Integrand size = 18, antiderivative size = 133 \[ \int (c+d x)^3 (a+b \coth (e+f x)) \, dx=\frac {a (c+d x)^4}{4 d}-\frac {b (c+d x)^4}{4 d}+\frac {b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{2 f^2}-\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^3}+\frac {3 b d^3 \operatorname {PolyLog}\left (4,e^{2 (e+f x)}\right )}{4 f^4} \] Output:
1/4*a*(d*x+c)^4/d-1/4*b*(d*x+c)^4/d+b*(d*x+c)^3*ln(1-exp(2*f*x+2*e))/f+3/2 *b*d*(d*x+c)^2*polylog(2,exp(2*f*x+2*e))/f^2-3/2*b*d^2*(d*x+c)*polylog(3,e xp(2*f*x+2*e))/f^3+3/4*b*d^3*polylog(4,exp(2*f*x+2*e))/f^4
Leaf count is larger than twice the leaf count of optimal. \(311\) vs. \(2(133)=266\).
Time = 0.14 (sec) , antiderivative size = 311, normalized size of antiderivative = 2.34 \[ \int (c+d x)^3 (a+b \coth (e+f x)) \, dx=a c^3 x+\frac {3}{2} a c^2 d x^2-\frac {3}{2} b c^2 d x^2+a c d^2 x^3-b c d^2 x^3+\frac {1}{4} a d^3 x^4-\frac {1}{4} b d^3 x^4+\frac {3 b c^2 d x \log \left (1-e^{2 e+2 f x}\right )}{f}+\frac {3 b c d^2 x^2 \log \left (1-e^{2 e+2 f x}\right )}{f}+\frac {b d^3 x^3 \log \left (1-e^{2 e+2 f x}\right )}{f}+\frac {b c^3 \log (\sinh (e+f x))}{f}+\frac {3 b c^2 d \operatorname {PolyLog}\left (2,e^{2 e+2 f x}\right )}{2 f^2}+\frac {3 b c d^2 x \operatorname {PolyLog}\left (2,e^{2 e+2 f x}\right )}{f^2}+\frac {3 b d^3 x^2 \operatorname {PolyLog}\left (2,e^{2 e+2 f x}\right )}{2 f^2}-\frac {3 b c d^2 \operatorname {PolyLog}\left (3,e^{2 e+2 f x}\right )}{2 f^3}-\frac {3 b d^3 x \operatorname {PolyLog}\left (3,e^{2 e+2 f x}\right )}{2 f^3}+\frac {3 b d^3 \operatorname {PolyLog}\left (4,e^{2 e+2 f x}\right )}{4 f^4} \] Input:
Integrate[(c + d*x)^3*(a + b*Coth[e + f*x]),x]
Output:
a*c^3*x + (3*a*c^2*d*x^2)/2 - (3*b*c^2*d*x^2)/2 + a*c*d^2*x^3 - b*c*d^2*x^ 3 + (a*d^3*x^4)/4 - (b*d^3*x^4)/4 + (3*b*c^2*d*x*Log[1 - E^(2*e + 2*f*x)]) /f + (3*b*c*d^2*x^2*Log[1 - E^(2*e + 2*f*x)])/f + (b*d^3*x^3*Log[1 - E^(2* e + 2*f*x)])/f + (b*c^3*Log[Sinh[e + f*x]])/f + (3*b*c^2*d*PolyLog[2, E^(2 *e + 2*f*x)])/(2*f^2) + (3*b*c*d^2*x*PolyLog[2, E^(2*e + 2*f*x)])/f^2 + (3 *b*d^3*x^2*PolyLog[2, E^(2*e + 2*f*x)])/(2*f^2) - (3*b*c*d^2*PolyLog[3, E^ (2*e + 2*f*x)])/(2*f^3) - (3*b*d^3*x*PolyLog[3, E^(2*e + 2*f*x)])/(2*f^3) + (3*b*d^3*PolyLog[4, E^(2*e + 2*f*x)])/(4*f^4)
Time = 0.52 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, 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.
| \(\displaystyle \int (c+d x)^3 (a+b \coth (e+f x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (c+d x)^3 \left (a-i b \tan \left (i e+i f x+\frac {\pi }{2}\right )\right )dx\) |
| \(\Big \downarrow \) 4205 |
| \(\displaystyle \int \left (a (c+d x)^3+b (c+d x)^3 \coth (e+f x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a (c+d x)^4}{4 d}-\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^3}+\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{2 f^2}+\frac {b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac {b (c+d x)^4}{4 d}+\frac {3 b d^3 \operatorname {PolyLog}\left (4,e^{2 (e+f x)}\right )}{4 f^4}\) |
Input:
Int[(c + d*x)^3*(a + b*Coth[e + f*x]),x]
Output:
(a*(c + d*x)^4)/(4*d) - (b*(c + d*x)^4)/(4*d) + (b*(c + d*x)^3*Log[1 - E^( 2*(e + f*x))])/f + (3*b*d*(c + d*x)^2*PolyLog[2, E^(2*(e + f*x))])/(2*f^2) - (3*b*d^2*(c + d*x)*PolyLog[3, E^(2*(e + f*x))])/(2*f^3) + (3*b*d^3*Poly Log[4, E^(2*(e + f*x))])/(4*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. \(765\) vs. \(2(123)=246\).
Time = 0.17 (sec) , antiderivative size = 766, normalized size of antiderivative = 5.76
| method | result | size |
| risch | \(\frac {d^{3} a \,x^{4}}{4}-\frac {d^{3} b \,x^{4}}{4}+\frac {a \,c^{4}}{4 d}+\frac {b \,c^{4}}{4 d}-\frac {6 b d \,c^{2} e x}{f}+\frac {6 b \,d^{2} c \,e^{2} x}{f^{2}}-\frac {3 b \,c^{2} d e \ln \left ({\mathrm e}^{f x +e}-1\right )}{f^{2}}+\frac {6 b \,c^{2} d e \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}+\frac {3 b d \,c^{2} \ln \left (1-{\mathrm e}^{f x +e}\right ) x}{f}+\frac {3 b d \,c^{2} \ln \left (1-{\mathrm e}^{f x +e}\right ) e}{f^{2}}+\frac {3 b d \,c^{2} \ln \left ({\mathrm e}^{f x +e}+1\right ) x}{f}+\frac {6 b \,d^{2} c \operatorname {polylog}\left (2, {\mathrm e}^{f x +e}\right ) x}{f^{2}}+\frac {3 b \,d^{2} c \ln \left ({\mathrm e}^{f x +e}+1\right ) x^{2}}{f}+\frac {6 b \,d^{2} c \operatorname {polylog}\left (2, -{\mathrm e}^{f x +e}\right ) x}{f^{2}}+\frac {3 b \,d^{2} c \ln \left (1-{\mathrm e}^{f x +e}\right ) x^{2}}{f}-\frac {3 b \,d^{2} c \ln \left (1-{\mathrm e}^{f x +e}\right ) e^{2}}{f^{3}}+\frac {3 b c \,d^{2} e^{2} \ln \left ({\mathrm e}^{f x +e}-1\right )}{f^{3}}-\frac {6 b c \,d^{2} e^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{f^{3}}-\frac {2 b \,d^{3} e^{3} x}{f^{3}}-\frac {3 b d \,c^{2} e^{2}}{f^{2}}+\frac {4 b \,d^{2} c \,e^{3}}{f^{3}}+\frac {b \,d^{3} \ln \left (1-{\mathrm e}^{f x +e}\right ) e^{3}}{f^{4}}+\frac {3 b \,d^{3} \operatorname {polylog}\left (2, -{\mathrm e}^{f x +e}\right ) x^{2}}{f^{2}}-\frac {6 b \,d^{3} \operatorname {polylog}\left (3, -{\mathrm e}^{f x +e}\right ) x}{f^{3}}+\frac {3 b d \,c^{2} \operatorname {polylog}\left (2, {\mathrm e}^{f x +e}\right )}{f^{2}}+\frac {3 b d \,c^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{f x +e}\right )}{f^{2}}-\frac {6 b \,d^{2} c \operatorname {polylog}\left (3, {\mathrm e}^{f x +e}\right )}{f^{3}}-\frac {6 b \,d^{2} c \operatorname {polylog}\left (3, -{\mathrm e}^{f x +e}\right )}{f^{3}}+\frac {b \,d^{3} \ln \left (1-{\mathrm e}^{f x +e}\right ) x^{3}}{f}+\frac {3 b \,d^{3} \operatorname {polylog}\left (2, {\mathrm e}^{f x +e}\right ) x^{2}}{f^{2}}-\frac {6 b \,d^{3} \operatorname {polylog}\left (3, {\mathrm e}^{f x +e}\right ) x}{f^{3}}+\frac {b \,d^{3} \ln \left ({\mathrm e}^{f x +e}+1\right ) x^{3}}{f}-\frac {b \,d^{3} e^{3} \ln \left ({\mathrm e}^{f x +e}-1\right )}{f^{4}}+\frac {2 b \,d^{3} e^{3} \ln \left ({\mathrm e}^{f x +e}\right )}{f^{4}}+d^{2} a c \,x^{3}-d^{2} b c \,x^{3}+\frac {3 d a \,c^{2} x^{2}}{2}-\frac {3 d b \,c^{2} x^{2}}{2}+a \,c^{3} x +b \,c^{3} x -\frac {3 b \,d^{3} e^{4}}{2 f^{4}}+\frac {b \,c^{3} \ln \left ({\mathrm e}^{f x +e}-1\right )}{f}-\frac {2 b \,c^{3} \ln \left ({\mathrm e}^{f x +e}\right )}{f}+\frac {b \,c^{3} \ln \left ({\mathrm e}^{f x +e}+1\right )}{f}+\frac {6 b \,d^{3} \operatorname {polylog}\left (4, {\mathrm e}^{f x +e}\right )}{f^{4}}+\frac {6 b \,d^{3} \operatorname {polylog}\left (4, -{\mathrm e}^{f x +e}\right )}{f^{4}}\) | \(766\) |
Input:
int((d*x+c)^3*(a+b*coth(f*x+e)),x,method=_RETURNVERBOSE)
Output:
1/4*d^3*a*x^4-1/4*d^3*b*x^4+1/4/d*a*c^4+1/4/d*b*c^4-6/f*b*d*c^2*e*x+6/f^2* b*d^2*c*e^2*x-3/f^2*b*c^2*d*e*ln(exp(f*x+e)-1)+6/f^2*b*c^2*d*e*ln(exp(f*x+ e))+3/f*b*d*c^2*ln(1-exp(f*x+e))*x+3/f^2*b*d*c^2*ln(1-exp(f*x+e))*e+3/f*b* d*c^2*ln(exp(f*x+e)+1)*x+6/f^2*b*d^2*c*polylog(2,exp(f*x+e))*x+3/f*b*d^2*c *ln(exp(f*x+e)+1)*x^2+6/f^2*b*d^2*c*polylog(2,-exp(f*x+e))*x+3/f*b*d^2*c*l n(1-exp(f*x+e))*x^2-3/f^3*b*d^2*c*ln(1-exp(f*x+e))*e^2+3/f^3*b*c*d^2*e^2*l n(exp(f*x+e)-1)-6/f^3*b*c*d^2*e^2*ln(exp(f*x+e))-2/f^3*b*d^3*e^3*x-3/f^2*b *d*c^2*e^2+4/f^3*b*d^2*c*e^3+1/f^4*b*d^3*ln(1-exp(f*x+e))*e^3+3/f^2*b*d^3* polylog(2,-exp(f*x+e))*x^2-6/f^3*b*d^3*polylog(3,-exp(f*x+e))*x+3/f^2*b*d* c^2*polylog(2,exp(f*x+e))+3/f^2*b*d*c^2*polylog(2,-exp(f*x+e))-6/f^3*b*d^2 *c*polylog(3,exp(f*x+e))-6/f^3*b*d^2*c*polylog(3,-exp(f*x+e))+1/f*b*d^3*ln (1-exp(f*x+e))*x^3+3/f^2*b*d^3*polylog(2,exp(f*x+e))*x^2-6/f^3*b*d^3*polyl og(3,exp(f*x+e))*x+1/f*b*d^3*ln(exp(f*x+e)+1)*x^3-1/f^4*b*d^3*e^3*ln(exp(f *x+e)-1)+2/f^4*b*d^3*e^3*ln(exp(f*x+e))+d^2*a*c*x^3-d^2*b*c*x^3+3/2*d*a*c^ 2*x^2-3/2*d*b*c^2*x^2+a*c^3*x+b*c^3*x-3/2/f^4*b*d^3*e^4+1/f*b*c^3*ln(exp(f *x+e)-1)-2/f*b*c^3*ln(exp(f*x+e))+1/f*b*c^3*ln(exp(f*x+e)+1)+6/f^4*b*d^3*p olylog(4,exp(f*x+e))+6/f^4*b*d^3*polylog(4,-exp(f*x+e))
Leaf count of result is larger than twice the leaf count of optimal. 488 vs. \(2 (122) = 244\).
Time = 0.13 (sec) , antiderivative size = 488, normalized size of antiderivative = 3.67 \[ \int (c+d x)^3 (a+b \coth (e+f x)) \, dx=\frac {{\left (a - b\right )} d^{3} f^{4} x^{4} + 4 \, {\left (a - b\right )} c d^{2} f^{4} x^{3} + 6 \, {\left (a - b\right )} c^{2} d f^{4} x^{2} + 4 \, {\left (a - b\right )} c^{3} f^{4} x + 24 \, b d^{3} {\rm polylog}\left (4, \cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right ) + 24 \, b d^{3} {\rm polylog}\left (4, -\cosh \left (f x + e\right ) - \sinh \left (f x + e\right )\right ) + 12 \, {\left (b d^{3} f^{2} x^{2} + 2 \, b c d^{2} f^{2} x + b c^{2} d f^{2}\right )} {\rm Li}_2\left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right ) + 12 \, {\left (b d^{3} f^{2} x^{2} + 2 \, b c d^{2} f^{2} x + b c^{2} d f^{2}\right )} {\rm Li}_2\left (-\cosh \left (f x + e\right ) - \sinh \left (f x + e\right )\right ) + 4 \, {\left (b d^{3} f^{3} x^{3} + 3 \, b c d^{2} f^{3} x^{2} + 3 \, b c^{2} d f^{3} x + b c^{3} f^{3}\right )} \log \left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right ) + 1\right ) - 4 \, {\left (b d^{3} e^{3} - 3 \, b c d^{2} e^{2} f + 3 \, b c^{2} d e f^{2} - b c^{3} f^{3}\right )} \log \left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right ) - 1\right ) + 4 \, {\left (b d^{3} f^{3} x^{3} + 3 \, b c d^{2} f^{3} x^{2} + 3 \, b c^{2} d f^{3} x + b d^{3} e^{3} - 3 \, b c d^{2} e^{2} f + 3 \, b c^{2} d e f^{2}\right )} \log \left (-\cosh \left (f x + e\right ) - \sinh \left (f x + e\right ) + 1\right ) - 24 \, {\left (b d^{3} f x + b c d^{2} f\right )} {\rm polylog}\left (3, \cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right ) - 24 \, {\left (b d^{3} f x + b c d^{2} f\right )} {\rm polylog}\left (3, -\cosh \left (f x + e\right ) - \sinh \left (f x + e\right )\right )}{4 \, f^{4}} \] Input:
integrate((d*x+c)^3*(a+b*coth(f*x+e)),x, algorithm="fricas")
Output:
1/4*((a - b)*d^3*f^4*x^4 + 4*(a - b)*c*d^2*f^4*x^3 + 6*(a - b)*c^2*d*f^4*x ^2 + 4*(a - b)*c^3*f^4*x + 24*b*d^3*polylog(4, cosh(f*x + e) + sinh(f*x + e)) + 24*b*d^3*polylog(4, -cosh(f*x + e) - sinh(f*x + e)) + 12*(b*d^3*f^2* x^2 + 2*b*c*d^2*f^2*x + b*c^2*d*f^2)*dilog(cosh(f*x + e) + sinh(f*x + e)) + 12*(b*d^3*f^2*x^2 + 2*b*c*d^2*f^2*x + b*c^2*d*f^2)*dilog(-cosh(f*x + e) - sinh(f*x + e)) + 4*(b*d^3*f^3*x^3 + 3*b*c*d^2*f^3*x^2 + 3*b*c^2*d*f^3*x + b*c^3*f^3)*log(cosh(f*x + e) + sinh(f*x + e) + 1) - 4*(b*d^3*e^3 - 3*b*c *d^2*e^2*f + 3*b*c^2*d*e*f^2 - b*c^3*f^3)*log(cosh(f*x + e) + sinh(f*x + e ) - 1) + 4*(b*d^3*f^3*x^3 + 3*b*c*d^2*f^3*x^2 + 3*b*c^2*d*f^3*x + b*d^3*e^ 3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d*e*f^2)*log(-cosh(f*x + e) - sinh(f*x + e) + 1) - 24*(b*d^3*f*x + b*c*d^2*f)*polylog(3, cosh(f*x + e) + sinh(f*x + e) ) - 24*(b*d^3*f*x + b*c*d^2*f)*polylog(3, -cosh(f*x + e) - sinh(f*x + e))) /f^4
\[ \int (c+d x)^3 (a+b \coth (e+f x)) \, dx=\int \left (a + b \coth {\left (e + f x \right )}\right ) \left (c + d x\right )^{3}\, dx \] Input:
integrate((d*x+c)**3*(a+b*coth(f*x+e)),x)
Output:
Integral((a + b*coth(e + f*x))*(c + d*x)**3, x)
Leaf count of result is larger than twice the leaf count of optimal. 419 vs. \(2 (122) = 244\).
Time = 0.17 (sec) , antiderivative size = 419, normalized size of antiderivative = 3.15 \[ \int (c+d x)^3 (a+b \coth (e+f x)) \, dx=\frac {1}{4} \, a d^{3} x^{4} + \frac {1}{4} \, b d^{3} x^{4} + a c d^{2} x^{3} + b c d^{2} x^{3} + \frac {3}{2} \, a c^{2} d x^{2} + \frac {3}{2} \, b c^{2} d x^{2} + a c^{3} x + \frac {b c^{3} \log \left (\sinh \left (f x + e\right )\right )}{f} + \frac {3 \, {\left (f x \log \left (e^{\left (f x + e\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (f x + e\right )}\right )\right )} b c^{2} d}{f^{2}} + \frac {3 \, {\left (f x \log \left (-e^{\left (f x + e\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (f x + e\right )}\right )\right )} b c^{2} d}{f^{2}} + \frac {3 \, {\left (f^{2} x^{2} \log \left (e^{\left (f x + e\right )} + 1\right ) + 2 \, f x {\rm Li}_2\left (-e^{\left (f x + e\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (f x + e\right )})\right )} b c d^{2}}{f^{3}} + \frac {3 \, {\left (f^{2} x^{2} \log \left (-e^{\left (f x + e\right )} + 1\right ) + 2 \, f x {\rm Li}_2\left (e^{\left (f x + e\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (f x + e\right )})\right )} b c d^{2}}{f^{3}} + \frac {{\left (f^{3} x^{3} \log \left (e^{\left (f x + e\right )} + 1\right ) + 3 \, f^{2} x^{2} {\rm Li}_2\left (-e^{\left (f x + e\right )}\right ) - 6 \, f x {\rm Li}_{3}(-e^{\left (f x + e\right )}) + 6 \, {\rm Li}_{4}(-e^{\left (f x + e\right )})\right )} b d^{3}}{f^{4}} + \frac {{\left (f^{3} x^{3} \log \left (-e^{\left (f x + e\right )} + 1\right ) + 3 \, f^{2} x^{2} {\rm Li}_2\left (e^{\left (f x + e\right )}\right ) - 6 \, f x {\rm Li}_{3}(e^{\left (f x + e\right )}) + 6 \, {\rm Li}_{4}(e^{\left (f x + e\right )})\right )} b d^{3}}{f^{4}} - \frac {b d^{3} f^{4} x^{4} + 4 \, b c d^{2} f^{4} x^{3} + 6 \, b c^{2} d f^{4} x^{2}}{2 \, f^{4}} \] Input:
integrate((d*x+c)^3*(a+b*coth(f*x+e)),x, algorithm="maxima")
Output:
1/4*a*d^3*x^4 + 1/4*b*d^3*x^4 + a*c*d^2*x^3 + b*c*d^2*x^3 + 3/2*a*c^2*d*x^ 2 + 3/2*b*c^2*d*x^2 + a*c^3*x + b*c^3*log(sinh(f*x + e))/f + 3*(f*x*log(e^ (f*x + e) + 1) + dilog(-e^(f*x + e)))*b*c^2*d/f^2 + 3*(f*x*log(-e^(f*x + e ) + 1) + dilog(e^(f*x + e)))*b*c^2*d/f^2 + 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)))*b*c*d^2/f^3 + 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)))*b*c*d^2/f^3 + (f^3*x^3*log(e^(f*x + e) + 1) + 3*f^2*x^2*dilo g(-e^(f*x + e)) - 6*f*x*polylog(3, -e^(f*x + e)) + 6*polylog(4, -e^(f*x + e)))*b*d^3/f^4 + (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)))*b*d^3/f^ 4 - 1/2*(b*d^3*f^4*x^4 + 4*b*c*d^2*f^4*x^3 + 6*b*c^2*d*f^4*x^2)/f^4
\[ \int (c+d x)^3 (a+b \coth (e+f x)) \, dx=\int { {\left (d x + c\right )}^{3} {\left (b \coth \left (f x + e\right ) + a\right )} \,d x } \] Input:
integrate((d*x+c)^3*(a+b*coth(f*x+e)),x, algorithm="giac")
Output:
integrate((d*x + c)^3*(b*coth(f*x + e) + a), x)
Timed out. \[ \int (c+d x)^3 (a+b \coth (e+f x)) \, dx=\int \left (a+b\,\mathrm {coth}\left (e+f\,x\right )\right )\,{\left (c+d\,x\right )}^3 \,d x \] Input:
int((a + b*coth(e + f*x))*(c + d*x)^3,x)
Output:
int((a + b*coth(e + f*x))*(c + d*x)^3, x)
\[ \int (c+d x)^3 (a+b \coth (e+f x)) \, dx=\frac {8 e^{2 e} \left (\int \frac {e^{2 f x} x^{3}}{e^{2 f x +2 e}-1}d x \right ) b \,d^{3} f +24 e^{2 e} \left (\int \frac {e^{2 f x} x^{2}}{e^{2 f x +2 e}-1}d x \right ) b c \,d^{2} f +24 e^{2 e} \left (\int \frac {e^{2 f x} x}{e^{2 f x +2 e}-1}d x \right ) b \,c^{2} d f +4 \,\mathrm {log}\left (e^{f x +e}-1\right ) b \,c^{3}+4 \,\mathrm {log}\left (e^{f x +e}+1\right ) b \,c^{3}+4 a \,c^{3} f x +6 a \,c^{2} d f \,x^{2}+4 a c \,d^{2} f \,x^{3}+a \,d^{3} f \,x^{4}-4 b \,c^{3} f x -6 b \,c^{2} d f \,x^{2}-4 b c \,d^{2} f \,x^{3}-b \,d^{3} f \,x^{4}}{4 f} \] Input:
int((d*x+c)^3*(a+b*coth(f*x+e)),x)
Output:
(8*e**(2*e)*int((e**(2*f*x)*x**3)/(e**(2*e + 2*f*x) - 1),x)*b*d**3*f + 24* e**(2*e)*int((e**(2*f*x)*x**2)/(e**(2*e + 2*f*x) - 1),x)*b*c*d**2*f + 24*e **(2*e)*int((e**(2*f*x)*x)/(e**(2*e + 2*f*x) - 1),x)*b*c**2*d*f + 4*log(e* *(e + f*x) - 1)*b*c**3 + 4*log(e**(e + f*x) + 1)*b*c**3 + 4*a*c**3*f*x + 6 *a*c**2*d*f*x**2 + 4*a*c*d**2*f*x**3 + a*d**3*f*x**4 - 4*b*c**3*f*x - 6*b* c**2*d*f*x**2 - 4*b*c*d**2*f*x**3 - b*d**3*f*x**4)/(4*f)