3.1.2 \(\int \tanh ^5(a+b x) \, dx\) [2]

Optimal. Leaf size=42 \[ \frac {\log (\cosh (a+b x))}{b}-\frac {\tanh ^2(a+b x)}{2 b}-\frac {\tanh ^4(a+b x)}{4 b} \]

[Out]

ln(cosh(b*x+a))/b-1/2*tanh(b*x+a)^2/b-1/4*tanh(b*x+a)^4/b

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Rubi [A]
time = 0.02, antiderivative size = 42, 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 = {3554, 3556} \begin {gather*} -\frac {\tanh ^4(a+b x)}{4 b}-\frac {\tanh ^2(a+b x)}{2 b}+\frac {\log (\cosh (a+b x))}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tanh[a + b*x]^5,x]

[Out]

Log[Cosh[a + b*x]]/b - Tanh[a + b*x]^2/(2*b) - Tanh[a + b*x]^4/(4*b)

Rule 3554

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 3556

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 \tanh ^5(a+b x) \, dx &=-\frac {\tanh ^4(a+b x)}{4 b}+\int \tanh ^3(a+b x) \, dx\\ &=-\frac {\tanh ^2(a+b x)}{2 b}-\frac {\tanh ^4(a+b x)}{4 b}+\int \tanh (a+b x) \, dx\\ &=\frac {\log (\cosh (a+b x))}{b}-\frac {\tanh ^2(a+b x)}{2 b}-\frac {\tanh ^4(a+b x)}{4 b}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 37, normalized size = 0.88 \begin {gather*} \frac {4 \log (\cosh (a+b x))-2 \tanh ^2(a+b x)-\tanh ^4(a+b x)}{4 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Tanh[a + b*x]^5,x]

[Out]

(4*Log[Cosh[a + b*x]] - 2*Tanh[a + b*x]^2 - Tanh[a + b*x]^4)/(4*b)

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Maple [A]
time = 0.26, size = 48, normalized size = 1.14

method	result	size
derivativedivides	\(\frac {-\frac {\left (\tanh ^{4}\left (b x +a \right )\right )}{4}-\frac {\left (\tanh ^{2}\left (b x +a \right )\right )}{2}-\frac {\ln \left (-1+\tanh \left (b x +a \right )\right )}{2}-\frac {\ln \left (\tanh \left (b x +a \right )+1\right )}{2}}{b}\)	\(48\)
default	\(\frac {-\frac {\left (\tanh ^{4}\left (b x +a \right )\right )}{4}-\frac {\left (\tanh ^{2}\left (b x +a \right )\right )}{2}-\frac {\ln \left (-1+\tanh \left (b x +a \right )\right )}{2}-\frac {\ln \left (\tanh \left (b x +a \right )+1\right )}{2}}{b}\)	\(48\)
risch	\(-x -\frac {2 a}{b}+\frac {4 \,{\mathrm e}^{2 b x +2 a} \left ({\mathrm e}^{4 b x +4 a}+{\mathrm e}^{2 b x +2 a}+1\right )}{b \left ({\mathrm e}^{2 b x +2 a}+1\right )^{4}}+\frac {\ln \left ({\mathrm e}^{2 b x +2 a}+1\right )}{b}\)	\(74\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(b*x+a)^5,x,method=_RETURNVERBOSE)

[Out]

1/b*(-1/4*tanh(b*x+a)^4-1/2*tanh(b*x+a)^2-1/2*ln(-1+tanh(b*x+a))-1/2*ln(tanh(b*x+a)+1))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (38) = 76\).
time = 0.47, size = 102, normalized size = 2.43 \begin {gather*} x + \frac {a}{b} + \frac {\log \left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )}{b} + \frac {4 \, {\left (e^{\left (-2 \, b x - 2 \, a\right )} + e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )}\right )}}{b {\left (4 \, e^{\left (-2 \, b x - 2 \, a\right )} + 6 \, e^{\left (-4 \, b x - 4 \, a\right )} + 4 \, e^{\left (-6 \, b x - 6 \, a\right )} + e^{\left (-8 \, b x - 8 \, a\right )} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(b*x+a)^5,x, algorithm="maxima")

[Out]

x + a/b + log(e^(-2*b*x - 2*a) + 1)/b + 4*(e^(-2*b*x - 2*a) + e^(-4*b*x - 4*a) + e^(-6*b*x - 6*a))/(b*(4*e^(-2
*b*x - 2*a) + 6*e^(-4*b*x - 4*a) + 4*e^(-6*b*x - 6*a) + e^(-8*b*x - 8*a) + 1))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 968 vs. \(2 (38) = 76\).
time = 0.37, size = 968, normalized size = 23.05 \begin {gather*} -\frac {b x \cosh \left (b x + a\right )^{8} + 8 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{7} + b x \sinh \left (b x + a\right )^{8} + 4 \, {\left (b x - 1\right )} \cosh \left (b x + a\right )^{6} + 4 \, {\left (7 \, b x \cosh \left (b x + a\right )^{2} + b x - 1\right )} \sinh \left (b x + a\right )^{6} + 8 \, {\left (7 \, b x \cosh \left (b x + a\right )^{3} + 3 \, {\left (b x - 1\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{5} + 2 \, {\left (3 \, b x - 2\right )} \cosh \left (b x + a\right )^{4} + 2 \, {\left (35 \, b x \cosh \left (b x + a\right )^{4} + 30 \, {\left (b x - 1\right )} \cosh \left (b x + a\right )^{2} + 3 \, b x - 2\right )} \sinh \left (b x + a\right )^{4} + 8 \, {\left (7 \, b x \cosh \left (b x + a\right )^{5} + 10 \, {\left (b x - 1\right )} \cosh \left (b x + a\right )^{3} + {\left (3 \, b x - 2\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 4 \, {\left (b x - 1\right )} \cosh \left (b x + a\right )^{2} + 4 \, {\left (7 \, b x \cosh \left (b x + a\right )^{6} + 15 \, {\left (b x - 1\right )} \cosh \left (b x + a\right )^{4} + 3 \, {\left (3 \, b x - 2\right )} \cosh \left (b x + a\right )^{2} + b x - 1\right )} \sinh \left (b x + a\right )^{2} + b x - {\left (\cosh \left (b x + a\right )^{8} + 8 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{7} + \sinh \left (b x + a\right )^{8} + 4 \, {\left (7 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{6} + 4 \, \cosh \left (b x + a\right )^{6} + 8 \, {\left (7 \, \cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{5} + 2 \, {\left (35 \, \cosh \left (b x + a\right )^{4} + 30 \, \cosh \left (b x + a\right )^{2} + 3\right )} \sinh \left (b x + a\right )^{4} + 6 \, \cosh \left (b x + a\right )^{4} + 8 \, {\left (7 \, \cosh \left (b x + a\right )^{5} + 10 \, \cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 4 \, {\left (7 \, \cosh \left (b x + a\right )^{6} + 15 \, \cosh \left (b x + a\right )^{4} + 9 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{2} + 4 \, \cosh \left (b x + a\right )^{2} + 8 \, {\left (\cosh \left (b x + a\right )^{7} + 3 \, \cosh \left (b x + a\right )^{5} + 3 \, \cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \log \left (\frac {2 \, \cosh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right ) + 8 \, {\left (b x \cosh \left (b x + a\right )^{7} + 3 \, {\left (b x - 1\right )} \cosh \left (b x + a\right )^{5} + {\left (3 \, b x - 2\right )} \cosh \left (b x + a\right )^{3} + {\left (b x - 1\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{b \cosh \left (b x + a\right )^{8} + 8 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{7} + b \sinh \left (b x + a\right )^{8} + 4 \, b \cosh \left (b x + a\right )^{6} + 4 \, {\left (7 \, b \cosh \left (b x + a\right )^{2} + b\right )} \sinh \left (b x + a\right )^{6} + 8 \, {\left (7 \, b \cosh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{5} + 6 \, b \cosh \left (b x + a\right )^{4} + 2 \, {\left (35 \, b \cosh \left (b x + a\right )^{4} + 30 \, b \cosh \left (b x + a\right )^{2} + 3 \, b\right )} \sinh \left (b x + a\right )^{4} + 8 \, {\left (7 \, b \cosh \left (b x + a\right )^{5} + 10 \, b \cosh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 4 \, b \cosh \left (b x + a\right )^{2} + 4 \, {\left (7 \, b \cosh \left (b x + a\right )^{6} + 15 \, b \cosh \left (b x + a\right )^{4} + 9 \, b \cosh \left (b x + a\right )^{2} + b\right )} \sinh \left (b x + a\right )^{2} + 8 \, {\left (b \cosh \left (b x + a\right )^{7} + 3 \, b \cosh \left (b x + a\right )^{5} + 3 \, b \cosh \left (b x + a\right )^{3} + b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(b*x+a)^5,x, algorithm="fricas")

[Out]

-(b*x*cosh(b*x + a)^8 + 8*b*x*cosh(b*x + a)*sinh(b*x + a)^7 + b*x*sinh(b*x + a)^8 + 4*(b*x - 1)*cosh(b*x + a)^
6 + 4*(7*b*x*cosh(b*x + a)^2 + b*x - 1)*sinh(b*x + a)^6 + 8*(7*b*x*cosh(b*x + a)^3 + 3*(b*x - 1)*cosh(b*x + a)
)*sinh(b*x + a)^5 + 2*(3*b*x - 2)*cosh(b*x + a)^4 + 2*(35*b*x*cosh(b*x + a)^4 + 30*(b*x - 1)*cosh(b*x + a)^2 +
3*b*x - 2)*sinh(b*x + a)^4 + 8*(7*b*x*cosh(b*x + a)^5 + 10*(b*x - 1)*cosh(b*x + a)^3 + (3*b*x - 2)*cosh(b*x +
a))*sinh(b*x + a)^3 + 4*(b*x - 1)*cosh(b*x + a)^2 + 4*(7*b*x*cosh(b*x + a)^6 + 15*(b*x - 1)*cosh(b*x + a)^4 +
3*(3*b*x - 2)*cosh(b*x + a)^2 + b*x - 1)*sinh(b*x + a)^2 + b*x - (cosh(b*x + a)^8 + 8*cosh(b*x + a)*sinh(b*x
+ a)^7 + sinh(b*x + a)^8 + 4*(7*cosh(b*x + a)^2 + 1)*sinh(b*x + a)^6 + 4*cosh(b*x + a)^6 + 8*(7*cosh(b*x + a)^
3 + 3*cosh(b*x + a))*sinh(b*x + a)^5 + 2*(35*cosh(b*x + a)^4 + 30*cosh(b*x + a)^2 + 3)*sinh(b*x + a)^4 + 6*cos
h(b*x + a)^4 + 8*(7*cosh(b*x + a)^5 + 10*cosh(b*x + a)^3 + 3*cosh(b*x + a))*sinh(b*x + a)^3 + 4*(7*cosh(b*x +
a)^6 + 15*cosh(b*x + a)^4 + 9*cosh(b*x + a)^2 + 1)*sinh(b*x + a)^2 + 4*cosh(b*x + a)^2 + 8*(cosh(b*x + a)^7 +
3*cosh(b*x + a)^5 + 3*cosh(b*x + a)^3 + cosh(b*x + a))*sinh(b*x + a) + 1)*log(2*cosh(b*x + a)/(cosh(b*x + a) -
sinh(b*x + a))) + 8*(b*x*cosh(b*x + a)^7 + 3*(b*x - 1)*cosh(b*x + a)^5 + (3*b*x - 2)*cosh(b*x + a)^3 + (b*x -
1)*cosh(b*x + a))*sinh(b*x + a))/(b*cosh(b*x + a)^8 + 8*b*cosh(b*x + a)*sinh(b*x + a)^7 + b*sinh(b*x + a)^8 +
4*b*cosh(b*x + a)^6 + 4*(7*b*cosh(b*x + a)^2 + b)*sinh(b*x + a)^6 + 8*(7*b*cosh(b*x + a)^3 + 3*b*cosh(b*x + a
))*sinh(b*x + a)^5 + 6*b*cosh(b*x + a)^4 + 2*(35*b*cosh(b*x + a)^4 + 30*b*cosh(b*x + a)^2 + 3*b)*sinh(b*x + a)
^4 + 8*(7*b*cosh(b*x + a)^5 + 10*b*cosh(b*x + a)^3 + 3*b*cosh(b*x + a))*sinh(b*x + a)^3 + 4*b*cosh(b*x + a)^2
+ 4*(7*b*cosh(b*x + a)^6 + 15*b*cosh(b*x + a)^4 + 9*b*cosh(b*x + a)^2 + b)*sinh(b*x + a)^2 + 8*(b*cosh(b*x + a
)^7 + 3*b*cosh(b*x + a)^5 + 3*b*cosh(b*x + a)^3 + b*cosh(b*x + a))*sinh(b*x + a) + b)

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Sympy [A]
time = 0.12, size = 42, normalized size = 1.00 \begin {gather*} \begin {cases} x - \frac {\log {\left (\tanh {\left (a + b x \right )} + 1 \right )}}{b} - \frac {\tanh ^{4}{\left (a + b x \right )}}{4 b} - \frac {\tanh ^{2}{\left (a + b x \right )}}{2 b} & \text {for}\: b \neq 0 \\x \tanh ^{5}{\left (a \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(b*x+a)**5,x)

[Out]

Piecewise((x - log(tanh(a + b*x) + 1)/b - tanh(a + b*x)**4/(4*b) - tanh(a + b*x)**2/(2*b), Ne(b, 0)), (x*tanh(
a)**5, True))

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Giac [A]
time = 0.42, size = 67, normalized size = 1.60 \begin {gather*} -\frac {b x + a - \frac {4 \, {\left (e^{\left (6 \, b x + 6 \, a\right )} + e^{\left (4 \, b x + 4 \, a\right )} + e^{\left (2 \, b x + 2 \, a\right )}\right )}}{{\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}^{4}} - \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(b*x+a)^5,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.12, size = 37, normalized size = 0.88 \begin {gather*} x-\frac {\ln \left (\mathrm {tanh}\left (a+b\,x\right )+1\right )+\frac {{\mathrm {tanh}\left (a+b\,x\right )}^2}{2}+\frac {{\mathrm {tanh}\left (a+b\,x\right )}^4}{4}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(a + b*x)^5,x)

[Out]

x - (log(tanh(a + b*x) + 1) + tanh(a + b*x)^2/2 + tanh(a + b*x)^4/4)/b

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