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

Optimal. Leaf size=261 \[ \frac{3 a^2 b d \text{PolyLog}\left (2,e^{2 (e+f x)}\right )}{2 f^2}+\frac{b^3 d \text{PolyLog}\left (2,e^{2 (e+f x)}\right )}{2 f^2}+\frac{3 a^2 b (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac{3 a^2 b (c+d x)^2}{2 d}+\frac{a^3 (c+d x)^2}{2 d}-\frac{3 a b^2 (c+d x) \coth (e+f x)}{f}+3 a b^2 c x+\frac{3 a b^2 d \log (\sinh (e+f x))}{f^2}+\frac{3}{2} a b^2 d x^2+\frac{b^3 (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac{b^3 (c+d x) \coth ^2(e+f x)}{2 f}-\frac{b^3 (c+d x)^2}{2 d}-\frac{b^3 d \coth (e+f x)}{2 f^2}+\frac{b^3 d x}{2 f} \]

[Out]

3*a*b^2*c*x + (b^3*d*x)/(2*f) + (3*a*b^2*d*x^2)/2 + (a^3*(c + d*x)^2)/(2*d) - (3*a^2*b*(c + d*x)^2)/(2*d) - (b
^3*(c + d*x)^2)/(2*d) - (b^3*d*Coth[e + f*x])/(2*f^2) - (3*a*b^2*(c + d*x)*Coth[e + f*x])/f - (b^3*(c + d*x)*C
oth[e + f*x]^2)/(2*f) + (3*a^2*b*(c + d*x)*Log[1 - E^(2*(e + f*x))])/f + (b^3*(c + d*x)*Log[1 - E^(2*(e + f*x)
)])/f + (3*a*b^2*d*Log[Sinh[e + f*x]])/f^2 + (3*a^2*b*d*PolyLog[2, E^(2*(e + f*x))])/(2*f^2) + (b^3*d*PolyLog[
2, E^(2*(e + f*x))])/(2*f^2)

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Rubi [A]  time = 0.350042, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 9, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3722, 3716, 2190, 2279, 2391, 3720, 3475, 3473, 8} \[ \frac{3 a^2 b d \text{PolyLog}\left (2,e^{2 (e+f x)}\right )}{2 f^2}+\frac{b^3 d \text{PolyLog}\left (2,e^{2 (e+f x)}\right )}{2 f^2}+\frac{3 a^2 b (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac{3 a^2 b (c+d x)^2}{2 d}+\frac{a^3 (c+d x)^2}{2 d}-\frac{3 a b^2 (c+d x) \coth (e+f x)}{f}+3 a b^2 c x+\frac{3 a b^2 d \log (\sinh (e+f x))}{f^2}+\frac{3}{2} a b^2 d x^2+\frac{b^3 (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac{b^3 (c+d x) \coth ^2(e+f x)}{2 f}-\frac{b^3 (c+d x)^2}{2 d}-\frac{b^3 d \coth (e+f x)}{2 f^2}+\frac{b^3 d x}{2 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

3*a*b^2*c*x + (b^3*d*x)/(2*f) + (3*a*b^2*d*x^2)/2 + (a^3*(c + d*x)^2)/(2*d) - (3*a^2*b*(c + d*x)^2)/(2*d) - (b
^3*(c + d*x)^2)/(2*d) - (b^3*d*Coth[e + f*x])/(2*f^2) - (3*a*b^2*(c + d*x)*Coth[e + f*x])/f - (b^3*(c + d*x)*C
oth[e + f*x]^2)/(2*f) + (3*a^2*b*(c + d*x)*Log[1 - E^(2*(e + f*x))])/f + (b^3*(c + d*x)*Log[1 - E^(2*(e + f*x)
)])/f + (3*a*b^2*d*Log[Sinh[e + f*x]])/f^2 + (3*a^2*b*d*PolyLog[2, E^(2*(e + f*x))])/(2*f^2) + (b^3*d*PolyLog[
2, E^(2*(e + f*x))])/(2*f^2)

Rule 3722

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 3716

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)))/(E^(2*I*k*Pi)*(1 + E^(2*
(-(I*e) + f*fz*x))/E^(2*I*k*Pi))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2279

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

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 3720

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(c + d*x)^m*(b*Tan[e
+ f*x])^(n - 1))/(f*(n - 1)), x] + (-Dist[(b*d*m)/(f*(n - 1)), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1)
, x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n,
1] && GtQ[m, 0]

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int (c+d x) (a+b \coth (e+f x))^3 \, dx &=\int \left (a^3 (c+d x)+3 a^2 b (c+d x) \coth (e+f x)+3 a b^2 (c+d x) \coth ^2(e+f x)+b^3 (c+d x) \coth ^3(e+f x)\right ) \, dx\\ &=\frac{a^3 (c+d x)^2}{2 d}+\left (3 a^2 b\right ) \int (c+d x) \coth (e+f x) \, dx+\left (3 a b^2\right ) \int (c+d x) \coth ^2(e+f x) \, dx+b^3 \int (c+d x) \coth ^3(e+f x) \, dx\\ &=\frac{a^3 (c+d x)^2}{2 d}-\frac{3 a^2 b (c+d x)^2}{2 d}-\frac{3 a b^2 (c+d x) \coth (e+f x)}{f}-\frac{b^3 (c+d x) \coth ^2(e+f x)}{2 f}-\left (6 a^2 b\right ) \int \frac{e^{2 (e+f x)} (c+d x)}{1-e^{2 (e+f x)}} \, dx+\left (3 a b^2\right ) \int (c+d x) \, dx+b^3 \int (c+d x) \coth (e+f x) \, dx+\frac{\left (3 a b^2 d\right ) \int \coth (e+f x) \, dx}{f}+\frac{\left (b^3 d\right ) \int \coth ^2(e+f x) \, dx}{2 f}\\ &=3 a b^2 c x+\frac{3}{2} a b^2 d x^2+\frac{a^3 (c+d x)^2}{2 d}-\frac{3 a^2 b (c+d x)^2}{2 d}-\frac{b^3 (c+d x)^2}{2 d}-\frac{b^3 d \coth (e+f x)}{2 f^2}-\frac{3 a b^2 (c+d x) \coth (e+f x)}{f}-\frac{b^3 (c+d x) \coth ^2(e+f x)}{2 f}+\frac{3 a^2 b (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac{3 a b^2 d \log (\sinh (e+f x))}{f^2}-\left (2 b^3\right ) \int \frac{e^{2 (e+f x)} (c+d x)}{1-e^{2 (e+f x)}} \, dx-\frac{\left (3 a^2 b d\right ) \int \log \left (1-e^{2 (e+f x)}\right ) \, dx}{f}+\frac{\left (b^3 d\right ) \int 1 \, dx}{2 f}\\ &=3 a b^2 c x+\frac{b^3 d x}{2 f}+\frac{3}{2} a b^2 d x^2+\frac{a^3 (c+d x)^2}{2 d}-\frac{3 a^2 b (c+d x)^2}{2 d}-\frac{b^3 (c+d x)^2}{2 d}-\frac{b^3 d \coth (e+f x)}{2 f^2}-\frac{3 a b^2 (c+d x) \coth (e+f x)}{f}-\frac{b^3 (c+d x) \coth ^2(e+f x)}{2 f}+\frac{3 a^2 b (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac{b^3 (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac{3 a b^2 d \log (\sinh (e+f x))}{f^2}-\frac{\left (3 a^2 b d\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{2 f^2}-\frac{\left (b^3 d\right ) \int \log \left (1-e^{2 (e+f x)}\right ) \, dx}{f}\\ &=3 a b^2 c x+\frac{b^3 d x}{2 f}+\frac{3}{2} a b^2 d x^2+\frac{a^3 (c+d x)^2}{2 d}-\frac{3 a^2 b (c+d x)^2}{2 d}-\frac{b^3 (c+d x)^2}{2 d}-\frac{b^3 d \coth (e+f x)}{2 f^2}-\frac{3 a b^2 (c+d x) \coth (e+f x)}{f}-\frac{b^3 (c+d x) \coth ^2(e+f x)}{2 f}+\frac{3 a^2 b (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac{b^3 (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac{3 a b^2 d \log (\sinh (e+f x))}{f^2}+\frac{3 a^2 b d \text{Li}_2\left (e^{2 (e+f x)}\right )}{2 f^2}-\frac{\left (b^3 d\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{2 f^2}\\ &=3 a b^2 c x+\frac{b^3 d x}{2 f}+\frac{3}{2} a b^2 d x^2+\frac{a^3 (c+d x)^2}{2 d}-\frac{3 a^2 b (c+d x)^2}{2 d}-\frac{b^3 (c+d x)^2}{2 d}-\frac{b^3 d \coth (e+f x)}{2 f^2}-\frac{3 a b^2 (c+d x) \coth (e+f x)}{f}-\frac{b^3 (c+d x) \coth ^2(e+f x)}{2 f}+\frac{3 a^2 b (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac{b^3 (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac{3 a b^2 d \log (\sinh (e+f x))}{f^2}+\frac{3 a^2 b d \text{Li}_2\left (e^{2 (e+f x)}\right )}{2 f^2}+\frac{b^3 d \text{Li}_2\left (e^{2 (e+f x)}\right )}{2 f^2}\\ \end{align*}

Mathematica [A]  time = 3.45699, size = 266, normalized size = 1.02 \[ \frac{\sinh (e+f x) (a+b \coth (e+f x))^3 \left (-b d \left (3 a^2+b^2\right ) \sinh ^2(e+f x) \text{PolyLog}\left (2,e^{-2 (e+f x)}\right )+\sinh ^2(e+f x) \left (-(e+f x) \left (-3 a^2 b d (e+f x)+a^3 (d (e-f x)-2 c f)+3 a b^2 (d (e-f x)-2 c f)-b^3 d (e+f x)\right )-2 b \log (\sinh (e+f x)) \left (3 a^2 (d e-c f)-3 a b d+b^2 (d e-c f)\right )+2 b d \left (3 a^2+b^2\right ) (e+f x) \log \left (1-e^{-2 (e+f x)}\right )\right )-\frac{1}{2} b^2 (\sinh (2 (e+f x)) (6 a f (c+d x)+b d)+2 b f (c+d x))\right )}{2 f^2 (a \sinh (e+f x)+b \cosh (e+f x))^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*Coth[e + f*x])^3*Sinh[e + f*x]*((-((e + f*x)*(-3*a^2*b*d*(e + f*x) - b^3*d*(e + f*x) + a^3*(-2*c*f + d
*(e - f*x)) + 3*a*b^2*(-2*c*f + d*(e - f*x)))) + 2*b*(3*a^2 + b^2)*d*(e + f*x)*Log[1 - E^(-2*(e + f*x))] - 2*b
*(-3*a*b*d + 3*a^2*(d*e - c*f) + b^2*(d*e - c*f))*Log[Sinh[e + f*x]])*Sinh[e + f*x]^2 - b*(3*a^2 + b^2)*d*Poly
Log[2, E^(-2*(e + f*x))]*Sinh[e + f*x]^2 - (b^2*(2*b*f*(c + d*x) + (b*d + 6*a*f*(c + d*x))*Sinh[2*(e + f*x)]))
/2))/(2*f^2*(b*Cosh[e + f*x] + a*Sinh[e + f*x])^3)

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Maple [B]  time = 0.121, size = 651, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.50365, size = 713, normalized size = 2.73 \begin{align*} \frac{1}{2} \, a^{3} d x^{2} + a^{3} c x + \frac{3 \, a^{2} b c \log \left (\sinh \left (f x + e\right )\right )}{f} -{\left (3 \, a^{2} b d + b^{3} d\right )} x^{2} - \frac{2 \,{\left (b^{3} c f + 3 \, a b^{2} d\right )} x}{f} + \frac{12 \, a b^{2} c f + 2 \, b^{3} d +{\left (3 \, a^{2} b d f^{2} + 3 \, a b^{2} d f^{2} + b^{3} d f^{2}\right )} x^{2} + 2 \,{\left (b^{3} c f^{2} + 3 \,{\left (c f^{2} + 2 \, d f\right )} a b^{2}\right )} x +{\left ({\left (3 \, a^{2} b d f^{2} e^{\left (4 \, e\right )} + 3 \, a b^{2} d f^{2} e^{\left (4 \, e\right )} + b^{3} d f^{2} e^{\left (4 \, e\right )}\right )} x^{2} + 2 \,{\left (3 \, a b^{2} c f^{2} e^{\left (4 \, e\right )} + b^{3} c f^{2} e^{\left (4 \, e\right )}\right )} x\right )} e^{\left (4 \, f x\right )} - 2 \,{\left (6 \, a b^{2} c f e^{\left (2 \, e\right )} +{\left (2 \, c f e^{\left (2 \, e\right )} + d e^{\left (2 \, e\right )}\right )} b^{3} +{\left (3 \, a^{2} b d f^{2} e^{\left (2 \, e\right )} + 3 \, a b^{2} d f^{2} e^{\left (2 \, e\right )} + b^{3} d f^{2} e^{\left (2 \, e\right )}\right )} x^{2} + 2 \,{\left (3 \,{\left (c f^{2} e^{\left (2 \, e\right )} + d f e^{\left (2 \, e\right )}\right )} a b^{2} +{\left (c f^{2} e^{\left (2 \, e\right )} + d f e^{\left (2 \, e\right )}\right )} b^{3}\right )} x\right )} e^{\left (2 \, f x\right )}}{2 \,{\left (f^{2} e^{\left (4 \, f x + 4 \, e\right )} - 2 \, f^{2} e^{\left (2 \, f x + 2 \, e\right )} + f^{2}\right )}} + \frac{{\left (3 \, a^{2} b d + b^{3} d\right )}{\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 )}}{f^{2}} + \frac{{\left (3 \, a^{2} b d + b^{3} d\right )}{\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 )}}{f^{2}} + \frac{{\left (b^{3} c f + 3 \, a b^{2} d\right )} \log \left (e^{\left (f x + e\right )} + 1\right )}{f^{2}} + \frac{{\left (b^{3} c f + 3 \, a b^{2} d\right )} \log \left (e^{\left (f x + e\right )} - 1\right )}{f^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^3*d*x^2 + a^3*c*x + 3*a^2*b*c*log(sinh(f*x + e))/f - (3*a^2*b*d + b^3*d)*x^2 - 2*(b^3*c*f + 3*a*b^2*d)*x
/f + 1/2*(12*a*b^2*c*f + 2*b^3*d + (3*a^2*b*d*f^2 + 3*a*b^2*d*f^2 + b^3*d*f^2)*x^2 + 2*(b^3*c*f^2 + 3*(c*f^2 +
2*d*f)*a*b^2)*x + ((3*a^2*b*d*f^2*e^(4*e) + 3*a*b^2*d*f^2*e^(4*e) + b^3*d*f^2*e^(4*e))*x^2 + 2*(3*a*b^2*c*f^2
*e^(4*e) + b^3*c*f^2*e^(4*e))*x)*e^(4*f*x) - 2*(6*a*b^2*c*f*e^(2*e) + (2*c*f*e^(2*e) + d*e^(2*e))*b^3 + (3*a^2
*b*d*f^2*e^(2*e) + 3*a*b^2*d*f^2*e^(2*e) + b^3*d*f^2*e^(2*e))*x^2 + 2*(3*(c*f^2*e^(2*e) + d*f*e^(2*e))*a*b^2 +
(c*f^2*e^(2*e) + d*f*e^(2*e))*b^3)*x)*e^(2*f*x))/(f^2*e^(4*f*x + 4*e) - 2*f^2*e^(2*f*x + 2*e) + f^2) + (3*a^2
*b*d + b^3*d)*(f*x*log(e^(f*x + e) + 1) + dilog(-e^(f*x + e)))/f^2 + (3*a^2*b*d + b^3*d)*(f*x*log(-e^(f*x + e)
+ 1) + dilog(e^(f*x + e)))/f^2 + (b^3*c*f + 3*a*b^2*d)*log(e^(f*x + e) + 1)/f^2 + (b^3*c*f + 3*a*b^2*d)*log(e
^(f*x + e) - 1)/f^2

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Fricas [B]  time = 2.6298, size = 6711, normalized size = 25.71 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*f^2*x^2 - 12*a*b^2*d*e + 2*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c*f^2*x + ((
a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*f^2*x^2 - 12*a*b^2*d*e + 2*(3*a^2*b + b^3)*d*e^2 - 4*(3*a^2*b + b^3)*c*e*f -
2*(6*a*b^2*d*f - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c*f^2)*x)*cosh(f*x + e)^4 + 4*((a^3 - 3*a^2*b + 3*a*b^2 - b^3
)*d*f^2*x^2 - 12*a*b^2*d*e + 2*(3*a^2*b + b^3)*d*e^2 - 4*(3*a^2*b + b^3)*c*e*f - 2*(6*a*b^2*d*f - (a^3 - 3*a^2
*b + 3*a*b^2 - b^3)*c*f^2)*x)*cosh(f*x + e)*sinh(f*x + e)^3 + ((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*f^2*x^2 - 12*
a*b^2*d*e + 2*(3*a^2*b + b^3)*d*e^2 - 4*(3*a^2*b + b^3)*c*e*f - 2*(6*a*b^2*d*f - (a^3 - 3*a^2*b + 3*a*b^2 - b^
3)*c*f^2)*x)*sinh(f*x + e)^4 + 2*b^3*d + 2*(3*a^2*b + b^3)*d*e^2 - 2*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*f^2*x^
2 - 12*a*b^2*d*e + b^3*d + 2*(3*a^2*b + b^3)*d*e^2 - 2*(2*(3*a^2*b + b^3)*c*e - (3*a*b^2 + b^3)*c)*f + 2*((a^3
- 3*a^2*b + 3*a*b^2 - b^3)*c*f^2 - (3*a*b^2 - b^3)*d*f)*x)*cosh(f*x + e)^2 - 2*((a^3 - 3*a^2*b + 3*a*b^2 - b^
3)*d*f^2*x^2 - 12*a*b^2*d*e + b^3*d + 2*(3*a^2*b + b^3)*d*e^2 - 3*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*f^2*x^2 -
12*a*b^2*d*e + 2*(3*a^2*b + b^3)*d*e^2 - 4*(3*a^2*b + b^3)*c*e*f - 2*(6*a*b^2*d*f - (a^3 - 3*a^2*b + 3*a*b^2
- b^3)*c*f^2)*x)*cosh(f*x + e)^2 - 2*(2*(3*a^2*b + b^3)*c*e - (3*a*b^2 + b^3)*c)*f + 2*((a^3 - 3*a^2*b + 3*a*b
^2 - b^3)*c*f^2 - (3*a*b^2 - b^3)*d*f)*x)*sinh(f*x + e)^2 + 4*(3*a*b^2*c - (3*a^2*b + b^3)*c*e)*f + 2*((3*a^2*
b + b^3)*d*cosh(f*x + e)^4 + 4*(3*a^2*b + b^3)*d*cosh(f*x + e)*sinh(f*x + e)^3 + (3*a^2*b + b^3)*d*sinh(f*x +
e)^4 - 2*(3*a^2*b + b^3)*d*cosh(f*x + e)^2 + 2*(3*(3*a^2*b + b^3)*d*cosh(f*x + e)^2 - (3*a^2*b + b^3)*d)*sinh(
f*x + e)^2 + (3*a^2*b + b^3)*d + 4*((3*a^2*b + b^3)*d*cosh(f*x + e)^3 - (3*a^2*b + b^3)*d*cosh(f*x + e))*sinh(
f*x + e))*dilog(cosh(f*x + e) + sinh(f*x + e)) + 2*((3*a^2*b + b^3)*d*cosh(f*x + e)^4 + 4*(3*a^2*b + b^3)*d*co
sh(f*x + e)*sinh(f*x + e)^3 + (3*a^2*b + b^3)*d*sinh(f*x + e)^4 - 2*(3*a^2*b + b^3)*d*cosh(f*x + e)^2 + 2*(3*(
3*a^2*b + b^3)*d*cosh(f*x + e)^2 - (3*a^2*b + b^3)*d)*sinh(f*x + e)^2 + (3*a^2*b + b^3)*d + 4*((3*a^2*b + b^3)
*d*cosh(f*x + e)^3 - (3*a^2*b + b^3)*d*cosh(f*x + e))*sinh(f*x + e))*dilog(-cosh(f*x + e) - sinh(f*x + e)) + 2
*((3*a*b^2*d + (3*a^2*b + b^3)*d*f*x + (3*a^2*b + b^3)*c*f)*cosh(f*x + e)^4 + 4*(3*a*b^2*d + (3*a^2*b + b^3)*d
*f*x + (3*a^2*b + b^3)*c*f)*cosh(f*x + e)*sinh(f*x + e)^3 + (3*a*b^2*d + (3*a^2*b + b^3)*d*f*x + (3*a^2*b + b^
3)*c*f)*sinh(f*x + e)^4 + 3*a*b^2*d + (3*a^2*b + b^3)*d*f*x + (3*a^2*b + b^3)*c*f - 2*(3*a*b^2*d + (3*a^2*b +
b^3)*d*f*x + (3*a^2*b + b^3)*c*f)*cosh(f*x + e)^2 - 2*(3*a*b^2*d + (3*a^2*b + b^3)*d*f*x + (3*a^2*b + b^3)*c*f
- 3*(3*a*b^2*d + (3*a^2*b + b^3)*d*f*x + (3*a^2*b + b^3)*c*f)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 4*((3*a*b^2*
d + (3*a^2*b + b^3)*d*f*x + (3*a^2*b + b^3)*c*f)*cosh(f*x + e)^3 - (3*a*b^2*d + (3*a^2*b + b^3)*d*f*x + (3*a^2
*b + b^3)*c*f)*cosh(f*x + e))*sinh(f*x + e))*log(cosh(f*x + e) + sinh(f*x + e) + 1) + 2*((3*a*b^2*d - (3*a^2*b
+ b^3)*d*e + (3*a^2*b + b^3)*c*f)*cosh(f*x + e)^4 + 4*(3*a*b^2*d - (3*a^2*b + b^3)*d*e + (3*a^2*b + b^3)*c*f)
*cosh(f*x + e)*sinh(f*x + e)^3 + (3*a*b^2*d - (3*a^2*b + b^3)*d*e + (3*a^2*b + b^3)*c*f)*sinh(f*x + e)^4 + 3*a
*b^2*d - (3*a^2*b + b^3)*d*e + (3*a^2*b + b^3)*c*f - 2*(3*a*b^2*d - (3*a^2*b + b^3)*d*e + (3*a^2*b + b^3)*c*f)
*cosh(f*x + e)^2 - 2*(3*a*b^2*d - (3*a^2*b + b^3)*d*e + (3*a^2*b + b^3)*c*f - 3*(3*a*b^2*d - (3*a^2*b + b^3)*d
*e + (3*a^2*b + b^3)*c*f)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 4*((3*a*b^2*d - (3*a^2*b + b^3)*d*e + (3*a^2*b +
b^3)*c*f)*cosh(f*x + e)^3 - (3*a*b^2*d - (3*a^2*b + b^3)*d*e + (3*a^2*b + b^3)*c*f)*cosh(f*x + e))*sinh(f*x +
e))*log(cosh(f*x + e) + sinh(f*x + e) - 1) + 2*(((3*a^2*b + b^3)*d*f*x + (3*a^2*b + b^3)*d*e)*cosh(f*x + e)^4
+ 4*((3*a^2*b + b^3)*d*f*x + (3*a^2*b + b^3)*d*e)*cosh(f*x + e)*sinh(f*x + e)^3 + ((3*a^2*b + b^3)*d*f*x + (3*
a^2*b + b^3)*d*e)*sinh(f*x + e)^4 + (3*a^2*b + b^3)*d*f*x + (3*a^2*b + b^3)*d*e - 2*((3*a^2*b + b^3)*d*f*x + (
3*a^2*b + b^3)*d*e)*cosh(f*x + e)^2 - 2*((3*a^2*b + b^3)*d*f*x + (3*a^2*b + b^3)*d*e - 3*((3*a^2*b + b^3)*d*f*
x + (3*a^2*b + b^3)*d*e)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 4*(((3*a^2*b + b^3)*d*f*x + (3*a^2*b + b^3)*d*e)*c
osh(f*x + e)^3 - ((3*a^2*b + b^3)*d*f*x + (3*a^2*b + b^3)*d*e)*cosh(f*x + e))*sinh(f*x + e))*log(-cosh(f*x + e
) - sinh(f*x + e) + 1) + 4*(((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*f^2*x^2 - 12*a*b^2*d*e + 2*(3*a^2*b + b^3)*d*e^
2 - 4*(3*a^2*b + b^3)*c*e*f - 2*(6*a*b^2*d*f - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c*f^2)*x)*cosh(f*x + e)^3 - ((a
^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*f^2*x^2 - 12*a*b^2*d*e + b^3*d + 2*(3*a^2*b + b^3)*d*e^2 - 2*(2*(3*a^2*b + b^3
)*c*e - (3*a*b^2 + b^3)*c)*f + 2*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c*f^2 - (3*a*b^2 - b^3)*d*f)*x)*cosh(f*x + e
))*sinh(f*x + e))/(f^2*cosh(f*x + e)^4 + 4*f^2*cosh(f*x + e)*sinh(f*x + e)^3 + f^2*sinh(f*x + e)^4 - 2*f^2*cos
h(f*x + e)^2 + 2*(3*f^2*cosh(f*x + e)^2 - f^2)*sinh(f*x + e)^2 + f^2 + 4*(f^2*cosh(f*x + e)^3 - f^2*cosh(f*x +
e))*sinh(f*x + e))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}{\left (b \coth \left (f x + e\right ) + a\right )}^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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