i b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, d \,a^{5}}+\frac {9 i b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, d \,a^{3}}-\frac {6 i b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, d \,a^{5}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d \,a^{3}}-\frac {6 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{a^{5} d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d \,a^{3}}+\frac {6 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{a^{5} d}\)	\(679\)

[In]

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

[Out]

1/d*(1/4/a^4*(1/2*tan(1/2*d*x+1/2*c)^2*a-6*b*tan(1/2*d*x+1/2*c))+4/a^5*(((-3/4*a^3*b+2*a*b^3)*tan(1/2*d*x+1/2*
c)^3+(-1/2*a^4+3/4*a^2*b^2+7/2*b^4)*tan(1/2*d*x+1/2*c)^2-5/4*a*b*(a^2-4*b^2)*tan(1/2*d*x+1/2*c)-1/4*a^2*(2*a^2
-7*b^2))/(tan(1/2*d*x+1/2*c)^2*a+2*b*tan(1/2*d*x+1/2*c)+a)^2+3/4*b*(3*a^2-4*b^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)))-1/8/a^3/tan(1/2*d*x+1/2*c)^2+1/4/a^5*(-6*a^2+24*b^2)*ln(tan(1/2*d
*x+1/2*c))+3/2*b/a^4/tan(1/2*d*x+1/2*c))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 738 vs. \(2 (205) = 410\).

Time = 0.46 (sec) , antiderivative size = 1560, normalized size of antiderivative = 7.16 \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/4*(4*(a^6 - 10*a^4*b^2 + 9*a^2*b^4)*cos(d*x + c)^3 + 3*(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 + 2*(3*a^3*b^2 - 4*a*b^4 - (3*a^3*b^2 - 4*a*b^4)*co
s(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^6 - 7*a^4*b^2 + 6*a^2*b^4)*cos(d*x + c) + 3*(a^6 - 4*a^4*b^2 - a^2*b^4 + 4*b^6 + (a^4*
b^2 - 5*a^2*b^4 + 4*b^6)*cos(d*x + c)^4 - (a^6 - 3*a^4*b^2 - 6*a^2*b^4 + 8*b^6)*cos(d*x + c)^2 + 2*(a^5*b - 5*
a^3*b^3 + 4*a*b^5 - (a^5*b - 5*a^3*b^3 + 4*a*b^5)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) -
3*(a^6 - 4*a^4*b^2 - a^2*b^4 + 4*b^6 + (a^4*b^2 - 5*a^2*b^4 + 4*b^6)*cos(d*x + c)^4 - (a^6 - 3*a^4*b^2 - 6*a^2
*b^4 + 8*b^6)*cos(d*x + c)^2 + 2*(a^5*b - 5*a^3*b^3 + 4*a*b^5 - (a^5*b - 5*a^3*b^3 + 4*a*b^5)*cos(d*x + c)^2)*
sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2) + 2*((a^5*b - 13*a^3*b^3 + 12*a*b^5)*cos(d*x + c)^3 + 3*(a^5*b + 3*
a^3*b^3 - 4*a*b^5)*cos(d*x + c))*sin(d*x + c))/((a^7*b^2 - a^5*b^4)*d*cos(d*x + c)^4 - (a^9 + a^7*b^2 - 2*a^5*
b^4)*d*cos(d*x + c)^2 + (a^9 - a^5*b^4)*d - 2*((a^8*b - a^6*b^3)*d*cos(d*x + c)^2 - (a^8*b - a^6*b^3)*d)*sin(d
*x + c)), 1/4*(4*(a^6 - 10*a^4*b^2 + 9*a^2*b^4)*cos(d*x + c)^3 - 6*(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 + 2*(3*a^3*b^2 - 4*a*b^4 - (3*a^3*b^2 - 4*
a*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
))) - 6*(a^6 - 7*a^4*b^2 + 6*a^2*b^4)*cos(d*x + c) + 3*(a^6 - 4*a^4*b^2 - a^2*b^4 + 4*b^6 + (a^4*b^2 - 5*a^2*b
^4 + 4*b^6)*cos(d*x + c)^4 - (a^6 - 3*a^4*b^2 - 6*a^2*b^4 + 8*b^6)*cos(d*x + c)^2 + 2*(a^5*b - 5*a^3*b^3 + 4*a
*b^5 - (a^5*b - 5*a^3*b^3 + 4*a*b^5)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) - 3*(a^6 - 4*a^
4*b^2 - a^2*b^4 + 4*b^6 + (a^4*b^2 - 5*a^2*b^4 + 4*b^6)*cos(d*x + c)^4 - (a^6 - 3*a^4*b^2 - 6*a^2*b^4 + 8*b^6)
*cos(d*x + c)^2 + 2*(a^5*b - 5*a^3*b^3 + 4*a*b^5 - (a^5*b - 5*a^3*b^3 + 4*a*b^5)*cos(d*x + c)^2)*sin(d*x + c))
*log(-1/2*cos(d*x + c) + 1/2) + 2*((a^5*b - 13*a^3*b^3 + 12*a*b^5)*cos(d*x + c)^3 + 3*(a^5*b + 3*a^3*b^3 - 4*a
*b^5)*cos(d*x + c))*sin(d*x + c))/((a^7*b^2 - a^5*b^4)*d*cos(d*x + c)^4 - (a^9 + a^7*b^2 - 2*a^5*b^4)*d*cos(d*
x + c)^2 + (a^9 - a^5*b^4)*d - 2*((a^8*b - a^6*b^3)*d*cos(d*x + c)^2 - (a^8*b - a^6*b^3)*d)*sin(d*x + c))]

Sympy [F]

\[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\int \frac {\cos ^{4}{\left (c + d x \right )} \csc ^{3}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{3}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(cos(d*x+c)^4*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 de

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.81 \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {\frac {12 \, {\left (a^{2} - 4 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{5}} - \frac {24 \, {\left (3 \, a^{2} b - 4 \, b^{3}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} a^{5}} - \frac {a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}} - \frac {6 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 24 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 12 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 32 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 5 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 48 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 16 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 4 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 112 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 76 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{4}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}^{2} a^{5}}}{8 \, d} \]

[In]

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

[Out]

-1/8*(12*(a^2 - 4*b^2)*log(abs(tan(1/2*d*x + 1/2*c)))/a^5 - 24*(3*a^2*b - 4*b^3)*(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) - (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 - (6*a^4*tan(1/2*d*x + 1/2*c)^6 - 24*a^2*b^2*tan(1/2*d*x + 1/2*
c)^6 + 12*a^3*b*tan(1/2*d*x + 1/2*c)^5 - 32*a*b^3*tan(1/2*d*x + 1/2*c)^5 - 5*a^4*tan(1/2*d*x + 1/2*c)^4 + 48*a
^2*b^2*tan(1/2*d*x + 1/2*c)^4 + 16*b^4*tan(1/2*d*x + 1/2*c)^4 + 4*a^3*b*tan(1/2*d*x + 1/2*c)^3 + 112*a*b^3*tan
(1/2*d*x + 1/2*c)^3 - 12*a^4*tan(1/2*d*x + 1/2*c)^2 + 76*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*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))/d

Mupad [B] (verification not implemented)

Time = 10.93 (sec) , antiderivative size = 1100, normalized size of antiderivative = 5.05 \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

int(cos(c + d*x)^4/(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) + (tan(c/2 + (d*x)/2)^2*(50*a*b^2 - 9*a^3) - tan(c/2 + (d*x)/2)^5*(6*a^2*b - 32
*b^3) - tan(c/2 + (d*x)/2)^3*(10*a^2*b - 104*b^3) - a^3/2 + 4*a^2*b*tan(c/2 + (d*x)/2) + (tan(c/2 + (d*x)/2)^4
*(112*b^4 - 17*a^4 + 72*a^2*b^2))/(2*a))/(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*ta
n(c/2 + (d*x)/2))/(2*a^4*d) - (log(tan(c/2 + (d*x)/2))*(3*a^2 - 12*b^2))/(2*a^5*d) - (b*atan(((b*(-(a + b)*(a
- b))^(1/2)*(3*a^2 - 4*b^2)*((12*a^7*b - 24*a^5*b^3)/a^8 - (tan(c/2 + (d*x)/2)*(3*a^7 + 48*a^3*b^4 - 36*a^5*b^
2))/a^7 + (3*b*(-(a + b)*(a - b))^(1/2)*(2*a^2*b - (tan(c/2 + (d*x)/2)*(6*a^10 - 8*a^8*b^2))/a^7)*(3*a^2 - 4*b
^2))/(2*(a^7 - a^5*b^2)))*3i)/(2*(a^7 - a^5*b^2)) - (b*(-(a + b)*(a - b))^(1/2)*(3*a^2 - 4*b^2)*((tan(c/2 + (d
*x)/2)*(3*a^7 + 48*a^3*b^4 - 36*a^5*b^2))/a^7 - (12*a^7*b - 24*a^5*b^3)/a^8 + (3*b*(-(a + b)*(a - b))^(1/2)*(2
*a^2*b - (tan(c/2 + (d*x)/2)*(6*a^10 - 8*a^8*b^2))/a^7)*(3*a^2 - 4*b^2))/(2*(a^7 - a^5*b^2)))*3i)/(2*(a^7 - a^
5*b^2)))/((27*a^4*b + 144*b^5 - 144*a^2*b^3)/a^8 + (2*tan(c/2 + (d*x)/2)*(72*b^4 - 54*a^2*b^2))/a^7 + (3*b*(-(
a + b)*(a - b))^(1/2)*(3*a^2 - 4*b^2)*((12*a^7*b - 24*a^5*b^3)/a^8 - (tan(c/2 + (d*x)/2)*(3*a^7 + 48*a^3*b^4 -
36*a^5*b^2))/a^7 + (3*b*(-(a + b)*(a - b))^(1/2)*(2*a^2*b - (tan(c/2 + (d*x)/2)*(6*a^10 - 8*a^8*b^2))/a^7)*(3
*a^2 - 4*b^2))/(2*(a^7 - a^5*b^2))))/(2*(a^7 - a^5*b^2)) + (3*b*(-(a + b)*(a - b))^(1/2)*(3*a^2 - 4*b^2)*((tan
(c/2 + (d*x)/2)*(3*a^7 + 48*a^3*b^4 - 36*a^5*b^2))/a^7 - (12*a^7*b - 24*a^5*b^3)/a^8 + (3*b*(-(a + b)*(a - b))
^(1/2)*(2*a^2*b - (tan(c/2 + (d*x)/2)*(6*a^10 - 8*a^8*b^2))/a^7)*(3*a^2 - 4*b^2))/(2*(a^7 - a^5*b^2))))/(2*(a^
7 - a^5*b^2))))*(-(a + b)*(a - b))^(1/2)*(3*a^2 - 4*b^2)*3i)/(d*(a^7 - a^5*b^2))