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

Optimal. Leaf size=36 \[ -\frac{(a+b) \tanh (c+d x)}{d}+x (a+b)-\frac{b \tanh ^3(c+d x)}{3 d} \]

[Out]

(a + b)*x - ((a + b)*Tanh[c + d*x])/d - (b*Tanh[c + d*x]^3)/(3*d)

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Rubi [A]  time = 0.0396052, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3631, 3473, 8} \[ -\frac{(a+b) \tanh (c+d x)}{d}+x (a+b)-\frac{b \tanh ^3(c+d x)}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[Tanh[c + d*x]^2*(a + b*Tanh[c + d*x]^2),x]

[Out]

(a + b)*x - ((a + b)*Tanh[c + d*x])/d - (b*Tanh[c + d*x]^3)/(3*d)

Rule 3631

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp
[(C*(a + b*Tan[e + f*x])^(m + 1))/(b*f*(m + 1)), x] + Dist[A - C, Int[(a + b*Tan[e + f*x])^m, x], x] /; FreeQ[
{a, b, e, f, A, C, m}, x] && NeQ[A*b^2 + a^2*C, 0] &&  !LeQ[m, -1]

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 8

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

Rubi steps

\begin{align*} \int \tanh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx &=-\frac{b \tanh ^3(c+d x)}{3 d}-(-a-b) \int \tanh ^2(c+d x) \, dx\\ &=-\frac{(a+b) \tanh (c+d x)}{d}-\frac{b \tanh ^3(c+d x)}{3 d}-(-a-b) \int 1 \, dx\\ &=(a+b) x-\frac{(a+b) \tanh (c+d x)}{d}-\frac{b \tanh ^3(c+d x)}{3 d}\\ \end{align*}

Mathematica [A]  time = 0.0248043, size = 65, normalized size = 1.81 \[ \frac{a \tanh ^{-1}(\tanh (c+d x))}{d}-\frac{a \tanh (c+d x)}{d}-\frac{b \tanh ^3(c+d x)}{3 d}+\frac{b \tanh ^{-1}(\tanh (c+d x))}{d}-\frac{b \tanh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Tanh[c + d*x]^2*(a + b*Tanh[c + d*x]^2),x]

[Out]

(a*ArcTanh[Tanh[c + d*x]])/d + (b*ArcTanh[Tanh[c + d*x]])/d - (a*Tanh[c + d*x])/d - (b*Tanh[c + d*x])/d - (b*T
anh[c + d*x]^3)/(3*d)

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Maple [B]  time = 0.005, size = 100, normalized size = 2.8 \begin{align*} -{\frac{b \left ( \tanh \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-{\frac{a\tanh \left ( dx+c \right ) }{d}}-{\frac{b\tanh \left ( dx+c \right ) }{d}}-{\frac{\ln \left ( \tanh \left ( dx+c \right ) -1 \right ) a}{2\,d}}-{\frac{\ln \left ( \tanh \left ( dx+c \right ) -1 \right ) b}{2\,d}}+{\frac{\ln \left ( \tanh \left ( dx+c \right ) +1 \right ) a}{2\,d}}+{\frac{\ln \left ( \tanh \left ( dx+c \right ) +1 \right ) b}{2\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*b*tanh(d*x+c)^3/d-a*tanh(d*x+c)/d-b*tanh(d*x+c)/d-1/2/d*ln(tanh(d*x+c)-1)*a-1/2/d*ln(tanh(d*x+c)-1)*b+1/2
/d*ln(tanh(d*x+c)+1)*a+1/2/d*ln(tanh(d*x+c)+1)*b

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Maxima [B]  time = 1.13756, size = 142, normalized size = 3.94 \begin{align*} \frac{1}{3} \, b{\left (3 \, x + \frac{3 \, c}{d} - \frac{4 \,{\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + 2\right )}}{d{\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}}\right )} + a{\left (x + \frac{c}{d} - \frac{2}{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(tanh(d*x+c)^2*(a+b*tanh(d*x+c)^2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.83019, size = 428, normalized size = 11.89 \begin{align*} \frac{{\left (3 \,{\left (a + b\right )} d x + 3 \, a + 4 \, b\right )} \cosh \left (d x + c\right )^{3} + 3 \,{\left (3 \,{\left (a + b\right )} d x + 3 \, a + 4 \, b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} -{\left (3 \, a + 4 \, b\right )} \sinh \left (d x + c\right )^{3} + 3 \,{\left (3 \,{\left (a + b\right )} d x + 3 \, a + 4 \, b\right )} \cosh \left (d x + c\right ) - 3 \,{\left ({\left (3 \, a + 4 \, b\right )} \cosh \left (d x + c\right )^{2} + a\right )} \sinh \left (d x + c\right )}{3 \,{\left (d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 3 \, d \cosh \left (d x + c\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*((3*(a + b)*d*x + 3*a + 4*b)*cosh(d*x + c)^3 + 3*(3*(a + b)*d*x + 3*a + 4*b)*cosh(d*x + c)*sinh(d*x + c)^2
- (3*a + 4*b)*sinh(d*x + c)^3 + 3*(3*(a + b)*d*x + 3*a + 4*b)*cosh(d*x + c) - 3*((3*a + 4*b)*cosh(d*x + c)^2
+ a)*sinh(d*x + c))/(d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c)*sinh(d*x + c)^2 + 3*d*cosh(d*x + c))

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Sympy [A]  time = 0.381292, size = 54, normalized size = 1.5 \begin{align*} \begin{cases} a x - \frac{a \tanh{\left (c + d x \right )}}{d} + b x - \frac{b \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac{b \tanh{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a + b \tanh ^{2}{\left (c \right )}\right ) \tanh ^{2}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*x - a*tanh(c + d*x)/d + b*x - b*tanh(c + d*x)**3/(3*d) - b*tanh(c + d*x)/d, Ne(d, 0)), (x*(a + b*
tanh(c)**2)*tanh(c)**2, True))

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Giac [B]  time = 1.16641, size = 116, normalized size = 3.22 \begin{align*} \frac{{\left (d x + c\right )}{\left (a + b\right )}}{d} + \frac{2 \,{\left (3 \, a e^{\left (4 \, d x + 4 \, c\right )} + 6 \, b e^{\left (4 \, d x + 4 \, c\right )} + 6 \, a e^{\left (2 \, d x + 2 \, c\right )} + 6 \, b e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a + 4 \, b\right )}}{3 \, d{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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