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

Optimal. Leaf size=74 \[ -\frac{c+d x}{2 f (a \coth (e+f x)+a)}+\frac{(c+d x)^2}{4 a d}-\frac{d}{4 f^2 (a \coth (e+f x)+a)}+\frac{d x}{4 a f} \]

[Out]

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

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Rubi [A]  time = 0.051007, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3723, 3479, 8} \[ -\frac{c+d x}{2 f (a \coth (e+f x)+a)}+\frac{(c+d x)^2}{4 a d}-\frac{d}{4 f^2 (a \coth (e+f x)+a)}+\frac{d x}{4 a f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*x)/(4*a*f) + (c + d*x)^2/(4*a*d) - d/(4*f^2*(a + a*Coth[e + f*x])) - (c + d*x)/(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}{a+a \coth (e+f x)} \, dx &=\frac{(c+d x)^2}{4 a d}-\frac{c+d x}{2 f (a+a \coth (e+f x))}+\frac{d \int \frac{1}{a+a \coth (e+f x)} \, dx}{2 f}\\ &=\frac{(c+d x)^2}{4 a d}-\frac{d}{4 f^2 (a+a \coth (e+f x))}-\frac{c+d x}{2 f (a+a \coth (e+f x))}+\frac{d \int 1 \, dx}{4 a f}\\ &=\frac{d x}{4 a f}+\frac{(c+d x)^2}{4 a d}-\frac{d}{4 f^2 (a+a \coth (e+f x))}-\frac{c+d x}{2 f (a+a \coth (e+f x))}\\ \end{align*}

Mathematica [A]  time = 0.262145, size = 81, normalized size = 1.09 \[ \frac{\left (2 c f (2 f x+1)+d \left (2 f^2 x^2+2 f x+1\right )\right ) \coth (e+f x)+2 c f (2 f x-1)+d \left (2 f^2 x^2-2 f x-1\right )}{8 a f^2 (\coth (e+f x)+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*c*f*(-1 + 2*f*x) + d*(-1 - 2*f*x + 2*f^2*x^2) + (2*c*f*(1 + 2*f*x) + d*(1 + 2*f*x + 2*f^2*x^2))*Coth[e + f*
x])/(8*a*f^2*(1 + Coth[e + f*x]))

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Maple [B]  time = 0.058, size = 175, normalized size = 2.4 \begin{align*}{\frac{1}{af} \left ( -{\frac{d}{f} \left ({\frac{ \left ( fx+e \right ) \cosh \left ( fx+e \right ) \sinh \left ( fx+e \right ) }{2}}-{\frac{ \left ( fx+e \right ) ^{2}}{4}}-{\frac{ \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}{4}} \right ) }+{\frac{d}{f} \left ({\frac{ \left ( fx+e \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}{2}}-{\frac{\cosh \left ( fx+e \right ) \sinh \left ( fx+e \right ) }{4}}-{\frac{fx}{4}}-{\frac{e}{4}} \right ) }+{\frac{de}{f} \left ({\frac{\cosh \left ( fx+e \right ) \sinh \left ( fx+e \right ) }{2}}-{\frac{fx}{2}}-{\frac{e}{2}} \right ) }-{\frac{de \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}{2\,f}}-c \left ({\frac{\cosh \left ( fx+e \right ) \sinh \left ( fx+e \right ) }{2}}-{\frac{fx}{2}}-{\frac{e}{2}} \right ) +{\frac{c \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.16564, size = 97, normalized size = 1.31 \begin{align*} \frac{1}{4} \, c{\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 )} d e^{\left (-2 \, e\right )}}{8 \, a f^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.11715, size = 239, normalized size = 3.23 \begin{align*} \frac{{\left (2 \, d f^{2} x^{2} + 2 \, c f + 2 \,{\left (2 \, c f^{2} + d f\right )} x + d\right )} \cosh \left (f x + e\right ) +{\left (2 \, d f^{2} x^{2} - 2 \, c f + 2 \,{\left (2 \, c f^{2} - d f\right )} x - d\right )} \sinh \left (f x + e\right )}{8 \,{\left (a f^{2} \cosh \left (f x + e\right ) + a f^{2} \sinh \left (f x + e\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.45016, size = 250, normalized size = 3.38 \begin{align*} \begin{cases} \frac{2 c f^{2} x \tanh{\left (e + f x \right )}}{4 a f^{2} \tanh{\left (e + f x \right )} + 4 a f^{2}} + \frac{2 c f^{2} x}{4 a f^{2} \tanh{\left (e + f x \right )} + 4 a f^{2}} + \frac{2 c f}{4 a f^{2} \tanh{\left (e + f x \right )} + 4 a f^{2}} + \frac{d f^{2} x^{2} \tanh{\left (e + f x \right )}}{4 a f^{2} \tanh{\left (e + f x \right )} + 4 a f^{2}} + \frac{d f^{2} x^{2}}{4 a f^{2} \tanh{\left (e + f x \right )} + 4 a f^{2}} - \frac{d f x \tanh{\left (e + f x \right )}}{4 a f^{2} \tanh{\left (e + f x \right )} + 4 a f^{2}} + \frac{d f x}{4 a f^{2} \tanh{\left (e + f x \right )} + 4 a f^{2}} + \frac{d}{4 a f^{2} \tanh{\left (e + f x \right )} + 4 a f^{2}} & \text{for}\: f \neq 0 \\\frac{c x + \frac{d x^{2}}{2}}{a \coth{\left (e \right )} + a} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.18953, size = 88, normalized size = 1.19 \begin{align*} \frac{{\left (2 \, d f^{2} x^{2} e^{\left (2 \, f x + 2 \, e\right )} + 4 \, c f^{2} x e^{\left (2 \, f x + 2 \, e\right )} + 2 \, d f x + 2 \, c f + d\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{8 \, a f^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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