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

Optimal. Leaf size=38 \[ x \left (a^2+b^2\right )+\frac{2 a b \log (\cosh (c+d x))}{d}-\frac{b^2 \tanh (c+d x)}{d} \]

[Out]

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

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Rubi [A]  time = 0.0226947, antiderivative size = 38, 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} \[ x \left (a^2+b^2\right )+\frac{2 a b \log (\cosh (c+d x))}{d}-\frac{b^2 \tanh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Tanh[c + d*x])^2,x]

[Out]

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

Mathematica [A]  time = 0.0951131, size = 54, normalized size = 1.42 \[ \frac{(a-b)^2 \log (\tanh (c+d x)+1)-(a+b)^2 \log (1-\tanh (c+d x))-2 b^2 \tanh (c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Tanh[c + d*x])^2,x]

[Out]

(-((a + b)^2*Log[1 - Tanh[c + d*x]]) + (a - b)^2*Log[1 + Tanh[c + d*x]] - 2*b^2*Tanh[c + d*x])/(2*d)

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Maple [B]  time = 0.004, size = 116, normalized size = 3.1 \begin{align*} -{\frac{{b}^{2}\tanh \left ( dx+c \right ) }{d}}-{\frac{{a}^{2}\ln \left ( \tanh \left ( dx+c \right ) -1 \right ) }{2\,d}}-{\frac{\ln \left ( \tanh \left ( dx+c \right ) -1 \right ) ab}{d}}-{\frac{\ln \left ( \tanh \left ( dx+c \right ) -1 \right ){b}^{2}}{2\,d}}+{\frac{\ln \left ( \tanh \left ( dx+c \right ) +1 \right ){a}^{2}}{2\,d}}-{\frac{\ln \left ( \tanh \left ( dx+c \right ) +1 \right ) ab}{d}}+{\frac{\ln \left ( \tanh \left ( dx+c \right ) +1 \right ){b}^{2}}{2\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.12727, size = 66, normalized size = 1.74 \begin{align*} b^{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 b \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+b*tanh(d*x+c))^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.21507, size = 535, normalized size = 14.08 \begin{align*} \frac{{\left (a^{2} - 2 \, a b + b^{2}\right )} d x \cosh \left (d x + c\right )^{2} + 2 \,{\left (a^{2} - 2 \, a b + b^{2}\right )} d x \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) +{\left (a^{2} - 2 \, a b + b^{2}\right )} d x \sinh \left (d x + c\right )^{2} +{\left (a^{2} - 2 \, a b + b^{2}\right )} d x + 2 \, b^{2} + 2 \,{\left (a b \cosh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a b \sinh \left (d x + c\right )^{2} + a b\right )} \log \left (\frac{2 \, \cosh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\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+b*tanh(d*x+c))^2,x, algorithm="fricas")

[Out]

((a^2 - 2*a*b + b^2)*d*x*cosh(d*x + c)^2 + 2*(a^2 - 2*a*b + b^2)*d*x*cosh(d*x + c)*sinh(d*x + c) + (a^2 - 2*a*
b + b^2)*d*x*sinh(d*x + c)^2 + (a^2 - 2*a*b + b^2)*d*x + 2*b^2 + 2*(a*b*cosh(d*x + c)^2 + 2*a*b*cosh(d*x + c)*
sinh(d*x + c) + a*b*sinh(d*x + c)^2 + a*b)*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 = 0.224003, size = 54, normalized size = 1.42 \begin{align*} \begin{cases} a^{2} x + 2 a b x - \frac{2 a b \log{\left (\tanh{\left (c + d x \right )} + 1 \right )}}{d} + b^{2} x - \frac{b^{2} \tanh{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a + b \tanh{\left (c \right )}\right )^{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.23799, size = 82, normalized size = 2.16 \begin{align*} \frac{2 \, a b \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}{d} + \frac{{\left (a^{2} - 2 \, a b + b^{2}\right )}{\left (d x + c\right )}}{d} + \frac{2 \, b^{2}}{d{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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