3.1.52 \(\int \frac {(c+d x)^3}{a+b \coth (e+f x)} \, dx\) [52]

Optimal. Leaf size=210 \[ \frac {(c+d x)^4}{4 (a+b) d}-\frac {b (c+d x)^3 \log \left (1-\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{\left (a^2-b^2\right ) f}+\frac {3 b d (c+d x)^2 \text {PolyLog}\left (2,\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{2 \left (a^2-b^2\right ) f^2}+\frac {3 b d^2 (c+d x) \text {PolyLog}\left (3,\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{2 \left (a^2-b^2\right ) f^3}+\frac {3 b d^3 \text {PolyLog}\left (4,\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{4 \left (a^2-b^2\right ) f^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.25, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3812, 2221, 2611, 6744, 2320, 6724} \begin {gather*} \frac {3 b d^2 (c+d x) \text {Li}_3\left (\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{2 f^3 \left (a^2-b^2\right )}+\frac {3 b d (c+d x)^2 \text {Li}_2\left (\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{2 f^2 \left (a^2-b^2\right )}-\frac {b (c+d x)^3 \log \left (1-\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{f \left (a^2-b^2\right )}+\frac {3 b d^3 \text {Li}_4\left (\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{4 f^4 \left (a^2-b^2\right )}+\frac {(c+d x)^4}{4 d (a+b)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c + d*x)^4/(4*(a + b)*d) - (b*(c + d*x)^3*Log[1 - (a - b)/((a + b)*E^(2*(e + f*x)))])/((a^2 - b^2)*f) + (3*b*
d*(c + d*x)^2*PolyLog[2, (a - b)/((a + b)*E^(2*(e + f*x)))])/(2*(a^2 - b^2)*f^2) + (3*b*d^2*(c + d*x)*PolyLog[
3, (a - b)/((a + b)*E^(2*(e + f*x)))])/(2*(a^2 - b^2)*f^3) + (3*b*d^3*PolyLog[4, (a - b)/((a + b)*E^(2*(e + f*
x)))])/(4*(a^2 - b^2)*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 3812

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c + d*x)^
(m + 1)/(d*(m + 1)*(a + I*b)), x] + Dist[2*I*b, Int[(c + d*x)^m*E^(2*I*k*Pi)*(E^Simp[2*I*(e + f*x), x]/((a + I
*b)^2 + (a^2 + b^2)*E^(2*I*k*Pi)*E^Simp[2*I*(e + f*x), x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && Integer
Q[4*k] && NeQ[a^2 + b^2, 0] && IGtQ[m, 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*} \int \frac {(c+d x)^3}{a+b \coth (e+f x)} \, dx &=\frac {(c+d x)^4}{4 (a+b) d}-(2 b) \int \frac {e^{-2 (e+f x)} (c+d x)^3}{(a+b)^2+\left (-a^2+b^2\right ) e^{-2 (e+f x)}} \, dx\\ &=\frac {(c+d x)^4}{4 (a+b) d}-\frac {b (c+d x)^3 \log \left (1-\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{\left (a^2-b^2\right ) f}+\frac {(3 b d) \int (c+d x)^2 \log \left (1+\frac {\left (-a^2+b^2\right ) e^{-2 (e+f x)}}{(a+b)^2}\right ) \, dx}{\left (a^2-b^2\right ) f}\\ &=\frac {(c+d x)^4}{4 (a+b) d}-\frac {b (c+d x)^3 \log \left (1-\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{\left (a^2-b^2\right ) f}+\frac {3 b d (c+d x)^2 \text {Li}_2\left (\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{2 \left (a^2-b^2\right ) f^2}-\frac {\left (3 b d^2\right ) \int (c+d x) \text {Li}_2\left (-\frac {\left (-a^2+b^2\right ) e^{-2 (e+f x)}}{(a+b)^2}\right ) \, dx}{\left (a^2-b^2\right ) f^2}\\ &=\frac {(c+d x)^4}{4 (a+b) d}-\frac {b (c+d x)^3 \log \left (1-\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{\left (a^2-b^2\right ) f}+\frac {3 b d (c+d x)^2 \text {Li}_2\left (\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{2 \left (a^2-b^2\right ) f^2}+\frac {3 b d^2 (c+d x) \text {Li}_3\left (\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{2 \left (a^2-b^2\right ) f^3}-\frac {\left (3 b d^3\right ) \int \text {Li}_3\left (-\frac {\left (-a^2+b^2\right ) e^{-2 (e+f x)}}{(a+b)^2}\right ) \, dx}{2 \left (a^2-b^2\right ) f^3}\\ &=\frac {(c+d x)^4}{4 (a+b) d}-\frac {b (c+d x)^3 \log \left (1-\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{\left (a^2-b^2\right ) f}+\frac {3 b d (c+d x)^2 \text {Li}_2\left (\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{2 \left (a^2-b^2\right ) f^2}+\frac {3 b d^2 (c+d x) \text {Li}_3\left (\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{2 \left (a^2-b^2\right ) f^3}+\frac {\left (3 b d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (\frac {(a-b) x}{a+b}\right )}{x} \, dx,x,e^{-2 (e+f x)}\right )}{4 \left (a^2-b^2\right ) f^4}\\ &=\frac {(c+d x)^4}{4 (a+b) d}-\frac {b (c+d x)^3 \log \left (1-\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{\left (a^2-b^2\right ) f}+\frac {3 b d (c+d x)^2 \text {Li}_2\left (\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{2 \left (a^2-b^2\right ) f^2}+\frac {3 b d^2 (c+d x) \text {Li}_3\left (\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{2 \left (a^2-b^2\right ) f^3}+\frac {3 b d^3 \text {Li}_4\left (\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{4 \left (a^2-b^2\right ) f^4}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 2.29, size = 349, normalized size = 1.66 \begin {gather*} \frac {4 a c^3 f^4 x+4 b c^3 f^4 x+6 a c^2 d f^4 x^2+6 b c^2 d f^4 x^2+4 a c d^2 f^4 x^3+4 b c d^2 f^4 x^3+a d^3 f^4 x^4+b d^3 f^4 x^4-4 b c^3 f^3 \log \left (1+\frac {(a+b) e^{2 (e+f x)}}{-a+b}\right )-12 b c^2 d f^3 x \log \left (1+\frac {(a+b) e^{2 (e+f x)}}{-a+b}\right )-12 b c d^2 f^3 x^2 \log \left (1+\frac {(a+b) e^{2 (e+f x)}}{-a+b}\right )-4 b d^3 f^3 x^3 \log \left (1+\frac {(a+b) e^{2 (e+f x)}}{-a+b}\right )-6 b d f^2 (c+d x)^2 \text {PolyLog}\left (2,\frac {(a+b) e^{2 (e+f x)}}{a-b}\right )+6 b d^2 f (c+d x) \text {PolyLog}\left (3,\frac {(a+b) e^{2 (e+f x)}}{a-b}\right )-3 b d^3 \text {PolyLog}\left (4,\frac {(a+b) e^{2 (e+f x)}}{a-b}\right )}{4 (a-b) (a+b) f^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1157\) vs. \(2(209)=418\).
time = 5.36, size = 1158, normalized size = 5.51

method	result	size
risch	\(\text {Expression too large to display}\)	\(1158\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 478 vs. \(2 (205) = 410\).
time = 0.41, size = 478, normalized size = 2.28 \begin {gather*} -\frac {3 \, {\left (2 \, f x \log \left (-\frac {{\left (a + b\right )} e^{\left (2 \, f x + 2 \, e\right )}}{a - b} + 1\right ) + {\rm Li}_2\left (\frac {{\left (a + b\right )} e^{\left (2 \, f x + 2 \, e\right )}}{a - b}\right )\right )} b c^{2} d}{2 \, {\left (a^{2} f^{2} - b^{2} f^{2}\right )}} - \frac {3 \, {\left (2 \, f^{2} x^{2} \log \left (-\frac {{\left (a + b\right )} e^{\left (2 \, f x + 2 \, e\right )}}{a - b} + 1\right ) + 2 \, f x {\rm Li}_2\left (\frac {{\left (a + b\right )} e^{\left (2 \, f x + 2 \, e\right )}}{a - b}\right ) - {\rm Li}_{3}(\frac {{\left (a + b\right )} e^{\left (2 \, f x + 2 \, e\right )}}{a - b})\right )} b c d^{2}}{2 \, {\left (a^{2} f^{3} - b^{2} f^{3}\right )}} - \frac {{\left (4 \, f^{3} x^{3} \log \left (-\frac {{\left (a + b\right )} e^{\left (2 \, f x + 2 \, e\right )}}{a - b} + 1\right ) + 6 \, f^{2} x^{2} {\rm Li}_2\left (\frac {{\left (a + b\right )} e^{\left (2 \, f x + 2 \, e\right )}}{a - b}\right ) - 6 \, f x {\rm Li}_{3}(\frac {{\left (a + b\right )} e^{\left (2 \, f x + 2 \, e\right )}}{a - b}) + 3 \, {\rm Li}_{4}(\frac {{\left (a + b\right )} e^{\left (2 \, f x + 2 \, e\right )}}{a - b})\right )} b d^{3}}{3 \, {\left (a^{2} f^{4} - b^{2} f^{4}\right )}} - c^{3} {\left (\frac {b \log \left (-{\left (a - b\right )} e^{\left (-2 \, f x - 2 \, e\right )} + a + b\right )}{{\left (a^{2} - b^{2}\right )} f} - \frac {f x + e}{{\left (a + b\right )} f}\right )} + \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 \, {\left (a^{2} f^{4} - b^{2} f^{4}\right )}} + \frac {d^{3} x^{4} + 4 \, c d^{2} x^{3} + 6 \, c^{2} d x^{2}}{4 \, {\left (a + b\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*(2*f*x*log(-(a + b)*e^(2*f*x + 2*e)/(a - b) + 1) + dilog((a + b)*e^(2*f*x + 2*e)/(a - b)))*b*c^2*d/(a^2*f
^2 - b^2*f^2) - 3/2*(2*f^2*x^2*log(-(a + b)*e^(2*f*x + 2*e)/(a - b) + 1) + 2*f*x*dilog((a + b)*e^(2*f*x + 2*e)
/(a - b)) - polylog(3, (a + b)*e^(2*f*x + 2*e)/(a - b)))*b*c*d^2/(a^2*f^3 - b^2*f^3) - 1/3*(4*f^3*x^3*log(-(a
+ b)*e^(2*f*x + 2*e)/(a - b) + 1) + 6*f^2*x^2*dilog((a + b)*e^(2*f*x + 2*e)/(a - b)) - 6*f*x*polylog(3, (a + b
)*e^(2*f*x + 2*e)/(a - b)) + 3*polylog(4, (a + b)*e^(2*f*x + 2*e)/(a - b)))*b*d^3/(a^2*f^4 - b^2*f^4) - c^3*(b
*log(-(a - b)*e^(-2*f*x - 2*e) + a + b)/((a^2 - b^2)*f) - (f*x + e)/((a + b)*f)) + 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)/(a^2*f^4 - b^2*f^4) + 1/4*(d^3*x^4 + 4*c*d^2*x^3 + 6*c^2*d*x^2)/(a + b)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1060 vs. \(2 (205) = 410\).
time = 0.41, size = 1060, normalized size = 5.05 \begin {gather*} \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, \sqrt {\frac {a + b}{a - b}} {\left (\cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )\right )}\right ) - 24 \, b d^{3} {\rm polylog}\left (4, -\sqrt {\frac {a + b}{a - b}} {\left (\cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )\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 (\sqrt {\frac {a + b}{a - b}} {\left (\cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )\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 (-\sqrt {\frac {a + b}{a - b}} {\left (\cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )\right )}\right ) - 4 \, {\left (b c^{3} f^{3} - 3 \, b c^{2} d f^{2} \cosh \left (1\right ) + 3 \, b c d^{2} f \cosh \left (1\right )^{2} - b d^{3} \cosh \left (1\right )^{3} - b d^{3} \sinh \left (1\right )^{3} + 3 \, {\left (b c d^{2} f - b d^{3} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} - 3 \, {\left (b c^{2} d f^{2} - 2 \, b c d^{2} f \cosh \left (1\right ) + b d^{3} \cosh \left (1\right )^{2}\right )} \sinh \left (1\right )\right )} \log \left (2 \, {\left (a + b\right )} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + 2 \, {\left (a + b\right )} \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + 2 \, {\left (a - b\right )} \sqrt {\frac {a + b}{a - b}}\right ) - 4 \, {\left (b c^{3} f^{3} - 3 \, b c^{2} d f^{2} \cosh \left (1\right ) + 3 \, b c d^{2} f \cosh \left (1\right )^{2} - b d^{3} \cosh \left (1\right )^{3} - b d^{3} \sinh \left (1\right )^{3} + 3 \, {\left (b c d^{2} f - b d^{3} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} - 3 \, {\left (b c^{2} d f^{2} - 2 \, b c d^{2} f \cosh \left (1\right ) + b d^{3} \cosh \left (1\right )^{2}\right )} \sinh \left (1\right )\right )} \log \left (2 \, {\left (a + b\right )} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + 2 \, {\left (a + b\right )} \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) - 2 \, {\left (a - b\right )} \sqrt {\frac {a + b}{a - b}}\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 + 3 \, b c^{2} d f^{2} \cosh \left (1\right ) - 3 \, b c d^{2} f \cosh \left (1\right )^{2} + b d^{3} \cosh \left (1\right )^{3} + b d^{3} \sinh \left (1\right )^{3} - 3 \, {\left (b c d^{2} f - b d^{3} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} + 3 \, {\left (b c^{2} d f^{2} - 2 \, b c d^{2} f \cosh \left (1\right ) + b d^{3} \cosh \left (1\right )^{2}\right )} \sinh \left (1\right )\right )} \log \left (\sqrt {\frac {a + b}{a - b}} {\left (\cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )\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 + 3 \, b c^{2} d f^{2} \cosh \left (1\right ) - 3 \, b c d^{2} f \cosh \left (1\right )^{2} + b d^{3} \cosh \left (1\right )^{3} + b d^{3} \sinh \left (1\right )^{3} - 3 \, {\left (b c d^{2} f - b d^{3} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} + 3 \, {\left (b c^{2} d f^{2} - 2 \, b c d^{2} f \cosh \left (1\right ) + b d^{3} \cosh \left (1\right )^{2}\right )} \sinh \left (1\right )\right )} \log \left (-\sqrt {\frac {a + b}{a - b}} {\left (\cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )\right )} + 1\right ) + 24 \, {\left (b d^{3} f x + b c d^{2} f\right )} {\rm polylog}\left (3, \sqrt {\frac {a + b}{a - b}} {\left (\cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )\right )}\right ) + 24 \, {\left (b d^{3} f x + b c d^{2} f\right )} {\rm polylog}\left (3, -\sqrt {\frac {a + b}{a - b}} {\left (\cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )\right )}\right )}{4 \, {\left (a^{2} - b^{2}\right )} f^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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, sqrt((a + b)/(a - b))*(cosh(f*x + cosh(1) + sinh(1)) + sinh(f*x + cosh(1) + sinh(1)))) - 24*b*d^3*p
olylog(4, -sqrt((a + b)/(a - b))*(cosh(f*x + cosh(1) + sinh(1)) + sinh(f*x + cosh(1) + sinh(1)))) - 12*(b*d^3*
f^2*x^2 + 2*b*c*d^2*f^2*x + b*c^2*d*f^2)*dilog(sqrt((a + b)/(a - b))*(cosh(f*x + cosh(1) + sinh(1)) + sinh(f*x
+ cosh(1) + sinh(1)))) - 12*(b*d^3*f^2*x^2 + 2*b*c*d^2*f^2*x + b*c^2*d*f^2)*dilog(-sqrt((a + b)/(a - b))*(cos
h(f*x + cosh(1) + sinh(1)) + sinh(f*x + cosh(1) + sinh(1)))) - 4*(b*c^3*f^3 - 3*b*c^2*d*f^2*cosh(1) + 3*b*c*d^
2*f*cosh(1)^2 - b*d^3*cosh(1)^3 - b*d^3*sinh(1)^3 + 3*(b*c*d^2*f - b*d^3*cosh(1))*sinh(1)^2 - 3*(b*c^2*d*f^2 -
2*b*c*d^2*f*cosh(1) + b*d^3*cosh(1)^2)*sinh(1))*log(2*(a + b)*cosh(f*x + cosh(1) + sinh(1)) + 2*(a + b)*sinh(
f*x + cosh(1) + sinh(1)) + 2*(a - b)*sqrt((a + b)/(a - b))) - 4*(b*c^3*f^3 - 3*b*c^2*d*f^2*cosh(1) + 3*b*c*d^2
*f*cosh(1)^2 - b*d^3*cosh(1)^3 - b*d^3*sinh(1)^3 + 3*(b*c*d^2*f - b*d^3*cosh(1))*sinh(1)^2 - 3*(b*c^2*d*f^2 -
2*b*c*d^2*f*cosh(1) + b*d^3*cosh(1)^2)*sinh(1))*log(2*(a + b)*cosh(f*x + cosh(1) + sinh(1)) + 2*(a + b)*sinh(f
*x + cosh(1) + sinh(1)) - 2*(a - b)*sqrt((a + b)/(a - b))) - 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 + 3*b*c^2*d*f^2*cosh(1) - 3*b*c*d^2*f*cosh(1)^2 + b*d^3*cosh(1)^3 + b*d^3*sinh(1)^3 - 3*(b*c*d^2*f - b*d
^3*cosh(1))*sinh(1)^2 + 3*(b*c^2*d*f^2 - 2*b*c*d^2*f*cosh(1) + b*d^3*cosh(1)^2)*sinh(1))*log(sqrt((a + b)/(a -
b))*(cosh(f*x + cosh(1) + sinh(1)) + sinh(f*x + cosh(1) + sinh(1))) + 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 + 3*b*c^2*d*f^2*cosh(1) - 3*b*c*d^2*f*cosh(1)^2 + b*d^3*cosh(1)^3 + b*d^3*sinh(1)^3 - 3*(
b*c*d^2*f - b*d^3*cosh(1))*sinh(1)^2 + 3*(b*c^2*d*f^2 - 2*b*c*d^2*f*cosh(1) + b*d^3*cosh(1)^2)*sinh(1))*log(-s
qrt((a + b)/(a - b))*(cosh(f*x + cosh(1) + sinh(1)) + sinh(f*x + cosh(1) + sinh(1))) + 1) + 24*(b*d^3*f*x + b*
c*d^2*f)*polylog(3, sqrt((a + b)/(a - b))*(cosh(f*x + cosh(1) + sinh(1)) + sinh(f*x + cosh(1) + sinh(1)))) + 2
4*(b*d^3*f*x + b*c*d^2*f)*polylog(3, -sqrt((a + b)/(a - b))*(cosh(f*x + cosh(1) + sinh(1)) + sinh(f*x + cosh(1
) + sinh(1)))))/((a^2 - b^2)*f^4)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c + d x\right )^{3}}{a + b \coth {\left (e + f x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^3}{a+b\,\mathrm {coth}\left (e+f\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________