Integrand size = 20, antiderivative size = 227 \[ \int (c+d x)^2 (a+b \cot (e+f x))^2 \, dx=-\frac {i b^2 (c+d x)^2}{f}+\frac {a^2 (c+d x)^3}{3 d}-\frac {2 i a b (c+d x)^3}{3 d}-\frac {b^2 (c+d x)^3}{3 d}-\frac {b^2 (c+d x)^2 \cot (e+f x)}{f}+\frac {2 b^2 d (c+d x) \log \left (1-e^{2 i (e+f x)}\right )}{f^2}+\frac {2 a b (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac {i b^2 d^2 \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^3}-\frac {2 i a b d (c+d x) \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^2}+\frac {a b d^2 \operatorname {PolyLog}\left (3,e^{2 i (e+f x)}\right )}{f^3} \] Output:
-I*b^2*(d*x+c)^2/f+1/3*a^2*(d*x+c)^3/d-2/3*I*a*b*(d*x+c)^3/d-1/3*b^2*(d*x+ c)^3/d-b^2*(d*x+c)^2*cot(f*x+e)/f+2*b^2*d*(d*x+c)*ln(1-exp(2*I*(f*x+e)))/f ^2+2*a*b*(d*x+c)^2*ln(1-exp(2*I*(f*x+e)))/f-I*b^2*d^2*polylog(2,exp(2*I*(f *x+e)))/f^3-2*I*a*b*d*(d*x+c)*polylog(2,exp(2*I*(f*x+e)))/f^2+a*b*d^2*poly log(3,exp(2*I*(f*x+e)))/f^3
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(737\) vs. \(2(227)=454\).
Time = 6.78 (sec) , antiderivative size = 737, normalized size of antiderivative = 3.25 \[ \int (c+d x)^2 (a+b \cot (e+f x))^2 \, dx =\text {Too large to display} \] Input:
Integrate[(c + d*x)^2*(a + b*Cot[e + f*x])^2,x]
Output:
-1/3*(a*b*d^2*E^(I*e)*Csc[e]*((2*f^3*x^3)/E^((2*I)*e) + (3*I)*(1 - E^((-2* I)*e))*f^2*x^2*Log[1 - E^((-I)*(e + f*x))] + (3*I)*(1 - E^((-2*I)*e))*f^2* x^2*Log[1 + E^((-I)*(e + f*x))] - 6*(1 - E^((-2*I)*e))*f*x*PolyLog[2, -E^( (-I)*(e + f*x))] - 6*(1 - E^((-2*I)*e))*f*x*PolyLog[2, E^((-I)*(e + f*x))] + (6*I)*(1 - E^((-2*I)*e))*PolyLog[3, -E^((-I)*(e + f*x))] + (6*I)*(1 - E ^((-2*I)*e))*PolyLog[3, E^((-I)*(e + f*x))]))/f^3 + (x*(3*c^2 + 3*c*d*x + d^2*x^2)*Csc[e]*(2*a*b*Cos[e] + a^2*Sin[e] - b^2*Sin[e]))/3 + (2*b^2*c*d*C sc[e]*(-(f*x*Cos[e]) + Log[Cos[f*x]*Sin[e] + Cos[e]*Sin[f*x]]*Sin[e]))/(f^ 2*(Cos[e]^2 + Sin[e]^2)) + (2*a*b*c^2*Csc[e]*(-(f*x*Cos[e]) + Log[Cos[f*x] *Sin[e] + Cos[e]*Sin[f*x]]*Sin[e]))/(f*(Cos[e]^2 + Sin[e]^2)) + (Csc[e]*Cs c[e + f*x]*(b^2*c^2*Sin[f*x] + 2*b^2*c*d*x*Sin[f*x] + b^2*d^2*x^2*Sin[f*x] ))/f - (b^2*d^2*Csc[e]*Sec[e]*(E^(I*ArcTan[Tan[e]])*f^2*x^2 + ((I*f*x*(-Pi + 2*ArcTan[Tan[e]]) - Pi*Log[1 + E^((-2*I)*f*x)] - 2*(f*x + ArcTan[Tan[e] ])*Log[1 - E^((2*I)*(f*x + ArcTan[Tan[e]]))] + Pi*Log[Cos[f*x]] + 2*ArcTan [Tan[e]]*Log[Sin[f*x + ArcTan[Tan[e]]]] + I*PolyLog[2, E^((2*I)*(f*x + Arc Tan[Tan[e]]))])*Tan[e])/Sqrt[1 + Tan[e]^2]))/(f^3*Sqrt[Sec[e]^2*(Cos[e]^2 + Sin[e]^2)]) - (2*a*b*c*d*Csc[e]*Sec[e]*(E^(I*ArcTan[Tan[e]])*f^2*x^2 + ( (I*f*x*(-Pi + 2*ArcTan[Tan[e]]) - Pi*Log[1 + E^((-2*I)*f*x)] - 2*(f*x + Ar cTan[Tan[e]])*Log[1 - E^((2*I)*(f*x + ArcTan[Tan[e]]))] + Pi*Log[Cos[f*x]] + 2*ArcTan[Tan[e]]*Log[Sin[f*x + ArcTan[Tan[e]]]] + I*PolyLog[2, E^((2...
Time = 0.65 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3042, 4205, 2009}
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.
\(\displaystyle \int (c+d x)^2 (a+b \cot (e+f x))^2 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (c+d x)^2 \left (a-b \tan \left (e+f x+\frac {\pi }{2}\right )\right )^2dx\) |
\(\Big \downarrow \) 4205 |
\(\displaystyle \int \left (a^2 (c+d x)^2+2 a b (c+d x)^2 \cot (e+f x)+b^2 (c+d x)^2 \cot ^2(e+f x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^2 (c+d x)^3}{3 d}-\frac {2 i a b d (c+d x) \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^2}+\frac {2 a b (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac {2 i a b (c+d x)^3}{3 d}+\frac {a b d^2 \operatorname {PolyLog}\left (3,e^{2 i (e+f x)}\right )}{f^3}+\frac {2 b^2 d (c+d x) \log \left (1-e^{2 i (e+f x)}\right )}{f^2}-\frac {b^2 (c+d x)^2 \cot (e+f x)}{f}-\frac {i b^2 (c+d x)^2}{f}-\frac {b^2 (c+d x)^3}{3 d}-\frac {i b^2 d^2 \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^3}\) |
Input:
Int[(c + d*x)^2*(a + b*Cot[e + f*x])^2,x]
Output:
((-I)*b^2*(c + d*x)^2)/f + (a^2*(c + d*x)^3)/(3*d) - (((2*I)/3)*a*b*(c + d *x)^3)/d - (b^2*(c + d*x)^3)/(3*d) - (b^2*(c + d*x)^2*Cot[e + f*x])/f + (2 *b^2*d*(c + d*x)*Log[1 - E^((2*I)*(e + f*x))])/f^2 + (2*a*b*(c + d*x)^2*Lo g[1 - E^((2*I)*(e + f*x))])/f - (I*b^2*d^2*PolyLog[2, E^((2*I)*(e + f*x))] )/f^3 - ((2*I)*a*b*d*(c + d*x)*PolyLog[2, E^((2*I)*(e + f*x))])/f^2 + (a*b *d^2*PolyLog[3, E^((2*I)*(e + f*x))])/f^3
Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.) , x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, (a + b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[m, 0] && IGtQ[n, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 931 vs. \(2 (207 ) = 414\).
Time = 1.07 (sec) , antiderivative size = 932, normalized size of antiderivative = 4.11
Input:
int((d*x+c)^2*(a+b*cot(f*x+e))^2,x,method=_RETURNVERBOSE)
Output:
-8*I/f*b*d*a*c*e*x-4/f*b*a*c^2*ln(exp(I*(f*x+e)))+2/f*b*a*c^2*ln(exp(I*(f* x+e))+1)+2/f^2*b^2*c*d*ln(exp(I*(f*x+e))-1)-4/f^2*b^2*c*d*ln(exp(I*(f*x+e) ))+2/f^2*b^2*c*d*ln(exp(I*(f*x+e))+1)-2*I/f^3*b^2*d^2*polylog(2,-exp(I*(f* x+e)))-2*I/f*b^2*d^2*x^2-2*I/f^3*b^2*d^2*e^2-2*I/f^3*b^2*d^2*polylog(2,exp (I*(f*x+e)))-2/3*I*d^2*a*b*x^3-2/f^3*b^2*e*d^2*ln(exp(I*(f*x+e))-1)+4/f^3* b^2*e*d^2*ln(exp(I*(f*x+e)))+2/f^3*b^2*d^2*ln(1-exp(I*(f*x+e)))*e+2/f^2*b^ 2*d^2*ln(exp(I*(f*x+e))+1)*x+2/f^2*b^2*d^2*ln(1-exp(I*(f*x+e)))*x+4/f^3*b* a*d^2*polylog(3,exp(I*(f*x+e)))+4/f^3*b*a*d^2*polylog(3,-exp(I*(f*x+e)))+2 /f*b*a*c^2*ln(exp(I*(f*x+e))-1)+d*a^2*c*x^2+a^2*c^2*x-d*b^2*c*x^2-2*I*b^2* (d^2*x^2+2*c*d*x+c^2)/f/(exp(2*I*(f*x+e))-1)+2*I*a*b*c^2*x+2/3*I/d*a*b*c^3 -4/f^2*b*e*a*c*d*ln(exp(I*(f*x+e))-1)+8/f^2*b*e*a*c*d*ln(exp(I*(f*x+e)))+4 /f^2*b*d*a*c*ln(1-exp(I*(f*x+e)))*e+4/f*b*d*a*c*ln(1-exp(I*(f*x+e)))*x+4/f *b*d*a*c*ln(exp(I*(f*x+e))+1)*x-4*I/f^2*b*d*a*c*e^2-4*I/f^2*b*d*a*c*polylo g(2,exp(I*(f*x+e)))-4*I/f^2*b*d*a*c*polylog(2,-exp(I*(f*x+e)))+4*I/f^2*b*a *d^2*e^2*x-4*I/f^2*b*a*d^2*polylog(2,exp(I*(f*x+e)))*x-4*I/f^2*b*a*d^2*pol ylog(2,-exp(I*(f*x+e)))*x-1/3*d^2*b^2*x^3-b^2*c^2*x-1/3/d*b^2*c^3+1/3*d^2* a^2*x^3+1/3/d*a^2*c^3-2*I*d*a*b*c*x^2+2/f*b*a*d^2*ln(1-exp(I*(f*x+e)))*x^2 -2/f^3*b*a*d^2*ln(1-exp(I*(f*x+e)))*e^2+2/f*b*a*d^2*ln(exp(I*(f*x+e))+1)*x ^2+2/f^3*b*e^2*a*d^2*ln(exp(I*(f*x+e))-1)-4/f^3*b*e^2*a*d^2*ln(exp(I*(f*x+ e)))+8/3*I/f^3*b*a*d^2*e^3-4*I/f^2*b^2*d^2*e*x
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 715 vs. \(2 (201) = 402\).
Time = 0.12 (sec) , antiderivative size = 715, normalized size of antiderivative = 3.15 \[ \int (c+d x)^2 (a+b \cot (e+f x))^2 \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)^2*(a+b*cot(f*x+e))^2,x, algorithm="fricas")
Output:
-1/6*(6*b^2*d^2*f^2*x^2 + 12*b^2*c*d*f^2*x + 6*b^2*c^2*f^2 - 3*a*b*d^2*pol ylog(3, cos(2*f*x + 2*e) + I*sin(2*f*x + 2*e))*sin(2*f*x + 2*e) - 3*a*b*d^ 2*polylog(3, cos(2*f*x + 2*e) - I*sin(2*f*x + 2*e))*sin(2*f*x + 2*e) + 3*( 2*I*a*b*d^2*f*x + 2*I*a*b*c*d*f + I*b^2*d^2)*dilog(cos(2*f*x + 2*e) + I*si n(2*f*x + 2*e))*sin(2*f*x + 2*e) + 3*(-2*I*a*b*d^2*f*x - 2*I*a*b*c*d*f - I *b^2*d^2)*dilog(cos(2*f*x + 2*e) - I*sin(2*f*x + 2*e))*sin(2*f*x + 2*e) - 6*(a*b*d^2*e^2 + a*b*c^2*f^2 - b^2*d^2*e - (2*a*b*c*d*e - b^2*c*d)*f)*log( -1/2*cos(2*f*x + 2*e) + 1/2*I*sin(2*f*x + 2*e) + 1/2)*sin(2*f*x + 2*e) - 6 *(a*b*d^2*e^2 + a*b*c^2*f^2 - b^2*d^2*e - (2*a*b*c*d*e - b^2*c*d)*f)*log(- 1/2*cos(2*f*x + 2*e) - 1/2*I*sin(2*f*x + 2*e) + 1/2)*sin(2*f*x + 2*e) - 6* (a*b*d^2*f^2*x^2 - a*b*d^2*e^2 + 2*a*b*c*d*e*f + b^2*d^2*e + (2*a*b*c*d*f^ 2 + b^2*d^2*f)*x)*log(-cos(2*f*x + 2*e) + I*sin(2*f*x + 2*e) + 1)*sin(2*f* x + 2*e) - 6*(a*b*d^2*f^2*x^2 - a*b*d^2*e^2 + 2*a*b*c*d*e*f + b^2*d^2*e + (2*a*b*c*d*f^2 + b^2*d^2*f)*x)*log(-cos(2*f*x + 2*e) - I*sin(2*f*x + 2*e) + 1)*sin(2*f*x + 2*e) + 6*(b^2*d^2*f^2*x^2 + 2*b^2*c*d*f^2*x + b^2*c^2*f^2 )*cos(2*f*x + 2*e) - 2*((a^2 - b^2)*d^2*f^3*x^3 + 3*(a^2 - b^2)*c*d*f^3*x^ 2 + 3*(a^2 - b^2)*c^2*f^3*x)*sin(2*f*x + 2*e))/(f^3*sin(2*f*x + 2*e))
\[ \int (c+d x)^2 (a+b \cot (e+f x))^2 \, dx=\int \left (a + b \cot {\left (e + f x \right )}\right )^{2} \left (c + d x\right )^{2}\, dx \] Input:
integrate((d*x+c)**2*(a+b*cot(f*x+e))**2,x)
Output:
Integral((a + b*cot(e + f*x))**2*(c + d*x)**2, x)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1948 vs. \(2 (201) = 402\).
Time = 0.33 (sec) , antiderivative size = 1948, normalized size of antiderivative = 8.58 \[ \int (c+d x)^2 (a+b \cot (e+f x))^2 \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)^2*(a+b*cot(f*x+e))^2,x, algorithm="maxima")
Output:
1/3*(3*(f*x + e)*a^2*c^2 + (f*x + e)^3*a^2*d^2/f^2 - 3*(f*x + e)^2*a^2*d^2 *e/f^2 + 3*(f*x + e)*a^2*d^2*e^2/f^2 + 3*(f*x + e)^2*a^2*c*d/f - 6*(f*x + e)*a^2*c*d*e/f + 6*a*b*c^2*log(sin(f*x + e)) + 6*a*b*d^2*e^2*log(sin(f*x + e))/f^2 - 12*a*b*c*d*e*log(sin(f*x + e))/f + 3*((2*a*b - I*b^2)*(f*x + e) ^3*d^2 - 6*b^2*d^2*e^2 + 12*b^2*c*d*e*f - 6*b^2*c^2*f^2 - 3*((2*a*b - I*b^ 2)*d^2*e - (2*a*b - I*b^2)*c*d*f)*(f*x + e)^2 - 3*(I*b^2*d^2*e^2 - 2*I*b^2 *c*d*e*f + I*b^2*c^2*f^2)*(f*x + e) - 6*((f*x + e)^2*a*b*d^2 - b^2*d^2*e + b^2*c*d*f - (2*a*b*d^2*e - 2*a*b*c*d*f - b^2*d^2)*(f*x + e) - ((f*x + e)^ 2*a*b*d^2 - b^2*d^2*e + b^2*c*d*f - (2*a*b*d^2*e - 2*a*b*c*d*f - b^2*d^2)* (f*x + e))*cos(2*f*x + 2*e) + (-I*(f*x + e)^2*a*b*d^2 + I*b^2*d^2*e - I*b^ 2*c*d*f + (2*I*a*b*d^2*e - 2*I*a*b*c*d*f - I*b^2*d^2)*(f*x + e))*sin(2*f*x + 2*e))*arctan2(sin(f*x + e), cos(f*x + e) + 1) + 6*(b^2*d^2*e - b^2*c*d* f - (b^2*d^2*e - b^2*c*d*f)*cos(2*f*x + 2*e) - (I*b^2*d^2*e - I*b^2*c*d*f) *sin(2*f*x + 2*e))*arctan2(sin(f*x + e), cos(f*x + e) - 1) + 6*((f*x + e)^ 2*a*b*d^2 - (2*a*b*d^2*e - 2*a*b*c*d*f - b^2*d^2)*(f*x + e) - ((f*x + e)^2 *a*b*d^2 - (2*a*b*d^2*e - 2*a*b*c*d*f - b^2*d^2)*(f*x + e))*cos(2*f*x + 2* e) - (I*(f*x + e)^2*a*b*d^2 + (-2*I*a*b*d^2*e + 2*I*a*b*c*d*f + I*b^2*d^2) *(f*x + e))*sin(2*f*x + 2*e))*arctan2(sin(f*x + e), -cos(f*x + e) + 1) - ( (2*a*b - I*b^2)*(f*x + e)^3*d^2 + 3*(2*b^2*d^2 - (2*a*b - I*b^2)*d^2*e + ( 2*a*b - I*b^2)*c*d*f)*(f*x + e)^2 + 3*(-I*b^2*d^2*e^2 - I*b^2*c^2*f^2 -...
\[ \int (c+d x)^2 (a+b \cot (e+f x))^2 \, dx=\int { {\left (d x + c\right )}^{2} {\left (b \cot \left (f x + e\right ) + a\right )}^{2} \,d x } \] Input:
integrate((d*x+c)^2*(a+b*cot(f*x+e))^2,x, algorithm="giac")
Output:
integrate((d*x + c)^2*(b*cot(f*x + e) + a)^2, x)
Timed out. \[ \int (c+d x)^2 (a+b \cot (e+f x))^2 \, dx=\int {\left (a+b\,\mathrm {cot}\left (e+f\,x\right )\right )}^2\,{\left (c+d\,x\right )}^2 \,d x \] Input:
int((a + b*cot(e + f*x))^2*(c + d*x)^2,x)
Output:
int((a + b*cot(e + f*x))^2*(c + d*x)^2, x)
\[ \int (c+d x)^2 (a+b \cot (e+f x))^2 \, dx=\frac {-6 \cos \left (f x +e \right ) b^{2} c d f x -3 \cot \left (f x +e \right ) \sin \left (f x +e \right ) b^{2} c^{2} f +6 \left (\int \cot \left (f x +e \right ) x^{2}d x \right ) \sin \left (f x +e \right ) a b \,d^{2} f^{2}+12 \left (\int \cot \left (f x +e \right ) x d x \right ) \sin \left (f x +e \right ) a b c d \,f^{2}+3 \left (\int \cot \left (f x +e \right )^{2} x^{2}d x \right ) \sin \left (f x +e \right ) b^{2} d^{2} f^{2}-6 \,\mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+1\right ) \sin \left (f x +e \right ) a b \,c^{2} f -6 \,\mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+1\right ) \sin \left (f x +e \right ) b^{2} c d +6 \,\mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \sin \left (f x +e \right ) a b \,c^{2} f +6 \,\mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \sin \left (f x +e \right ) b^{2} c d +3 \sin \left (f x +e \right ) a^{2} c^{2} f^{2} x +3 \sin \left (f x +e \right ) a^{2} c d \,f^{2} x^{2}+\sin \left (f x +e \right ) a^{2} d^{2} f^{2} x^{3}-3 \sin \left (f x +e \right ) b^{2} c^{2} f^{2} x -3 \sin \left (f x +e \right ) b^{2} c d \,f^{2} x^{2}}{3 \sin \left (f x +e \right ) f^{2}} \] Input:
int((d*x+c)^2*(a+b*cot(f*x+e))^2,x)
Output:
( - 6*cos(e + f*x)*b**2*c*d*f*x - 3*cot(e + f*x)*sin(e + f*x)*b**2*c**2*f + 6*int(cot(e + f*x)*x**2,x)*sin(e + f*x)*a*b*d**2*f**2 + 12*int(cot(e + f *x)*x,x)*sin(e + f*x)*a*b*c*d*f**2 + 3*int(cot(e + f*x)**2*x**2,x)*sin(e + f*x)*b**2*d**2*f**2 - 6*log(tan((e + f*x)/2)**2 + 1)*sin(e + f*x)*a*b*c** 2*f - 6*log(tan((e + f*x)/2)**2 + 1)*sin(e + f*x)*b**2*c*d + 6*log(tan((e + f*x)/2))*sin(e + f*x)*a*b*c**2*f + 6*log(tan((e + f*x)/2))*sin(e + f*x)* b**2*c*d + 3*sin(e + f*x)*a**2*c**2*f**2*x + 3*sin(e + f*x)*a**2*c*d*f**2* x**2 + sin(e + f*x)*a**2*d**2*f**2*x**3 - 3*sin(e + f*x)*b**2*c**2*f**2*x - 3*sin(e + f*x)*b**2*c*d*f**2*x**2)/(3*sin(e + f*x)*f**2)