\(\int \cot ^6(c+d x) \csc (c+d x) (a+b \sin (c+d x))^2 \, dx\) [1250]

Optimal result

Integrand size = 27, antiderivative size = 175 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+b \sin (c+d x))^2 \, dx=-2 a b x+\frac {5 \left (a^2-6 b^2\right ) \text {arctanh}(\cos (c+d x))}{16 d}+\frac {b^2 \cos (c+d x)}{d}-\frac {2 a b \cot (c+d x)}{d}+\frac {2 a b \cot ^3(c+d x)}{3 d}-\frac {2 a b \cot ^5(c+d x)}{5 d}-\frac {\left (11 a^2-18 b^2\right ) \cot (c+d x) \csc (c+d x)}{16 d}+\frac {\left (13 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d} \]

[Out]

-2*a*b*x+5/16*(a^2-6*b^2)*arctanh(cos(d*x+c))/d+b^2*cos(d*x+c)/d-2*a*b*cot(d*x+c)/d+2/3*a*b*cot(d*x+c)^3/d-2/5
*a*b*cot(d*x+c)^5/d-1/16*(11*a^2-18*b^2)*cot(d*x+c)*csc(d*x+c)/d+1/24*(13*a^2-6*b^2)*cot(d*x+c)*csc(d*x+c)^3/d
-1/6*a^2*cot(d*x+c)*csc(d*x+c)^5/d

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2990, 3554, 8, 4451, 466, 1828, 1171, 396, 212} \[ \int \cot ^6(c+d x) \csc (c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {5 \left (a^2-6 b^2\right ) \text {arctanh}(\cos (c+d x))}{16 d}+\frac {\left (13 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {\left (11 a^2-18 b^2\right ) \cot (c+d x) \csc (c+d x)}{16 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac {2 a b \cot ^5(c+d x)}{5 d}+\frac {2 a b \cot ^3(c+d x)}{3 d}-\frac {2 a b \cot (c+d x)}{d}-2 a b x+\frac {b^2 \cos (c+d x)}{d} \]

[In]

Int[Cot[c + d*x]^6*Csc[c + d*x]*(a + b*Sin[c + d*x])^2,x]

[Out]

-2*a*b*x + (5*(a^2 - 6*b^2)*ArcTanh[Cos[c + d*x]])/(16*d) + (b^2*Cos[c + d*x])/d - (2*a*b*Cot[c + d*x])/d + (2
*a*b*Cot[c + d*x]^3)/(3*d) - (2*a*b*Cot[c + d*x]^5)/(5*d) - ((11*a^2 - 18*b^2)*Cot[c + d*x]*Csc[c + d*x])/(16*
d) + ((13*a^2 - 6*b^2)*Cot[c + d*x]*Csc[c + d*x]^3)/(24*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(6*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 212

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

Rule 396

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(
p + 1) + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 466

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x
*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p + 1))), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[(a + b*x^2)^(p + 1)*E
xpandToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d))/(a + b*x^2)]
- (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[
m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])

Rule 1171

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2,
x], x, 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1
)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] &&
NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rule 1828

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*
g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

Rule 2990

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^2, x_Symbol] :> Dist[2*a*(b/d), Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n + 1), x], x] + Int[(g*Cos[e
+ f*x])^p*(d*Sin[e + f*x])^n*(a^2 + b^2*Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && NeQ[a^2 -
b^2, 0]

Rule 3554

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 4451

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))]^(n_), x_Symbol] :> With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, Dist
[-d/(b*c), Subst[Int[SubstFor[(1 - d^2*x^2)^((n - 1)/2), Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d]
, x] /; FunctionOfQ[Cos[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u] &&
(EqQ[F, Sin] || EqQ[F, sin])

Rubi steps \begin{align*} \text {integral}& = (2 a b) \int \cot ^6(c+d x) \, dx+\int \cot ^6(c+d x) \csc (c+d x) \left (a^2+b^2 \sin ^2(c+d x)\right ) \, dx \\ & = -\frac {2 a b \cot ^5(c+d x)}{5 d}-(2 a b) \int \cot ^4(c+d x) \, dx-\frac {\text {Subst}\left (\int \frac {x^6 \left (a^2+b^2-b^2 x^2\right )}{\left (1-x^2\right )^4} \, dx,x,\cos (c+d x)\right )}{d} \\ & = \frac {2 a b \cot ^3(c+d x)}{3 d}-\frac {2 a b \cot ^5(c+d x)}{5 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}+(2 a b) \int \cot ^2(c+d x) \, dx+\frac {\text {Subst}\left (\int \frac {a^2+6 a^2 x^2+6 a^2 x^4-6 b^2 x^6}{\left (1-x^2\right )^3} \, dx,x,\cos (c+d x)\right )}{6 d} \\ & = -\frac {2 a b \cot (c+d x)}{d}+\frac {2 a b \cot ^3(c+d x)}{3 d}-\frac {2 a b \cot ^5(c+d x)}{5 d}+\frac {\left (13 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}-(2 a b) \int 1 \, dx-\frac {\text {Subst}\left (\int \frac {3 \left (3 a^2-2 b^2\right )+24 \left (a^2-b^2\right ) x^2-24 b^2 x^4}{\left (1-x^2\right )^2} \, dx,x,\cos (c+d x)\right )}{24 d} \\ & = -2 a b x-\frac {2 a b \cot (c+d x)}{d}+\frac {2 a b \cot ^3(c+d x)}{3 d}-\frac {2 a b \cot ^5(c+d x)}{5 d}-\frac {\left (11 a^2-18 b^2\right ) \cot (c+d x) \csc (c+d x)}{16 d}+\frac {\left (13 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac {\text {Subst}\left (\int \frac {3 \left (5 a^2-14 b^2\right )-48 b^2 x^2}{1-x^2} \, dx,x,\cos (c+d x)\right )}{48 d} \\ & = -2 a b x+\frac {b^2 \cos (c+d x)}{d}-\frac {2 a b \cot (c+d x)}{d}+\frac {2 a b \cot ^3(c+d x)}{3 d}-\frac {2 a b \cot ^5(c+d x)}{5 d}-\frac {\left (11 a^2-18 b^2\right ) \cot (c+d x) \csc (c+d x)}{16 d}+\frac {\left (13 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac {\left (5 \left (a^2-6 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{16 d} \\ & = -2 a b x+\frac {5 \left (a^2-6 b^2\right ) \text {arctanh}(\cos (c+d x))}{16 d}+\frac {b^2 \cos (c+d x)}{d}-\frac {2 a b \cot (c+d x)}{d}+\frac {2 a b \cot ^3(c+d x)}{3 d}-\frac {2 a b \cot ^5(c+d x)}{5 d}-\frac {\left (11 a^2-18 b^2\right ) \cot (c+d x) \csc (c+d x)}{16 d}+\frac {\left (13 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(384\) vs. \(2(175)=350\).

Time = 1.63 (sec) , antiderivative size = 384, normalized size of antiderivative = 2.19 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {-3840 a b c-3840 a b d x+1920 b^2 \cos (c+d x)-2944 a b \cot \left (\frac {1}{2} (c+d x)\right )-330 a^2 \csc ^2\left (\frac {1}{2} (c+d x)\right )+540 b^2 \csc ^2\left (\frac {1}{2} (c+d x)\right )+60 a^2 \csc ^4\left (\frac {1}{2} (c+d x)\right )-30 b^2 \csc ^4\left (\frac {1}{2} (c+d x)\right )-5 a^2 \csc ^6\left (\frac {1}{2} (c+d x)\right )+600 a^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-3600 b^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-600 a^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+3600 b^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+330 a^2 \sec ^2\left (\frac {1}{2} (c+d x)\right )-540 b^2 \sec ^2\left (\frac {1}{2} (c+d x)\right )-60 a^2 \sec ^4\left (\frac {1}{2} (c+d x)\right )+30 b^2 \sec ^4\left (\frac {1}{2} (c+d x)\right )+5 a^2 \sec ^6\left (\frac {1}{2} (c+d x)\right )-2624 a b \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+768 a b \csc ^5(c+d x) \sin ^6\left (\frac {1}{2} (c+d x)\right )+164 a b \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-12 a b \csc ^6\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)+2944 a b \tan \left (\frac {1}{2} (c+d x)\right )}{1920 d} \]

[In]

Integrate[Cot[c + d*x]^6*Csc[c + d*x]*(a + b*Sin[c + d*x])^2,x]

[Out]

(-3840*a*b*c - 3840*a*b*d*x + 1920*b^2*Cos[c + d*x] - 2944*a*b*Cot[(c + d*x)/2] - 330*a^2*Csc[(c + d*x)/2]^2 +
540*b^2*Csc[(c + d*x)/2]^2 + 60*a^2*Csc[(c + d*x)/2]^4 - 30*b^2*Csc[(c + d*x)/2]^4 - 5*a^2*Csc[(c + d*x)/2]^6
+ 600*a^2*Log[Cos[(c + d*x)/2]] - 3600*b^2*Log[Cos[(c + d*x)/2]] - 600*a^2*Log[Sin[(c + d*x)/2]] + 3600*b^2*L
og[Sin[(c + d*x)/2]] + 330*a^2*Sec[(c + d*x)/2]^2 - 540*b^2*Sec[(c + d*x)/2]^2 - 60*a^2*Sec[(c + d*x)/2]^4 + 3
0*b^2*Sec[(c + d*x)/2]^4 + 5*a^2*Sec[(c + d*x)/2]^6 - 2624*a*b*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 768*a*b*Csc
[c + d*x]^5*Sin[(c + d*x)/2]^6 + 164*a*b*Csc[(c + d*x)/2]^4*Sin[c + d*x] - 12*a*b*Csc[(c + d*x)/2]^6*Sin[c + d
*x] + 2944*a*b*Tan[(c + d*x)/2])/(1920*d)

Maple [A] (verified)

Time = 0.65 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.36

method	result	size
derivativedivides	\(\frac {a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )+2 a b \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )+b^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{8 \sin \left (d x +c \right )^{2}}+\frac {3 \left (\cos ^{5}\left (d x +c \right )\right )}{8}+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {15 \cos \left (d x +c \right )}{8}+\frac {15 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )}{d}\)	\(238\)
default	\(\frac {a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )+2 a b \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )+b^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{8 \sin \left (d x +c \right )^{2}}+\frac {3 \left (\cos ^{5}\left (d x +c \right )\right )}{8}+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {15 \cos \left (d x +c \right )}{8}+\frac {15 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )}{d}\)	\(238\)
parallelrisch	\(\frac {\left (-1228800 a^{2}+7372800 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2640 \left (\cos \left (5 d x +5 c \right )+\frac {15 \cos \left (6 d x +6 c \right )}{176}+\frac {30 \cos \left (d x +c \right )}{11}+\frac {225 \cos \left (2 d x +2 c \right )}{176}+\frac {5 \cos \left (3 d x +3 c \right )}{33}-\frac {45 \cos \left (4 d x +4 c \right )}{88}-\frac {75}{88}\right ) \left (\sec ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} \left (\csc ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-23552 b \left (\cos \left (5 d x +5 c \right )+\frac {50 \cos \left (d x +c \right )}{23}-\frac {25 \cos \left (3 d x +3 c \right )}{23}\right ) \left (\sec ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \left (\csc ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+15360 b^{2} \left (\cos \left (5 d x +5 c \right )+\frac {5 \cos \left (d x +c \right )}{2}+10 \cos \left (2 d x +2 c \right )-\frac {15 \cos \left (3 d x +3 c \right )}{2}-\frac {5 \cos \left (4 d x +4 c \right )}{2}-\frac {15}{2}\right ) \left (\sec ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7864320 a b x d}{3932160 d}\)	\(259\)
risch	\(-2 a b x +\frac {b^{2} {\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {{\mathrm e}^{-i \left (d x +c \right )} b^{2}}{2 d}+\frac {-7360 i a b \,{\mathrm e}^{6 i \left (d x +c \right )}+165 a^{2} {\mathrm e}^{11 i \left (d x +c \right )}-270 b^{2} {\mathrm e}^{11 i \left (d x +c \right )}-1440 i a b \,{\mathrm e}^{10 i \left (d x +c \right )}+25 a^{2} {\mathrm e}^{9 i \left (d x +c \right )}+570 b^{2} {\mathrm e}^{9 i \left (d x +c \right )}+736 i a b +450 a^{2} {\mathrm e}^{7 i \left (d x +c \right )}-300 b^{2} {\mathrm e}^{7 i \left (d x +c \right )}+4320 i a b \,{\mathrm e}^{8 i \left (d x +c \right )}+450 a^{2} {\mathrm e}^{5 i \left (d x +c \right )}-300 b^{2} {\mathrm e}^{5 i \left (d x +c \right )}-2976 i a b \,{\mathrm e}^{2 i \left (d x +c \right )}+25 a^{2} {\mathrm e}^{3 i \left (d x +c \right )}+570 b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+6720 i a b \,{\mathrm e}^{4 i \left (d x +c \right )}+165 a^{2} {\mathrm e}^{i \left (d x +c \right )}-270 b^{2} {\mathrm e}^{i \left (d x +c \right )}}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}+\frac {5 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d}-\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{8 d}-\frac {5 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d}+\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{8 d}\)	\(383\)
norman	\(\frac {-\frac {a^{2}}{384 d}+\frac {a^{2} \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d}-\frac {7 \left (a^{2}-3 b^{2}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d}+\frac {7 \left (a^{2}-3 b^{2}\right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d}+\frac {\left (7 a^{2}-6 b^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d}-\frac {\left (7 a^{2}-6 b^{2}\right ) \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d}-\frac {\left (27 a^{2}-190 b^{2}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {\left (27 a^{2}-190 b^{2}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{80 d}+\frac {29 a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{240 d}-\frac {263 a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{240 d}-\frac {59 a b \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48 d}+\frac {59 a b \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48 d}+\frac {263 a b \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{240 d}-\frac {29 a b \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{240 d}+\frac {a b \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d}-2 a b x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a b x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a b x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {5 \left (a^{2}-6 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}\)	\(429\)

[In]

int(cos(d*x+c)^6*csc(d*x+c)^7*(a+b*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(a^2*(-1/6/sin(d*x+c)^6*cos(d*x+c)^7+1/24/sin(d*x+c)^4*cos(d*x+c)^7-1/16/sin(d*x+c)^2*cos(d*x+c)^7-1/16*co
s(d*x+c)^5-5/48*cos(d*x+c)^3-5/16*cos(d*x+c)-5/16*ln(csc(d*x+c)-cot(d*x+c)))+2*a*b*(-1/5*cot(d*x+c)^5+1/3*cot(
d*x+c)^3-cot(d*x+c)-d*x-c)+b^2*(-1/4/sin(d*x+c)^4*cos(d*x+c)^7+3/8/sin(d*x+c)^2*cos(d*x+c)^7+3/8*cos(d*x+c)^5+
5/8*cos(d*x+c)^3+15/8*cos(d*x+c)+15/8*ln(csc(d*x+c)-cot(d*x+c))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (163) = 326\).

Time = 0.37 (sec) , antiderivative size = 360, normalized size of antiderivative = 2.06 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {960 \, a b d x \cos \left (d x + c\right )^{6} - 480 \, b^{2} \cos \left (d x + c\right )^{7} - 2880 \, a b d x \cos \left (d x + c\right )^{4} + 2880 \, a b d x \cos \left (d x + c\right )^{2} - 330 \, {\left (a^{2} - 6 \, b^{2}\right )} \cos \left (d x + c\right )^{5} - 960 \, a b d x + 400 \, {\left (a^{2} - 6 \, b^{2}\right )} \cos \left (d x + c\right )^{3} - 150 \, {\left (a^{2} - 6 \, b^{2}\right )} \cos \left (d x + c\right ) - 75 \, {\left ({\left (a^{2} - 6 \, b^{2}\right )} \cos \left (d x + c\right )^{6} - 3 \, {\left (a^{2} - 6 \, b^{2}\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (a^{2} - 6 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - a^{2} + 6 \, b^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 75 \, {\left ({\left (a^{2} - 6 \, b^{2}\right )} \cos \left (d x + c\right )^{6} - 3 \, {\left (a^{2} - 6 \, b^{2}\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (a^{2} - 6 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - a^{2} + 6 \, b^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 64 \, {\left (23 \, a b \cos \left (d x + c\right )^{5} - 35 \, a b \cos \left (d x + c\right )^{3} + 15 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{480 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^7*(a+b*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/480*(960*a*b*d*x*cos(d*x + c)^6 - 480*b^2*cos(d*x + c)^7 - 2880*a*b*d*x*cos(d*x + c)^4 + 2880*a*b*d*x*cos(d
*x + c)^2 - 330*(a^2 - 6*b^2)*cos(d*x + c)^5 - 960*a*b*d*x + 400*(a^2 - 6*b^2)*cos(d*x + c)^3 - 150*(a^2 - 6*b
^2)*cos(d*x + c) - 75*((a^2 - 6*b^2)*cos(d*x + c)^6 - 3*(a^2 - 6*b^2)*cos(d*x + c)^4 + 3*(a^2 - 6*b^2)*cos(d*x
+ c)^2 - a^2 + 6*b^2)*log(1/2*cos(d*x + c) + 1/2) + 75*((a^2 - 6*b^2)*cos(d*x + c)^6 - 3*(a^2 - 6*b^2)*cos(d*
x + c)^4 + 3*(a^2 - 6*b^2)*cos(d*x + c)^2 - a^2 + 6*b^2)*log(-1/2*cos(d*x + c) + 1/2) - 64*(23*a*b*cos(d*x + c
)^5 - 35*a*b*cos(d*x + c)^3 + 15*a*b*cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^6 - 3*d*cos(d*x + c)^4 + 3*d*
cos(d*x + c)^2 - d)

Sympy [F(-1)]

Timed out. \[ \int \cot ^6(c+d x) \csc (c+d x) (a+b \sin (c+d x))^2 \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**6*csc(d*x+c)**7*(a+b*sin(d*x+c))**2,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.25 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {64 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a b - 5 \, a^{2} {\left (\frac {2 \, {\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 30 \, b^{2} {\left (\frac {2 \, {\left (9 \, \cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 16 \, \cos \left (d x + c\right ) + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{480 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^7*(a+b*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/480*(64*(15*d*x + 15*c + (15*tan(d*x + c)^4 - 5*tan(d*x + c)^2 + 3)/tan(d*x + c)^5)*a*b - 5*a^2*(2*(33*cos(
d*x + c)^5 - 40*cos(d*x + c)^3 + 15*cos(d*x + c))/(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1) +
15*log(cos(d*x + c) + 1) - 15*log(cos(d*x + c) - 1)) + 30*b^2*(2*(9*cos(d*x + c)^3 - 7*cos(d*x + c))/(cos(d*x
+ c)^4 - 2*cos(d*x + c)^2 + 1) - 16*cos(d*x + c) + 15*log(cos(d*x + c) + 1) - 15*log(cos(d*x + c) - 1)))/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 337 vs. \(2 (163) = 326\).

Time = 0.42 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.93 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {5 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 24 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 45 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 30 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 280 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 225 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 480 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3840 \, {\left (d x + c\right )} a b + 2640 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 600 \, {\left (a^{2} - 6 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {3840 \, b^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + \frac {1470 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 8820 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 2640 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 225 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 480 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 280 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 45 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 30 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 24 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}}}{1920 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^7*(a+b*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/1920*(5*a^2*tan(1/2*d*x + 1/2*c)^6 + 24*a*b*tan(1/2*d*x + 1/2*c)^5 - 45*a^2*tan(1/2*d*x + 1/2*c)^4 + 30*b^2*
tan(1/2*d*x + 1/2*c)^4 - 280*a*b*tan(1/2*d*x + 1/2*c)^3 + 225*a^2*tan(1/2*d*x + 1/2*c)^2 - 480*b^2*tan(1/2*d*x
+ 1/2*c)^2 - 3840*(d*x + c)*a*b + 2640*a*b*tan(1/2*d*x + 1/2*c) - 600*(a^2 - 6*b^2)*log(abs(tan(1/2*d*x + 1/2
*c))) + 3840*b^2/(tan(1/2*d*x + 1/2*c)^2 + 1) + (1470*a^2*tan(1/2*d*x + 1/2*c)^6 - 8820*b^2*tan(1/2*d*x + 1/2*
c)^6 - 2640*a*b*tan(1/2*d*x + 1/2*c)^5 - 225*a^2*tan(1/2*d*x + 1/2*c)^4 + 480*b^2*tan(1/2*d*x + 1/2*c)^4 + 280
*a*b*tan(1/2*d*x + 1/2*c)^3 + 45*a^2*tan(1/2*d*x + 1/2*c)^2 - 30*b^2*tan(1/2*d*x + 1/2*c)^2 - 24*a*b*tan(1/2*d
*x + 1/2*c) - 5*a^2)/tan(1/2*d*x + 1/2*c)^6)/d

Mupad [B] (verification not implemented)

Time = 15.00 (sec) , antiderivative size = 985, normalized size of antiderivative = 5.63 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+b \sin (c+d x))^2 \, dx=\text {Too large to display} \]

[In]

int((cos(c + d*x)^6*(a + b*sin(c + d*x))^2)/sin(c + d*x)^7,x)

[Out]

(5*a^2*sin(c/2 + (d*x)/2)^14 - 5*a^2*cos(c/2 + (d*x)/2)^14 - 40*a^2*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^12
+ 180*a^2*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^10 + 225*a^2*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^8 - 22
5*a^2*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^6 - 180*a^2*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^4 + 40*a^2*
cos(c/2 + (d*x)/2)^12*sin(c/2 + (d*x)/2)^2 + 30*b^2*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^12 - 450*b^2*cos(c
/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^10 - 480*b^2*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^8 + 4320*b^2*cos(c/2 +
(d*x)/2)^8*sin(c/2 + (d*x)/2)^6 + 450*b^2*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^4 - 30*b^2*cos(c/2 + (d*x)
/2)^12*sin(c/2 + (d*x)/2)^2 + 24*a*b*cos(c/2 + (d*x)/2)*sin(c/2 + (d*x)/2)^13 - 24*a*b*cos(c/2 + (d*x)/2)^13*s
in(c/2 + (d*x)/2) - 600*a^2*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)
^8 - 600*a^2*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^6 + 3600*b^2*l
og(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^8 + 3600*b^2*log(sin(c/2 + (
d*x)/2)/cos(c/2 + (d*x)/2))*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^6 - 256*a*b*cos(c/2 + (d*x)/2)^3*sin(c/2 +
(d*x)/2)^11 + 2360*a*b*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^9 - 2360*a*b*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d
*x)/2)^5 + 256*a*b*cos(c/2 + (d*x)/2)^11*sin(c/2 + (d*x)/2)^3 + 7680*a*b*atan((5*a^2*sin(c/2 + (d*x)/2) - 30*b
^2*sin(c/2 + (d*x)/2) + 32*a*b*cos(c/2 + (d*x)/2))/(30*b^2*cos(c/2 + (d*x)/2) - 5*a^2*cos(c/2 + (d*x)/2) + 32*
a*b*sin(c/2 + (d*x)/2)))*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^8 + 7680*a*b*atan((5*a^2*sin(c/2 + (d*x)/2) -
30*b^2*sin(c/2 + (d*x)/2) + 32*a*b*cos(c/2 + (d*x)/2))/(30*b^2*cos(c/2 + (d*x)/2) - 5*a^2*cos(c/2 + (d*x)/2)
+ 32*a*b*sin(c/2 + (d*x)/2)))*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^6)/(1920*d*cos(c/2 + (d*x)/2)^6*sin(c/2
+ (d*x)/2)^6*(cos(c/2 + (d*x)/2)^2 + sin(c/2 + (d*x)/2)^2))