3.2.10 \(\int \cosh ^3(a+b x) \coth ^3(a+b x) \, dx\) [110]

Optimal. Leaf size=66 \[ -\frac {5 \tanh ^{-1}(\cosh (a+b x))}{2 b}+\frac {5 \cosh (a+b x)}{2 b}+\frac {5 \cosh ^3(a+b x)}{6 b}-\frac {\cosh ^3(a+b x) \coth ^2(a+b x)}{2 b} \]

[Out]

-5/2*arctanh(cosh(b*x+a))/b+5/2*cosh(b*x+a)/b+5/6*cosh(b*x+a)^3/b-1/2*cosh(b*x+a)^3*coth(b*x+a)^2/b

________________________________________________________________________________________

Rubi [A]
time = 0.04, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2672, 294, 308, 212} \begin {gather*} \frac {5 \cosh ^3(a+b x)}{6 b}+\frac {5 \cosh (a+b x)}{2 b}-\frac {\cosh ^3(a+b x) \coth ^2(a+b x)}{2 b}-\frac {5 \tanh ^{-1}(\cosh (a+b x))}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*ArcTanh[Cosh[a + b*x]])/(2*b) + (5*Cosh[a + b*x])/(2*b) + (5*Cosh[a + b*x]^3)/(6*b) - (Cosh[a + b*x]^3*Cot
h[a + b*x]^2)/(2*b)

Rule 212

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

Rule 294

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^
n)^(p + 1)/(b*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 308

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 2672

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 ^3(a+b x) \, dx &=\frac {\text {Subst}\left (\int \frac {x^6}{\left (1-x^2\right )^2} \, dx,x,\cosh (a+b x)\right )}{b}\\ &=-\frac {\cosh ^3(a+b x) \coth ^2(a+b x)}{2 b}-\frac {5 \text {Subst}\left (\int \frac {x^4}{1-x^2} \, dx,x,\cosh (a+b x)\right )}{2 b}\\ &=-\frac {\cosh ^3(a+b x) \coth ^2(a+b x)}{2 b}-\frac {5 \text {Subst}\left (\int \left (-1-x^2+\frac {1}{1-x^2}\right ) \, dx,x,\cosh (a+b x)\right )}{2 b}\\ &=\frac {5 \cosh (a+b x)}{2 b}+\frac {5 \cosh ^3(a+b x)}{6 b}-\frac {\cosh ^3(a+b x) \coth ^2(a+b x)}{2 b}-\frac {5 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (a+b x)\right )}{2 b}\\ &=-\frac {5 \tanh ^{-1}(\cosh (a+b x))}{2 b}+\frac {5 \cosh (a+b x)}{2 b}+\frac {5 \cosh ^3(a+b x)}{6 b}-\frac {\cosh ^3(a+b x) \coth ^2(a+b x)}{2 b}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.04, size = 85, normalized size = 1.29 \begin {gather*} \frac {9 \cosh (a+b x)}{4 b}+\frac {\cosh (3 (a+b x))}{12 b}-\frac {\text {csch}^2\left (\frac {1}{2} (a+b x)\right )}{8 b}+\frac {5 \log \left (\tanh \left (\frac {1}{2} (a+b x)\right )\right )}{2 b}-\frac {\text {sech}^2\left (\frac {1}{2} (a+b x)\right )}{8 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(9*Cosh[a + b*x])/(4*b) + Cosh[3*(a + b*x)]/(12*b) - Csch[(a + b*x)/2]^2/(8*b) + (5*Log[Tanh[(a + b*x)/2]])/(2
*b) - Sech[(a + b*x)/2]^2/(8*b)

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(117\) vs. \(2(58)=116\).
time = 1.83, size = 118, normalized size = 1.79

method	result	size
risch	\(\frac {{\mathrm e}^{3 b x +3 a}}{24 b}+\frac {9 \,{\mathrm e}^{b x +a}}{8 b}+\frac {9 \,{\mathrm e}^{-b x -a}}{8 b}+\frac {{\mathrm e}^{-3 b x -3 a}}{24 b}-\frac {{\mathrm e}^{b x +a} \left ({\mathrm e}^{2 b x +2 a}+1\right )}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}-\frac {5 \ln \left ({\mathrm e}^{b x +a}+1\right )}{2 b}+\frac {5 \ln \left ({\mathrm e}^{b x +a}-1\right )}{2 b}\)	\(118\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*exp(3*b*x+3*a)/b+9/8*exp(b*x+a)/b+9/8*exp(-b*x-a)/b+1/24*exp(-3*b*x-3*a)/b-exp(b*x+a)*(exp(2*b*x+2*a)+1)/
b/(exp(2*b*x+2*a)-1)^2-5/2/b*ln(exp(b*x+a)+1)+5/2/b*ln(exp(b*x+a)-1)

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (58) = 116\).
time = 0.26, size = 133, normalized size = 2.02 \begin {gather*} \frac {27 \, e^{\left (-b x - a\right )} + e^{\left (-3 \, b x - 3 \, a\right )}}{24 \, b} - \frac {5 \, \log \left (e^{\left (-b x - a\right )} + 1\right )}{2 \, b} + \frac {5 \, \log \left (e^{\left (-b x - a\right )} - 1\right )}{2 \, b} + \frac {25 \, e^{\left (-2 \, b x - 2 \, a\right )} - 77 \, e^{\left (-4 \, b x - 4 \, a\right )} + 3 \, e^{\left (-6 \, b x - 6 \, a\right )} + 1}{24 \, b {\left (e^{\left (-3 \, b x - 3 \, a\right )} - 2 \, e^{\left (-5 \, b x - 5 \, a\right )} + e^{\left (-7 \, b x - 7 \, a\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(27*e^(-b*x - a) + e^(-3*b*x - 3*a))/b - 5/2*log(e^(-b*x - a) + 1)/b + 5/2*log(e^(-b*x - a) - 1)/b + 1/24
*(25*e^(-2*b*x - 2*a) - 77*e^(-4*b*x - 4*a) + 3*e^(-6*b*x - 6*a) + 1)/(b*(e^(-3*b*x - 3*a) - 2*e^(-5*b*x - 5*a
) + e^(-7*b*x - 7*a)))

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1077 vs. \(2 (58) = 116\).
time = 0.41, size = 1077, normalized size = 16.32 \begin {gather*} \frac {\cosh \left (b x + a\right )^{10} + 10 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{9} + \sinh \left (b x + a\right )^{10} + 5 \, {\left (9 \, \cosh \left (b x + a\right )^{2} + 5\right )} \sinh \left (b x + a\right )^{8} + 25 \, \cosh \left (b x + a\right )^{8} + 40 \, {\left (3 \, \cosh \left (b x + a\right )^{3} + 5 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{7} + 10 \, {\left (21 \, \cosh \left (b x + a\right )^{4} + 70 \, \cosh \left (b x + a\right )^{2} - 5\right )} \sinh \left (b x + a\right )^{6} - 50 \, \cosh \left (b x + a\right )^{6} + 4 \, {\left (63 \, \cosh \left (b x + a\right )^{5} + 350 \, \cosh \left (b x + a\right )^{3} - 75 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{5} + 10 \, {\left (21 \, \cosh \left (b x + a\right )^{6} + 175 \, \cosh \left (b x + a\right )^{4} - 75 \, \cosh \left (b x + a\right )^{2} - 5\right )} \sinh \left (b x + a\right )^{4} - 50 \, \cosh \left (b x + a\right )^{4} + 40 \, {\left (3 \, \cosh \left (b x + a\right )^{7} + 35 \, \cosh \left (b x + a\right )^{5} - 25 \, \cosh \left (b x + a\right )^{3} - 5 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 5 \, {\left (9 \, \cosh \left (b x + a\right )^{8} + 140 \, \cosh \left (b x + a\right )^{6} - 150 \, \cosh \left (b x + a\right )^{4} - 60 \, \cosh \left (b x + a\right )^{2} + 5\right )} \sinh \left (b x + a\right )^{2} + 25 \, \cosh \left (b x + a\right )^{2} - 60 \, {\left (\cosh \left (b x + a\right )^{7} + 7 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{6} + \sinh \left (b x + a\right )^{7} + {\left (21 \, \cosh \left (b x + a\right )^{2} - 2\right )} \sinh \left (b x + a\right )^{5} - 2 \, \cosh \left (b x + a\right )^{5} + 5 \, {\left (7 \, \cosh \left (b x + a\right )^{3} - 2 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{4} + {\left (35 \, \cosh \left (b x + a\right )^{4} - 20 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )^{3} + {\left (21 \, \cosh \left (b x + a\right )^{5} - 20 \, \cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} + {\left (7 \, \cosh \left (b x + a\right )^{6} - 10 \, \cosh \left (b x + a\right )^{4} + 3 \, \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + 60 \, {\left (\cosh \left (b x + a\right )^{7} + 7 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{6} + \sinh \left (b x + a\right )^{7} + {\left (21 \, \cosh \left (b x + a\right )^{2} - 2\right )} \sinh \left (b x + a\right )^{5} - 2 \, \cosh \left (b x + a\right )^{5} + 5 \, {\left (7 \, \cosh \left (b x + a\right )^{3} - 2 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{4} + {\left (35 \, \cosh \left (b x + a\right )^{4} - 20 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )^{3} + {\left (21 \, \cosh \left (b x + a\right )^{5} - 20 \, \cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} + {\left (7 \, \cosh \left (b x + a\right )^{6} - 10 \, \cosh \left (b x + a\right )^{4} + 3 \, \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 10 \, {\left (\cosh \left (b x + a\right )^{9} + 20 \, \cosh \left (b x + a\right )^{7} - 30 \, \cosh \left (b x + a\right )^{5} - 20 \, \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 )^{7} + 7 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{6} + b \sinh \left (b x + a\right )^{7} - 2 \, b \cosh \left (b x + a\right )^{5} + {\left (21 \, b \cosh \left (b x + a\right )^{2} - 2 \, b\right )} \sinh \left (b x + a\right )^{5} + 5 \, {\left (7 \, b \cosh \left (b x + a\right )^{3} - 2 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{4} + b \cosh \left (b x + a\right )^{3} + {\left (35 \, b \cosh \left (b x + a\right )^{4} - 20 \, b \cosh \left (b x + a\right )^{2} + b\right )} \sinh \left (b x + a\right )^{3} + {\left (21 \, b \cosh \left (b x + a\right )^{5} - 20 \, b \cosh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} + {\left (7 \, b \cosh \left (b x + a\right )^{6} - 10 \, b \cosh \left (b x + a\right )^{4} + 3 \, b \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(cosh(b*x + a)^10 + 10*cosh(b*x + a)*sinh(b*x + a)^9 + sinh(b*x + a)^10 + 5*(9*cosh(b*x + a)^2 + 5)*sinh(
b*x + a)^8 + 25*cosh(b*x + a)^8 + 40*(3*cosh(b*x + a)^3 + 5*cosh(b*x + a))*sinh(b*x + a)^7 + 10*(21*cosh(b*x +
a)^4 + 70*cosh(b*x + a)^2 - 5)*sinh(b*x + a)^6 - 50*cosh(b*x + a)^6 + 4*(63*cosh(b*x + a)^5 + 350*cosh(b*x +
a)^3 - 75*cosh(b*x + a))*sinh(b*x + a)^5 + 10*(21*cosh(b*x + a)^6 + 175*cosh(b*x + a)^4 - 75*cosh(b*x + a)^2 -
5)*sinh(b*x + a)^4 - 50*cosh(b*x + a)^4 + 40*(3*cosh(b*x + a)^7 + 35*cosh(b*x + a)^5 - 25*cosh(b*x + a)^3 - 5
*cosh(b*x + a))*sinh(b*x + a)^3 + 5*(9*cosh(b*x + a)^8 + 140*cosh(b*x + a)^6 - 150*cosh(b*x + a)^4 - 60*cosh(b
*x + a)^2 + 5)*sinh(b*x + a)^2 + 25*cosh(b*x + a)^2 - 60*(cosh(b*x + a)^7 + 7*cosh(b*x + a)*sinh(b*x + a)^6 +
sinh(b*x + a)^7 + (21*cosh(b*x + a)^2 - 2)*sinh(b*x + a)^5 - 2*cosh(b*x + a)^5 + 5*(7*cosh(b*x + a)^3 - 2*cosh
(b*x + a))*sinh(b*x + a)^4 + (35*cosh(b*x + a)^4 - 20*cosh(b*x + a)^2 + 1)*sinh(b*x + a)^3 + cosh(b*x + a)^3 +
(21*cosh(b*x + a)^5 - 20*cosh(b*x + a)^3 + 3*cosh(b*x + a))*sinh(b*x + a)^2 + (7*cosh(b*x + a)^6 - 10*cosh(b*
x + a)^4 + 3*cosh(b*x + a)^2)*sinh(b*x + a))*log(cosh(b*x + a) + sinh(b*x + a) + 1) + 60*(cosh(b*x + a)^7 + 7*
cosh(b*x + a)*sinh(b*x + a)^6 + sinh(b*x + a)^7 + (21*cosh(b*x + a)^2 - 2)*sinh(b*x + a)^5 - 2*cosh(b*x + a)^5
+ 5*(7*cosh(b*x + a)^3 - 2*cosh(b*x + a))*sinh(b*x + a)^4 + (35*cosh(b*x + a)^4 - 20*cosh(b*x + a)^2 + 1)*sin
h(b*x + a)^3 + cosh(b*x + a)^3 + (21*cosh(b*x + a)^5 - 20*cosh(b*x + a)^3 + 3*cosh(b*x + a))*sinh(b*x + a)^2 +
(7*cosh(b*x + a)^6 - 10*cosh(b*x + a)^4 + 3*cosh(b*x + a)^2)*sinh(b*x + a))*log(cosh(b*x + a) + sinh(b*x + a)
- 1) + 10*(cosh(b*x + a)^9 + 20*cosh(b*x + a)^7 - 30*cosh(b*x + a)^5 - 20*cosh(b*x + a)^3 + 5*cosh(b*x + a))*
sinh(b*x + a) + 1)/(b*cosh(b*x + a)^7 + 7*b*cosh(b*x + a)*sinh(b*x + a)^6 + b*sinh(b*x + a)^7 - 2*b*cosh(b*x +
a)^5 + (21*b*cosh(b*x + a)^2 - 2*b)*sinh(b*x + a)^5 + 5*(7*b*cosh(b*x + a)^3 - 2*b*cosh(b*x + a))*sinh(b*x +
a)^4 + b*cosh(b*x + a)^3 + (35*b*cosh(b*x + a)^4 - 20*b*cosh(b*x + a)^2 + b)*sinh(b*x + a)^3 + (21*b*cosh(b*x
+ a)^5 - 20*b*cosh(b*x + a)^3 + 3*b*cosh(b*x + a))*sinh(b*x + a)^2 + (7*b*cosh(b*x + a)^6 - 10*b*cosh(b*x + a)
^4 + 3*b*cosh(b*x + a)^2)*sinh(b*x + a))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (58) = 116\).
time = 0.43, size = 123, normalized size = 1.86 \begin {gather*} \frac {{\left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}\right )}^{3} - \frac {24 \, {\left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}\right )}}{{\left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}\right )}^{2} - 4} + 24 \, e^{\left (b x + a\right )} + 24 \, e^{\left (-b x - a\right )} - 30 \, \log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} + 2\right ) + 30 \, \log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} - 2\right )}{24 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*((e^(b*x + a) + e^(-b*x - a))^3 - 24*(e^(b*x + a) + e^(-b*x - a))/((e^(b*x + a) + e^(-b*x - a))^2 - 4) +
24*e^(b*x + a) + 24*e^(-b*x - a) - 30*log(e^(b*x + a) + e^(-b*x - a) + 2) + 30*log(e^(b*x + a) + e^(-b*x - a)
- 2))/b

________________________________________________________________________________________

Mupad [B]
time = 1.49, size = 140, normalized size = 2.12 \begin {gather*} \frac {9\,{\mathrm {e}}^{a+b\,x}}{8\,b}-\frac {5\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^2}}{b}\right )}{\sqrt {-b^2}}+\frac {9\,{\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}-\frac {2\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-2\,{\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}-\frac {{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(9*exp(a + b*x))/(8*b) - (5*atan((exp(b*x)*exp(a)*(-b^2)^(1/2))/b))/(-b^2)^(1/2) + (9*exp(- a - b*x))/(8*b) +
exp(- 3*a - 3*b*x)/(24*b) + exp(3*a + 3*b*x)/(24*b) - (2*exp(a + b*x))/(b*(exp(4*a + 4*b*x) - 2*exp(2*a + 2*b*
x) + 1)) - exp(a + b*x)/(b*(exp(2*a + 2*b*x) - 1))

________________________________________________________________________________________