Integrand size = 27, antiderivative size = 175 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+b \sin (c+d x))^2 \, dx=-2 a b x+\frac {5 \left (a^2-6 b^2\right ) \text {arctanh}(\cos (c+d x))}{16 d}+\frac {b^2 \cos (c+d x)}{d}-\frac {2 a b \cot (c+d x)}{d}+\frac {2 a b \cot ^3(c+d x)}{3 d}-\frac {2 a b \cot ^5(c+d x)}{5 d}-\frac {\left (11 a^2-18 b^2\right ) \cot (c+d x) \csc (c+d x)}{16 d}+\frac {\left (13 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d} \] Output:
-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
Leaf count is larger than twice the leaf count of optimal. \(384\) vs. \(2(175)=350\).
Time = 2.62 (sec) , antiderivative size = 384, normalized size of antiderivative = 2.19 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {-3840 a b c-3840 a b d x+1920 b^2 \cos (c+d x)-2944 a b \cot \left (\frac {1}{2} (c+d x)\right )-330 a^2 \csc ^2\left (\frac {1}{2} (c+d x)\right )+540 b^2 \csc ^2\left (\frac {1}{2} (c+d x)\right )+60 a^2 \csc ^4\left (\frac {1}{2} (c+d x)\right )-30 b^2 \csc ^4\left (\frac {1}{2} (c+d x)\right )-5 a^2 \csc ^6\left (\frac {1}{2} (c+d x)\right )+600 a^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-3600 b^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-600 a^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+3600 b^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+330 a^2 \sec ^2\left (\frac {1}{2} (c+d x)\right )-540 b^2 \sec ^2\left (\frac {1}{2} (c+d x)\right )-60 a^2 \sec ^4\left (\frac {1}{2} (c+d x)\right )+30 b^2 \sec ^4\left (\frac {1}{2} (c+d x)\right )+5 a^2 \sec ^6\left (\frac {1}{2} (c+d x)\right )-2624 a b \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+768 a b \csc ^5(c+d x) \sin ^6\left (\frac {1}{2} (c+d x)\right )+164 a b \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-12 a b \csc ^6\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)+2944 a b \tan \left (\frac {1}{2} (c+d x)\right )}{1920 d} \] Input:
Integrate[Cot[c + d*x]^6*Csc[c + d*x]*(a + b*Sin[c + d*x])^2,x]
Output:
(-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*C sc[(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*Log[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 + 30*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*C sc[(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)
Time = 0.94 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.13, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.593, Rules used = {3042, 3390, 3042, 3954, 3042, 3954, 3042, 3954, 24, 4866, 360, 2345, 27, 1471, 299, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \cot ^6(c+d x) \csc (c+d x) (a+b \sin (c+d x))^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^6 (a+b \sin (c+d x))^2}{\sin (c+d x)^7}dx\) |
| \(\Big \downarrow \) 3390 |
| \(\displaystyle \int \cot ^6(c+d x) \csc (c+d x) \left (a^2+b^2 \sin ^2(c+d x)\right )dx+2 a b \int \cot ^6(c+d x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a^2+b^2 \sin (c+d x)^2}{\sin (c+d x) \tan (c+d x)^6}dx+2 a b \int \tan \left (c+d x+\frac {\pi }{2}\right )^6dx\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle \int \frac {a^2+b^2 \sin (c+d x)^2}{\sin (c+d x) \tan (c+d x)^6}dx+2 a b \left (-\int \cot ^4(c+d x)dx-\frac {\cot ^5(c+d x)}{5 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a^2+b^2 \sin (c+d x)^2}{\sin (c+d x) \tan (c+d x)^6}dx+2 a b \left (-\int \tan \left (c+d x+\frac {\pi }{2}\right )^4dx-\frac {\cot ^5(c+d x)}{5 d}\right )\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle \int \frac {a^2+b^2 \sin (c+d x)^2}{\sin (c+d x) \tan (c+d x)^6}dx+2 a b \left (\int \cot ^2(c+d x)dx-\frac {\cot ^5(c+d x)}{5 d}+\frac {\cot ^3(c+d x)}{3 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a^2+b^2 \sin (c+d x)^2}{\sin (c+d x) \tan (c+d x)^6}dx+2 a b \left (\int \tan \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {\cot ^5(c+d x)}{5 d}+\frac {\cot ^3(c+d x)}{3 d}\right )\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle \int \frac {a^2+b^2 \sin (c+d x)^2}{\sin (c+d x) \tan (c+d x)^6}dx+2 a b \left (-\int 1dx-\frac {\cot ^5(c+d x)}{5 d}+\frac {\cot ^3(c+d x)}{3 d}-\frac {\cot (c+d x)}{d}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \int \frac {a^2+b^2 \sin (c+d x)^2}{\sin (c+d x) \tan (c+d x)^6}dx+2 a b \left (-\frac {\cot ^5(c+d x)}{5 d}+\frac {\cot ^3(c+d x)}{3 d}-\frac {\cot (c+d x)}{d}-x\right )\) |
| \(\Big \downarrow \) 4866 |
| \(\displaystyle 2 a b \left (-\frac {\cot ^5(c+d x)}{5 d}+\frac {\cot ^3(c+d x)}{3 d}-\frac {\cot (c+d x)}{d}-x\right )-\frac {\int \frac {\cos ^6(c+d x) \left (a^2+b^2-b^2 \cos ^2(c+d x)\right )}{\left (1-\cos ^2(c+d x)\right )^4}d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 360 |
| \(\displaystyle 2 a b \left (-\frac {\cot ^5(c+d x)}{5 d}+\frac {\cot ^3(c+d x)}{3 d}-\frac {\cot (c+d x)}{d}-x\right )-\frac {\frac {a^2 \cos (c+d x)}{6 \left (1-\cos ^2(c+d x)\right )^3}-\frac {1}{6} \int \frac {-6 b^2 \cos ^6(c+d x)+6 a^2 \cos ^4(c+d x)+6 a^2 \cos ^2(c+d x)+a^2}{\left (1-\cos ^2(c+d x)\right )^3}d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle 2 a b \left (-\frac {\cot ^5(c+d x)}{5 d}+\frac {\cot ^3(c+d x)}{3 d}-\frac {\cot (c+d x)}{d}-x\right )-\frac {\frac {1}{6} \left (\frac {1}{4} \int \frac {3 \left (-8 b^2 \cos ^4(c+d x)+8 \left (a^2-b^2\right ) \cos ^2(c+d x)+3 a^2-2 b^2\right )}{\left (1-\cos ^2(c+d x)\right )^2}d\cos (c+d x)-\frac {\left (13 a^2-6 b^2\right ) \cos (c+d x)}{4 \left (1-\cos ^2(c+d x)\right )^2}\right )+\frac {a^2 \cos (c+d x)}{6 \left (1-\cos ^2(c+d x)\right )^3}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 a b \left (-\frac {\cot ^5(c+d x)}{5 d}+\frac {\cot ^3(c+d x)}{3 d}-\frac {\cot (c+d x)}{d}-x\right )-\frac {\frac {1}{6} \left (\frac {3}{4} \int \frac {-8 b^2 \cos ^4(c+d x)+8 \left (a^2-b^2\right ) \cos ^2(c+d x)+3 a^2-2 b^2}{\left (1-\cos ^2(c+d x)\right )^2}d\cos (c+d x)-\frac {\left (13 a^2-6 b^2\right ) \cos (c+d x)}{4 \left (1-\cos ^2(c+d x)\right )^2}\right )+\frac {a^2 \cos (c+d x)}{6 \left (1-\cos ^2(c+d x)\right )^3}}{d}\) |
| \(\Big \downarrow \) 1471 |
| \(\displaystyle 2 a b \left (-\frac {\cot ^5(c+d x)}{5 d}+\frac {\cot ^3(c+d x)}{3 d}-\frac {\cot (c+d x)}{d}-x\right )-\frac {\frac {1}{6} \left (\frac {3}{4} \left (\frac {\left (11 a^2-18 b^2\right ) \cos (c+d x)}{2 \left (1-\cos ^2(c+d x)\right )}-\frac {1}{2} \int \frac {5 a^2-14 b^2-16 b^2 \cos ^2(c+d x)}{1-\cos ^2(c+d x)}d\cos (c+d x)\right )-\frac {\left (13 a^2-6 b^2\right ) \cos (c+d x)}{4 \left (1-\cos ^2(c+d x)\right )^2}\right )+\frac {a^2 \cos (c+d x)}{6 \left (1-\cos ^2(c+d x)\right )^3}}{d}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle 2 a b \left (-\frac {\cot ^5(c+d x)}{5 d}+\frac {\cot ^3(c+d x)}{3 d}-\frac {\cot (c+d x)}{d}-x\right )-\frac {\frac {1}{6} \left (\frac {3}{4} \left (\frac {1}{2} \left (-5 \left (a^2-6 b^2\right ) \int \frac {1}{1-\cos ^2(c+d x)}d\cos (c+d x)-16 b^2 \cos (c+d x)\right )+\frac {\left (11 a^2-18 b^2\right ) \cos (c+d x)}{2 \left (1-\cos ^2(c+d x)\right )}\right )-\frac {\left (13 a^2-6 b^2\right ) \cos (c+d x)}{4 \left (1-\cos ^2(c+d x)\right )^2}\right )+\frac {a^2 \cos (c+d x)}{6 \left (1-\cos ^2(c+d x)\right )^3}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle 2 a b \left (-\frac {\cot ^5(c+d x)}{5 d}+\frac {\cot ^3(c+d x)}{3 d}-\frac {\cot (c+d x)}{d}-x\right )-\frac {\frac {1}{6} \left (\frac {3}{4} \left (\frac {1}{2} \left (-5 \left (a^2-6 b^2\right ) \text {arctanh}(\cos (c+d x))-16 b^2 \cos (c+d x)\right )+\frac {\left (11 a^2-18 b^2\right ) \cos (c+d x)}{2 \left (1-\cos ^2(c+d x)\right )}\right )-\frac {\left (13 a^2-6 b^2\right ) \cos (c+d x)}{4 \left (1-\cos ^2(c+d x)\right )^2}\right )+\frac {a^2 \cos (c+d x)}{6 \left (1-\cos ^2(c+d x)\right )^3}}{d}\) |
Input:
Int[Cot[c + d*x]^6*Csc[c + d*x]*(a + b*Sin[c + d*x])^2,x]
Output:
-(((a^2*Cos[c + d*x])/(6*(1 - Cos[c + d*x]^2)^3) + (-1/4*((13*a^2 - 6*b^2) *Cos[c + d*x])/(1 - Cos[c + d*x]^2)^2 + (3*((-5*(a^2 - 6*b^2)*ArcTanh[Cos[ c + d*x]] - 16*b^2*Cos[c + d*x])/2 + ((11*a^2 - 18*b^2)*Cos[c + d*x])/(2*( 1 - Cos[c + d*x]^2))))/4)/6)/d) + 2*a*b*(-x - Cot[c + d*x]/d + Cot[c + d*x ]^3/(3*d) - Cot[c + d*x]^5/(5*d))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
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] + Simp[1/(2*b^(m/2 + 1)*(p + 1)) Int[(a + b*x^2)^(p + 1)*Expan dToSum[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])
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(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] + Simp[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]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[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] + Simp[1/(2*a*(p + 1)) In t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[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]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d *x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2 Int[(b*Tan[c + d*x])^(n - 2), x] , x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))]^(n_), x_Symbol] :> With[{d = Free Factors[Cos[c*(a + b*x)], x]}, Simp[-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] && Intege rQ[(n - 1)/2] && NonsumQ[u] && (EqQ[F, Sin] || EqQ[F, sin])
Time = 1.04 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.36
| method | result | size |
| derivativedivides | \(\frac {a^{2} \left (-\frac {\cos \left (d x +c \right )^{7}}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{7}}{16 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{16}-\frac {5 \cos \left (d x +c \right )^{3}}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )+2 a b \left (-\frac {\cot \left (d x +c \right )^{5}}{5}+\frac {\cot \left (d x +c \right )^{3}}{3}-\cot \left (d x +c \right )-d x -c \right )+b^{2} \left (-\frac {\cos \left (d x +c \right )^{7}}{4 \sin \left (d x +c \right )^{4}}+\frac {3 \cos \left (d x +c \right )^{7}}{8 \sin \left (d x +c \right )^{2}}+\frac {3 \cos \left (d x +c \right )^{5}}{8}+\frac {5 \cos \left (d x +c \right )^{3}}{8}+\frac {15 \cos \left (d x +c \right )}{8}+\frac {15 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )}{d}\) | \(238\) |
| default | \(\frac {a^{2} \left (-\frac {\cos \left (d x +c \right )^{7}}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{7}}{16 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{16}-\frac {5 \cos \left (d x +c \right )^{3}}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )+2 a b \left (-\frac {\cot \left (d x +c \right )^{5}}{5}+\frac {\cot \left (d x +c \right )^{3}}{3}-\cot \left (d x +c \right )-d x -c \right )+b^{2} \left (-\frac {\cos \left (d x +c \right )^{7}}{4 \sin \left (d x +c \right )^{4}}+\frac {3 \cos \left (d x +c \right )^{7}}{8 \sin \left (d x +c \right )^{2}}+\frac {3 \cos \left (d x +c \right )^{5}}{8}+\frac {5 \cos \left (d x +c \right )^{3}}{8}+\frac {15 \cos \left (d x +c \right )}{8}+\frac {15 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )}{d}\) | \(238\) |
| risch | \(-2 a b x +\frac {{\mathrm e}^{i \left (d x +c \right )} b^{2}}{2 d}+\frac {{\mathrm e}^{-i \left (d x +c \right )} b^{2}}{2 d}+\frac {6720 i a b \,{\mathrm e}^{4 i \left (d x +c \right )}+165 a^{2} {\mathrm e}^{11 i \left (d x +c \right )}-270 b^{2} {\mathrm e}^{11 i \left (d x +c \right )}+736 i a b +25 a^{2} {\mathrm e}^{9 i \left (d x +c \right )}+570 b^{2} {\mathrm e}^{9 i \left (d x +c \right )}-7360 i a b \,{\mathrm e}^{6 i \left (d x +c \right )}+450 a^{2} {\mathrm e}^{7 i \left (d x +c \right )}-300 b^{2} {\mathrm e}^{7 i \left (d x +c \right )}-2976 i a b \,{\mathrm e}^{2 i \left (d x +c \right )}+450 a^{2} {\mathrm e}^{5 i \left (d x +c \right )}-300 b^{2} {\mathrm e}^{5 i \left (d x +c \right )}-1440 i a b \,{\mathrm e}^{10 i \left (d x +c \right )}+25 a^{2} {\mathrm e}^{3 i \left (d x +c \right )}+570 b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+4320 i a b \,{\mathrm e}^{8 i \left (d x +c \right )}+165 a^{2} {\mathrm e}^{i \left (d x +c \right )}-270 b^{2} {\mathrm e}^{i \left (d x +c \right )}}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}+\frac {5 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d}-\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{8 d}-\frac {5 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d}+\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{8 d}\) | \(383\) |
Input:
int(cot(d*x+c)^6*csc(d*x+c)*(a+b*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
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*cos(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))))
Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (163) = 326\).
Time = 0.11 (sec) , antiderivative size = 360, normalized size of antiderivative = 2.06 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {960 \, a b d x \cos \left (d x + c\right )^{6} - 480 \, b^{2} \cos \left (d x + c\right )^{7} - 2880 \, a b d x \cos \left (d x + c\right )^{4} + 2880 \, a b d x \cos \left (d x + c\right )^{2} - 330 \, {\left (a^{2} - 6 \, b^{2}\right )} \cos \left (d x + c\right )^{5} - 960 \, a b d x + 400 \, {\left (a^{2} - 6 \, b^{2}\right )} \cos \left (d x + c\right )^{3} - 150 \, {\left (a^{2} - 6 \, b^{2}\right )} \cos \left (d x + c\right ) - 75 \, {\left ({\left (a^{2} - 6 \, b^{2}\right )} \cos \left (d x + c\right )^{6} - 3 \, {\left (a^{2} - 6 \, b^{2}\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (a^{2} - 6 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - a^{2} + 6 \, b^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 75 \, {\left ({\left (a^{2} - 6 \, b^{2}\right )} \cos \left (d x + c\right )^{6} - 3 \, {\left (a^{2} - 6 \, b^{2}\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (a^{2} - 6 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - a^{2} + 6 \, b^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 64 \, {\left (23 \, a b \cos \left (d x + c\right )^{5} - 35 \, a b \cos \left (d x + c\right )^{3} + 15 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{480 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)*(a+b*sin(d*x+c))^2,x, algorithm="fricas" )
Output:
-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)
Timed out. \[ \int \cot ^6(c+d x) \csc (c+d x) (a+b \sin (c+d x))^2 \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**6*csc(d*x+c)*(a+b*sin(d*x+c))**2,x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.25 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {64 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a b - 5 \, a^{2} {\left (\frac {2 \, {\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 30 \, b^{2} {\left (\frac {2 \, {\left (9 \, \cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 16 \, \cos \left (d x + c\right ) + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{480 \, d} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)*(a+b*sin(d*x+c))^2,x, algorithm="maxima" )
Output:
-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*co s(d*x + c))/(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1) + 1 5*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
Leaf count of result is larger than twice the leaf count of optimal. 337 vs. \(2 (163) = 326\).
Time = 0.28 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.93 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {5 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 24 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 45 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 30 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 280 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 225 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 480 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3840 \, {\left (d x + c\right )} a b + 2640 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 600 \, {\left (a^{2} - 6 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {3840 \, b^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + \frac {1470 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 8820 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 2640 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 225 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 480 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 280 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 45 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 30 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 24 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}}}{1920 \, d} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)*(a+b*sin(d*x+c))^2,x, algorithm="giac")
Output:
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
Time = 24.41 (sec) , antiderivative size = 985, normalized size of antiderivative = 5.63 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+b \sin (c+d x))^2 \, dx=\text {Too large to display} \] Input:
int((cot(c + d*x)^6*(a + b*sin(c + d*x))^2)/sin(c + d*x),x)
Output:
(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 - 225* 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*sin(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*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 + 3600*b^2*log(sin(c/2 + (d*x)/2)/co s(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 + 2 56*a*b*cos(c/2 + (d*x)/2)^11*sin(c/2 + (d*x)/2)^3 + 7680*a*b*atan((5*a^2*s in(c/2 + (d*x)/2) - 30*b^2*sin(c/2 + (d*x)/2) + 32*a*b*cos(c/2 + (d*x)/...
Time = 0.17 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.47 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {1920 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} b^{2}-5888 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} a b -1320 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a^{2}+2160 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} b^{2}+2816 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a b +1040 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{2}-480 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} b^{2}-768 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a b -320 \cos \left (d x +c \right ) a^{2}-600 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{6} a^{2}+3600 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{6} b^{2}+225 \sin \left (d x +c \right )^{6} a^{2}-3840 \sin \left (d x +c \right )^{6} a b d x -2400 \sin \left (d x +c \right )^{6} b^{2}}{1920 \sin \left (d x +c \right )^{6} d} \] Input:
int(cot(d*x+c)^6*csc(d*x+c)*(a+b*sin(d*x+c))^2,x)
Output:
(1920*cos(c + d*x)*sin(c + d*x)**6*b**2 - 5888*cos(c + d*x)*sin(c + d*x)** 5*a*b - 1320*cos(c + d*x)*sin(c + d*x)**4*a**2 + 2160*cos(c + d*x)*sin(c + d*x)**4*b**2 + 2816*cos(c + d*x)*sin(c + d*x)**3*a*b + 1040*cos(c + d*x)* sin(c + d*x)**2*a**2 - 480*cos(c + d*x)*sin(c + d*x)**2*b**2 - 768*cos(c + d*x)*sin(c + d*x)*a*b - 320*cos(c + d*x)*a**2 - 600*log(tan((c + d*x)/2)) *sin(c + d*x)**6*a**2 + 3600*log(tan((c + d*x)/2))*sin(c + d*x)**6*b**2 + 225*sin(c + d*x)**6*a**2 - 3840*sin(c + d*x)**6*a*b*d*x - 2400*sin(c + d*x )**6*b**2)/(1920*sin(c + d*x)**6*d)