\(\int \frac {\cos (c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^3} \, dx\) [1088]

Optimal result

Integrand size = 25, antiderivative size = 154 \[ \int \frac {\cos (c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {b \left (3 a^2-2 b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^3 \left (a^2-b^2\right )^{3/2} d}-\frac {\text {arctanh}(\cos (c+d x))}{a^3 d}+\frac {\cos (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac {\left (a^2-2 b^2\right ) \cos (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {2968, 3135, 3080, 3855, 2739, 632, 210} \[ \int \frac {\cos (c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {\text {arctanh}(\cos (c+d x))}{a^3 d}+\frac {\left (a^2-2 b^2\right ) \cos (c+d x)}{2 a^2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))}-\frac {b \left (3 a^2-2 b^2\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^3 d \left (a^2-b^2\right )^{3/2}}+\frac {\cos (c+d x)}{2 a d (a+b \sin (c+d x))^2} \]

[In]

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

[Out]

-((b*(3*a^2 - 2*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^3*(a^2 - b^2)^(3/2)*d)) - ArcTanh[Co
s[c + d*x]]/(a^3*d) + Cos[c + d*x]/(2*a*d*(a + b*Sin[c + d*x])^2) + ((a^2 - 2*b^2)*Cos[c + d*x])/(2*a^2*(a^2 -
b^2)*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 2968

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

Rule 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 3135

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*s
in[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 + a^2*C))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c
+ d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2))), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)),
Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[a*(m + 1)*(b*c - a*d)*(A + C) + d*(A*b^2 + a^2*C
)*(m + n + 2) - (c*(A*b^2 + a^2*C) + b*(m + 1)*(b*c - a*d)*(A + C))*Sin[e + f*x] - d*(A*b^2 + a^2*C)*(m + n +
3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2,
0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && L
tQ[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}& = \int \frac {\csc (c+d x) \left (1-\sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^3} \, dx \\ & = \frac {\cos (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac {\int \frac {\csc (c+d x) \left (2 \left (a^2-b^2\right )-\left (a^2-b^2\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{2 a \left (a^2-b^2\right )} \\ & = \frac {\cos (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac {\left (a^2-2 b^2\right ) \cos (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\int \frac {\csc (c+d x) \left (2 \left (a^2-b^2\right )^2-a b \left (a^2-b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )^2} \\ & = \frac {\cos (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac {\left (a^2-2 b^2\right ) \cos (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\int \csc (c+d x) \, dx}{a^3}-\frac {\left (b \left (3 a^2-2 b^2\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{2 a^3 \left (a^2-b^2\right )} \\ & = -\frac {\text {arctanh}(\cos (c+d x))}{a^3 d}+\frac {\cos (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac {\left (a^2-2 b^2\right ) \cos (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}-\frac {\left (b \left (3 a^2-2 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^3 \left (a^2-b^2\right ) d} \\ & = -\frac {\text {arctanh}(\cos (c+d x))}{a^3 d}+\frac {\cos (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac {\left (a^2-2 b^2\right ) \cos (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\left (2 b \left (3 a^2-2 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^3 \left (a^2-b^2\right ) d} \\ & = -\frac {b \left (3 a^2-2 b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^3 \left (a^2-b^2\right )^{3/2} d}-\frac {\text {arctanh}(\cos (c+d x))}{a^3 d}+\frac {\cos (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac {\left (a^2-2 b^2\right ) \cos (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.92 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\frac {2 b \left (-3 a^2+2 b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}-2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {a \cos (c+d x) \left (2 a^3-3 a b^2+b \left (a^2-2 b^2\right ) \sin (c+d x)\right )}{(a-b) (a+b) (a+b \sin (c+d x))^2}}{2 a^3 d} \]

[In]

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

[Out]

((2*b*(-3*a^2 + 2*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) - 2*Log[Cos[(c + d*
x)/2]] + 2*Log[Sin[(c + d*x)/2]] + (a*Cos[c + d*x]*(2*a^3 - 3*a*b^2 + b*(a^2 - 2*b^2)*Sin[c + d*x]))/((a - b)*
(a + b)*(a + b*Sin[c + d*x])^2))/(2*a^3*d)

Maple [A] (verified)

Time = 0.66 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.64

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (\frac {-\frac {a b \left (3 a^{2}-4 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a^{2}-b^{2}\right )}-\frac {\left (2 a^{4}+a^{2} b^{2}-6 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a^{2}-b^{2}\right )}-\frac {a b \left (5 a^{2}-8 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a^{2}-b^{2}\right )}-\frac {a^{2} \left (2 a^{2}-3 b^{2}\right )}{2 \left (a^{2}-b^{2}\right )}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {b \left (3 a^{2}-2 b^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \left (a^{2}-b^{2}\right )^{\frac {3}{2}}}\right )}{a^{3}}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}}{d}\)	\(253\)
default	\(\frac {-\frac {2 \left (\frac {-\frac {a b \left (3 a^{2}-4 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a^{2}-b^{2}\right )}-\frac {\left (2 a^{4}+a^{2} b^{2}-6 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a^{2}-b^{2}\right )}-\frac {a b \left (5 a^{2}-8 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a^{2}-b^{2}\right )}-\frac {a^{2} \left (2 a^{2}-3 b^{2}\right )}{2 \left (a^{2}-b^{2}\right )}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {b \left (3 a^{2}-2 b^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \left (a^{2}-b^{2}\right )^{\frac {3}{2}}}\right )}{a^{3}}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}}{d}\)	\(253\)
risch	\(-\frac {i \left (i a \,b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+4 i a^{3} b \,{\mathrm e}^{i \left (d x +c \right )}-7 i a \,b^{3} {\mathrm e}^{i \left (d x +c \right )}+2 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-3 b^{2} a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-2 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-a^{2} b^{2}+2 b^{4}\right )}{\left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +2 a \,{\mathrm e}^{i \left (d x +c \right )}\right )^{2} a^{2} \left (a^{2}-b^{2}\right ) d b}+\frac {3 i b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d a}-\frac {i b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{3}}-\frac {3 i b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d a}+\frac {i b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{3}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d \,a^{3}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d \,a^{3}}\)	\(539\)

[In]

int(cos(d*x+c)^2*csc(d*x+c)/(a+b*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(-2/a^3*((-1/2*a*b*(3*a^2-4*b^2)/(a^2-b^2)*tan(1/2*d*x+1/2*c)^3-1/2*(2*a^4+a^2*b^2-6*b^4)/(a^2-b^2)*tan(1/
2*d*x+1/2*c)^2-1/2*a*b*(5*a^2-8*b^2)/(a^2-b^2)*tan(1/2*d*x+1/2*c)-1/2*a^2*(2*a^2-3*b^2)/(a^2-b^2))/(tan(1/2*d*
x+1/2*c)^2*a+2*b*tan(1/2*d*x+1/2*c)+a)^2+1/2*b*(3*a^2-2*b^2)/(a^2-b^2)^(3/2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c
)+2*b)/(a^2-b^2)^(1/2)))+1/a^3*ln(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. 454 vs. \(2 (145) = 290\).

Time = 0.48 (sec) , antiderivative size = 996, normalized size of antiderivative = 6.47 \[ \int \frac {\cos (c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[-1/4*(2*(a^5*b - 3*a^3*b^3 + 2*a*b^5)*cos(d*x + c)*sin(d*x + c) - (3*a^4*b + a^2*b^3 - 2*b^5 - (3*a^2*b^3 - 2
*b^5)*cos(d*x + c)^2 + 2*(3*a^3*b^2 - 2*a*b^4)*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)) + 2*(2*a^6 - 5*a^4*b^2 + 3*a^2*b^4)*cos(d*x + c) - 2*(a^6 -
a^4*b^2 - a^2*b^4 + b^6 - (a^4*b^2 - 2*a^2*b^4 + b^6)*cos(d*x + c)^2 + 2*(a^5*b - 2*a^3*b^3 + a*b^5)*sin(d*x +
c))*log(1/2*cos(d*x + c) + 1/2) + 2*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - (a^4*b^2 - 2*a^2*b^4 + b^6)*cos(d*x + c)
^2 + 2*(a^5*b - 2*a^3*b^3 + a*b^5)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2))/((a^7*b^2 - 2*a^5*b^4 + a^3*b^6
)*d*cos(d*x + c)^2 - 2*(a^8*b - 2*a^6*b^3 + a^4*b^5)*d*sin(d*x + c) - (a^9 - a^7*b^2 - a^5*b^4 + a^3*b^6)*d),
-1/2*((a^5*b - 3*a^3*b^3 + 2*a*b^5)*cos(d*x + c)*sin(d*x + c) + (3*a^4*b + a^2*b^3 - 2*b^5 - (3*a^2*b^3 - 2*b^
5)*cos(d*x + c)^2 + 2*(3*a^3*b^2 - 2*a*b^4)*sin(d*x + c))*sqrt(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a
^2 - b^2)*cos(d*x + c))) + (2*a^6 - 5*a^4*b^2 + 3*a^2*b^4)*cos(d*x + c) - (a^6 - a^4*b^2 - a^2*b^4 + b^6 - (a^
4*b^2 - 2*a^2*b^4 + b^6)*cos(d*x + c)^2 + 2*(a^5*b - 2*a^3*b^3 + a*b^5)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1
/2) + (a^6 - a^4*b^2 - a^2*b^4 + b^6 - (a^4*b^2 - 2*a^2*b^4 + b^6)*cos(d*x + c)^2 + 2*(a^5*b - 2*a^3*b^3 + a*b
^5)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2))/((a^7*b^2 - 2*a^5*b^4 + a^3*b^6)*d*cos(d*x + c)^2 - 2*(a^8*b -
2*a^6*b^3 + a^4*b^5)*d*sin(d*x + c) - (a^9 - a^7*b^2 - a^5*b^4 + a^3*b^6)*d)]

Sympy [F]

\[ \int \frac {\cos (c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\int \frac {\cos ^{2}{\left (c + d x \right )} \csc {\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{3}}\, dx \]

[In]

integrate(cos(d*x+c)**2*csc(d*x+c)/(a+b*sin(d*x+c))**3,x)

[Out]

Integral(cos(c + d*x)**2*csc(c + d*x)/(a + b*sin(c + d*x))**3, x)

Maxima [F(-2)]

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

[In]

integrate(cos(d*x+c)^2*csc(d*x+c)/(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 = 0.35 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.80 \[ \int \frac {\cos (c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {\frac {{\left (3 \, a^{2} b - 2 \, b^{3}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{5} - a^{3} b^{2}\right )} \sqrt {a^{2} - b^{2}}} - \frac {3 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 6 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 5 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, a^{4} - 3 \, a^{2} b^{2}}{{\left (a^{5} - a^{3} b^{2}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )}^{2}} - \frac {\log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}}}{d} \]

[In]

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

[Out]

-((3*a^2*b - 2*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)))/((a^5 - a^3*b^2)*sqrt(a^2 - b^2)) - (3*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 4*a*b^3*tan(1/2*d*x + 1/2*c)^3 +
2*a^4*tan(1/2*d*x + 1/2*c)^2 + a^2*b^2*tan(1/2*d*x + 1/2*c)^2 - 6*b^4*tan(1/2*d*x + 1/2*c)^2 + 5*a^3*b*tan(1/2
*d*x + 1/2*c) - 8*a*b^3*tan(1/2*d*x + 1/2*c) + 2*a^4 - 3*a^2*b^2)/((a^5 - a^3*b^2)*(a*tan(1/2*d*x + 1/2*c)^2 +
2*b*tan(1/2*d*x + 1/2*c) + a)^2) - log(abs(tan(1/2*d*x + 1/2*c)))/a^3)/d

Mupad [B] (verification not implemented)

Time = 14.14 (sec) , antiderivative size = 1610, normalized size of antiderivative = 10.45 \[ \int \frac {\cos (c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

int(cos(c + d*x)^2/(sin(c + d*x)*(a + b*sin(c + d*x))^3),x)

[Out]

log(tan(c/2 + (d*x)/2))/(a^3*d) + ((2*a^2 - 3*b^2)/(a*(a^2 - b^2)) + (tan(c/2 + (d*x)/2)^2*(2*a^4 - 6*b^4 + a^
2*b^2))/(a^3*(a^2 - b^2)) + (tan(c/2 + (d*x)/2)*(5*a^2*b - 8*b^3))/(a^2*(a^2 - b^2)) + (b*tan(c/2 + (d*x)/2)^3
*(3*a^2 - 4*b^2))/(a^2*(a^2 - b^2)))/(d*(tan(c/2 + (d*x)/2)^2*(2*a^2 + 4*b^2) + a^2*tan(c/2 + (d*x)/2)^4 + a^2
+ 4*a*b*tan(c/2 + (d*x)/2)^3 + 4*a*b*tan(c/2 + (d*x)/2))) + (b*atan(((b*(3*a^2 - 2*b^2)*(-(a + b)^3*(a - b)^3
)^(1/2)*((5*a^5*b - 4*a^3*b^3)/(a^6 - a^4*b^2) + (tan(c/2 + (d*x)/2)*(8*a*b^6 - 2*a^7 - 20*a^3*b^4 + 14*a^5*b^
2))/(a^7 + a^3*b^4 - 2*a^5*b^2) - (b*((2*a^8*b - 2*a^6*b^3)/(a^6 - a^4*b^2) - (tan(c/2 + (d*x)/2)*(6*a^10 - 8*
a^4*b^6 + 22*a^6*b^4 - 20*a^8*b^2))/(a^7 + a^3*b^4 - 2*a^5*b^2))*(3*a^2 - 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2))
/(2*(a^9 - a^3*b^6 + 3*a^5*b^4 - 3*a^7*b^2)))*1i)/(2*(a^9 - a^3*b^6 + 3*a^5*b^4 - 3*a^7*b^2)) + (b*(3*a^2 - 2*
b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((5*a^5*b - 4*a^3*b^3)/(a^6 - a^4*b^2) + (tan(c/2 + (d*x)/2)*(8*a*b^6 - 2*a^
7 - 20*a^3*b^4 + 14*a^5*b^2))/(a^7 + a^3*b^4 - 2*a^5*b^2) + (b*((2*a^8*b - 2*a^6*b^3)/(a^6 - a^4*b^2) - (tan(c
/2 + (d*x)/2)*(6*a^10 - 8*a^4*b^6 + 22*a^6*b^4 - 20*a^8*b^2))/(a^7 + a^3*b^4 - 2*a^5*b^2))*(3*a^2 - 2*b^2)*(-(
a + b)^3*(a - b)^3)^(1/2))/(2*(a^9 - a^3*b^6 + 3*a^5*b^4 - 3*a^7*b^2)))*1i)/(2*(a^9 - a^3*b^6 + 3*a^5*b^4 - 3*
a^7*b^2)))/((2*(3*a^2*b - 2*b^3))/(a^6 - a^4*b^2) + (2*tan(c/2 + (d*x)/2)*(2*b^4 - 3*a^2*b^2))/(a^7 + a^3*b^4
- 2*a^5*b^2) - (b*(3*a^2 - 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((5*a^5*b - 4*a^3*b^3)/(a^6 - a^4*b^2) + (tan(c
/2 + (d*x)/2)*(8*a*b^6 - 2*a^7 - 20*a^3*b^4 + 14*a^5*b^2))/(a^7 + a^3*b^4 - 2*a^5*b^2) - (b*((2*a^8*b - 2*a^6*
b^3)/(a^6 - a^4*b^2) - (tan(c/2 + (d*x)/2)*(6*a^10 - 8*a^4*b^6 + 22*a^6*b^4 - 20*a^8*b^2))/(a^7 + a^3*b^4 - 2*
a^5*b^2))*(3*a^2 - 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2))/(2*(a^9 - a^3*b^6 + 3*a^5*b^4 - 3*a^7*b^2))))/(2*(a^9
- a^3*b^6 + 3*a^5*b^4 - 3*a^7*b^2)) + (b*(3*a^2 - 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((5*a^5*b - 4*a^3*b^3)/(
a^6 - a^4*b^2) + (tan(c/2 + (d*x)/2)*(8*a*b^6 - 2*a^7 - 20*a^3*b^4 + 14*a^5*b^2))/(a^7 + a^3*b^4 - 2*a^5*b^2)
+ (b*((2*a^8*b - 2*a^6*b^3)/(a^6 - a^4*b^2) - (tan(c/2 + (d*x)/2)*(6*a^10 - 8*a^4*b^6 + 22*a^6*b^4 - 20*a^8*b^
2))/(a^7 + a^3*b^4 - 2*a^5*b^2))*(3*a^2 - 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2))/(2*(a^9 - a^3*b^6 + 3*a^5*b^4 -
3*a^7*b^2))))/(2*(a^9 - a^3*b^6 + 3*a^5*b^4 - 3*a^7*b^2))))*(3*a^2 - 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*1i)/
(d*(a^9 - a^3*b^6 + 3*a^5*b^4 - 3*a^7*b^2))