3.3 \(\int \tanh ^4(a+b x) \, dx\)

Optimal. Leaf size=28 \[ -\frac {\tanh ^3(a+b x)}{3 b}-\frac {\tanh (a+b x)}{b}+x \]

[Out]

x-tanh(b*x+a)/b-1/3*tanh(b*x+a)^3/b

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Rubi [A]  time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3473, 8} \[ -\frac {\tanh ^3(a+b x)}{3 b}-\frac {\tanh (a+b x)}{b}+x \]

Antiderivative was successfully verified.

[In]

Int[Tanh[a + b*x]^4,x]

[Out]

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

Rule 8

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

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]

Rubi steps

\begin {align*} \int \tanh ^4(a+b x) \, dx &=-\frac {\tanh ^3(a+b x)}{3 b}+\int \tanh ^2(a+b x) \, dx\\ &=-\frac {\tanh (a+b x)}{b}-\frac {\tanh ^3(a+b x)}{3 b}+\int 1 \, dx\\ &=x-\frac {\tanh (a+b x)}{b}-\frac {\tanh ^3(a+b x)}{3 b}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 38, normalized size = 1.36 \[ \frac {\tanh ^{-1}(\tanh (a+b x))}{b}-\frac {\tanh ^3(a+b x)}{3 b}-\frac {\tanh (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Tanh[a + b*x]^4,x]

[Out]

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

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fricas [B]  time = 0.51, size = 119, normalized size = 4.25 \[ \frac {{\left (3 \, b x + 4\right )} \cosh \left (b x + a\right )^{3} + 3 \, {\left (3 \, b x + 4\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 12 \, \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right ) - 4 \, \sinh \left (b x + a\right )^{3} + 3 \, {\left (3 \, b x + 4\right )} \cosh \left (b x + a\right )}{3 \, {\left (b \cosh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + 3 \, b \cosh \left (b x + a\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(b*x+a)^4,x, algorithm="fricas")

[Out]

1/3*((3*b*x + 4)*cosh(b*x + a)^3 + 3*(3*b*x + 4)*cosh(b*x + a)*sinh(b*x + a)^2 - 12*cosh(b*x + a)^2*sinh(b*x +
a) - 4*sinh(b*x + a)^3 + 3*(3*b*x + 4)*cosh(b*x + a))/(b*cosh(b*x + a)^3 + 3*b*cosh(b*x + a)*sinh(b*x + a)^2
+ 3*b*cosh(b*x + a))

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giac [A]  time = 0.14, size = 52, normalized size = 1.86 \[ \frac {3 \, b x + 3 \, a + \frac {4 \, {\left (3 \, e^{\left (4 \, b x + 4 \, a\right )} + 3 \, e^{\left (2 \, b x + 2 \, a\right )} + 2\right )}}{{\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}^{3}}}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(b*x+a)^4,x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.01, size = 54, normalized size = 1.93 \[ -\frac {\tanh ^{3}\left (b x +a \right )}{3 b}-\frac {\tanh \left (b x +a \right )}{b}-\frac {\ln \left (-1+\tanh \left (b x +a \right )\right )}{2 b}+\frac {\ln \left (1+\tanh \left (b x +a \right )\right )}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(b*x+a)^4,x)

[Out]

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

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maxima [B]  time = 0.31, size = 71, normalized size = 2.54 \[ x + \frac {a}{b} - \frac {4 \, {\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} + 3 \, e^{\left (-4 \, b x - 4 \, a\right )} + 2\right )}}{3 \, b {\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} + 3 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(b*x+a)^4,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.01, size = 24, normalized size = 0.86 \[ x-\frac {\frac {{\mathrm {tanh}\left (a+b\,x\right )}^3}{3}+\mathrm {tanh}\left (a+b\,x\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(a + b*x)^4,x)

[Out]

x - (tanh(a + b*x) + tanh(a + b*x)^3/3)/b

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sympy [A]  time = 0.26, size = 27, normalized size = 0.96 \[ \begin {cases} x - \frac {\tanh ^{3}{\left (a + b x \right )}}{3 b} - \frac {\tanh {\left (a + b x \right )}}{b} & \text {for}\: b \neq 0 \\x \tanh ^{4}{\relax (a )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(b*x+a)**4,x)

[Out]

Piecewise((x - tanh(a + b*x)**3/(3*b) - tanh(a + b*x)/b, Ne(b, 0)), (x*tanh(a)**4, True))

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