\(\int \frac {\cos ^3(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx\) [1130]

Optimal result

Integrand size = 27, antiderivative size = 137 \[ \int \frac {\cos ^3(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {2 a x}{b^3}+\frac {2 \sqrt {a^2-b^2} \left (2 a^2+b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^2 b^3 d}-\frac {\text {arctanh}(\cos (c+d x))}{a^2 d}-\frac {\cos (c+d x)}{b^2 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x)}{a b^2 d (a+b \sin (c+d x))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2971, 3136, 2739, 632, 210, 3855} \[ \int \frac {\cos ^3(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {2 \sqrt {a^2-b^2} \left (2 a^2+b^2\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^2 b^3 d}-\frac {\text {arctanh}(\cos (c+d x))}{a^2 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x)}{a b^2 d (a+b \sin (c+d x))}-\frac {2 a x}{b^3}-\frac {\cos (c+d x)}{b^2 d} \]

[In]

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

[Out]

(-2*a*x)/b^3 + (2*Sqrt[a^2 - b^2]*(2*a^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2*b^3*d)
- ArcTanh[Cos[c + d*x]]/(a^2*d) - Cos[c + d*x]/(b^2*d) - ((a^2 - b^2)*Cos[c + d*x])/(a*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 2971

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

Rule 3136

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2)/(((a_) + (b_.)*sin[(e_.) + (f_.)
*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[C*(x/(b*d)), x] + (Dist[(A*b^2 - a*b*B + a
^2*C)/(b*(b*c - a*d)), Int[1/(a + b*Sin[e + f*x]), x], x] - Dist[(c^2*C - B*c*d + A*d^2)/(d*(b*c - a*d)), Int[
1/(c + d*Sin[e + f*x]), x], x]) /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2
, 0] && NeQ[c^2 - d^2, 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 {\cos (c+d x)}{b^2 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x)}{a b^2 d (a+b \sin (c+d x))}+\frac {\int \frac {\csc (c+d x) \left (b^2-a b \sin (c+d x)-2 a^2 \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{a b^2} \\ & = -\frac {2 a x}{b^3}-\frac {\cos (c+d x)}{b^2 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x)}{a b^2 d (a+b \sin (c+d x))}+\frac {\int \csc (c+d x) \, dx}{a^2}-\frac {\left (-2 a^4+a^2 b^2+b^4\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^2 b^3} \\ & = -\frac {2 a x}{b^3}-\frac {\text {arctanh}(\cos (c+d x))}{a^2 d}-\frac {\cos (c+d x)}{b^2 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x)}{a b^2 d (a+b \sin (c+d x))}-\frac {\left (2 \left (-2 a^4+a^2 b^2+b^4\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^2 b^3 d} \\ & = -\frac {2 a x}{b^3}-\frac {\text {arctanh}(\cos (c+d x))}{a^2 d}-\frac {\cos (c+d x)}{b^2 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x)}{a b^2 d (a+b \sin (c+d x))}+\frac {\left (4 \left (-2 a^4+a^2 b^2+b^4\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^2 b^3 d} \\ & = -\frac {2 a x}{b^3}+\frac {2 \left (2 a^4-a^2 b^2-b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^2 b^3 \sqrt {a^2-b^2} d}-\frac {\text {arctanh}(\cos (c+d x))}{a^2 d}-\frac {\cos (c+d x)}{b^2 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x)}{a b^2 d (a+b \sin (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.76 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.18 \[ \int \frac {\cos ^3(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {-\frac {2 a (c+d x)}{b^3}+\frac {2 \left (2 a^4-a^2 b^2-b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^2 b^3 \sqrt {a^2-b^2}}-\frac {\cos (c+d x)}{b^2}-\frac {\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{a^2}+\frac {\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{a^2}+\frac {\left (-a^2+b^2\right ) \cos (c+d x)}{a b^2 (a+b \sin (c+d x))}}{d} \]

[In]

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

[Out]

((-2*a*(c + d*x))/b^3 + (2*(2*a^4 - a^2*b^2 - b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2*b^3*
Sqrt[a^2 - b^2]) - Cos[c + d*x]/b^2 - Log[Cos[(c + d*x)/2]]/a^2 + Log[Sin[(c + d*x)/2]]/a^2 + ((-a^2 + b^2)*Co
s[c + d*x])/(a*b^2*(a + b*Sin[c + d*x])))/d

Maple [A] (verified)

Time = 0.87 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.41

method	result	size
derivativedivides	\(\frac {\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}-\frac {4 \left (\frac {b}{2+2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+a \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{b^{3}}+\frac {\frac {4 \left (-\frac {b^{2} \left (a^{2}-b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {a^{3} b}{2}+\frac {a \,b^{3}}{2}\right )}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {2 \left (2 a^{4}-a^{2} b^{2}-b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}}{a^{2} b^{3}}}{d}\)	\(193\)
default	\(\frac {\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}-\frac {4 \left (\frac {b}{2+2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+a \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{b^{3}}+\frac {\frac {4 \left (-\frac {b^{2} \left (a^{2}-b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {a^{3} b}{2}+\frac {a \,b^{3}}{2}\right )}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {2 \left (2 a^{4}-a^{2} b^{2}-b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}}{a^{2} b^{3}}}{d}\)	\(193\)
risch	\(-\frac {2 a x}{b^{3}}-\frac {{\mathrm e}^{i \left (d x +c \right )}}{2 b^{2} d}-\frac {{\mathrm e}^{-i \left (d x +c \right )}}{2 b^{2} d}+\frac {2 i \left (-a^{2}+b^{2}\right ) \left (-i a \,{\mathrm e}^{i \left (d x +c \right )}+b \right )}{b^{3} d a \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}-b +2 i a \,{\mathrm e}^{i \left (d x +c \right )}\right )}+\frac {2 i \sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,b^{3}}+\frac {i \sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d b \,a^{2}}-\frac {2 i \sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,b^{3}}-\frac {i \sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d b \,a^{2}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d \,a^{2}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d \,a^{2}}\)	\(358\)

[In]

int(cos(d*x+c)^4*csc(d*x+c)/(a+b*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(1/a^2*ln(tan(1/2*d*x+1/2*c))-4/b^3*(1/2*b/(1+tan(1/2*d*x+1/2*c)^2)+a*arctan(tan(1/2*d*x+1/2*c)))+4/a^2/b^
3*((-1/2*b^2*(a^2-b^2)*tan(1/2*d*x+1/2*c)-1/2*a^3*b+1/2*a*b^3)/(tan(1/2*d*x+1/2*c)^2*a+2*b*tan(1/2*d*x+1/2*c)+
a)+1/2*(2*a^4-a^2*b^2-b^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))))

Fricas [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 516, normalized size of antiderivative = 3.77 \[ \int \frac {\cos ^3(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\left [-\frac {4 \, a^{4} d x - {\left (2 \, a^{3} + a b^{2} + {\left (2 \, a^{2} b + b^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (-\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} - 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) + 2 \, {\left (2 \, a^{3} b - a b^{3}\right )} \cos \left (d x + c\right ) + {\left (b^{4} \sin \left (d x + c\right ) + a b^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - {\left (b^{4} \sin \left (d x + c\right ) + a b^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, {\left (2 \, a^{3} b d x + a^{2} b^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left (a^{2} b^{4} d \sin \left (d x + c\right ) + a^{3} b^{3} d\right )}}, -\frac {4 \, a^{4} d x + 2 \, {\left (2 \, a^{3} + a b^{2} + {\left (2 \, a^{2} b + b^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) + 2 \, {\left (2 \, a^{3} b - a b^{3}\right )} \cos \left (d x + c\right ) + {\left (b^{4} \sin \left (d x + c\right ) + a b^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - {\left (b^{4} \sin \left (d x + c\right ) + a b^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, {\left (2 \, a^{3} b d x + a^{2} b^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left (a^{2} b^{4} d \sin \left (d x + c\right ) + a^{3} b^{3} d\right )}}\right ] \]

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)/(a+b*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

[-1/2*(4*a^4*d*x - (2*a^3 + a*b^2 + (2*a^2*b + b^3)*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^3*b - a*b^3)*cos(d*x + c) + (b^4*sin(d*x + c)
+ a*b^3)*log(1/2*cos(d*x + c) + 1/2) - (b^4*sin(d*x + c) + a*b^3)*log(-1/2*cos(d*x + c) + 1/2) + 2*(2*a^3*b*d*
x + a^2*b^2*cos(d*x + c))*sin(d*x + c))/(a^2*b^4*d*sin(d*x + c) + a^3*b^3*d), -1/2*(4*a^4*d*x + 2*(2*a^3 + a*b
^2 + (2*a^2*b + b^3)*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*(2*a^3*b - a*b^3)*cos(d*x + c) + (b^4*sin(d*x + c) + a*b^3)*log(1/2*cos(d*x + c) + 1/2) - (b^4*sin(d*x +
c) + a*b^3)*log(-1/2*cos(d*x + c) + 1/2) + 2*(2*a^3*b*d*x + a^2*b^2*cos(d*x + c))*sin(d*x + c))/(a^2*b^4*d*si
n(d*x + c) + a^3*b^3*d)]

Sympy [F]

\[ \int \frac {\cos ^3(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\int \frac {\cos ^{4}{\left (c + d x \right )} \csc {\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{2}}\, dx \]

[In]

integrate(cos(d*x+c)**4*csc(d*x+c)/(a+b*sin(d*x+c))**2,x)

[Out]

Integral(cos(c + d*x)**4*csc(c + d*x)/(a + b*sin(c + d*x))**2, x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cos ^3(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)/(a+b*sin(d*x+c))^2,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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 286 vs. \(2 (132) = 264\).

Time = 0.34 (sec) , antiderivative size = 286, normalized size of antiderivative = 2.09 \[ \int \frac {\cos ^3(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {\frac {2 \, {\left (d x + c\right )} a}{b^{3}} - \frac {\log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac {2 \, {\left (2 \, a^{4} - a^{2} b^{2} - b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} a^{2} b^{3}} + \frac {2 \, {\left (a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, a^{3} - a b^{2}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )} a^{2} b^{2}}}{d} \]

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)/(a+b*sin(d*x+c))^2,x, algorithm="giac")

[Out]

-(2*(d*x + c)*a/b^3 - log(abs(tan(1/2*d*x + 1/2*c)))/a^2 - 2*(2*a^4 - a^2*b^2 - b^4)*(pi*floor(1/2*(d*x + c)/p
i + 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^2*b^3) + 2*(a^2*b*t
an(1/2*d*x + 1/2*c)^3 - b^3*tan(1/2*d*x + 1/2*c)^3 + 2*a^3*tan(1/2*d*x + 1/2*c)^2 - a*b^2*tan(1/2*d*x + 1/2*c)
^2 + 3*a^2*b*tan(1/2*d*x + 1/2*c) - b^3*tan(1/2*d*x + 1/2*c) + 2*a^3 - a*b^2)/((a*tan(1/2*d*x + 1/2*c)^4 + 2*b
*tan(1/2*d*x + 1/2*c)^3 + 2*a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)*a^2*b^2))/d

Mupad [B] (verification not implemented)

Time = 12.75 (sec) , antiderivative size = 2773, normalized size of antiderivative = 20.24 \[ \int \frac {\cos ^3(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

int(cos(c + d*x)^4/(sin(c + d*x)*(a + b*sin(c + d*x))^2),x)

[Out]

log(tan(c/2 + (d*x)/2))/(a^2*d) + (atan((((2*(b^2 - a^2)^(1/2))/b^3 + (b^2 - a^2)^(1/2)/(a^2*b))*(((2*(b^2 - a
^2)^(1/2))/b^3 + (b^2 - a^2)^(1/2)/(a^2*b))*(((2*(b^2 - a^2)^(1/2))/b^3 + (b^2 - a^2)^(1/2)/(a^2*b))*(((32*(4*
a^4*b^10 - 3*a^6*b^8))/(a^2*b^5) + (32*tan(c/2 + (d*x)/2)*(16*a^4*b^14 - 17*a^6*b^12 + 2*a^8*b^10))/(a^3*b^8))
*((2*(b^2 - a^2)^(1/2))/b^3 + (b^2 - a^2)^(1/2)/(a^2*b)) + (32*(8*a^2*b^10 - 7*a^6*b^6))/(a^2*b^5) + (32*tan(c
/2 + (d*x)/2)*(16*a^2*b^14 - 5*a^4*b^12 - 18*a^6*b^10 + 8*a^8*b^8))/(a^3*b^8)) + (32*(4*b^10 + 3*a^2*b^8 - 7*a
^4*b^6 + 16*a^6*b^4 - 12*a^8*b^2))/(a^2*b^5) + (32*tan(c/2 + (d*x)/2)*(6*a^4*b^10 - 11*a^2*b^12 + 101*a^6*b^8
- 100*a^8*b^6 + 8*a^10*b^4))/(a^3*b^8)) - (32*(28*a^8 + 2*a^2*b^6 - 28*a^4*b^4 - 6*a^6*b^2))/(a^2*b^5) + (32*t
an(c/2 + (d*x)/2)*(b^12 + 2*a^2*b^10 + 61*a^4*b^8 - 20*a^6*b^6 - 72*a^8*b^4 + 32*a^10*b^2))/(a^3*b^8))*1i - ((
2*(b^2 - a^2)^(1/2))/b^3 + (b^2 - a^2)^(1/2)/(a^2*b))*(((2*(b^2 - a^2)^(1/2))/b^3 + (b^2 - a^2)^(1/2)/(a^2*b))
*((32*(4*b^10 + 3*a^2*b^8 - 7*a^4*b^6 + 16*a^6*b^4 - 12*a^8*b^2))/(a^2*b^5) - ((2*(b^2 - a^2)^(1/2))/b^3 + (b^
2 - a^2)^(1/2)/(a^2*b))*((32*(8*a^2*b^10 - 7*a^6*b^6))/(a^2*b^5) - ((32*(4*a^4*b^10 - 3*a^6*b^8))/(a^2*b^5) +
(32*tan(c/2 + (d*x)/2)*(16*a^4*b^14 - 17*a^6*b^12 + 2*a^8*b^10))/(a^3*b^8))*((2*(b^2 - a^2)^(1/2))/b^3 + (b^2
- a^2)^(1/2)/(a^2*b)) + (32*tan(c/2 + (d*x)/2)*(16*a^2*b^14 - 5*a^4*b^12 - 18*a^6*b^10 + 8*a^8*b^8))/(a^3*b^8)
) + (32*tan(c/2 + (d*x)/2)*(6*a^4*b^10 - 11*a^2*b^12 + 101*a^6*b^8 - 100*a^8*b^6 + 8*a^10*b^4))/(a^3*b^8)) + (
32*(28*a^8 + 2*a^2*b^6 - 28*a^4*b^4 - 6*a^6*b^2))/(a^2*b^5) - (32*tan(c/2 + (d*x)/2)*(b^12 + 2*a^2*b^10 + 61*a
^4*b^8 - 20*a^6*b^6 - 72*a^8*b^4 + 32*a^10*b^2))/(a^3*b^8))*1i)/(((2*(b^2 - a^2)^(1/2))/b^3 + (b^2 - a^2)^(1/2
)/(a^2*b))*(((2*(b^2 - a^2)^(1/2))/b^3 + (b^2 - a^2)^(1/2)/(a^2*b))*(((2*(b^2 - a^2)^(1/2))/b^3 + (b^2 - a^2)^
(1/2)/(a^2*b))*(((32*(4*a^4*b^10 - 3*a^6*b^8))/(a^2*b^5) + (32*tan(c/2 + (d*x)/2)*(16*a^4*b^14 - 17*a^6*b^12 +
2*a^8*b^10))/(a^3*b^8))*((2*(b^2 - a^2)^(1/2))/b^3 + (b^2 - a^2)^(1/2)/(a^2*b)) + (32*(8*a^2*b^10 - 7*a^6*b^6
))/(a^2*b^5) + (32*tan(c/2 + (d*x)/2)*(16*a^2*b^14 - 5*a^4*b^12 - 18*a^6*b^10 + 8*a^8*b^8))/(a^3*b^8)) + (32*(
4*b^10 + 3*a^2*b^8 - 7*a^4*b^6 + 16*a^6*b^4 - 12*a^8*b^2))/(a^2*b^5) + (32*tan(c/2 + (d*x)/2)*(6*a^4*b^10 - 11
*a^2*b^12 + 101*a^6*b^8 - 100*a^8*b^6 + 8*a^10*b^4))/(a^3*b^8)) - (32*(28*a^8 + 2*a^2*b^6 - 28*a^4*b^4 - 6*a^6
*b^2))/(a^2*b^5) + (32*tan(c/2 + (d*x)/2)*(b^12 + 2*a^2*b^10 + 61*a^4*b^8 - 20*a^6*b^6 - 72*a^8*b^4 + 32*a^10*
b^2))/(a^3*b^8)) + ((2*(b^2 - a^2)^(1/2))/b^3 + (b^2 - a^2)^(1/2)/(a^2*b))*(((2*(b^2 - a^2)^(1/2))/b^3 + (b^2
- a^2)^(1/2)/(a^2*b))*((32*(4*b^10 + 3*a^2*b^8 - 7*a^4*b^6 + 16*a^6*b^4 - 12*a^8*b^2))/(a^2*b^5) - ((2*(b^2 -
a^2)^(1/2))/b^3 + (b^2 - a^2)^(1/2)/(a^2*b))*((32*(8*a^2*b^10 - 7*a^6*b^6))/(a^2*b^5) - ((32*(4*a^4*b^10 - 3*a
^6*b^8))/(a^2*b^5) + (32*tan(c/2 + (d*x)/2)*(16*a^4*b^14 - 17*a^6*b^12 + 2*a^8*b^10))/(a^3*b^8))*((2*(b^2 - a^
2)^(1/2))/b^3 + (b^2 - a^2)^(1/2)/(a^2*b)) + (32*tan(c/2 + (d*x)/2)*(16*a^2*b^14 - 5*a^4*b^12 - 18*a^6*b^10 +
8*a^8*b^8))/(a^3*b^8)) + (32*tan(c/2 + (d*x)/2)*(6*a^4*b^10 - 11*a^2*b^12 + 101*a^6*b^8 - 100*a^8*b^6 + 8*a^10
*b^4))/(a^3*b^8)) + (32*(28*a^8 + 2*a^2*b^6 - 28*a^4*b^4 - 6*a^6*b^2))/(a^2*b^5) - (32*tan(c/2 + (d*x)/2)*(b^1
2 + 2*a^2*b^10 + 61*a^4*b^8 - 20*a^6*b^6 - 72*a^8*b^4 + 32*a^10*b^2))/(a^3*b^8)) - (64*(28*a^6 + 2*b^6 - 12*a^
2*b^4 - 18*a^4*b^2))/(a^2*b^5) - (64*tan(c/2 + (d*x)/2)*(128*a^10 + 48*a^4*b^6 - 16*a^6*b^4 - 160*a^8*b^2))/(a
^3*b^8)))*(((b^2 - a^2)^(1/2)*4i)/b^3 + ((b^2 - a^2)^(1/2)*2i)/(a^2*b)))/d - ((2*(2*a^2 - b^2))/(a*b^2) + (2*t
an(c/2 + (d*x)/2)*(3*a^2 - b^2))/(a^2*b) + (2*tan(c/2 + (d*x)/2)^3*(a^2 - b^2))/(a^2*b) + (2*tan(c/2 + (d*x)/2
)^2*(2*a^2 - b^2))/(a*b^2))/(d*(a + 2*b*tan(c/2 + (d*x)/2) + 2*a*tan(c/2 + (d*x)/2)^2 + a*tan(c/2 + (d*x)/2)^4
+ 2*b*tan(c/2 + (d*x)/2)^3)) - (4*a*atan((256*tan(c/2 + (d*x)/2))/((128*b^2)/a^2 - (384*a^2)/b^2 + (256*a*tan
(c/2 + (d*x)/2))/b + (512*a^3*tan(c/2 + (d*x)/2))/b^3 - (768*a^5*tan(c/2 + (d*x)/2))/b^5 + 256) - (512*a^3)/(2
56*b^3 - 384*a^2*b + (128*b^5)/a^2 + 512*a^3*tan(c/2 + (d*x)/2) + 256*a*b^2*tan(c/2 + (d*x)/2) - (768*a^5*tan(
c/2 + (d*x)/2))/b^2) + (768*a^5)/(256*b^5 - 384*a^2*b^3 + (128*b^7)/a^2 - 768*a^5*tan(c/2 + (d*x)/2) + 256*a*b
^4*tan(c/2 + (d*x)/2) + 512*a^3*b^2*tan(c/2 + (d*x)/2)) - (256*a)/(256*b + 256*a*tan(c/2 + (d*x)/2) - (384*a^2
)/b + (128*b^3)/a^2 + (512*a^3*tan(c/2 + (d*x)/2))/b^2 - (768*a^5*tan(c/2 + (d*x)/2))/b^4) - (384*a^2*tan(c/2
+ (d*x)/2))/(256*b^2 - 384*a^2 + (128*b^4)/a^2 + (512*a^3*tan(c/2 + (d*x)/2))/b - (768*a^5*tan(c/2 + (d*x)/2))
/b^3 + 256*a*b*tan(c/2 + (d*x)/2)) + (128*b*tan(c/2 + (d*x)/2))/(128*b + (256*a^2)/b - (384*a^4)/b^3 + (256*a^
3*tan(c/2 + (d*x)/2))/b^2 + (512*a^5*tan(c/2 + (d*x)/2))/b^4 - (768*a^7*tan(c/2 + (d*x)/2))/b^6)))/(b^3*d)