\(\int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^2 \, dx\) [1110]

Optimal result

Integrand size = 27, antiderivative size = 189 \[ \int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=-3 a b x+\frac {\left (3 a^2-2 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 d}-\frac {\left (4 a^2-23 b^2\right ) \cos (c+d x)}{6 d}-\frac {b \left (a^2-3 b^2\right ) \cos (c+d x) \sin (c+d x)}{3 a d}-\frac {\left (2 a^2-3 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{6 a^2 d}-\frac {b \cot (c+d x) (a+b \sin (c+d x))^3}{2 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 a d} \]

[Out]

-3*a*b*x+1/2*(3*a^2-2*b^2)*arctanh(cos(d*x+c))/d-1/6*(4*a^2-23*b^2)*cos(d*x+c)/d-1/3*b*(a^2-3*b^2)*cos(d*x+c)*
sin(d*x+c)/a/d-1/6*(2*a^2-3*b^2)*cos(d*x+c)*(a+b*sin(d*x+c))^2/a^2/d-1/2*b*cot(d*x+c)*(a+b*sin(d*x+c))^3/a^2/d
-1/2*cot(d*x+c)*csc(d*x+c)*(a+b*sin(d*x+c))^3/a/d

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 189, 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 = {2972, 3128, 3112, 3102, 2814, 3855} \[ \int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {\left (3 a^2-2 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 d}-\frac {\left (4 a^2-23 b^2\right ) \cos (c+d x)}{6 d}-\frac {\left (2 a^2-3 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{6 a^2 d}-\frac {b \left (a^2-3 b^2\right ) \sin (c+d x) \cos (c+d x)}{3 a d}-\frac {b \cot (c+d x) (a+b \sin (c+d x))^3}{2 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 a d}-3 a b x \]

[In]

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

[Out]

-3*a*b*x + ((3*a^2 - 2*b^2)*ArcTanh[Cos[c + d*x]])/(2*d) - ((4*a^2 - 23*b^2)*Cos[c + d*x])/(6*d) - (b*(a^2 - 3
*b^2)*Cos[c + d*x]*Sin[c + d*x])/(3*a*d) - ((2*a^2 - 3*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^2)/(6*a^2*d) - (
b*Cot[c + d*x]*(a + b*Sin[c + d*x])^3)/(2*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^3)/(2*a*d)

Rule 2814

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

Rule 2972

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]*(a + b*Sin[e + f*x])^(m + 1)*((d*Sin[e + f*x])^(n + 1)/(a*d*f*(n + 1))), x] +
(-Dist[1/(a^2*d^2*(n + 1)*(n + 2)), Int[(a + b*Sin[e + f*x])^m*(d*Sin[e + f*x])^(n + 2)*Simp[a^2*n*(n + 2) -
b^2*(m + n + 2)*(m + n + 3) + a*b*m*Sin[e + f*x] - (a^2*(n + 1)*(n + 2) - b^2*(m + n + 2)*(m + n + 4))*Sin[e +
f*x]^2, x], x], x] - Simp[b*(m + n + 2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((d*Sin[e + f*x])^(n + 2)/(
a^2*d^2*f*(n + 1)*(n + 2))), x]) /; FreeQ[{a, b, d, e, f, m}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || Intege
rsQ[2*m, 2*n]) &&  !m < -1 && LtQ[n, -1] && (LtQ[n, -2] || EqQ[m + n + 4, 0])

Rule 3102

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Dist[1/(
b*(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x],
x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rule 3112

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

Rule 3128

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)
*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e
+ f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Dist[1/(d*(m + n + 2)), Int[(a + b*Sin[e + f*
x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n +
2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*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] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 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 {b \cot (c+d x) (a+b \sin (c+d x))^3}{2 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 a d}-\frac {\int \csc (c+d x) (a+b \sin (c+d x))^2 \left (3 a^2-2 b^2+2 a b \sin (c+d x)-\left (2 a^2-3 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{2 a^2} \\ & = -\frac {\left (2 a^2-3 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{6 a^2 d}-\frac {b \cot (c+d x) (a+b \sin (c+d x))^3}{2 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 a d}-\frac {\int \csc (c+d x) (a+b \sin (c+d x)) \left (3 a \left (3 a^2-2 b^2\right )+11 a^2 b \sin (c+d x)-4 a \left (a^2-3 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{6 a^2} \\ & = -\frac {b \left (a^2-3 b^2\right ) \cos (c+d x) \sin (c+d x)}{3 a d}-\frac {\left (2 a^2-3 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{6 a^2 d}-\frac {b \cot (c+d x) (a+b \sin (c+d x))^3}{2 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 a d}-\frac {\int \csc (c+d x) \left (6 a^2 \left (3 a^2-2 b^2\right )+36 a^3 b \sin (c+d x)-2 a^2 \left (4 a^2-23 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{12 a^2} \\ & = -\frac {\left (4 a^2-23 b^2\right ) \cos (c+d x)}{6 d}-\frac {b \left (a^2-3 b^2\right ) \cos (c+d x) \sin (c+d x)}{3 a d}-\frac {\left (2 a^2-3 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{6 a^2 d}-\frac {b \cot (c+d x) (a+b \sin (c+d x))^3}{2 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 a d}-\frac {\int \csc (c+d x) \left (6 a^2 \left (3 a^2-2 b^2\right )+36 a^3 b \sin (c+d x)\right ) \, dx}{12 a^2} \\ & = -3 a b x-\frac {\left (4 a^2-23 b^2\right ) \cos (c+d x)}{6 d}-\frac {b \left (a^2-3 b^2\right ) \cos (c+d x) \sin (c+d x)}{3 a d}-\frac {\left (2 a^2-3 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{6 a^2 d}-\frac {b \cot (c+d x) (a+b \sin (c+d x))^3}{2 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 a d}-\frac {1}{2} \left (3 a^2-2 b^2\right ) \int \csc (c+d x) \, dx \\ & = -3 a b x+\frac {\left (3 a^2-2 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 d}-\frac {\left (4 a^2-23 b^2\right ) \cos (c+d x)}{6 d}-\frac {b \left (a^2-3 b^2\right ) \cos (c+d x) \sin (c+d x)}{3 a d}-\frac {\left (2 a^2-3 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{6 a^2 d}-\frac {b \cot (c+d x) (a+b \sin (c+d x))^3}{2 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.15 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.01 \[ \int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {-6 \left (4 a^2-5 b^2\right ) \cos (c+d x)+2 b^2 \cos (3 (c+d x))+3 \left (-24 a b c-24 a b d x-8 a b \cot \left (\frac {1}{2} (c+d x)\right )-a^2 \csc ^2\left (\frac {1}{2} (c+d x)\right )+12 a^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-8 b^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-12 a^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+8 b^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+a^2 \sec ^2\left (\frac {1}{2} (c+d x)\right )-4 a b \sin (2 (c+d x))+8 a b \tan \left (\frac {1}{2} (c+d x)\right )\right )}{24 d} \]

[In]

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

[Out]

(-6*(4*a^2 - 5*b^2)*Cos[c + d*x] + 2*b^2*Cos[3*(c + d*x)] + 3*(-24*a*b*c - 24*a*b*d*x - 8*a*b*Cot[(c + d*x)/2]
- a^2*Csc[(c + d*x)/2]^2 + 12*a^2*Log[Cos[(c + d*x)/2]] - 8*b^2*Log[Cos[(c + d*x)/2]] - 12*a^2*Log[Sin[(c + d
*x)/2]] + 8*b^2*Log[Sin[(c + d*x)/2]] + a^2*Sec[(c + d*x)/2]^2 - 4*a*b*Sin[2*(c + d*x)] + 8*a*b*Tan[(c + d*x)/
2]))/(24*d)

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(\frac {a^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{2}-\frac {3 \cos \left (d x +c \right )}{2}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+2 a b \left (-\frac {\cos ^{5}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+b^{2} \left (\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )}{d}\)	\(157\)
default	\(\frac {a^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{2}-\frac {3 \cos \left (d x +c \right )}{2}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+2 a b \left (-\frac {\cos ^{5}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+b^{2} \left (\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )}{d}\)	\(157\)
parallelrisch	\(\frac {\left (-288 a^{2}+192 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-36 \left (\cos \left (d x +c \right )-\frac {11 \cos \left (2 d x +2 c \right )}{12}-\frac {\cos \left (3 d x +3 c \right )}{3}+\frac {11}{12}\right ) \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-216 b \left (\cos \left (d x +c \right )-\frac {\cos \left (3 d x +3 c \right )}{9}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right ) a \csc \left (\frac {d x}{2}+\frac {c}{2}\right )-576 a b x d +240 \cos \left (d x +c \right ) b^{2}+16 \cos \left (3 d x +3 c \right ) b^{2}+256 b^{2}}{192 d}\)	\(162\)
risch	\(-3 a b x +\frac {b^{2} {\mathrm e}^{3 i \left (d x +c \right )}}{24 d}+\frac {i a b \,{\mathrm e}^{2 i \left (d x +c \right )}}{4 d}-\frac {a^{2} {\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {5 b^{2} {\mathrm e}^{i \left (d x +c \right )}}{8 d}-\frac {a^{2} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {5 \,{\mathrm e}^{-i \left (d x +c \right )} b^{2}}{8 d}-\frac {i a b \,{\mathrm e}^{-2 i \left (d x +c \right )}}{4 d}+\frac {b^{2} {\mathrm e}^{-3 i \left (d x +c \right )}}{24 d}-\frac {i a \left (i a \,{\mathrm e}^{3 i \left (d x +c \right )}+i a \,{\mathrm e}^{i \left (d x +c \right )}+4 b \,{\mathrm e}^{2 i \left (d x +c \right )}-4 b \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}+\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{d}-\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{d}\)	\(284\)
norman	\(\frac {\frac {a b \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a^{2}}{8 d}+\frac {a^{2} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {\left (23 a^{2}-32 b^{2}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {\left (33 a^{2}-32 b^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {\left (43 a^{2}-32 b^{2}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {4 a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {4 a b \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-3 a b x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-9 a b x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-9 a b x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 a b x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {\left (3 a^{2}-2 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}\)	\(297\)

[In]

int(cos(d*x+c)^4*csc(d*x+c)^3*(a+b*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(a^2*(-1/2/sin(d*x+c)^2*cos(d*x+c)^5-1/2*cos(d*x+c)^3-3/2*cos(d*x+c)-3/2*ln(csc(d*x+c)-cot(d*x+c)))+2*a*b*
(-1/sin(d*x+c)*cos(d*x+c)^5-(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)-3/2*d*x-3/2*c)+b^2*(1/3*cos(d*x+c)^3+cos(
d*x+c)+ln(csc(d*x+c)-cot(d*x+c))))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.11 \[ \int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {4 \, b^{2} \cos \left (d x + c\right )^{5} - 36 \, a b d x \cos \left (d x + c\right )^{2} + 36 \, a b d x - 4 \, {\left (3 \, a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{3} + 6 \, {\left (3 \, a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right ) + 3 \, {\left ({\left (3 \, a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 3 \, a^{2} + 2 \, b^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 3 \, {\left ({\left (3 \, a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 3 \, a^{2} + 2 \, b^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 12 \, {\left (a b \cos \left (d x + c\right )^{3} - 3 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{12 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \]

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^3*(a+b*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

1/12*(4*b^2*cos(d*x + c)^5 - 36*a*b*d*x*cos(d*x + c)^2 + 36*a*b*d*x - 4*(3*a^2 - 2*b^2)*cos(d*x + c)^3 + 6*(3*
a^2 - 2*b^2)*cos(d*x + c) + 3*((3*a^2 - 2*b^2)*cos(d*x + c)^2 - 3*a^2 + 2*b^2)*log(1/2*cos(d*x + c) + 1/2) - 3
*((3*a^2 - 2*b^2)*cos(d*x + c)^2 - 3*a^2 + 2*b^2)*log(-1/2*cos(d*x + c) + 1/2) - 12*(a*b*cos(d*x + c)^3 - 3*a*
b*cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^2 - d)

Sympy [F(-1)]

Timed out. \[ \int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**4*csc(d*x+c)**3*(a+b*sin(d*x+c))**2,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.79 \[ \int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {12 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} + 2}{\tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a b - 2 \, {\left (2 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right ) - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} b^{2} - 3 \, a^{2} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - 4 \, \cos \left (d x + c\right ) + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{12 \, d} \]

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^3*(a+b*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/12*(12*(3*d*x + 3*c + (3*tan(d*x + c)^2 + 2)/(tan(d*x + c)^3 + tan(d*x + c)))*a*b - 2*(2*cos(d*x + c)^3 + 6
*cos(d*x + c) - 3*log(cos(d*x + c) + 1) + 3*log(cos(d*x + c) - 1))*b^2 - 3*a^2*(2*cos(d*x + c)/(cos(d*x + c)^2
- 1) - 4*cos(d*x + c) + 3*log(cos(d*x + c) + 1) - 3*log(cos(d*x + c) - 1)))/d

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.33 \[ \int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 72 \, {\left (d x + c\right )} a b + 24 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 12 \, {\left (3 \, a^{2} - 2 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {3 \, {\left (18 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{2}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} + \frac {16 \, {\left (3 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 6 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 6 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 6 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a^{2} + 4 \, b^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{24 \, d} \]

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^3*(a+b*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/24*(3*a^2*tan(1/2*d*x + 1/2*c)^2 - 72*(d*x + c)*a*b + 24*a*b*tan(1/2*d*x + 1/2*c) - 12*(3*a^2 - 2*b^2)*log(a
bs(tan(1/2*d*x + 1/2*c))) + 3*(18*a^2*tan(1/2*d*x + 1/2*c)^2 - 12*b^2*tan(1/2*d*x + 1/2*c)^2 - 8*a*b*tan(1/2*d
*x + 1/2*c) - a^2)/tan(1/2*d*x + 1/2*c)^2 + 16*(3*a*b*tan(1/2*d*x + 1/2*c)^5 - 3*a^2*tan(1/2*d*x + 1/2*c)^4 +
6*b^2*tan(1/2*d*x + 1/2*c)^4 - 6*a^2*tan(1/2*d*x + 1/2*c)^2 + 6*b^2*tan(1/2*d*x + 1/2*c)^2 - 3*a*b*tan(1/2*d*x
+ 1/2*c) - 3*a^2 + 4*b^2)/(tan(1/2*d*x + 1/2*c)^2 + 1)^3)/d

Mupad [B] (verification not implemented)

Time = 11.23 (sec) , antiderivative size = 397, normalized size of antiderivative = 2.10 \[ \int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {3\,a^2}{2}-b^2\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {17\,a^2}{2}-16\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {35\,a^2}{2}-16\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {19\,a^2}{2}-\frac {32\,b^2}{3}\right )+\frac {a^2}{2}+20\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+12\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-4\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+4\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}+\frac {6\,a\,b\,\mathrm {atan}\left (\frac {36\,a^2\,b^2}{-18\,a^3\,b+36\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b^2+12\,a\,b^3}-\frac {12\,a\,b^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{-18\,a^3\,b+36\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b^2+12\,a\,b^3}+\frac {18\,a^3\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{-18\,a^3\,b+36\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b^2+12\,a\,b^3}\right )}{d}+\frac {a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d} \]

[In]

int((cos(c + d*x)^4*(a + b*sin(c + d*x))^2)/sin(c + d*x)^3,x)

[Out]

(a^2*tan(c/2 + (d*x)/2)^2)/(8*d) - (log(tan(c/2 + (d*x)/2))*((3*a^2)/2 - b^2))/d - (tan(c/2 + (d*x)/2)^6*((17*
a^2)/2 - 16*b^2) + tan(c/2 + (d*x)/2)^4*((35*a^2)/2 - 16*b^2) + tan(c/2 + (d*x)/2)^2*((19*a^2)/2 - (32*b^2)/3)
+ a^2/2 + 20*a*b*tan(c/2 + (d*x)/2)^3 + 12*a*b*tan(c/2 + (d*x)/2)^5 - 4*a*b*tan(c/2 + (d*x)/2)^7 + 4*a*b*tan(
c/2 + (d*x)/2))/(d*(4*tan(c/2 + (d*x)/2)^2 + 12*tan(c/2 + (d*x)/2)^4 + 12*tan(c/2 + (d*x)/2)^6 + 4*tan(c/2 + (
d*x)/2)^8)) + (6*a*b*atan((36*a^2*b^2)/(12*a*b^3 - 18*a^3*b + 36*a^2*b^2*tan(c/2 + (d*x)/2)) - (12*a*b^3*tan(c
/2 + (d*x)/2))/(12*a*b^3 - 18*a^3*b + 36*a^2*b^2*tan(c/2 + (d*x)/2)) + (18*a^3*b*tan(c/2 + (d*x)/2))/(12*a*b^3
- 18*a^3*b + 36*a^2*b^2*tan(c/2 + (d*x)/2))))/d + (a*b*tan(c/2 + (d*x)/2))/d