\(\int (c+d x)^3 (a+b \coth (e+f x)) \, dx\) [37]

Optimal result

Integrand size = 18, antiderivative size = 133 \[ \int (c+d x)^3 (a+b \coth (e+f x)) \, dx=\frac {a (c+d x)^4}{4 d}-\frac {b (c+d x)^4}{4 d}+\frac {b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{2 f^2}-\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^3}+\frac {3 b d^3 \operatorname {PolyLog}\left (4,e^{2 (e+f x)}\right )}{4 f^4} \]

[Out]

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,exp(2*f*x+2*e))/f^3+3/4*b*d^3*polylog(4,exp(2*f*x+2*e))/f^4

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {3803, 3797, 2221, 2611, 6744, 2320, 6724} \[ \int (c+d x)^3 (a+b \coth (e+f x)) \, dx=\frac {a (c+d x)^4}{4 d}-\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^3}+\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{2 f^2}+\frac {b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac {b (c+d x)^4}{4 d}+\frac {3 b d^3 \operatorname {PolyLog}\left (4,e^{2 (e+f x)}\right )}{4 f^4} \]

[In]

Int[(c + d*x)^3*(a + b*Coth[e + f*x]),x]

[Out]

(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*Po
lyLog[4, E^(2*(e + f*x))])/(4*f^4)

Rule 2221

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]
- Dist[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]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

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] + Dist[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]

Rule 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((
c + d*x)^(m + 1)/(d*(m + 1))), x] + Dist[2*I, Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 3803

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]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> 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]

Rule 6744

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Dist[f*(m/(b*c*p*Log[F])), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[m,
0]

Rubi steps \begin{align*} \text {integral}& = \int \left (a (c+d x)^3+b (c+d x)^3 \coth (e+f x)\right ) \, dx \\ & = \frac {a (c+d x)^4}{4 d}+b \int (c+d x)^3 \coth (e+f x) \, dx \\ & = \frac {a (c+d x)^4}{4 d}-\frac {b (c+d x)^4}{4 d}-(2 b) \int \frac {e^{2 (e+f x)} (c+d x)^3}{1-e^{2 (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) \int (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right ) \, dx}{f} \\ & = \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 {\left (3 b d^2\right ) \int (c+d x) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right ) \, dx}{f^2} \\ & = \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 {\left (3 b d^3\right ) \int \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right ) \, dx}{2 f^3} \\ & = \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 {\left (3 b d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{4 f^4} \\ & = \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.87 \[ \int (c+d x)^3 (a+b \coth (e+f x)) \, dx=\frac {1}{4} \left (4 a c^3 x+6 a c^2 d x^2-6 b c^2 d x^2+4 a c d^2 x^3-4 b c d^2 x^3+a d^3 x^4-b d^3 x^4+\frac {12 b c^2 d x \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {12 b c d^2 x^2 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {4 b d^3 x^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {4 b c^3 \log (\cosh (e+f x))}{f}+\frac {4 b c^3 \log (\tanh (e+f x))}{f}+\frac {6 b d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}-\frac {6 b d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )}{f^3}+\frac {3 b d^3 \operatorname {PolyLog}\left (4,e^{2 (e+f x)}\right )}{f^4}\right ) \]

[In]

Integrate[(c + d*x)^3*(a + b*Coth[e + f*x]),x]

[Out]

(4*a*c^3*x + 6*a*c^2*d*x^2 - 6*b*c^2*d*x^2 + 4*a*c*d^2*x^3 - 4*b*c*d^2*x^3 + a*d^3*x^4 - b*d^3*x^4 + (12*b*c^2
*d*x*Log[1 - E^(2*(e + f*x))])/f + (12*b*c*d^2*x^2*Log[1 - E^(2*(e + f*x))])/f + (4*b*d^3*x^3*Log[1 - E^(2*(e
+ f*x))])/f + (4*b*c^3*Log[Cosh[e + f*x]])/f + (4*b*c^3*Log[Tanh[e + f*x]])/f + (6*b*d*(c + d*x)^2*PolyLog[2,
E^(2*(e + f*x))])/f^2 - (6*b*d^2*(c + d*x)*PolyLog[3, E^(2*(e + f*x))])/f^3 + (3*b*d^3*PolyLog[4, E^(2*(e + f*
x))])/f^4)/4

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(765\) vs. \(2(123)=246\).

Time = 0.42 (sec) , antiderivative size = 766, normalized size of antiderivative = 5.76

method	result	size
risch	\(-\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 {2 b \,d^{3} e^{3} \ln \left ({\mathrm e}^{f x +e}\right )}{f^{4}}+a \,d^{2} c \,x^{3}+\frac {3 a d \,c^{2} x^{2}}{2}+a \,c^{3} x +\frac {b \,c^{4}}{4 d}-\frac {d^{3} b \,x^{4}}{4}+\frac {a \,d^{3} x^{4}}{4}+\frac {a \,c^{4}}{4 d}+\frac {6 b \,d^{2} c \,e^{2} x}{f^{2}}-d^{2} b c \,x^{3}-\frac {3 d b \,c^{2} x^{2}}{2}+b \,c^{3} x -\frac {6 b d \,c^{2} e x}{f}-\frac {2 b \,c^{3} \ln \left ({\mathrm e}^{f x +e}\right )}{f}-\frac {3 b \,d^{3} e^{4}}{2 f^{4}}-\frac {6 b \,d^{2} c \,e^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{f^{3}}+\frac {6 b d \,c^{2} e \ln \left ({\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 {3 b d \,c^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{f x +e}\right )}{f^{2}}+\frac {b \,d^{3} \ln \left (1-{\mathrm e}^{f x +e}\right ) e^{3}}{f^{4}}+\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^{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 {6 b \,d^{3} \operatorname {polylog}\left (3, {\mathrm e}^{f x +e}\right ) x}{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} e^{3} \ln \left ({\mathrm e}^{f x +e}-1\right )}{f^{4}}+\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 (1+{\mathrm e}^{f x +e}\right ) x}{f}+\frac {3 b \,d^{2} c \,e^{2} \ln \left ({\mathrm e}^{f x +e}-1\right )}{f^{3}}+\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 {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 {6 b \,d^{2} c \operatorname {polylog}\left (2, -{\mathrm e}^{f x +e}\right ) x}{f^{2}}+\frac {b \,c^{3} \ln \left ({\mathrm e}^{f x +e}-1\right )}{f}+\frac {b \,c^{3} \ln \left (1+{\mathrm e}^{f x +e}\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}}-\frac {3 b d \,c^{2} e \ln \left ({\mathrm e}^{f x +e}-1\right )}{f^{2}}\)	\(766\)

[In]

int((d*x+c)^3*(a+b*coth(f*x+e)),x,method=_RETURNVERBOSE)

[Out]

-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+2/f^4*b*d^3*e^3*ln(exp(f*x+e))+a*d^2*c*x^3+3/2*a*d*c^2*
x^2+a*c^3*x+1/4/d*b*c^4-1/4*d^3*b*x^4+1/4*a*d^3*x^4+1/4*a/d*c^4+6/f^2*b*d^2*c*e^2*x-d^2*b*c*x^3-3/2*d*b*c^2*x^
2+b*c^3*x-6/f*b*d*c^2*e*x-2/f*b*c^3*ln(exp(f*x+e))-3/2/f^4*b*d^3*e^4-6/f^3*b*d^2*c*e^2*ln(exp(f*x+e))+6/f^2*b*
d*c^2*e*ln(exp(f*x+e))+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))+1/f^4*b*d^3*ln
(1-exp(f*x+e))*e^3+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^2*c*polylog(
3,exp(f*x+e))-6/f^3*b*d^2*c*polylog(3,-exp(f*x+e))-6/f^3*b*d^3*polylog(3,exp(f*x+e))*x+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*polylog(3,-exp(f*x+e))*x-1/f^4*b*d^3*e^3*ln(exp(f*x
+e)-1)+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(1+exp(f*x+e))*x+3/f^3*b*
d^2*c*e^2*ln(exp(f*x+e)-1)+3/f*b*d^2*c*ln(1-exp(f*x+e))*x^2-3/f^3*b*d^2*c*ln(1-exp(f*x+e))*e^2+6/f^2*b*d^2*c*p
olylog(2,exp(f*x+e))*x+3/f*b*d^2*c*ln(1+exp(f*x+e))*x^2+6/f^2*b*d^2*c*polylog(2,-exp(f*x+e))*x+1/f*b*c^3*ln(ex
p(f*x+e)-1)+1/f*b*c^3*ln(1+exp(f*x+e))+6/f^4*b*d^3*polylog(4,-exp(f*x+e))+6/f^4*b*d^3*polylog(4,exp(f*x+e))-3/
f^2*b*d*c^2*e*ln(exp(f*x+e)-1)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 488 vs. \(2 (122) = 244\).

Time = 0.26 (sec) , antiderivative size = 488, normalized size of antiderivative = 3.67 \[ \int (c+d x)^3 (a+b \coth (e+f x)) \, dx=\frac {{\left (a - b\right )} d^{3} f^{4} x^{4} + 4 \, {\left (a - b\right )} c d^{2} f^{4} x^{3} + 6 \, {\left (a - b\right )} c^{2} d f^{4} x^{2} + 4 \, {\left (a - b\right )} c^{3} f^{4} x + 24 \, b d^{3} {\rm polylog}\left (4, \cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right ) + 24 \, b d^{3} {\rm polylog}\left (4, -\cosh \left (f x + e\right ) - \sinh \left (f x + e\right )\right ) + 12 \, {\left (b d^{3} f^{2} x^{2} + 2 \, b c d^{2} f^{2} x + b c^{2} d f^{2}\right )} {\rm Li}_2\left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right ) + 12 \, {\left (b d^{3} f^{2} x^{2} + 2 \, b c d^{2} f^{2} x + b c^{2} d f^{2}\right )} {\rm Li}_2\left (-\cosh \left (f x + e\right ) - \sinh \left (f x + e\right )\right ) + 4 \, {\left (b d^{3} f^{3} x^{3} + 3 \, b c d^{2} f^{3} x^{2} + 3 \, b c^{2} d f^{3} x + b c^{3} f^{3}\right )} \log \left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right ) + 1\right ) - 4 \, {\left (b d^{3} e^{3} - 3 \, b c d^{2} e^{2} f + 3 \, b c^{2} d e f^{2} - b c^{3} f^{3}\right )} \log \left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right ) - 1\right ) + 4 \, {\left (b d^{3} f^{3} x^{3} + 3 \, b c d^{2} f^{3} x^{2} + 3 \, b c^{2} d f^{3} x + b d^{3} e^{3} - 3 \, b c d^{2} e^{2} f + 3 \, b c^{2} d e f^{2}\right )} \log \left (-\cosh \left (f x + e\right ) - \sinh \left (f x + e\right ) + 1\right ) - 24 \, {\left (b d^{3} f x + b c d^{2} f\right )} {\rm polylog}\left (3, \cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right ) - 24 \, {\left (b d^{3} f x + b c d^{2} f\right )} {\rm polylog}\left (3, -\cosh \left (f x + e\right ) - \sinh \left (f x + e\right )\right )}{4 \, f^{4}} \]

[In]

integrate((d*x+c)^3*(a+b*coth(f*x+e)),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int (c+d x)^3 (a+b \coth (e+f x)) \, dx=\int \left (a + b \coth {\left (e + f x \right )}\right ) \left (c + d x\right )^{3}\, dx \]

[In]

integrate((d*x+c)**3*(a+b*coth(f*x+e)),x)

[Out]

Integral((a + b*coth(e + f*x))*(c + d*x)**3, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 419 vs. \(2 (122) = 244\).

Time = 0.26 (sec) , antiderivative size = 419, normalized size of antiderivative = 3.15 \[ \int (c+d x)^3 (a+b \coth (e+f x)) \, dx=\frac {1}{4} \, a d^{3} x^{4} + \frac {1}{4} \, b d^{3} x^{4} + a c d^{2} x^{3} + b c d^{2} x^{3} + \frac {3}{2} \, a c^{2} d x^{2} + \frac {3}{2} \, b c^{2} d x^{2} + a c^{3} x + \frac {b c^{3} \log \left (\sinh \left (f x + e\right )\right )}{f} + \frac {3 \, {\left (f x \log \left (e^{\left (f x + e\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (f x + e\right )}\right )\right )} b c^{2} d}{f^{2}} + \frac {3 \, {\left (f x \log \left (-e^{\left (f x + e\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (f x + e\right )}\right )\right )} b c^{2} d}{f^{2}} + \frac {3 \, {\left (f^{2} x^{2} \log \left (e^{\left (f x + e\right )} + 1\right ) + 2 \, f x {\rm Li}_2\left (-e^{\left (f x + e\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (f x + e\right )})\right )} b c d^{2}}{f^{3}} + \frac {3 \, {\left (f^{2} x^{2} \log \left (-e^{\left (f x + e\right )} + 1\right ) + 2 \, f x {\rm Li}_2\left (e^{\left (f x + e\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (f x + e\right )})\right )} b c d^{2}}{f^{3}} + \frac {{\left (f^{3} x^{3} \log \left (e^{\left (f x + e\right )} + 1\right ) + 3 \, f^{2} x^{2} {\rm Li}_2\left (-e^{\left (f x + e\right )}\right ) - 6 \, f x {\rm Li}_{3}(-e^{\left (f x + e\right )}) + 6 \, {\rm Li}_{4}(-e^{\left (f x + e\right )})\right )} b d^{3}}{f^{4}} + \frac {{\left (f^{3} x^{3} \log \left (-e^{\left (f x + e\right )} + 1\right ) + 3 \, f^{2} x^{2} {\rm Li}_2\left (e^{\left (f x + e\right )}\right ) - 6 \, f x {\rm Li}_{3}(e^{\left (f x + e\right )}) + 6 \, {\rm Li}_{4}(e^{\left (f x + e\right )})\right )} b d^{3}}{f^{4}} - \frac {b d^{3} f^{4} x^{4} + 4 \, b c d^{2} f^{4} x^{3} + 6 \, b c^{2} d f^{4} x^{2}}{2 \, f^{4}} \]

[In]

integrate((d*x+c)^3*(a+b*coth(f*x+e)),x, algorithm="maxima")

[Out]

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*p
olylog(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*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 + (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

Giac [F]

\[ \int (c+d x)^3 (a+b \coth (e+f x)) \, dx=\int { {\left (d x + c\right )}^{3} {\left (b \coth \left (f x + e\right ) + a\right )} \,d x } \]

[In]

integrate((d*x+c)^3*(a+b*coth(f*x+e)),x, algorithm="giac")

[Out]

integrate((d*x + c)^3*(b*coth(f*x + e) + a), x)

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^3 (a+b \coth (e+f x)) \, dx=\int \left (a+b\,\mathrm {coth}\left (e+f\,x\right )\right )\,{\left (c+d\,x\right )}^3 \,d x \]

[In]

int((a + b*coth(e + f*x))*(c + d*x)^3,x)

[Out]

int((a + b*coth(e + f*x))*(c + d*x)^3, x)