Integrand size = 20, antiderivative size = 271 \[ \int (c+d x)^3 (a+b \coth (e+f x))^2 \, dx=-\frac {b^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}-\frac {a b (c+d x)^4}{2 d}+\frac {b^2 (c+d x)^4}{4 d}-\frac {b^2 (c+d x)^3 \coth (e+f x)}{f}+\frac {3 b^2 d (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f^2}+\frac {2 a b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {3 b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^3}+\frac {3 a b d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}-\frac {3 b^2 d^3 \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^4}-\frac {3 a b d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )}{f^3}+\frac {3 a b d^3 \operatorname {PolyLog}\left (4,e^{2 (e+f x)}\right )}{2 f^4} \] Output:
-b^2*(d*x+c)^3/f+1/4*a^2*(d*x+c)^4/d-1/2*a*b*(d*x+c)^4/d+1/4*b^2*(d*x+c)^4 /d-b^2*(d*x+c)^3*coth(f*x+e)/f+3*b^2*d*(d*x+c)^2*ln(1-exp(2*f*x+2*e))/f^2+ 2*a*b*(d*x+c)^3*ln(1-exp(2*f*x+2*e))/f+3*b^2*d^2*(d*x+c)*polylog(2,exp(2*f *x+2*e))/f^3+3*a*b*d*(d*x+c)^2*polylog(2,exp(2*f*x+2*e))/f^2-3/2*b^2*d^3*p olylog(3,exp(2*f*x+2*e))/f^4-3*a*b*d^2*(d*x+c)*polylog(3,exp(2*f*x+2*e))/f ^3+3/2*a*b*d^3*polylog(4,exp(2*f*x+2*e))/f^4
Leaf count is larger than twice the leaf count of optimal. \(843\) vs. \(2(271)=542\).
Time = 3.22 (sec) , antiderivative size = 843, normalized size of antiderivative = 3.11 \[ \int (c+d x)^3 (a+b \coth (e+f x))^2 \, dx =\text {Too large to display} \] Input:
Integrate[(c + d*x)^3*(a + b*Coth[e + f*x])^2,x]
Output:
(-2*b*c^2*(3*b*d + 2*a*c*f)*x)/f - (2*b^2*(c + d*x)^3)/((-1 + E^(2*e))*f) - (a*b*(c + d*x)^4)/(d*(-1 + E^(2*e))) + (6*b*c*d*(b*d + a*c*f)*x*Log[1 - E^(-e - f*x)])/f^2 + (3*b*d^2*(b*d + 2*a*c*f)*x^2*Log[1 - E^(-e - f*x)])/f ^2 + (2*a*b*d^3*x^3*Log[1 - E^(-e - f*x)])/f + (6*b*c*d*(b*d + a*c*f)*x*Lo g[1 + E^(-e - f*x)])/f^2 + (3*b*d^2*(b*d + 2*a*c*f)*x^2*Log[1 + E^(-e - f* x)])/f^2 + (2*a*b*d^3*x^3*Log[1 + E^(-e - f*x)])/f + (b*c^2*(3*b*d + 2*a*c *f)*Log[1 - E^(e + f*x)])/f^2 + (b*c^2*(3*b*d + 2*a*c*f)*Log[1 + E^(e + f* x)])/f^2 - (6*b*c*d*(b*d + a*c*f)*PolyLog[2, -E^(-e - f*x)])/f^3 - (6*b*d^ 2*(b*d + 2*a*c*f)*x*PolyLog[2, -E^(-e - f*x)])/f^3 - (6*a*b*d^3*x^2*PolyLo g[2, -E^(-e - f*x)])/f^2 - (6*b*c*d*(b*d + a*c*f)*PolyLog[2, E^(-e - f*x)] )/f^3 - (6*b*d^2*(b*d + 2*a*c*f)*x*PolyLog[2, E^(-e - f*x)])/f^3 - (6*a*b* d^3*x^2*PolyLog[2, E^(-e - f*x)])/f^2 - (6*b*d^2*(b*d + 2*a*c*f)*PolyLog[3 , -E^(-e - f*x)])/f^4 - (12*a*b*d^3*x*PolyLog[3, -E^(-e - f*x)])/f^3 - (6* b*d^2*(b*d + 2*a*c*f)*PolyLog[3, E^(-e - f*x)])/f^4 - (12*a*b*d^3*x*PolyLo g[3, E^(-e - f*x)])/f^3 - (12*a*b*d^3*PolyLog[4, -E^(-e - f*x)])/f^4 - (12 *a*b*d^3*PolyLog[4, E^(-e - f*x)])/f^4 + (Csch[e]*Csch[e + f*x]*(-((a^2 + b^2)*f*x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3)*Cosh[f*x]) + (a^2 + b ^2)*f*x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3)*Cosh[2*e + f*x] + 2*b* ((4*b*(c + d*x)^3 + a*f*x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3))*Sin h[f*x] + a*f*x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3)*Sinh[2*e + f...
Time = 0.82 (sec) , antiderivative size = 271, 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.
\(\displaystyle \int (c+d x)^3 (a+b \coth (e+f x))^2 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (c+d x)^3 \left (a-i b \tan \left (i e+i f x+\frac {\pi }{2}\right )\right )^2dx\) |
\(\Big \downarrow \) 4205 |
\(\displaystyle \int \left (a^2 (c+d x)^3+2 a b (c+d x)^3 \coth (e+f x)+b^2 (c+d x)^3 \coth ^2(e+f x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^2 (c+d x)^4}{4 d}-\frac {3 a b d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )}{f^3}+\frac {3 a b d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}+\frac {2 a b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac {a b (c+d x)^4}{2 d}+\frac {3 a b d^3 \operatorname {PolyLog}\left (4,e^{2 (e+f x)}\right )}{2 f^4}+\frac {3 b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^3}+\frac {3 b^2 d (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f^2}-\frac {b^2 (c+d x)^3 \coth (e+f x)}{f}-\frac {b^2 (c+d x)^3}{f}+\frac {b^2 (c+d x)^4}{4 d}-\frac {3 b^2 d^3 \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^4}\) |
Input:
Int[(c + d*x)^3*(a + b*Coth[e + f*x])^2,x]
Output:
-((b^2*(c + d*x)^3)/f) + (a^2*(c + d*x)^4)/(4*d) - (a*b*(c + d*x)^4)/(2*d) + (b^2*(c + d*x)^4)/(4*d) - (b^2*(c + d*x)^3*Coth[e + f*x])/f + (3*b^2*d* (c + d*x)^2*Log[1 - E^(2*(e + f*x))])/f^2 + (2*a*b*(c + d*x)^3*Log[1 - E^( 2*(e + f*x))])/f + (3*b^2*d^2*(c + d*x)*PolyLog[2, E^(2*(e + f*x))])/f^3 + (3*a*b*d*(c + d*x)^2*PolyLog[2, E^(2*(e + f*x))])/f^2 - (3*b^2*d^3*PolyLo g[3, E^(2*(e + f*x))])/(2*f^4) - (3*a*b*d^2*(c + d*x)*PolyLog[3, E^(2*(e + f*x))])/f^3 + (3*a*b*d^3*PolyLog[4, E^(2*(e + f*x))])/(2*f^4)
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]
Leaf count of result is larger than twice the leaf count of optimal. \(1424\) vs. \(2(261)=522\).
Time = 0.31 (sec) , antiderivative size = 1425, normalized size of antiderivative = 5.26
Input:
int((d*x+c)^3*(a+b*coth(f*x+e))^2,x,method=_RETURNVERBOSE)
Output:
-1/2*d^3*a*b*x^4+d^2*a^2*c*x^3+d^2*b^2*c*x^3+3/2*d*a^2*c^2*x^2+3/2*d*b^2*c ^2*x^2+a^2*c^3*x+b^2*c^3*x+1/2/d*a*b*c^4-2/f*b^2*d^3*x^3+4/f^4*b^2*d^3*e^3 -6/f^4*b^2*d^3*polylog(3,exp(f*x+e))-6/f^4*b^2*d^3*polylog(3,-exp(f*x+e))+ 1/4*d^3*a^2*x^4+1/4*d^3*b^2*x^4+1/4/d*a^2*c^4+1/4/d*b^2*c^4+8/f^3*b*a*c*d^ 2*e^3-6/f^2*b*a*c^2*d*e^2-12/f^2*b^2*c*d^2*e*x-4/f^3*b*a*d^3*e^3*x-12/f^3* b*a*c*d^2*polylog(3,exp(f*x+e))-2*d^2*a*b*c*x^3-3*d*a*b*c^2*x^2+2*a*b*c^3* x+6/f^3*b^2*d^3*e^2*x-3/f^4*b*a*d^3*e^4-6/f*b^2*c*d^2*x^2-6/f^3*b^2*c*d^2* e^2+3/f^2*b^2*d^3*ln(1-exp(f*x+e))*x^2-3/f^4*b^2*d^3*ln(1-exp(f*x+e))*e^2+ 6/f^3*b^2*d^3*polylog(2,exp(f*x+e))*x+3/f^2*b^2*d^3*ln(exp(f*x+e)+1)*x^2+6 /f^3*b^2*d^3*polylog(2,-exp(f*x+e))*x+3/f^2*b^2*c^2*d*ln(exp(f*x+e)-1)-6/f ^2*b^2*c^2*d*ln(exp(f*x+e))+3/f^2*b^2*c^2*d*ln(exp(f*x+e)+1)+12/f^4*b*a*d^ 3*polylog(4,exp(f*x+e))+12/f^4*b*a*d^3*polylog(4,-exp(f*x+e))+2/f*b*a*c^3* ln(exp(f*x+e)-1)-4/f*b*a*c^3*ln(exp(f*x+e))+2/f*b*a*c^3*ln(exp(f*x+e)+1)+3 /f^4*b^2*e^2*d^3*ln(exp(f*x+e)-1)-6/f^4*b^2*e^2*d^3*ln(exp(f*x+e))+6/f^3*b ^2*c*d^2*polylog(2,exp(f*x+e))+6/f^3*b^2*c*d^2*polylog(2,-exp(f*x+e))-12/f ^3*b*a*c*d^2*polylog(3,-exp(f*x+e))+6/f^2*b*a*c^2*d*polylog(2,exp(f*x+e))+ 6/f^2*b*a*c^2*d*polylog(2,-exp(f*x+e))-6/f^3*b^2*e*c*d^2*ln(exp(f*x+e)-1)+ 12/f^3*b^2*e*c*d^2*ln(exp(f*x+e))+6/f^2*b^2*c*d^2*ln(1-exp(f*x+e))*x+6/f^3 *b^2*c*d^2*ln(1-exp(f*x+e))*e+6/f^2*b^2*c*d^2*ln(exp(f*x+e)+1)*x+2/f^4*b*a *d^3*ln(1-exp(f*x+e))*e^3+2/f*b*a*d^3*ln(1-exp(f*x+e))*x^3+6/f^2*b*a*d^...
Leaf count of result is larger than twice the leaf count of optimal. 3239 vs. \(2 (259) = 518\).
Time = 0.15 (sec) , antiderivative size = 3239, normalized size of antiderivative = 11.95 \[ \int (c+d x)^3 (a+b \coth (e+f x))^2 \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)^3*(a+b*coth(f*x+e))^2,x, algorithm="fricas")
Output:
Too large to include
\[ \int (c+d x)^3 (a+b \coth (e+f x))^2 \, dx=\int \left (a + b \coth {\left (e + f x \right )}\right )^{2} \left (c + d x\right )^{3}\, dx \] Input:
integrate((d*x+c)**3*(a+b*coth(f*x+e))**2,x)
Output:
Integral((a + b*coth(e + f*x))**2*(c + d*x)**3, x)
Leaf count of result is larger than twice the leaf count of optimal. 781 vs. \(2 (259) = 518\).
Time = 0.20 (sec) , antiderivative size = 781, normalized size of antiderivative = 2.88 \[ \int (c+d x)^3 (a+b \coth (e+f x))^2 \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)^3*(a+b*coth(f*x+e))^2,x, algorithm="maxima")
Output:
1/4*a^2*d^3*x^4 + a^2*c*d^2*x^3 + 3/2*a^2*c^2*d*x^2 + a^2*c^3*x - 6*b^2*c^ 2*d*x/f + 2*a*b*c^3*log(sinh(f*x + e))/f + 3*b^2*c^2*d*log(e^(f*x + e) + 1 )/f^2 + 3*b^2*c^2*d*log(e^(f*x + e) - 1)/f^2 + 2*(f^3*x^3*log(e^(f*x + e) + 1) + 3*f^2*x^2*dilog(-e^(f*x + e)) - 6*f*x*polylog(3, -e^(f*x + e)) + 6* polylog(4, -e^(f*x + e)))*a*b*d^3/f^4 + 2*(f^3*x^3*log(-e^(f*x + e) + 1) + 3*f^2*x^2*dilog(e^(f*x + e)) - 6*f*x*polylog(3, e^(f*x + e)) + 6*polylog( 4, e^(f*x + e)))*a*b*d^3/f^4 - 1/4*(8*b^2*c^3 + (2*a*b*d^3*f + b^2*d^3*f)* x^4 + 4*(c^3*f + 6*c^2*d)*b^2*x + 4*(2*a*b*c*d^2*f + (c*d^2*f + 2*d^3)*b^2 )*x^3 + 6*(2*a*b*c^2*d*f + (c^2*d*f + 4*c*d^2)*b^2)*x^2 - (4*b^2*c^3*f*x*e ^(2*e) + (2*a*b*d^3*f*e^(2*e) + b^2*d^3*f*e^(2*e))*x^4 + 4*(2*a*b*c*d^2*f* e^(2*e) + b^2*c*d^2*f*e^(2*e))*x^3 + 6*(2*a*b*c^2*d*f*e^(2*e) + b^2*c^2*d* f*e^(2*e))*x^2)*e^(2*f*x))/(f*e^(2*f*x + 2*e) - f) + 6*(a*b*c^2*d*f + b^2* c*d^2)*(f*x*log(e^(f*x + e) + 1) + dilog(-e^(f*x + e)))/f^3 + 6*(a*b*c^2*d *f + b^2*c*d^2)*(f*x*log(-e^(f*x + e) + 1) + dilog(e^(f*x + e)))/f^3 + 3*( 2*a*b*c*d^2*f + b^2*d^3)*(f^2*x^2*log(e^(f*x + e) + 1) + 2*f*x*dilog(-e^(f *x + e)) - 2*polylog(3, -e^(f*x + e)))/f^4 + 3*(2*a*b*c*d^2*f + b^2*d^3)*( f^2*x^2*log(-e^(f*x + e) + 1) + 2*f*x*dilog(e^(f*x + e)) - 2*polylog(3, e^ (f*x + e)))/f^4 - (a*b*d^3*f^4*x^4 + 2*(2*a*b*c*d^2*f + b^2*d^3)*f^3*x^3 + 6*(a*b*c^2*d*f^2 + b^2*c*d^2*f)*f^2*x^2)/f^4
\[ \int (c+d x)^3 (a+b \coth (e+f x))^2 \, dx=\int { {\left (d x + c\right )}^{3} {\left (b \coth \left (f x + e\right ) + a\right )}^{2} \,d x } \] Input:
integrate((d*x+c)^3*(a+b*coth(f*x+e))^2,x, algorithm="giac")
Output:
integrate((d*x + c)^3*(b*coth(f*x + e) + a)^2, x)
Timed out. \[ \int (c+d x)^3 (a+b \coth (e+f x))^2 \, dx=\int {\left (a+b\,\mathrm {coth}\left (e+f\,x\right )\right )}^2\,{\left (c+d\,x\right )}^3 \,d x \] Input:
int((a + b*coth(e + f*x))^2*(c + d*x)^3,x)
Output:
int((a + b*coth(e + f*x))^2*(c + d*x)^3, x)
\[ \int (c+d x)^3 (a+b \coth (e+f x))^2 \, dx=\text {too large to display} \] Input:
int((d*x+c)^3*(a+b*coth(f*x+e))^2,x)
Output:
( - 16*e**(2*e + 2*f*x)*int(x**3/(e**(4*e + 4*f*x) - 2*e**(2*e + 2*f*x) + 1),x)*a*b*d**3*f**4 - 48*e**(2*e + 2*f*x)*int(x**2/(e**(4*e + 4*f*x) - 2*e **(2*e + 2*f*x) + 1),x)*a*b*c*d**2*f**4 - 24*e**(2*e + 2*f*x)*int(x**2/(e* *(4*e + 4*f*x) - 2*e**(2*e + 2*f*x) + 1),x)*a*b*d**3*f**3 - 24*e**(2*e + 2 *f*x)*int(x**2/(e**(4*e + 4*f*x) - 2*e**(2*e + 2*f*x) + 1),x)*b**2*d**3*f* *3 - 48*e**(2*e + 2*f*x)*int(x/(e**(4*e + 4*f*x) - 2*e**(2*e + 2*f*x) + 1) ,x)*a*b*c**2*d*f**4 - 48*e**(2*e + 2*f*x)*int(x/(e**(4*e + 4*f*x) - 2*e**( 2*e + 2*f*x) + 1),x)*a*b*c*d**2*f**3 - 24*e**(2*e + 2*f*x)*int(x/(e**(4*e + 4*f*x) - 2*e**(2*e + 2*f*x) + 1),x)*a*b*d**3*f**2 - 48*e**(2*e + 2*f*x)* int(x/(e**(4*e + 4*f*x) - 2*e**(2*e + 2*f*x) + 1),x)*b**2*c*d**2*f**3 - 24 *e**(2*e + 2*f*x)*int(x/(e**(4*e + 4*f*x) - 2*e**(2*e + 2*f*x) + 1),x)*b** 2*d**3*f**2 + 8*e**(2*e + 2*f*x)*log(e**(e + f*x) - 1)*a*b*c**3*f**3 + 12* e**(2*e + 2*f*x)*log(e**(e + f*x) - 1)*a*b*c**2*d*f**2 + 12*e**(2*e + 2*f* x)*log(e**(e + f*x) - 1)*a*b*c*d**2*f + 6*e**(2*e + 2*f*x)*log(e**(e + f*x ) - 1)*a*b*d**3 + 12*e**(2*e + 2*f*x)*log(e**(e + f*x) - 1)*b**2*c**2*d*f* *2 + 12*e**(2*e + 2*f*x)*log(e**(e + f*x) - 1)*b**2*c*d**2*f + 6*e**(2*e + 2*f*x)*log(e**(e + f*x) - 1)*b**2*d**3 + 8*e**(2*e + 2*f*x)*log(e**(e + f *x) + 1)*a*b*c**3*f**3 + 12*e**(2*e + 2*f*x)*log(e**(e + f*x) + 1)*a*b*c** 2*d*f**2 + 12*e**(2*e + 2*f*x)*log(e**(e + f*x) + 1)*a*b*c*d**2*f + 6*e**( 2*e + 2*f*x)*log(e**(e + f*x) + 1)*a*b*d**3 + 12*e**(2*e + 2*f*x)*log(e...