3.44 \(\int (a+a \tanh (c+d x))^2 \, dx\)

Optimal. Leaf size=36 \[ -\frac{a^2 \tanh (c+d x)}{d}+\frac{2 a^2 \log (\cosh (c+d x))}{d}+2 a^2 x \]

[Out]

2*a^2*x + (2*a^2*Log[Cosh[c + d*x]])/d - (a^2*Tanh[c + d*x])/d

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Rubi [A]  time = 0.0206018, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3477, 3475} \[ -\frac{a^2 \tanh (c+d x)}{d}+\frac{2 a^2 \log (\cosh (c+d x))}{d}+2 a^2 x \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Tanh[c + d*x])^2,x]

[Out]

2*a^2*x + (2*a^2*Log[Cosh[c + d*x]])/d - (a^2*Tanh[c + d*x])/d

Rule 3477

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

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 (a+a \tanh (c+d x))^2 \, dx &=2 a^2 x-\frac{a^2 \tanh (c+d x)}{d}+\left (2 a^2\right ) \int \tanh (c+d x) \, dx\\ &=2 a^2 x+\frac{2 a^2 \log (\cosh (c+d x))}{d}-\frac{a^2 \tanh (c+d x)}{d}\\ \end{align*}

Mathematica [A]  time = 0.5045, size = 58, normalized size = 1.61 \[ \frac{a^2 \text{sech}(c) \text{sech}(c+d x) (\cosh (d x) (\log (\cosh (c+d x))+d x)+\cosh (2 c+d x) (\log (\cosh (c+d x))+d x)-\sinh (d x))}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Tanh[c + d*x])^2,x]

[Out]

(a^2*Sech[c]*Sech[c + d*x]*(Cosh[d*x]*(d*x + Log[Cosh[c + d*x]]) + Cosh[2*c + d*x]*(d*x + Log[Cosh[c + d*x]])
- Sinh[d*x]))/d

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Maple [A]  time = 0.003, size = 33, normalized size = 0.9 \begin{align*} -{\frac{{a}^{2}\tanh \left ( dx+c \right ) }{d}}-2\,{\frac{{a}^{2}\ln \left ( \tanh \left ( dx+c \right ) -1 \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*tanh(d*x+c))^2,x)

[Out]

-a^2*tanh(d*x+c)/d-2*a^2/d*ln(tanh(d*x+c)-1)

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Maxima [A]  time = 1.07999, size = 68, normalized size = 1.89 \begin{align*} a^{2}{\left (x + \frac{c}{d} - \frac{2}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}}\right )} + a^{2} x + \frac{2 \, a^{2} \log \left (\cosh \left (d x + c\right )\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*tanh(d*x+c))^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.19895, size = 308, normalized size = 8.56 \begin{align*} \frac{2 \,{\left (a^{2} +{\left (a^{2} \cosh \left (d x + c\right )^{2} + 2 \, a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a^{2} \sinh \left (d x + c\right )^{2} + a^{2}\right )} \log \left (\frac{2 \, \cosh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right )\right )}}{d \cosh \left (d x + c\right )^{2} + 2 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + d \sinh \left (d x + c\right )^{2} + d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*tanh(d*x+c))^2,x, algorithm="fricas")

[Out]

2*(a^2 + (a^2*cosh(d*x + c)^2 + 2*a^2*cosh(d*x + c)*sinh(d*x + c) + a^2*sinh(d*x + c)^2 + a^2)*log(2*cosh(d*x
+ c)/(cosh(d*x + c) - sinh(d*x + c))))/(d*cosh(d*x + c)^2 + 2*d*cosh(d*x + c)*sinh(d*x + c) + d*sinh(d*x + c)^
2 + d)

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Sympy [A]  time = 0.296032, size = 44, normalized size = 1.22 \begin{align*} \begin{cases} 4 a^{2} x - \frac{2 a^{2} \log{\left (\tanh{\left (c + d x \right )} + 1 \right )}}{d} - \frac{a^{2} \tanh{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a \tanh{\left (c \right )} + a\right )^{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*tanh(d*x+c))**2,x)

[Out]

Piecewise((4*a**2*x - 2*a**2*log(tanh(c + d*x) + 1)/d - a**2*tanh(c + d*x)/d, Ne(d, 0)), (x*(a*tanh(c) + a)**2
, True))

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Giac [A]  time = 1.16192, size = 53, normalized size = 1.47 \begin{align*} 2 \, a^{2}{\left (\frac{\log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}{d} + \frac{1}{d{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*tanh(d*x+c))^2,x, algorithm="giac")

[Out]

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