3.7.45 \(\int (a \coth (x)+b \text {csch}(x))^4 \, dx\) [645]

Optimal. Leaf size=101 \[ a^4 x-\frac {1}{3} (b+a \cosh (x))^2 \left (a b+\left (3 a^2-2 b^2\right ) \cosh (x)\right ) \text {csch}(x)-\frac {1}{3} (b+a \cosh (x))^3 (a+b \cosh (x)) \text {csch}^3(x)+\frac {4}{3} a b \left (2 a^2-b^2\right ) \sinh (x)+\frac {1}{3} a^2 \left (3 a^2-2 b^2\right ) \cosh (x) \sinh (x) \]

[Out]

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

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Rubi [A]
time = 0.16, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {4477, 2770, 2940, 2813} \begin {gather*} a^4 x+\frac {4}{3} a b \left (2 a^2-b^2\right ) \sinh (x)+\frac {1}{3} a^2 \left (3 a^2-2 b^2\right ) \sinh (x) \cosh (x)-\frac {1}{3} \text {csch}(x) (a \cosh (x)+b)^2 \left (\left (3 a^2-2 b^2\right ) \cosh (x)+a b\right )-\frac {1}{3} \text {csch}^3(x) (a \cosh (x)+b)^3 (a+b \cosh (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*Coth[x] + b*Csch[x])^4,x]

[Out]

a^4*x - ((b + a*Cosh[x])^2*(a*b + (3*a^2 - 2*b^2)*Cosh[x])*Csch[x])/3 - ((b + a*Cosh[x])^3*(a + b*Cosh[x])*Csc
h[x]^3)/3 + (4*a*b*(2*a^2 - b^2)*Sinh[x])/3 + (a^2*(3*a^2 - 2*b^2)*Cosh[x]*Sinh[x])/3

Rule 2770

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(-(g*C
os[e + f*x])^(p + 1))*(a + b*Sin[e + f*x])^(m - 1)*((b + a*Sin[e + f*x])/(f*g*(p + 1))), x] + Dist[1/(g^2*(p +
1)), Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*(p + 2) + a*b*(m + p + 1)*S
in[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && LtQ[p, -1] && (Integers
Q[2*m, 2*p] || IntegerQ[m])

Rule 2813

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*a*c +
b*d)*(x/2), x] + (-Simp[(b*c + a*d)*(Cos[e + f*x]/f), x] - Simp[b*d*Cos[e + f*x]*(Sin[e + f*x]/(2*f)), x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]

Rule 2940

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[(-(g*Cos[e + f*x])^(p + 1))*(a + b*Sin[e + f*x])^m*((d + c*Sin[e + f*x])/(f
*g*(p + 1))), x] + Dist[1/(g^2*(p + 1)), Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 1)*Simp[a*c*(p
+ 2) + b*d*m + b*c*(m + p + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2,
0] && GtQ[m, 0] && LtQ[p, -1] && IntegerQ[2*m] &&  !(EqQ[m, 1] && NeQ[c^2 - d^2, 0] && SimplerQ[c + d*x, a + b
*x])

Rule 4477

Int[(cot[(c_.) + (d_.)*(x_)]^(n_.)*(a_.) + csc[(c_.) + (d_.)*(x_)]^(n_.)*(b_.))^(p_)*(u_.), x_Symbol] :> Int[A
ctivateTrig[u]*Csc[c + d*x]^(n*p)*(b + a*Cos[c + d*x]^n)^p, x] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p]

Rubi steps

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

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Mathematica [A]
time = 0.19, size = 95, normalized size = 0.94 \begin {gather*} -\frac {1}{12} \text {csch}^3(x) \left (-8 a^3 b+16 a b^3+6 b^2 \left (3 a^2+b^2\right ) \cosh (x)+24 a^3 b \cosh (2 x)+4 a^4 \cosh (3 x)+6 a^2 b^2 \cosh (3 x)-2 b^4 \cosh (3 x)+9 a^4 x \sinh (x)-3 a^4 x \sinh (3 x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*Coth[x] + b*Csch[x])^4,x]

[Out]

-1/12*(Csch[x]^3*(-8*a^3*b + 16*a*b^3 + 6*b^2*(3*a^2 + b^2)*Cosh[x] + 24*a^3*b*Cosh[2*x] + 4*a^4*Cosh[3*x] + 6
*a^2*b^2*Cosh[3*x] - 2*b^4*Cosh[3*x] + 9*a^4*x*Sinh[x] - 3*a^4*x*Sinh[3*x]))

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Maple [A]
time = 0.79, size = 113, normalized size = 1.12

method	result	size
risch	\(x \,a^{4}-\frac {4 \left (6 a^{3} b \,{\mathrm e}^{5 x}+3 a^{4} {\mathrm e}^{4 x}+9 \,{\mathrm e}^{4 x} a^{2} b^{2}-4 a^{3} b \,{\mathrm e}^{3 x}+8 a \,b^{3} {\mathrm e}^{3 x}-3 a^{4} {\mathrm e}^{2 x}+3 b^{4} {\mathrm e}^{2 x}+6 a^{3} b \,{\mathrm e}^{x}+2 a^{4}+3 a^{2} b^{2}-b^{4}\right )}{3 \left ({\mathrm e}^{2 x}-1\right )^{3}}\)	\(113\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*coth(x)+b*csch(x))^4,x,method=_RETURNVERBOSE)

[Out]

x*a^4-4/3*(6*a^3*b*exp(5*x)+3*a^4*exp(4*x)+9*exp(4*x)*a^2*b^2-4*a^3*b*exp(3*x)+8*a*b^3*exp(3*x)-3*a^4*exp(2*x)
+3*b^4*exp(2*x)+6*a^3*b*exp(x)+2*a^4+3*a^2*b^2-b^4)/(exp(2*x)-1)^3

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 214 vs. \(2 (93) = 186\).
time = 0.26, size = 214, normalized size = 2.12 \begin {gather*} -2 \, a^{2} b^{2} \coth \left (x\right )^{3} + \frac {1}{3} \, a^{4} {\left (3 \, x - \frac {4 \, {\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} - 2\right )}}{3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1}\right )} + \frac {8}{3} \, a^{3} b {\left (\frac {3 \, e^{\left (-x\right )}}{3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1} - \frac {2 \, e^{\left (-3 \, x\right )}}{3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1} + \frac {3 \, e^{\left (-5 \, x\right )}}{3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1}\right )} + \frac {4}{3} \, b^{4} {\left (\frac {3 \, e^{\left (-2 \, x\right )}}{3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1} - \frac {1}{3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1}\right )} + \frac {32 \, a b^{3}}{3 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a^2*b^2*coth(x)^3 + 1/3*a^4*(3*x - 4*(3*e^(-2*x) - 3*e^(-4*x) - 2)/(3*e^(-2*x) - 3*e^(-4*x) + e^(-6*x) - 1)
) + 8/3*a^3*b*(3*e^(-x)/(3*e^(-2*x) - 3*e^(-4*x) + e^(-6*x) - 1) - 2*e^(-3*x)/(3*e^(-2*x) - 3*e^(-4*x) + e^(-6
*x) - 1) + 3*e^(-5*x)/(3*e^(-2*x) - 3*e^(-4*x) + e^(-6*x) - 1)) + 4/3*b^4*(3*e^(-2*x)/(3*e^(-2*x) - 3*e^(-4*x)
+ e^(-6*x) - 1) - 1/(3*e^(-2*x) - 3*e^(-4*x) + e^(-6*x) - 1)) + 32/3*a*b^3/(e^(-x) - e^x)^3

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (93) = 186\).
time = 0.36, size = 209, normalized size = 2.07 \begin {gather*} -\frac {24 \, a^{3} b \cosh \left (x\right )^{2} - 8 \, a^{3} b + 16 \, a b^{3} + 2 \, {\left (2 \, a^{4} + 3 \, a^{2} b^{2} - b^{4}\right )} \cosh \left (x\right )^{3} - {\left (3 \, a^{4} x + 4 \, a^{4} + 6 \, a^{2} b^{2} - 2 \, b^{4}\right )} \sinh \left (x\right )^{3} + 6 \, {\left (4 \, a^{3} b + {\left (2 \, a^{4} + 3 \, a^{2} b^{2} - b^{4}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + 6 \, {\left (3 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right ) + 3 \, {\left (3 \, a^{4} x + 4 \, a^{4} + 6 \, a^{2} b^{2} - 2 \, b^{4} - {\left (3 \, a^{4} x + 4 \, a^{4} + 6 \, a^{2} b^{2} - 2 \, b^{4}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )}{3 \, {\left (\sinh \left (x\right )^{3} + 3 \, {\left (\cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(24*a^3*b*cosh(x)^2 - 8*a^3*b + 16*a*b^3 + 2*(2*a^4 + 3*a^2*b^2 - b^4)*cosh(x)^3 - (3*a^4*x + 4*a^4 + 6*a
^2*b^2 - 2*b^4)*sinh(x)^3 + 6*(4*a^3*b + (2*a^4 + 3*a^2*b^2 - b^4)*cosh(x))*sinh(x)^2 + 6*(3*a^2*b^2 + b^4)*co
sh(x) + 3*(3*a^4*x + 4*a^4 + 6*a^2*b^2 - 2*b^4 - (3*a^4*x + 4*a^4 + 6*a^2*b^2 - 2*b^4)*cosh(x)^2)*sinh(x))/(si
nh(x)^3 + 3*(cosh(x)^2 - 1)*sinh(x))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a \coth {\left (x \right )} + b \operatorname {csch}{\left (x \right )}\right )^{4}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*coth(x)+b*csch(x))**4,x)

[Out]

Integral((a*coth(x) + b*csch(x))**4, x)

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Giac [A]
time = 0.41, size = 112, normalized size = 1.11 \begin {gather*} a^{4} x - \frac {4 \, {\left (6 \, a^{3} b e^{\left (5 \, x\right )} + 3 \, a^{4} e^{\left (4 \, x\right )} + 9 \, a^{2} b^{2} e^{\left (4 \, x\right )} - 4 \, a^{3} b e^{\left (3 \, x\right )} + 8 \, a b^{3} e^{\left (3 \, x\right )} - 3 \, a^{4} e^{\left (2 \, x\right )} + 3 \, b^{4} e^{\left (2 \, x\right )} + 6 \, a^{3} b e^{x} + 2 \, a^{4} + 3 \, a^{2} b^{2} - b^{4}\right )}}{3 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^4*x - 4/3*(6*a^3*b*e^(5*x) + 3*a^4*e^(4*x) + 9*a^2*b^2*e^(4*x) - 4*a^3*b*e^(3*x) + 8*a*b^3*e^(3*x) - 3*a^4*e
^(2*x) + 3*b^4*e^(2*x) + 6*a^3*b*e^x + 2*a^4 + 3*a^2*b^2 - b^4)/(e^(2*x) - 1)^3

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Mupad [B]
time = 1.55, size = 146, normalized size = 1.45 \begin {gather*} a^4\,x-\frac {4\,a^4+8\,{\mathrm {e}}^x\,a^3\,b+12\,a^2\,b^2}{{\mathrm {e}}^{2\,x}-1}-\frac {{\mathrm {e}}^x\,\left (\frac {32\,a^3\,b}{3}+\frac {32\,a\,b^3}{3}\right )+4\,a^4+4\,b^4+24\,a^2\,b^2}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1}-\frac {{\mathrm {e}}^x\,\left (\frac {32\,a^3\,b}{3}+\frac {32\,a\,b^3}{3}\right )+\frac {8\,a^4}{3}+\frac {8\,b^4}{3}+16\,a^2\,b^2}{3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b/sinh(x) + a*coth(x))^4,x)

[Out]

a^4*x - (4*a^4 + 12*a^2*b^2 + 8*a^3*b*exp(x))/(exp(2*x) - 1) - (exp(x)*((32*a*b^3)/3 + (32*a^3*b)/3) + 4*a^4 +
4*b^4 + 24*a^2*b^2)/(exp(4*x) - 2*exp(2*x) + 1) - (exp(x)*((32*a*b^3)/3 + (32*a^3*b)/3) + (8*a^4)/3 + (8*b^4)
/3 + 16*a^2*b^2)/(3*exp(2*x) - 3*exp(4*x) + exp(6*x) - 1)

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