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

Optimal. Leaf size=123 \[ \frac{(c+d x) \tanh \left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f}+\frac{(c+d x) \tanh \left (\frac{e}{2}+\frac{f x}{2}\right ) \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{6 a^2 f}+\frac{d \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{6 a^2 f^2}-\frac{2 d \log \left (\cosh \left (\frac{e}{2}+\frac{f x}{2}\right )\right )}{3 a^2 f^2} \]

[Out]

(-2*d*Log[Cosh[e/2 + (f*x)/2]])/(3*a^2*f^2) + (d*Sech[e/2 + (f*x)/2]^2)/(6*a^2*f^2) + ((c + d*x)*Tanh[e/2 + (f
*x)/2])/(3*a^2*f) + ((c + d*x)*Sech[e/2 + (f*x)/2]^2*Tanh[e/2 + (f*x)/2])/(6*a^2*f)

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Rubi [A]  time = 0.0955596, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {3318, 4185, 4184, 3475} \[ \frac{(c+d x) \tanh \left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f}+\frac{(c+d x) \tanh \left (\frac{e}{2}+\frac{f x}{2}\right ) \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{6 a^2 f}+\frac{d \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{6 a^2 f^2}-\frac{2 d \log \left (\cosh \left (\frac{e}{2}+\frac{f x}{2}\right )\right )}{3 a^2 f^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d*Log[Cosh[e/2 + (f*x)/2]])/(3*a^2*f^2) + (d*Sech[e/2 + (f*x)/2]^2)/(6*a^2*f^2) + ((c + d*x)*Tanh[e/2 + (f
*x)/2])/(3*a^2*f) + ((c + d*x)*Sech[e/2 + (f*x)/2]^2*Tanh[e/2 + (f*x)/2])/(6*a^2*f)

Rule 3318

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[(2*a)^n, Int[(c
+ d*x)^m*Sin[(1*(e + (Pi*a)/(2*b)))/2 + (f*x)/2]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2
- b^2, 0] && IntegerQ[n] && (GtQ[n, 0] || IGtQ[m, 0])

Rule 4185

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> -Simp[(b^2*(c + d*x)*Cot[e + f*x]*
(b*Csc[e + f*x])^(n - 2))/(f*(n - 1)), x] + (Dist[(b^2*(n - 2))/(n - 1), Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2
), x], x] - Simp[(b^2*d*(b*Csc[e + f*x])^(n - 2))/(f^2*(n - 1)*(n - 2)), x]) /; FreeQ[{b, c, d, e, f}, x] && G
tQ[n, 1] && NeQ[n, 2]

Rule 4184

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp[((c + d*x)^m*Cot[e + f*x])/f, x]
+ Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && 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]

Rubi steps

\begin{align*} \int \frac{c+d x}{(a+a \cosh (e+f x))^2} \, dx &=\frac{\int (c+d x) \csc ^4\left (\frac{1}{2} (i e+\pi )+\frac{i f x}{2}\right ) \, dx}{4 a^2}\\ &=\frac{d \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{6 a^2 f^2}+\frac{(c+d x) \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right ) \tanh \left (\frac{e}{2}+\frac{f x}{2}\right )}{6 a^2 f}+\frac{\int (c+d x) \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right ) \, dx}{6 a^2}\\ &=\frac{d \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{6 a^2 f^2}+\frac{(c+d x) \tanh \left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f}+\frac{(c+d x) \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right ) \tanh \left (\frac{e}{2}+\frac{f x}{2}\right )}{6 a^2 f}-\frac{d \int \tanh \left (\frac{e}{2}+\frac{f x}{2}\right ) \, dx}{3 a^2 f}\\ &=-\frac{2 d \log \left (\cosh \left (\frac{e}{2}+\frac{f x}{2}\right )\right )}{3 a^2 f^2}+\frac{d \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{6 a^2 f^2}+\frac{(c+d x) \tanh \left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f}+\frac{(c+d x) \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right ) \tanh \left (\frac{e}{2}+\frac{f x}{2}\right )}{6 a^2 f}\\ \end{align*}

Mathematica [A]  time = 0.437002, size = 114, normalized size = 0.93 \[ \frac{\cosh \left (\frac{1}{2} (e+f x)\right ) \left (f (c+d x) \left (3 \sinh \left (\frac{1}{2} (e+f x)\right )+\sinh \left (\frac{3}{2} (e+f x)\right )\right )-2 d \cosh \left (\frac{3}{2} (e+f x)\right ) \log \left (\cosh \left (\frac{1}{2} (e+f x)\right )\right )+\cosh \left (\frac{1}{2} (e+f x)\right ) \left (2 d-6 d \log \left (\cosh \left (\frac{1}{2} (e+f x)\right )\right )\right )\right )}{3 a^2 f^2 (\cosh (e+f x)+1)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.06, size = 108, normalized size = 0.9 \begin{align*}{\frac{2\,dx}{3\,{a}^{2}f}}+{\frac{2\,de}{3\,{a}^{2}{f}^{2}}}-{\frac{6\,dfx{{\rm e}^{fx+e}}+6\,cf{{\rm e}^{fx+e}}+2\,dfx-2\,d{{\rm e}^{2\,fx+2\,e}}+2\,cf-2\,d{{\rm e}^{fx+e}}}{3\,{a}^{2}{f}^{2} \left ({{\rm e}^{fx+e}}+1 \right ) ^{3}}}-{\frac{2\,d\ln \left ({{\rm e}^{fx+e}}+1 \right ) }{3\,{a}^{2}{f}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.10989, size = 323, normalized size = 2.63 \begin{align*} \frac{2}{3} \, d{\left (\frac{f x e^{\left (3 \, f x + 3 \, e\right )} +{\left (3 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )} + e^{\left (f x + e\right )}}{a^{2} f^{2} e^{\left (3 \, f x + 3 \, e\right )} + 3 \, a^{2} f^{2} e^{\left (2 \, f x + 2 \, e\right )} + 3 \, a^{2} f^{2} e^{\left (f x + e\right )} + a^{2} f^{2}} - \frac{\log \left ({\left (e^{\left (f x + e\right )} + 1\right )} e^{\left (-e\right )}\right )}{a^{2} f^{2}}\right )} + \frac{2}{3} \, c{\left (\frac{3 \, e^{\left (-f x - e\right )}}{{\left (3 \, a^{2} e^{\left (-f x - e\right )} + 3 \, a^{2} e^{\left (-2 \, f x - 2 \, e\right )} + a^{2} e^{\left (-3 \, f x - 3 \, e\right )} + a^{2}\right )} f} + \frac{1}{{\left (3 \, a^{2} e^{\left (-f x - e\right )} + 3 \, a^{2} e^{\left (-2 \, f x - 2 \, e\right )} + a^{2} e^{\left (-3 \, f x - 3 \, e\right )} + a^{2}\right )} f}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.11928, size = 1002, normalized size = 8.15 \begin{align*} \frac{2 \,{\left (d f x \cosh \left (f x + e\right )^{3} + d f x \sinh \left (f x + e\right )^{3} +{\left (3 \, d f x + d\right )} \cosh \left (f x + e\right )^{2} +{\left (3 \, d f x \cosh \left (f x + e\right ) + 3 \, d f x + d\right )} \sinh \left (f x + e\right )^{2} - c f -{\left (3 \, c f - d\right )} \cosh \left (f x + e\right ) -{\left (d \cosh \left (f x + e\right )^{3} + d \sinh \left (f x + e\right )^{3} + 3 \, d \cosh \left (f x + e\right )^{2} + 3 \,{\left (d \cosh \left (f x + e\right ) + d\right )} \sinh \left (f x + e\right )^{2} + 3 \, d \cosh \left (f x + e\right ) + 3 \,{\left (d \cosh \left (f x + e\right )^{2} + 2 \, d \cosh \left (f x + e\right ) + d\right )} \sinh \left (f x + e\right ) + d\right )} \log \left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right ) + 1\right ) +{\left (3 \, d f x \cosh \left (f x + e\right )^{2} - 3 \, c f + 2 \,{\left (3 \, d f x + d\right )} \cosh \left (f x + e\right ) + d\right )} \sinh \left (f x + e\right )\right )}}{3 \,{\left (a^{2} f^{2} \cosh \left (f x + e\right )^{3} + a^{2} f^{2} \sinh \left (f x + e\right )^{3} + 3 \, a^{2} f^{2} \cosh \left (f x + e\right )^{2} + 3 \, a^{2} f^{2} \cosh \left (f x + e\right ) + a^{2} f^{2} + 3 \,{\left (a^{2} f^{2} \cosh \left (f x + e\right ) + a^{2} f^{2}\right )} \sinh \left (f x + e\right )^{2} + 3 \,{\left (a^{2} f^{2} \cosh \left (f x + e\right )^{2} + 2 \, a^{2} f^{2} \cosh \left (f x + e\right ) + a^{2} f^{2}\right )} \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*cosh(f*x+e))^2,x, algorithm="fricas")

[Out]

2/3*(d*f*x*cosh(f*x + e)^3 + d*f*x*sinh(f*x + e)^3 + (3*d*f*x + d)*cosh(f*x + e)^2 + (3*d*f*x*cosh(f*x + e) +
3*d*f*x + d)*sinh(f*x + e)^2 - c*f - (3*c*f - d)*cosh(f*x + e) - (d*cosh(f*x + e)^3 + d*sinh(f*x + e)^3 + 3*d*
cosh(f*x + e)^2 + 3*(d*cosh(f*x + e) + d)*sinh(f*x + e)^2 + 3*d*cosh(f*x + e) + 3*(d*cosh(f*x + e)^2 + 2*d*cos
h(f*x + e) + d)*sinh(f*x + e) + d)*log(cosh(f*x + e) + sinh(f*x + e) + 1) + (3*d*f*x*cosh(f*x + e)^2 - 3*c*f +
2*(3*d*f*x + d)*cosh(f*x + e) + d)*sinh(f*x + e))/(a^2*f^2*cosh(f*x + e)^3 + a^2*f^2*sinh(f*x + e)^3 + 3*a^2*
f^2*cosh(f*x + e)^2 + 3*a^2*f^2*cosh(f*x + e) + a^2*f^2 + 3*(a^2*f^2*cosh(f*x + e) + a^2*f^2)*sinh(f*x + e)^2
+ 3*(a^2*f^2*cosh(f*x + e)^2 + 2*a^2*f^2*cosh(f*x + e) + a^2*f^2)*sinh(f*x + e))

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Sympy [A]  time = 2.87245, size = 156, normalized size = 1.27 \begin{align*} \begin{cases} - \frac{c \tanh ^{3}{\left (\frac{e}{2} + \frac{f x}{2} \right )}}{6 a^{2} f} + \frac{c \tanh{\left (\frac{e}{2} + \frac{f x}{2} \right )}}{2 a^{2} f} - \frac{d x \tanh ^{3}{\left (\frac{e}{2} + \frac{f x}{2} \right )}}{6 a^{2} f} + \frac{d x \tanh{\left (\frac{e}{2} + \frac{f x}{2} \right )}}{2 a^{2} f} - \frac{d x}{3 a^{2} f} + \frac{2 d \log{\left (\tanh{\left (\frac{e}{2} + \frac{f x}{2} \right )} + 1 \right )}}{3 a^{2} f^{2}} - \frac{d \tanh ^{2}{\left (\frac{e}{2} + \frac{f x}{2} \right )}}{6 a^{2} f^{2}} & \text{for}\: f \neq 0 \\\frac{c x + \frac{d x^{2}}{2}}{\left (a \cosh{\left (e \right )} + a\right )^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-c*tanh(e/2 + f*x/2)**3/(6*a**2*f) + c*tanh(e/2 + f*x/2)/(2*a**2*f) - d*x*tanh(e/2 + f*x/2)**3/(6*a
**2*f) + d*x*tanh(e/2 + f*x/2)/(2*a**2*f) - d*x/(3*a**2*f) + 2*d*log(tanh(e/2 + f*x/2) + 1)/(3*a**2*f**2) - d*
tanh(e/2 + f*x/2)**2/(6*a**2*f**2), Ne(f, 0)), ((c*x + d*x**2/2)/(a*cosh(e) + a)**2, True))

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Giac [B]  time = 1.2571, size = 279, normalized size = 2.27 \begin{align*} \frac{2 \,{\left (d f x e^{\left (3 \, f x + 3 \, e\right )} + 3 \, d f x e^{\left (2 \, f x + 2 \, e\right )} - 3 \, c f e^{\left (f x + e\right )} - d e^{\left (3 \, f x + 3 \, e\right )} \log \left (e^{\left (f x + e\right )} + 1\right ) - 3 \, d e^{\left (2 \, f x + 2 \, e\right )} \log \left (e^{\left (f x + e\right )} + 1\right ) - 3 \, d e^{\left (f x + e\right )} \log \left (e^{\left (f x + e\right )} + 1\right ) - c f + d e^{\left (2 \, f x + 2 \, e\right )} + d e^{\left (f x + e\right )} - d \log \left (e^{\left (f x + e\right )} + 1\right )\right )}}{3 \,{\left (a^{2} f^{2} e^{\left (3 \, f x + 3 \, e\right )} + 3 \, a^{2} f^{2} e^{\left (2 \, f x + 2 \, e\right )} + 3 \, a^{2} f^{2} e^{\left (f x + e\right )} + a^{2} f^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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