3.108 \(\int \cosh ^3(a+b x) \coth (a+b x) \, dx\)

Optimal. Leaf size=38 \[ \frac {\cosh ^3(a+b x)}{3 b}+\frac {\cosh (a+b x)}{b}-\frac {\tanh ^{-1}(\cosh (a+b x))}{b} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2592, 302, 206} \[ \frac {\cosh ^3(a+b x)}{3 b}+\frac {\cosh (a+b x)}{b}-\frac {\tanh ^{-1}(\cosh (a+b x))}{b} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x]^3*Coth[a + b*x],x]

[Out]

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 2592

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> With[{ff = FreeFactors[S
in[e + f*x], x]}, Dist[ff/f, Subst[Int[(ff*x)^(m + n)/(a^2 - ff^2*x^2)^((n + 1)/2), x], x, (a*Sin[e + f*x])/ff
], x]] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 44, normalized size = 1.16 \[ \frac {5 \cosh (a+b x)}{4 b}+\frac {\cosh (3 (a+b x))}{12 b}+\frac {\log \left (\tanh \left (\frac {1}{2} (a+b x)\right )\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + b*x]^3*Coth[a + b*x],x]

[Out]

(5*Cosh[a + b*x])/(4*b) + Cosh[3*(a + b*x)]/(12*b) + Log[Tanh[(a + b*x)/2]]/b

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fricas [B]  time = 0.52, size = 357, normalized size = 9.39 \[ \frac {\cosh \left (b x + a\right )^{6} + 6 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + \sinh \left (b x + a\right )^{6} + 15 \, {\left (\cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{4} + 15 \, \cosh \left (b x + a\right )^{4} + 20 \, {\left (\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 15 \, {\left (\cosh \left (b x + a\right )^{4} + 6 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{2} + 15 \, \cosh \left (b x + a\right )^{2} - 24 \, {\left (\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right ) + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + 24 \, {\left (\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right ) + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 6 \, {\left (\cosh \left (b x + a\right )^{5} + 10 \, \cosh \left (b x + a\right )^{3} + 5 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1}{24 \, {\left (b \cosh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right ) + 3 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b \sinh \left (b x + a\right )^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^3*coth(b*x+a),x, algorithm="fricas")

[Out]

1/24*(cosh(b*x + a)^6 + 6*cosh(b*x + a)*sinh(b*x + a)^5 + sinh(b*x + a)^6 + 15*(cosh(b*x + a)^2 + 1)*sinh(b*x
+ a)^4 + 15*cosh(b*x + a)^4 + 20*(cosh(b*x + a)^3 + 3*cosh(b*x + a))*sinh(b*x + a)^3 + 15*(cosh(b*x + a)^4 + 6
*cosh(b*x + a)^2 + 1)*sinh(b*x + a)^2 + 15*cosh(b*x + a)^2 - 24*(cosh(b*x + a)^3 + 3*cosh(b*x + a)^2*sinh(b*x
+ a) + 3*cosh(b*x + a)*sinh(b*x + a)^2 + sinh(b*x + a)^3)*log(cosh(b*x + a) + sinh(b*x + a) + 1) + 24*(cosh(b*
x + a)^3 + 3*cosh(b*x + a)^2*sinh(b*x + a) + 3*cosh(b*x + a)*sinh(b*x + a)^2 + sinh(b*x + a)^3)*log(cosh(b*x +
a) + sinh(b*x + a) - 1) + 6*(cosh(b*x + a)^5 + 10*cosh(b*x + a)^3 + 5*cosh(b*x + a))*sinh(b*x + a) + 1)/(b*co
sh(b*x + a)^3 + 3*b*cosh(b*x + a)^2*sinh(b*x + a) + 3*b*cosh(b*x + a)*sinh(b*x + a)^2 + b*sinh(b*x + a)^3)

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giac [B]  time = 0.16, size = 77, normalized size = 2.03 \[ \frac {{\left (15 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )} e^{\left (-3 \, b x - 3 \, a\right )} + {\left (e^{\left (3 \, b x + 18 \, a\right )} + 15 \, e^{\left (b x + 16 \, a\right )}\right )} e^{\left (-15 \, a\right )} - 24 \, \log \left (e^{\left (b x + a\right )} + 1\right ) + 24 \, \log \left ({\left | e^{\left (b x + a\right )} - 1 \right |}\right )}{24 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^3*coth(b*x+a),x, algorithm="giac")

[Out]

1/24*((15*e^(2*b*x + 2*a) + 1)*e^(-3*b*x - 3*a) + (e^(3*b*x + 18*a) + 15*e^(b*x + 16*a))*e^(-15*a) - 24*log(e^
(b*x + a) + 1) + 24*log(abs(e^(b*x + a) - 1)))/b

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maple [A]  time = 0.12, size = 31, normalized size = 0.82 \[ \frac {\frac {\left (\cosh ^{3}\left (b x +a \right )\right )}{3}+\cosh \left (b x +a \right )-2 \arctanh \left ({\mathrm e}^{b x +a}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(b*x+a)^3*coth(b*x+a),x)

[Out]

1/b*(1/3*cosh(b*x+a)^3+cosh(b*x+a)-2*arctanh(exp(b*x+a)))

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maxima [B]  time = 0.32, size = 87, normalized size = 2.29 \[ \frac {{\left (15 \, e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )} e^{\left (3 \, b x + 3 \, a\right )}}{24 \, b} + \frac {15 \, e^{\left (-b x - a\right )} + e^{\left (-3 \, b x - 3 \, a\right )}}{24 \, b} - \frac {\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} + \frac {\log \left (e^{\left (-b x - a\right )} - 1\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^3*coth(b*x+a),x, algorithm="maxima")

[Out]

1/24*(15*e^(-2*b*x - 2*a) + 1)*e^(3*b*x + 3*a)/b + 1/24*(15*e^(-b*x - a) + e^(-3*b*x - 3*a))/b - log(e^(-b*x -
a) + 1)/b + log(e^(-b*x - a) - 1)/b

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mupad [B]  time = 1.44, size = 81, normalized size = 2.13 \[ \frac {5\,{\mathrm {e}}^{a+b\,x}}{8\,b}-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^2}}{b}\right )}{\sqrt {-b^2}}+\frac {5\,{\mathrm {e}}^{-a-b\,x}}{8\,b}+\frac {{\mathrm {e}}^{-3\,a-3\,b\,x}}{24\,b}+\frac {{\mathrm {e}}^{3\,a+3\,b\,x}}{24\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*x)^3*coth(a + b*x),x)

[Out]

(5*exp(a + b*x))/(8*b) - (2*atan((exp(b*x)*exp(a)*(-b^2)^(1/2))/b))/(-b^2)^(1/2) + (5*exp(- a - b*x))/(8*b) +
exp(- 3*a - 3*b*x)/(24*b) + exp(3*a + 3*b*x)/(24*b)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cosh ^{3}{\left (a + b x \right )} \coth {\left (a + b x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)**3*coth(b*x+a),x)

[Out]

Integral(cosh(a + b*x)**3*coth(a + b*x), x)

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