3.16 \(\int \frac{(c+d x)^3}{a+a \coth (e+f x)} \, dx\)

Optimal. Leaf size=169 \[ -\frac{3 d^2 (c+d x)}{4 f^3 (a \coth (e+f x)+a)}-\frac{3 d (c+d x)^2}{4 f^2 (a \coth (e+f x)+a)}-\frac{(c+d x)^3}{2 f (a \coth (e+f x)+a)}+\frac{3 d (c+d x)^2}{8 a f^2}+\frac{(c+d x)^3}{4 a f}+\frac{(c+d x)^4}{8 a d}-\frac{3 d^3}{8 f^4 (a \coth (e+f x)+a)}+\frac{3 d^3 x}{8 a f^3} \]

[Out]

(3*d^3*x)/(8*a*f^3) + (3*d*(c + d*x)^2)/(8*a*f^2) + (c + d*x)^3/(4*a*f) + (c + d*x)^4/(8*a*d) - (3*d^3)/(8*f^4
*(a + a*Coth[e + f*x])) - (3*d^2*(c + d*x))/(4*f^3*(a + a*Coth[e + f*x])) - (3*d*(c + d*x)^2)/(4*f^2*(a + a*Co
th[e + f*x])) - (c + d*x)^3/(2*f*(a + a*Coth[e + f*x]))

________________________________________________________________________________________

Rubi [A]  time = 0.185485, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {3723, 3479, 8} \[ -\frac{3 d^2 (c+d x)}{4 f^3 (a \coth (e+f x)+a)}-\frac{3 d (c+d x)^2}{4 f^2 (a \coth (e+f x)+a)}-\frac{(c+d x)^3}{2 f (a \coth (e+f x)+a)}+\frac{3 d (c+d x)^2}{8 a f^2}+\frac{(c+d x)^3}{4 a f}+\frac{(c+d x)^4}{8 a d}-\frac{3 d^3}{8 f^4 (a \coth (e+f x)+a)}+\frac{3 d^3 x}{8 a f^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*d^3*x)/(8*a*f^3) + (3*d*(c + d*x)^2)/(8*a*f^2) + (c + d*x)^3/(4*a*f) + (c + d*x)^4/(8*a*d) - (3*d^3)/(8*f^4
*(a + a*Coth[e + f*x])) - (3*d^2*(c + d*x))/(4*f^3*(a + a*Coth[e + f*x])) - (3*d*(c + d*x)^2)/(4*f^2*(a + a*Co
th[e + f*x])) - (c + d*x)^3/(2*f*(a + a*Coth[e + f*x]))

Rule 3723

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

Rule 3479

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(a*(a + b*Tan[c + d*x])^n)/(2*b*d*n), x] +
Dist[1/(2*a), Int[(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0] && LtQ[n
, 0]

Rule 8

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

Rubi steps

\begin{align*} \int \frac{(c+d x)^3}{a+a \coth (e+f x)} \, dx &=\frac{(c+d x)^4}{8 a d}-\frac{(c+d x)^3}{2 f (a+a \coth (e+f x))}+\frac{(3 d) \int \frac{(c+d x)^2}{a+a \coth (e+f x)} \, dx}{2 f}\\ &=\frac{(c+d x)^3}{4 a f}+\frac{(c+d x)^4}{8 a d}-\frac{3 d (c+d x)^2}{4 f^2 (a+a \coth (e+f x))}-\frac{(c+d x)^3}{2 f (a+a \coth (e+f x))}+\frac{\left (3 d^2\right ) \int \frac{c+d x}{a+a \coth (e+f x)} \, dx}{2 f^2}\\ &=\frac{3 d (c+d x)^2}{8 a f^2}+\frac{(c+d x)^3}{4 a f}+\frac{(c+d x)^4}{8 a d}-\frac{3 d^2 (c+d x)}{4 f^3 (a+a \coth (e+f x))}-\frac{3 d (c+d x)^2}{4 f^2 (a+a \coth (e+f x))}-\frac{(c+d x)^3}{2 f (a+a \coth (e+f x))}+\frac{\left (3 d^3\right ) \int \frac{1}{a+a \coth (e+f x)} \, dx}{4 f^3}\\ &=\frac{3 d (c+d x)^2}{8 a f^2}+\frac{(c+d x)^3}{4 a f}+\frac{(c+d x)^4}{8 a d}-\frac{3 d^3}{8 f^4 (a+a \coth (e+f x))}-\frac{3 d^2 (c+d x)}{4 f^3 (a+a \coth (e+f x))}-\frac{3 d (c+d x)^2}{4 f^2 (a+a \coth (e+f x))}-\frac{(c+d x)^3}{2 f (a+a \coth (e+f x))}+\frac{\left (3 d^3\right ) \int 1 \, dx}{8 a f^3}\\ &=\frac{3 d^3 x}{8 a f^3}+\frac{3 d (c+d x)^2}{8 a f^2}+\frac{(c+d x)^3}{4 a f}+\frac{(c+d x)^4}{8 a d}-\frac{3 d^3}{8 f^4 (a+a \coth (e+f x))}-\frac{3 d^2 (c+d x)}{4 f^3 (a+a \coth (e+f x))}-\frac{3 d (c+d x)^2}{4 f^2 (a+a \coth (e+f x))}-\frac{(c+d x)^3}{2 f (a+a \coth (e+f x))}\\ \end{align*}

Mathematica [A]  time = 0.400777, size = 244, normalized size = 1.44 \[ \frac{\text{csch}(e+f x) (\sinh (f x)+\cosh (f x)) \left (2 f^4 x \left (6 c^2 d x+4 c^3+4 c d^2 x^2+d^3 x^3\right ) (\sinh (e)+\cosh (e))+(\cosh (e)-\sinh (e)) \cosh (2 f x) \left (6 c^2 d f^2 (2 f x+1)+4 c^3 f^3+6 c d^2 f \left (2 f^2 x^2+2 f x+1\right )+d^3 \left (4 f^3 x^3+6 f^2 x^2+6 f x+3\right )\right )+(\sinh (e)-\cosh (e)) \sinh (2 f x) \left (6 c^2 d f^2 (2 f x+1)+4 c^3 f^3+6 c d^2 f \left (2 f^2 x^2+2 f x+1\right )+d^3 \left (4 f^3 x^3+6 f^2 x^2+6 f x+3\right )\right )\right )}{16 a f^4 (\coth (e+f x)+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Csch[e + f*x]*(Cosh[f*x] + Sinh[f*x])*((4*c^3*f^3 + 6*c^2*d*f^2*(1 + 2*f*x) + 6*c*d^2*f*(1 + 2*f*x + 2*f^2*x^
2) + d^3*(3 + 6*f*x + 6*f^2*x^2 + 4*f^3*x^3))*Cosh[2*f*x]*(Cosh[e] - Sinh[e]) + 2*f^4*x*(4*c^3 + 6*c^2*d*x + 4
*c*d^2*x^2 + d^3*x^3)*(Cosh[e] + Sinh[e]) + (4*c^3*f^3 + 6*c^2*d*f^2*(1 + 2*f*x) + 6*c*d^2*f*(1 + 2*f*x + 2*f^
2*x^2) + d^3*(3 + 6*f*x + 6*f^2*x^2 + 4*f^3*x^3))*(-Cosh[e] + Sinh[e])*Sinh[2*f*x]))/(16*a*f^4*(1 + Coth[e + f
*x]))

________________________________________________________________________________________

Maple [B]  time = 0.077, size = 959, normalized size = 5.7 \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)^3/(a+a*coth(f*x+e)),x)

[Out]

1/a/f*(-1/f^3*d^3*(1/2*(f*x+e)^3*cosh(f*x+e)*sinh(f*x+e)-1/8*(f*x+e)^4-3/4*(f*x+e)^2*cosh(f*x+e)^2+3/4*(f*x+e)
*cosh(f*x+e)*sinh(f*x+e)+3/8*(f*x+e)^2-3/8*cosh(f*x+e)^2)+1/f^3*d^3*(1/2*(f*x+e)^3*cosh(f*x+e)^2-3/4*(f*x+e)^2
*cosh(f*x+e)*sinh(f*x+e)-1/4*(f*x+e)^3+3/4*(f*x+e)*cosh(f*x+e)^2-3/8*cosh(f*x+e)*sinh(f*x+e)-3/8*f*x-3/8*e)+3/
f^3*d^3*e*(1/2*(f*x+e)^2*cosh(f*x+e)*sinh(f*x+e)-1/6*(f*x+e)^3-1/2*(f*x+e)*cosh(f*x+e)^2+1/4*cosh(f*x+e)*sinh(
f*x+e)+1/4*f*x+1/4*e)-3/f^3*d^3*e*(1/2*(f*x+e)^2*cosh(f*x+e)^2-1/2*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)-1/4*(f*x+e)
^2+1/4*cosh(f*x+e)^2)-3/f^2*d^2*c*(1/2*(f*x+e)^2*cosh(f*x+e)*sinh(f*x+e)-1/6*(f*x+e)^3-1/2*(f*x+e)*cosh(f*x+e)
^2+1/4*cosh(f*x+e)*sinh(f*x+e)+1/4*f*x+1/4*e)+3/f^2*d^2*c*(1/2*(f*x+e)^2*cosh(f*x+e)^2-1/2*(f*x+e)*cosh(f*x+e)
*sinh(f*x+e)-1/4*(f*x+e)^2+1/4*cosh(f*x+e)^2)-3/f^3*d^3*e^2*(1/2*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)-1/4*(f*x+e)^2
-1/4*cosh(f*x+e)^2)+3/f^3*d^3*e^2*(1/2*(f*x+e)*cosh(f*x+e)^2-1/4*cosh(f*x+e)*sinh(f*x+e)-1/4*f*x-1/4*e)+6/f^2*
d^2*e*c*(1/2*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)-1/4*(f*x+e)^2-1/4*cosh(f*x+e)^2)-6/f^2*d^2*e*c*(1/2*(f*x+e)*cosh(
f*x+e)^2-1/4*cosh(f*x+e)*sinh(f*x+e)-1/4*f*x-1/4*e)-3/f*d*c^2*(1/2*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)-1/4*(f*x+e)
^2-1/4*cosh(f*x+e)^2)+3/f*d*c^2*(1/2*(f*x+e)*cosh(f*x+e)^2-1/4*cosh(f*x+e)*sinh(f*x+e)-1/4*f*x-1/4*e)+d^3*e^3/
f^3*(1/2*cosh(f*x+e)*sinh(f*x+e)-1/2*f*x-1/2*e)-1/2*d^3*e^3/f^3*cosh(f*x+e)^2-3*d^2*e^2/f^2*c*(1/2*cosh(f*x+e)
*sinh(f*x+e)-1/2*f*x-1/2*e)+3/2*d^2*e^2/f^2*c*cosh(f*x+e)^2+3*d*e/f*c^2*(1/2*cosh(f*x+e)*sinh(f*x+e)-1/2*f*x-1
/2*e)-3/2*d*e/f*c^2*cosh(f*x+e)^2-c^3*(1/2*cosh(f*x+e)*sinh(f*x+e)-1/2*f*x-1/2*e)+1/2*c^3*cosh(f*x+e)^2)

________________________________________________________________________________________

Maxima [A]  time = 1.37541, size = 247, normalized size = 1.46 \begin{align*} \frac{1}{4} \, c^{3}{\left (\frac{2 \,{\left (f x + e\right )}}{a f} + \frac{e^{\left (-2 \, f x - 2 \, e\right )}}{a f}\right )} + \frac{3 \,{\left (2 \, f^{2} x^{2} e^{\left (2 \, e\right )} +{\left (2 \, f x + 1\right )} e^{\left (-2 \, f x\right )}\right )} c^{2} d e^{\left (-2 \, e\right )}}{8 \, a f^{2}} + \frac{{\left (4 \, f^{3} x^{3} e^{\left (2 \, e\right )} + 3 \,{\left (2 \, f^{2} x^{2} + 2 \, f x + 1\right )} e^{\left (-2 \, f x\right )}\right )} c d^{2} e^{\left (-2 \, e\right )}}{8 \, a f^{3}} + \frac{{\left (2 \, f^{4} x^{4} e^{\left (2 \, e\right )} +{\left (4 \, f^{3} x^{3} + 6 \, f^{2} x^{2} + 6 \, f x + 3\right )} e^{\left (-2 \, f x\right )}\right )} d^{3} e^{\left (-2 \, e\right )}}{16 \, a f^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*c^3*(2*(f*x + e)/(a*f) + e^(-2*f*x - 2*e)/(a*f)) + 3/8*(2*f^2*x^2*e^(2*e) + (2*f*x + 1)*e^(-2*f*x))*c^2*d*
e^(-2*e)/(a*f^2) + 1/8*(4*f^3*x^3*e^(2*e) + 3*(2*f^2*x^2 + 2*f*x + 1)*e^(-2*f*x))*c*d^2*e^(-2*e)/(a*f^3) + 1/1
6*(2*f^4*x^4*e^(2*e) + (4*f^3*x^3 + 6*f^2*x^2 + 6*f*x + 3)*e^(-2*f*x))*d^3*e^(-2*e)/(a*f^4)

________________________________________________________________________________________

Fricas [A]  time = 2.0833, size = 640, normalized size = 3.79 \begin{align*} \frac{{\left (2 \, d^{3} f^{4} x^{4} + 4 \, c^{3} f^{3} + 6 \, c^{2} d f^{2} + 6 \, c d^{2} f + 4 \,{\left (2 \, c d^{2} f^{4} + d^{3} f^{3}\right )} x^{3} + 3 \, d^{3} + 6 \,{\left (2 \, c^{2} d f^{4} + 2 \, c d^{2} f^{3} + d^{3} f^{2}\right )} x^{2} + 2 \,{\left (4 \, c^{3} f^{4} + 6 \, c^{2} d f^{3} + 6 \, c d^{2} f^{2} + 3 \, d^{3} f\right )} x\right )} \cosh \left (f x + e\right ) +{\left (2 \, d^{3} f^{4} x^{4} - 4 \, c^{3} f^{3} - 6 \, c^{2} d f^{2} - 6 \, c d^{2} f + 4 \,{\left (2 \, c d^{2} f^{4} - d^{3} f^{3}\right )} x^{3} - 3 \, d^{3} + 6 \,{\left (2 \, c^{2} d f^{4} - 2 \, c d^{2} f^{3} - d^{3} f^{2}\right )} x^{2} + 2 \,{\left (4 \, c^{3} f^{4} - 6 \, c^{2} d f^{3} - 6 \, c d^{2} f^{2} - 3 \, d^{3} f\right )} x\right )} \sinh \left (f x + e\right )}{16 \,{\left (a f^{4} \cosh \left (f x + e\right ) + a f^{4} \sinh \left (f x + e\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*((2*d^3*f^4*x^4 + 4*c^3*f^3 + 6*c^2*d*f^2 + 6*c*d^2*f + 4*(2*c*d^2*f^4 + d^3*f^3)*x^3 + 3*d^3 + 6*(2*c^2*
d*f^4 + 2*c*d^2*f^3 + d^3*f^2)*x^2 + 2*(4*c^3*f^4 + 6*c^2*d*f^3 + 6*c*d^2*f^2 + 3*d^3*f)*x)*cosh(f*x + e) + (2
*d^3*f^4*x^4 - 4*c^3*f^3 - 6*c^2*d*f^2 - 6*c*d^2*f + 4*(2*c*d^2*f^4 - d^3*f^3)*x^3 - 3*d^3 + 6*(2*c^2*d*f^4 -
2*c*d^2*f^3 - d^3*f^2)*x^2 + 2*(4*c^3*f^4 - 6*c^2*d*f^3 - 6*c*d^2*f^2 - 3*d^3*f)*x)*sinh(f*x + e))/(a*f^4*cosh
(f*x + e) + a*f^4*sinh(f*x + e))

________________________________________________________________________________________

Sympy [A]  time = 5.25127, size = 864, normalized size = 5.11 \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)**3/(a+a*coth(f*x+e)),x)

[Out]

Piecewise((4*c**3*f**4*x*tanh(e + f*x)/(8*a*f**4*tanh(e + f*x) + 8*a*f**4) + 4*c**3*f**4*x/(8*a*f**4*tanh(e +
f*x) + 8*a*f**4) + 4*c**3*f**3/(8*a*f**4*tanh(e + f*x) + 8*a*f**4) + 6*c**2*d*f**4*x**2*tanh(e + f*x)/(8*a*f**
4*tanh(e + f*x) + 8*a*f**4) + 6*c**2*d*f**4*x**2/(8*a*f**4*tanh(e + f*x) + 8*a*f**4) - 6*c**2*d*f**3*x*tanh(e
+ f*x)/(8*a*f**4*tanh(e + f*x) + 8*a*f**4) + 6*c**2*d*f**3*x/(8*a*f**4*tanh(e + f*x) + 8*a*f**4) + 6*c**2*d*f*
*2/(8*a*f**4*tanh(e + f*x) + 8*a*f**4) + 4*c*d**2*f**4*x**3*tanh(e + f*x)/(8*a*f**4*tanh(e + f*x) + 8*a*f**4)
+ 4*c*d**2*f**4*x**3/(8*a*f**4*tanh(e + f*x) + 8*a*f**4) - 6*c*d**2*f**3*x**2*tanh(e + f*x)/(8*a*f**4*tanh(e +
f*x) + 8*a*f**4) + 6*c*d**2*f**3*x**2/(8*a*f**4*tanh(e + f*x) + 8*a*f**4) - 6*c*d**2*f**2*x*tanh(e + f*x)/(8*
a*f**4*tanh(e + f*x) + 8*a*f**4) + 6*c*d**2*f**2*x/(8*a*f**4*tanh(e + f*x) + 8*a*f**4) + 6*c*d**2*f/(8*a*f**4*
tanh(e + f*x) + 8*a*f**4) + d**3*f**4*x**4*tanh(e + f*x)/(8*a*f**4*tanh(e + f*x) + 8*a*f**4) + d**3*f**4*x**4/
(8*a*f**4*tanh(e + f*x) + 8*a*f**4) - 2*d**3*f**3*x**3*tanh(e + f*x)/(8*a*f**4*tanh(e + f*x) + 8*a*f**4) + 2*d
**3*f**3*x**3/(8*a*f**4*tanh(e + f*x) + 8*a*f**4) - 3*d**3*f**2*x**2*tanh(e + f*x)/(8*a*f**4*tanh(e + f*x) + 8
*a*f**4) + 3*d**3*f**2*x**2/(8*a*f**4*tanh(e + f*x) + 8*a*f**4) - 3*d**3*f*x*tanh(e + f*x)/(8*a*f**4*tanh(e +
f*x) + 8*a*f**4) + 3*d**3*f*x/(8*a*f**4*tanh(e + f*x) + 8*a*f**4) + 3*d**3/(8*a*f**4*tanh(e + f*x) + 8*a*f**4)
, Ne(f, 0)), ((c**3*x + 3*c**2*d*x**2/2 + c*d**2*x**3 + d**3*x**4/4)/(a*coth(e) + a), True))

________________________________________________________________________________________

Giac [A]  time = 1.184, size = 261, normalized size = 1.54 \begin{align*} \frac{{\left (2 \, d^{3} f^{4} x^{4} e^{\left (2 \, f x + 2 \, e\right )} + 8 \, c d^{2} f^{4} x^{3} e^{\left (2 \, f x + 2 \, e\right )} + 12 \, c^{2} d f^{4} x^{2} e^{\left (2 \, f x + 2 \, e\right )} + 4 \, d^{3} f^{3} x^{3} + 8 \, c^{3} f^{4} x e^{\left (2 \, f x + 2 \, e\right )} + 12 \, c d^{2} f^{3} x^{2} + 12 \, c^{2} d f^{3} x + 6 \, d^{3} f^{2} x^{2} + 4 \, c^{3} f^{3} + 12 \, c d^{2} f^{2} x + 6 \, c^{2} d f^{2} + 6 \, d^{3} f x + 6 \, c d^{2} f + 3 \, d^{3}\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{16 \, a f^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(2*d^3*f^4*x^4*e^(2*f*x + 2*e) + 8*c*d^2*f^4*x^3*e^(2*f*x + 2*e) + 12*c^2*d*f^4*x^2*e^(2*f*x + 2*e) + 4*d
^3*f^3*x^3 + 8*c^3*f^4*x*e^(2*f*x + 2*e) + 12*c*d^2*f^3*x^2 + 12*c^2*d*f^3*x + 6*d^3*f^2*x^2 + 4*c^3*f^3 + 12*
c*d^2*f^2*x + 6*c^2*d*f^2 + 6*d^3*f*x + 6*c*d^2*f + 3*d^3)*e^(-2*f*x - 2*e)/(a*f^4)