Integrand size = 16, antiderivative size = 127 \[ \int (c+d x)^3 \cot ^2(a+b x) \, dx=-\frac {i (c+d x)^3}{b}-\frac {(c+d x)^4}{4 d}-\frac {(c+d x)^3 \cot (a+b x)}{b}+\frac {3 d (c+d x)^2 \log \left (1-e^{2 i (a+b x)}\right )}{b^2}-\frac {3 i d^2 (c+d x) \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^3}+\frac {3 d^3 \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{2 b^4} \] Output:
-I*(d*x+c)^3/b-1/4*(d*x+c)^4/d-(d*x+c)^3*cot(b*x+a)/b+3*d*(d*x+c)^2*ln(1-e xp(2*I*(b*x+a)))/b^2-3*I*d^2*(d*x+c)*polylog(2,exp(2*I*(b*x+a)))/b^3+3/2*d ^3*polylog(3,exp(2*I*(b*x+a)))/b^4
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(374\) vs. \(2(127)=254\).
Time = 4.15 (sec) , antiderivative size = 374, normalized size of antiderivative = 2.94 \[ \int (c+d x)^3 \cot ^2(a+b x) \, dx=-\frac {1}{4} x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )-\frac {3 c^2 d (b x \cot (a)-\log (\sin (a+b x)))}{b^2}+\frac {3 c d^2 \left (i b x (\pi -2 \arctan (\tan (a)))+\pi \log \left (1+e^{-2 i b x}\right )+2 (b x+\arctan (\tan (a))) \log \left (1-e^{2 i (b x+\arctan (\tan (a)))}\right )-\pi \log (\cos (b x))-2 \arctan (\tan (a)) \log (\sin (b x+\arctan (\tan (a))))-i \operatorname {PolyLog}\left (2,e^{2 i (b x+\arctan (\tan (a)))}\right )-b^2 e^{i \arctan (\tan (a))} x^2 \cot (a) \sqrt {\sec ^2(a)}\right )}{b^3}+\frac {d^3 e^{-i a} (i+\cot (a)) \left (i b^3 x^3-b^3 x^3 \cot (a)+3 b^2 x^2 \log \left (1-e^{-i (a+b x)}\right )+3 b^2 x^2 \log \left (1+e^{-i (a+b x)}\right )+6 i b x \operatorname {PolyLog}\left (2,-e^{-i (a+b x)}\right )+6 i b x \operatorname {PolyLog}\left (2,e^{-i (a+b x)}\right )+6 \operatorname {PolyLog}\left (3,-e^{-i (a+b x)}\right )+6 \operatorname {PolyLog}\left (3,e^{-i (a+b x)}\right )\right ) \sin (a)}{b^4}+\frac {(c+d x)^3 \csc (a) \csc (a+b x) \sin (b x)}{b} \] Input:
Integrate[(c + d*x)^3*Cot[a + b*x]^2,x]
Output:
-1/4*(x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3)) - (3*c^2*d*(b*x*Cot[a ] - Log[Sin[a + b*x]]))/b^2 + (3*c*d^2*(I*b*x*(Pi - 2*ArcTan[Tan[a]]) + Pi *Log[1 + E^((-2*I)*b*x)] + 2*(b*x + ArcTan[Tan[a]])*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] - Pi*Log[Cos[b*x]] - 2*ArcTan[Tan[a]]*Log[Sin[b*x + Ar cTan[Tan[a]]]] - I*PolyLog[2, E^((2*I)*(b*x + ArcTan[Tan[a]]))] - b^2*E^(I *ArcTan[Tan[a]])*x^2*Cot[a]*Sqrt[Sec[a]^2]))/b^3 + (d^3*(I + Cot[a])*(I*b^ 3*x^3 - b^3*x^3*Cot[a] + 3*b^2*x^2*Log[1 - E^((-I)*(a + b*x))] + 3*b^2*x^2 *Log[1 + E^((-I)*(a + b*x))] + (6*I)*b*x*PolyLog[2, -E^((-I)*(a + b*x))] + (6*I)*b*x*PolyLog[2, E^((-I)*(a + b*x))] + 6*PolyLog[3, -E^((-I)*(a + b*x ))] + 6*PolyLog[3, E^((-I)*(a + b*x))])*Sin[a])/(b^4*E^(I*a)) + ((c + d*x) ^3*Csc[a]*Csc[a + b*x]*Sin[b*x])/b
Time = 0.69 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.29, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {3042, 4203, 17, 25, 3042, 25, 4202, 2620, 3011, 2720, 7143}
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.
| \(\displaystyle \int (c+d x)^3 \cot ^2(a+b x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (c+d x)^3 \tan \left (a+b x+\frac {\pi }{2}\right )^2dx\) |
| \(\Big \downarrow \) 4203 |
| \(\displaystyle -\frac {3 d \int -(c+d x)^2 \cot (a+b x)dx}{b}-\int (c+d x)^3dx-\frac {(c+d x)^3 \cot (a+b x)}{b}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle -\frac {3 d \int -(c+d x)^2 \cot (a+b x)dx}{b}-\frac {(c+d x)^3 \cot (a+b x)}{b}-\frac {(c+d x)^4}{4 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 d \int (c+d x)^2 \cot (a+b x)dx}{b}-\frac {(c+d x)^3 \cot (a+b x)}{b}-\frac {(c+d x)^4}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 d \int -(c+d x)^2 \tan \left (a+b x+\frac {\pi }{2}\right )dx}{b}-\frac {(c+d x)^3 \cot (a+b x)}{b}-\frac {(c+d x)^4}{4 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {3 d \int (c+d x)^2 \tan \left (\frac {1}{2} (2 a+\pi )+b x\right )dx}{b}-\frac {(c+d x)^3 \cot (a+b x)}{b}-\frac {(c+d x)^4}{4 d}\) |
| \(\Big \downarrow \) 4202 |
| \(\displaystyle -\frac {3 d \left (\frac {i (c+d x)^3}{3 d}-2 i \int \frac {e^{i (2 a+2 b x+\pi )} (c+d x)^2}{1+e^{i (2 a+2 b x+\pi )}}dx\right )}{b}-\frac {(c+d x)^3 \cot (a+b x)}{b}-\frac {(c+d x)^4}{4 d}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {3 d \left (\frac {i (c+d x)^3}{3 d}-2 i \left (\frac {i d \int (c+d x) \log \left (1+e^{i (2 a+2 b x+\pi )}\right )dx}{b}-\frac {i (c+d x)^2 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )\right )}{b}-\frac {(c+d x)^3 \cot (a+b x)}{b}-\frac {(c+d x)^4}{4 d}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {3 d \left (\frac {i (c+d x)^3}{3 d}-2 i \left (\frac {i d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {i d \int \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )dx}{2 b}\right )}{b}-\frac {i (c+d x)^2 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )\right )}{b}-\frac {(c+d x)^3 \cot (a+b x)}{b}-\frac {(c+d x)^4}{4 d}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {3 d \left (\frac {i (c+d x)^3}{3 d}-2 i \left (\frac {i d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {d \int e^{-i (2 a+2 b x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )de^{i (2 a+2 b x+\pi )}}{4 b^2}\right )}{b}-\frac {i (c+d x)^2 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )\right )}{b}-\frac {(c+d x)^3 \cot (a+b x)}{b}-\frac {(c+d x)^4}{4 d}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -\frac {3 d \left (\frac {i (c+d x)^3}{3 d}-2 i \left (\frac {i d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {d \operatorname {PolyLog}\left (3,-e^{i (2 a+2 b x+\pi )}\right )}{4 b^2}\right )}{b}-\frac {i (c+d x)^2 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )\right )}{b}-\frac {(c+d x)^3 \cot (a+b x)}{b}-\frac {(c+d x)^4}{4 d}\) |
Input:
Int[(c + d*x)^3*Cot[a + b*x]^2,x]
Output:
-1/4*(c + d*x)^4/d - ((c + d*x)^3*Cot[a + b*x])/b - (3*d*(((I/3)*(c + d*x) ^3)/d - (2*I)*(((-1/2*I)*(c + d*x)^2*Log[1 + E^(I*(2*a + Pi + 2*b*x))])/b + (I*d*(((I/2)*(c + d*x)*PolyLog[2, -E^(I*(2*a + Pi + 2*b*x))])/b - (d*Pol yLog[3, -E^(I*(2*a + Pi + 2*b*x))])/(4*b^2)))/b)))/b
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I *((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I Int[(c + d*x)^m*(E^(2*I*( e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt Q[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symb ol] :> Simp[b*(c + d*x)^m*((b*Tan[e + f*x])^(n - 1)/(f*(n - 1))), x] + (-Si mp[b*d*(m/(f*(n - 1))) Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1), x] , x] - Simp[b^2 Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; Free Q[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 0]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 580 vs. \(2 (115 ) = 230\).
Time = 0.47 (sec) , antiderivative size = 581, normalized size of antiderivative = 4.57
| method | result | size |
| risch | \(-\frac {2 i \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}\right )}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}+\frac {3 d^{3} \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x^{2}}{b^{2}}+\frac {3 d^{3} \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x^{2}}{b^{2}}-\frac {6 d \,c^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}+\frac {3 d \,c^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{2}}+\frac {3 d \,c^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right )}{b^{2}}-\frac {6 d^{3} a^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}+\frac {3 d^{3} a^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{4}}-\frac {3 d^{3} \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) a^{2}}{b^{4}}-\frac {2 i d^{3} x^{3}}{b}+\frac {4 i d^{3} a^{3}}{b^{4}}-\frac {6 d^{2} c a \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{3}}-\frac {6 i d^{3} \operatorname {polylog}\left (2, {\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{3}}-\frac {6 i d^{2} c \,x^{2}}{b}-\frac {6 i d^{2} c \,a^{2}}{b^{3}}-\frac {6 i d^{2} c \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}-\frac {6 i d^{2} c \operatorname {polylog}\left (2, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {6 i d^{3} a^{2} x}{b^{3}}-\frac {6 i d^{3} \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{3}}+\frac {6 d^{2} c \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) a}{b^{3}}+\frac {6 d^{2} c \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x}{b^{2}}+\frac {6 d^{2} c \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{2}}+\frac {12 d^{2} c a \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}-\frac {d^{3} x^{4}}{4}-\frac {c^{4}}{4 d}-d^{2} c \,x^{3}-\frac {3 d \,c^{2} x^{2}}{2}-c^{3} x -\frac {12 i d^{2} c a x}{b^{2}}+\frac {6 d^{3} \operatorname {polylog}\left (3, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}+\frac {6 d^{3} \operatorname {polylog}\left (3, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}\) | \(581\) |
Input:
int((d*x+c)^3*cot(b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
-12*I*d^2/b^2*c*a*x-1/4*d^3*x^4-1/4/d*c^4-d^2*c*x^3-3/2*d*c^2*x^2-c^3*x-2* I*(d^3*x^3+3*c*d^2*x^2+3*c^2*d*x+c^3)/b/(exp(2*I*(b*x+a))-1)-6*d^2/b^3*c*a *ln(exp(I*(b*x+a))-1)-6*I*d^3/b^3*polylog(2,exp(I*(b*x+a)))*x-6*I*d^2/b*c* x^2-6*I*d^2/b^3*c*a^2-6*I*d^2/b^3*c*polylog(2,-exp(I*(b*x+a)))-6*I*d^2/b^3 *c*polylog(2,exp(I*(b*x+a)))+6*I*d^3/b^3*a^2*x-6*I*d^3/b^3*polylog(2,-exp( I*(b*x+a)))*x+6*d^2/b^3*c*ln(1-exp(I*(b*x+a)))*a+6*d^2/b^2*c*ln(exp(I*(b*x +a))+1)*x+6*d^2/b^2*c*ln(1-exp(I*(b*x+a)))*x+12*d^2/b^3*c*a*ln(exp(I*(b*x+ a)))+3*d^3/b^2*ln(exp(I*(b*x+a))+1)*x^2+3*d^3/b^2*ln(1-exp(I*(b*x+a)))*x^2 -6*d/b^2*c^2*ln(exp(I*(b*x+a)))+3*d/b^2*c^2*ln(exp(I*(b*x+a))-1)+3*d/b^2*c ^2*ln(exp(I*(b*x+a))+1)-6*d^3/b^4*a^2*ln(exp(I*(b*x+a)))+3*d^3/b^4*a^2*ln( exp(I*(b*x+a))-1)-3*d^3/b^4*ln(1-exp(I*(b*x+a)))*a^2-2*I*d^3/b*x^3+4*I*d^3 /b^4*a^3+6*d^3*polylog(3,-exp(I*(b*x+a)))/b^4+6*d^3*polylog(3,exp(I*(b*x+a )))/b^4
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 599 vs. \(2 (112) = 224\).
Time = 0.09 (sec) , antiderivative size = 599, normalized size of antiderivative = 4.72 \[ \int (c+d x)^3 \cot ^2(a+b x) \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)^3*cot(b*x+a)^2,x, algorithm="fricas")
Output:
-1/4*(4*b^3*d^3*x^3 + 12*b^3*c*d^2*x^2 + 12*b^3*c^2*d*x + 4*b^3*c^3 - 3*d^ 3*polylog(3, cos(2*b*x + 2*a) + I*sin(2*b*x + 2*a))*sin(2*b*x + 2*a) - 3*d ^3*polylog(3, cos(2*b*x + 2*a) - I*sin(2*b*x + 2*a))*sin(2*b*x + 2*a) + 6* (I*b*d^3*x + I*b*c*d^2)*dilog(cos(2*b*x + 2*a) + I*sin(2*b*x + 2*a))*sin(2 *b*x + 2*a) + 6*(-I*b*d^3*x - I*b*c*d^2)*dilog(cos(2*b*x + 2*a) - I*sin(2* b*x + 2*a))*sin(2*b*x + 2*a) - 6*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*log(- 1/2*cos(2*b*x + 2*a) + 1/2*I*sin(2*b*x + 2*a) + 1/2)*sin(2*b*x + 2*a) - 6* (b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*log(-1/2*cos(2*b*x + 2*a) - 1/2*I*sin( 2*b*x + 2*a) + 1/2)*sin(2*b*x + 2*a) - 6*(b^2*d^3*x^2 + 2*b^2*c*d^2*x + 2* a*b*c*d^2 - a^2*d^3)*log(-cos(2*b*x + 2*a) + I*sin(2*b*x + 2*a) + 1)*sin(2 *b*x + 2*a) - 6*(b^2*d^3*x^2 + 2*b^2*c*d^2*x + 2*a*b*c*d^2 - a^2*d^3)*log( -cos(2*b*x + 2*a) - I*sin(2*b*x + 2*a) + 1)*sin(2*b*x + 2*a) + 4*(b^3*d^3* x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*cos(2*b*x + 2*a) + (b^4*d ^3*x^4 + 4*b^4*c*d^2*x^3 + 6*b^4*c^2*d*x^2 + 4*b^4*c^3*x)*sin(2*b*x + 2*a) )/(b^4*sin(2*b*x + 2*a))
\[ \int (c+d x)^3 \cot ^2(a+b x) \, dx=\int \left (c + d x\right )^{3} \cot ^{2}{\left (a + b x \right )}\, dx \] Input:
integrate((d*x+c)**3*cot(b*x+a)**2,x)
Output:
Integral((c + d*x)**3*cot(a + b*x)**2, x)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1953 vs. \(2 (112) = 224\).
Time = 0.30 (sec) , antiderivative size = 1953, normalized size of antiderivative = 15.38 \[ \int (c+d x)^3 \cot ^2(a+b x) \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)^3*cot(b*x+a)^2,x, algorithm="maxima")
Output:
-1/2*(2*(b*x + a + 1/tan(b*x + a))*c^3 - 6*(b*x + a + 1/tan(b*x + a))*a*c^ 2*d/b + 6*(b*x + a + 1/tan(b*x + a))*a^2*c*d^2/b^2 - 2*(b*x + a + 1/tan(b* x + a))*a^3*d^3/b^3 + 3*((b*x + a)^2*cos(2*b*x + 2*a)^2 + (b*x + a)^2*sin( 2*b*x + 2*a)^2 - 2*(b*x + a)^2*cos(2*b*x + 2*a) + (b*x + a)^2 - (cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 - 2*cos(2*b*x + 2*a) + 1)*log(cos(b*x + a)^ 2 + sin(b*x + a)^2 + 2*cos(b*x + a) + 1) - (cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 - 2*cos(2*b*x + 2*a) + 1)*log(cos(b*x + a)^2 + sin(b*x + a)^2 - 2*cos(b*x + a) + 1) + 4*(b*x + a)*sin(2*b*x + 2*a))*c^2*d/((cos(2*b*x + 2* a)^2 + sin(2*b*x + 2*a)^2 - 2*cos(2*b*x + 2*a) + 1)*b) - 6*((b*x + a)^2*co s(2*b*x + 2*a)^2 + (b*x + a)^2*sin(2*b*x + 2*a)^2 - 2*(b*x + a)^2*cos(2*b* x + 2*a) + (b*x + a)^2 - (cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 - 2*cos( 2*b*x + 2*a) + 1)*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*cos(b*x + a) + 1 ) - (cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 - 2*cos(2*b*x + 2*a) + 1)*log (cos(b*x + a)^2 + sin(b*x + a)^2 - 2*cos(b*x + a) + 1) + 4*(b*x + a)*sin(2 *b*x + 2*a))*a*c*d^2/((cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 - 2*cos(2*b *x + 2*a) + 1)*b^2) + 3*((b*x + a)^2*cos(2*b*x + 2*a)^2 + (b*x + a)^2*sin( 2*b*x + 2*a)^2 - 2*(b*x + a)^2*cos(2*b*x + 2*a) + (b*x + a)^2 - (cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 - 2*cos(2*b*x + 2*a) + 1)*log(cos(b*x + a)^ 2 + sin(b*x + a)^2 + 2*cos(b*x + a) + 1) - (cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 - 2*cos(2*b*x + 2*a) + 1)*log(cos(b*x + a)^2 + sin(b*x + a)^2...
\[ \int (c+d x)^3 \cot ^2(a+b x) \, dx=\int { {\left (d x + c\right )}^{3} \cot \left (b x + a\right )^{2} \,d x } \] Input:
integrate((d*x+c)^3*cot(b*x+a)^2,x, algorithm="giac")
Output:
integrate((d*x + c)^3*cot(b*x + a)^2, x)
Timed out. \[ \int (c+d x)^3 \cot ^2(a+b x) \, dx=\int {\mathrm {cot}\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^3 \,d x \] Input:
int(cot(a + b*x)^2*(c + d*x)^3,x)
Output:
int(cot(a + b*x)^2*(c + d*x)^3, x)
\[ \int (c+d x)^3 \cot ^2(a+b x) \, dx=\frac {-4 \cos \left (b x +a \right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right ) b \,c^{3}-12 \cos \left (b x +a \right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right ) b \,c^{2} d x -6 \cos \left (b x +a \right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right ) b c \,d^{2} x^{2}-2 \cos \left (b x +a \right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right ) b \,d^{3} x^{3}+6 \left (\int \frac {x^{2}}{\tan \left (\frac {b x}{2}+\frac {a}{2}\right )}d x \right ) \sin \left (b x +a \right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right ) b \,d^{3}+12 \left (\int \frac {x}{\tan \left (\frac {b x}{2}+\frac {a}{2}\right )}d x \right ) \sin \left (b x +a \right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right ) b c \,d^{2}-6 \left (\int \tan \left (\frac {b x}{2}+\frac {a}{2}\right ) x^{2}d x \right ) \sin \left (b x +a \right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right ) b \,d^{3}-12 \left (\int \tan \left (\frac {b x}{2}+\frac {a}{2}\right ) x d x \right ) \sin \left (b x +a \right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right ) b c \,d^{2}-12 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+1\right ) \sin \left (b x +a \right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right ) c^{2} d +12 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \sin \left (b x +a \right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right ) c^{2} d -4 \sin \left (b x +a \right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right ) b^{2} c^{3} x -6 \sin \left (b x +a \right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right ) b^{2} c^{2} d \,x^{2}-4 \sin \left (b x +a \right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right ) b^{2} c \,d^{2} x^{3}-\sin \left (b x +a \right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right ) b^{2} d^{3} x^{4}-6 \sin \left (b x +a \right ) b c \,d^{2} x^{2}-2 \sin \left (b x +a \right ) b \,d^{3} x^{3}+6 \tan \left (\frac {b x}{2}+\frac {a}{2}\right ) b c \,d^{2} x^{2}+2 \tan \left (\frac {b x}{2}+\frac {a}{2}\right ) b \,d^{3} x^{3}}{4 \sin \left (b x +a \right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right ) b^{2}} \] Input:
int((d*x+c)^3*cot(b*x+a)^2,x)
Output:
( - 4*cos(a + b*x)*tan((a + b*x)/2)*b*c**3 - 12*cos(a + b*x)*tan((a + b*x) /2)*b*c**2*d*x - 6*cos(a + b*x)*tan((a + b*x)/2)*b*c*d**2*x**2 - 2*cos(a + b*x)*tan((a + b*x)/2)*b*d**3*x**3 + 6*int(x**2/tan((a + b*x)/2),x)*sin(a + b*x)*tan((a + b*x)/2)*b*d**3 + 12*int(x/tan((a + b*x)/2),x)*sin(a + b*x) *tan((a + b*x)/2)*b*c*d**2 - 6*int(tan((a + b*x)/2)*x**2,x)*sin(a + b*x)*t an((a + b*x)/2)*b*d**3 - 12*int(tan((a + b*x)/2)*x,x)*sin(a + b*x)*tan((a + b*x)/2)*b*c*d**2 - 12*log(tan((a + b*x)/2)**2 + 1)*sin(a + b*x)*tan((a + b*x)/2)*c**2*d + 12*log(tan((a + b*x)/2))*sin(a + b*x)*tan((a + b*x)/2)*c **2*d - 4*sin(a + b*x)*tan((a + b*x)/2)*b**2*c**3*x - 6*sin(a + b*x)*tan(( a + b*x)/2)*b**2*c**2*d*x**2 - 4*sin(a + b*x)*tan((a + b*x)/2)*b**2*c*d**2 *x**3 - sin(a + b*x)*tan((a + b*x)/2)*b**2*d**3*x**4 - 6*sin(a + b*x)*b*c* d**2*x**2 - 2*sin(a + b*x)*b*d**3*x**3 + 6*tan((a + b*x)/2)*b*c*d**2*x**2 + 2*tan((a + b*x)/2)*b*d**3*x**3)/(4*sin(a + b*x)*tan((a + b*x)/2)*b**2)