3.17 \(\int \frac {(c+d x)^2}{a+a \coth (e+f x)} \, dx\)

Optimal. Leaf size=122 \[ -\frac {d (c+d x)}{2 f^2 (a \coth (e+f x)+a)}-\frac {(c+d x)^2}{2 f (a \coth (e+f x)+a)}+\frac {(c+d x)^2}{4 a f}+\frac {(c+d x)^3}{6 a d}-\frac {d^2}{4 f^3 (a \coth (e+f x)+a)}+\frac {d^2 x}{4 a f^2} \]

[Out]

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

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Rubi [A]  time = 0.12, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3723, 3479, 8} \[ -\frac {d (c+d x)}{2 f^2 (a \coth (e+f x)+a)}-\frac {(c+d x)^2}{2 f (a \coth (e+f x)+a)}+\frac {(c+d x)^2}{4 a f}+\frac {(c+d x)^3}{6 a d}-\frac {d^2}{4 f^3 (a \coth (e+f x)+a)}+\frac {d^2 x}{4 a f^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2/(a + a*Coth[e + f*x]),x]

[Out]

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

Rule 8

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

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 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]

Rubi steps

\begin {align*} \int \frac {(c+d x)^2}{a+a \coth (e+f x)} \, dx &=\frac {(c+d x)^3}{6 a d}-\frac {(c+d x)^2}{2 f (a+a \coth (e+f x))}+\frac {d \int \frac {c+d x}{a+a \coth (e+f x)} \, dx}{f}\\ &=\frac {(c+d x)^2}{4 a f}+\frac {(c+d x)^3}{6 a d}-\frac {d (c+d x)}{2 f^2 (a+a \coth (e+f x))}-\frac {(c+d x)^2}{2 f (a+a \coth (e+f x))}+\frac {d^2 \int \frac {1}{a+a \coth (e+f x)} \, dx}{2 f^2}\\ &=\frac {(c+d x)^2}{4 a f}+\frac {(c+d x)^3}{6 a d}-\frac {d^2}{4 f^3 (a+a \coth (e+f x))}-\frac {d (c+d x)}{2 f^2 (a+a \coth (e+f x))}-\frac {(c+d x)^2}{2 f (a+a \coth (e+f x))}+\frac {d^2 \int 1 \, dx}{4 a f^2}\\ &=\frac {d^2 x}{4 a f^2}+\frac {(c+d x)^2}{4 a f}+\frac {(c+d x)^3}{6 a d}-\frac {d^2}{4 f^3 (a+a \coth (e+f x))}-\frac {d (c+d x)}{2 f^2 (a+a \coth (e+f x))}-\frac {(c+d x)^2}{2 f (a+a \coth (e+f x))}\\ \end {align*}

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Mathematica [A]  time = 0.28, size = 169, normalized size = 1.39 \[ \frac {\text {csch}(e+f x) (\sinh (f x)+\cosh (f x)) \left (\frac {4}{3} f^3 x \left (3 c^2+3 c d x+d^2 x^2\right ) (\sinh (e)+\cosh (e))+(\cosh (e)-\sinh (e)) \cosh (2 f x) \left (2 c^2 f^2+2 c d f (2 f x+1)+d^2 \left (2 f^2 x^2+2 f x+1\right )\right )+(\sinh (e)-\cosh (e)) \sinh (2 f x) \left (2 c^2 f^2+2 c d f (2 f x+1)+d^2 \left (2 f^2 x^2+2 f x+1\right )\right )\right )}{8 a f^3 (\coth (e+f x)+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^2/(a + a*Coth[e + f*x]),x]

[Out]

(Csch[e + f*x]*(Cosh[f*x] + Sinh[f*x])*((2*c^2*f^2 + 2*c*d*f*(1 + 2*f*x) + d^2*(1 + 2*f*x + 2*f^2*x^2))*Cosh[2
*f*x]*(Cosh[e] - Sinh[e]) + (4*f^3*x*(3*c^2 + 3*c*d*x + d^2*x^2)*(Cosh[e] + Sinh[e]))/3 + (2*c^2*f^2 + 2*c*d*f
*(1 + 2*f*x) + d^2*(1 + 2*f*x + 2*f^2*x^2))*(-Cosh[e] + Sinh[e])*Sinh[2*f*x]))/(8*a*f^3*(1 + Coth[e + f*x]))

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fricas [A]  time = 0.41, size = 192, normalized size = 1.57 \[ \frac {{\left (4 \, d^{2} f^{3} x^{3} + 6 \, c^{2} f^{2} + 6 \, c d f + 6 \, {\left (2 \, c d f^{3} + d^{2} f^{2}\right )} x^{2} + 3 \, d^{2} + 6 \, {\left (2 \, c^{2} f^{3} + 2 \, c d f^{2} + d^{2} f\right )} x\right )} \cosh \left (f x + e\right ) + {\left (4 \, d^{2} f^{3} x^{3} - 6 \, c^{2} f^{2} - 6 \, c d f + 6 \, {\left (2 \, c d f^{3} - d^{2} f^{2}\right )} x^{2} - 3 \, d^{2} + 6 \, {\left (2 \, c^{2} f^{3} - 2 \, c d f^{2} - d^{2} f\right )} x\right )} \sinh \left (f x + e\right )}{24 \, {\left (a f^{3} \cosh \left (f x + e\right ) + a f^{3} \sinh \left (f x + e\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/(a+a*coth(f*x+e)),x, algorithm="fricas")

[Out]

1/24*((4*d^2*f^3*x^3 + 6*c^2*f^2 + 6*c*d*f + 6*(2*c*d*f^3 + d^2*f^2)*x^2 + 3*d^2 + 6*(2*c^2*f^3 + 2*c*d*f^2 +
d^2*f)*x)*cosh(f*x + e) + (4*d^2*f^3*x^3 - 6*c^2*f^2 - 6*c*d*f + 6*(2*c*d*f^3 - d^2*f^2)*x^2 - 3*d^2 + 6*(2*c^
2*f^3 - 2*c*d*f^2 - d^2*f)*x)*sinh(f*x + e))/(a*f^3*cosh(f*x + e) + a*f^3*sinh(f*x + e))

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giac [A]  time = 0.12, size = 123, normalized size = 1.01 \[ \frac {{\left (4 \, d^{2} f^{3} x^{3} e^{\left (2 \, f x + 2 \, e\right )} + 12 \, c d f^{3} x^{2} e^{\left (2 \, f x + 2 \, e\right )} + 12 \, c^{2} f^{3} x e^{\left (2 \, f x + 2 \, e\right )} + 6 \, d^{2} f^{2} x^{2} + 12 \, c d f^{2} x + 6 \, c^{2} f^{2} + 6 \, d^{2} f x + 6 \, c d f + 3 \, d^{2}\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{24 \, a f^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/(a+a*coth(f*x+e)),x, algorithm="giac")

[Out]

1/24*(4*d^2*f^3*x^3*e^(2*f*x + 2*e) + 12*c*d*f^3*x^2*e^(2*f*x + 2*e) + 12*c^2*f^3*x*e^(2*f*x + 2*e) + 6*d^2*f^
2*x^2 + 12*c*d*f^2*x + 6*c^2*f^2 + 6*d^2*f*x + 6*c*d*f + 3*d^2)*e^(-2*f*x - 2*e)/(a*f^3)

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maple [B]  time = 0.39, size = 469, normalized size = 3.84 \[ \frac {-\frac {d^{2} \left (\frac {\left (f x +e \right )^{2} \cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {\left (f x +e \right )^{3}}{6}-\frac {\left (f x +e \right ) \left (\cosh ^{2}\left (f x +e \right )\right )}{2}+\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{4}+\frac {f x}{4}+\frac {e}{4}\right )}{f^{2}}+\frac {d^{2} \left (\frac {\left (f x +e \right )^{2} \left (\cosh ^{2}\left (f x +e \right )\right )}{2}-\frac {\left (f x +e \right ) \cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {\left (f x +e \right )^{2}}{4}+\frac {\left (\cosh ^{2}\left (f x +e \right )\right )}{4}\right )}{f^{2}}+\frac {2 d^{2} e \left (\frac {\left (f x +e \right ) \cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {\left (f x +e \right )^{2}}{4}-\frac {\left (\cosh ^{2}\left (f x +e \right )\right )}{4}\right )}{f^{2}}-\frac {2 d^{2} e \left (\frac {\left (f x +e \right ) \left (\cosh ^{2}\left (f x +e \right )\right )}{2}-\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{4}-\frac {f x}{4}-\frac {e}{4}\right )}{f^{2}}-\frac {2 d c \left (\frac {\left (f x +e \right ) \cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {\left (f x +e \right )^{2}}{4}-\frac {\left (\cosh ^{2}\left (f x +e \right )\right )}{4}\right )}{f}+\frac {2 d c \left (\frac {\left (f x +e \right ) \left (\cosh ^{2}\left (f x +e \right )\right )}{2}-\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{4}-\frac {f x}{4}-\frac {e}{4}\right )}{f}-\frac {d^{2} e^{2} \left (\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {f x}{2}-\frac {e}{2}\right )}{f^{2}}+\frac {d^{2} e^{2} \left (\cosh ^{2}\left (f x +e \right )\right )}{2 f^{2}}+\frac {2 d e c \left (\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {f x}{2}-\frac {e}{2}\right )}{f}-\frac {d e c \left (\cosh ^{2}\left (f x +e \right )\right )}{f}-c^{2} \left (\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {f x}{2}-\frac {e}{2}\right )+\frac {c^{2} \left (\cosh ^{2}\left (f x +e \right )\right )}{2}}{a f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^2/(a+a*coth(f*x+e)),x)

[Out]

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

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maxima [A]  time = 0.41, size = 124, normalized size = 1.02 \[ \frac {1}{4} \, c^{2} {\left (\frac {2 \, {\left (f x + e\right )}}{a f} + \frac {e^{\left (-2 \, f x - 2 \, e\right )}}{a f}\right )} + \frac {{\left (2 \, f^{2} x^{2} e^{\left (2 \, e\right )} + {\left (2 \, f x + 1\right )} e^{\left (-2 \, f x\right )}\right )} c d e^{\left (-2 \, e\right )}}{4 \, a f^{2}} + \frac {{\left (4 \, f^{3} x^{3} e^{\left (2 \, e\right )} + 3 \, {\left (2 \, f^{2} x^{2} + 2 \, f x + 1\right )} e^{\left (-2 \, f x\right )}\right )} d^{2} e^{\left (-2 \, e\right )}}{24 \, a f^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/(a+a*coth(f*x+e)),x, algorithm="maxima")

[Out]

1/4*c^2*(2*(f*x + e)/(a*f) + e^(-2*f*x - 2*e)/(a*f)) + 1/4*(2*f^2*x^2*e^(2*e) + (2*f*x + 1)*e^(-2*f*x))*c*d*e^
(-2*e)/(a*f^2) + 1/24*(4*f^3*x^3*e^(2*e) + 3*(2*f^2*x^2 + 2*f*x + 1)*e^(-2*f*x))*d^2*e^(-2*e)/(a*f^3)

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mupad [B]  time = 1.30, size = 186, normalized size = 1.52 \[ \frac {{\mathrm {e}}^{-2\,e-2\,f\,x}\,\left (12\,c^2\,x\,{\mathrm {e}}^{2\,e+2\,f\,x}+4\,d^2\,x^3\,{\mathrm {e}}^{2\,e+2\,f\,x}+12\,c\,d\,x^2\,{\mathrm {e}}^{2\,e+2\,f\,x}\right )}{24\,a}+\frac {\frac {{\mathrm {e}}^{-2\,e-2\,f\,x}\,\left (3\,d^2+3\,d^2\,{\mathrm {e}}^{2\,e+2\,f\,x}\right )}{24}+\frac {f\,{\mathrm {e}}^{-2\,e-2\,f\,x}\,\left (6\,c\,d+6\,d^2\,x+6\,c\,d\,{\mathrm {e}}^{2\,e+2\,f\,x}\right )}{24}+\frac {f^2\,{\mathrm {e}}^{-2\,e-2\,f\,x}\,\left (6\,c^2+6\,c^2\,{\mathrm {e}}^{2\,e+2\,f\,x}+6\,d^2\,x^2+12\,c\,d\,x\right )}{24}}{a\,f^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^2/(a + a*coth(e + f*x)),x)

[Out]

(exp(- 2*e - 2*f*x)*(12*c^2*x*exp(2*e + 2*f*x) + 4*d^2*x^3*exp(2*e + 2*f*x) + 12*c*d*x^2*exp(2*e + 2*f*x)))/(2
4*a) + ((exp(- 2*e - 2*f*x)*(3*d^2 + 3*d^2*exp(2*e + 2*f*x)))/24 + (f*exp(- 2*e - 2*f*x)*(6*c*d + 6*d^2*x + 6*
c*d*exp(2*e + 2*f*x)))/24 + (f^2*exp(- 2*e - 2*f*x)*(6*c^2 + 6*c^2*exp(2*e + 2*f*x) + 6*d^2*x^2 + 12*c*d*x))/2
4)/(a*f^3)

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sympy [A]  time = 1.25, size = 522, normalized size = 4.28 \[ \begin {cases} \frac {6 c^{2} f^{3} x \tanh {\left (e + f x \right )}}{12 a f^{3} \tanh {\left (e + f x \right )} + 12 a f^{3}} + \frac {6 c^{2} f^{3} x}{12 a f^{3} \tanh {\left (e + f x \right )} + 12 a f^{3}} + \frac {6 c^{2} f^{2}}{12 a f^{3} \tanh {\left (e + f x \right )} + 12 a f^{3}} + \frac {6 c d f^{3} x^{2} \tanh {\left (e + f x \right )}}{12 a f^{3} \tanh {\left (e + f x \right )} + 12 a f^{3}} + \frac {6 c d f^{3} x^{2}}{12 a f^{3} \tanh {\left (e + f x \right )} + 12 a f^{3}} - \frac {6 c d f^{2} x \tanh {\left (e + f x \right )}}{12 a f^{3} \tanh {\left (e + f x \right )} + 12 a f^{3}} + \frac {6 c d f^{2} x}{12 a f^{3} \tanh {\left (e + f x \right )} + 12 a f^{3}} + \frac {6 c d f}{12 a f^{3} \tanh {\left (e + f x \right )} + 12 a f^{3}} + \frac {2 d^{2} f^{3} x^{3} \tanh {\left (e + f x \right )}}{12 a f^{3} \tanh {\left (e + f x \right )} + 12 a f^{3}} + \frac {2 d^{2} f^{3} x^{3}}{12 a f^{3} \tanh {\left (e + f x \right )} + 12 a f^{3}} - \frac {3 d^{2} f^{2} x^{2} \tanh {\left (e + f x \right )}}{12 a f^{3} \tanh {\left (e + f x \right )} + 12 a f^{3}} + \frac {3 d^{2} f^{2} x^{2}}{12 a f^{3} \tanh {\left (e + f x \right )} + 12 a f^{3}} - \frac {3 d^{2} f x \tanh {\left (e + f x \right )}}{12 a f^{3} \tanh {\left (e + f x \right )} + 12 a f^{3}} + \frac {3 d^{2} f x}{12 a f^{3} \tanh {\left (e + f x \right )} + 12 a f^{3}} + \frac {3 d^{2}}{12 a f^{3} \tanh {\left (e + f x \right )} + 12 a f^{3}} & \text {for}\: f \neq 0 \\\frac {c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}}{a \coth {\relax (e )} + a} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**2/(a+a*coth(f*x+e)),x)

[Out]

Piecewise((6*c**2*f**3*x*tanh(e + f*x)/(12*a*f**3*tanh(e + f*x) + 12*a*f**3) + 6*c**2*f**3*x/(12*a*f**3*tanh(e
+ f*x) + 12*a*f**3) + 6*c**2*f**2/(12*a*f**3*tanh(e + f*x) + 12*a*f**3) + 6*c*d*f**3*x**2*tanh(e + f*x)/(12*a
*f**3*tanh(e + f*x) + 12*a*f**3) + 6*c*d*f**3*x**2/(12*a*f**3*tanh(e + f*x) + 12*a*f**3) - 6*c*d*f**2*x*tanh(e
+ f*x)/(12*a*f**3*tanh(e + f*x) + 12*a*f**3) + 6*c*d*f**2*x/(12*a*f**3*tanh(e + f*x) + 12*a*f**3) + 6*c*d*f/(
12*a*f**3*tanh(e + f*x) + 12*a*f**3) + 2*d**2*f**3*x**3*tanh(e + f*x)/(12*a*f**3*tanh(e + f*x) + 12*a*f**3) +
2*d**2*f**3*x**3/(12*a*f**3*tanh(e + f*x) + 12*a*f**3) - 3*d**2*f**2*x**2*tanh(e + f*x)/(12*a*f**3*tanh(e + f*
x) + 12*a*f**3) + 3*d**2*f**2*x**2/(12*a*f**3*tanh(e + f*x) + 12*a*f**3) - 3*d**2*f*x*tanh(e + f*x)/(12*a*f**3
*tanh(e + f*x) + 12*a*f**3) + 3*d**2*f*x/(12*a*f**3*tanh(e + f*x) + 12*a*f**3) + 3*d**2/(12*a*f**3*tanh(e + f*
x) + 12*a*f**3), Ne(f, 0)), ((c**2*x + c*d*x**2 + d**2*x**3/3)/(a*coth(e) + a), True))

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