3.123 \(\int \coth ^4(a+b x) \text {csch}(a+b x) \, dx\)

Optimal. Leaf size=55 \[ -\frac {3 \tanh ^{-1}(\cosh (a+b x))}{8 b}-\frac {\coth ^3(a+b x) \text {csch}(a+b x)}{4 b}-\frac {3 \coth (a+b x) \text {csch}(a+b x)}{8 b} \]

[Out]

-3/8*arctanh(cosh(b*x+a))/b-3/8*coth(b*x+a)*csch(b*x+a)/b-1/4*coth(b*x+a)^3*csch(b*x+a)/b

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Rubi [A]  time = 0.06, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2611, 3770} \[ -\frac {3 \tanh ^{-1}(\cosh (a+b x))}{8 b}-\frac {\coth ^3(a+b x) \text {csch}(a+b x)}{4 b}-\frac {3 \coth (a+b x) \text {csch}(a+b x)}{8 b} \]

Antiderivative was successfully verified.

[In]

Int[Coth[a + b*x]^4*Csch[a + b*x],x]

[Out]

(-3*ArcTanh[Cosh[a + b*x]])/(8*b) - (3*Coth[a + b*x]*Csch[a + b*x])/(8*b) - (Coth[a + b*x]^3*Csch[a + b*x])/(4
*b)

Rule 2611

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a*Sec[e
+ f*x])^m*(b*Tan[e + f*x])^(n - 1))/(f*(m + n - 1)), x] - Dist[(b^2*(n - 1))/(m + n - 1), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin {align*} \int \coth ^4(a+b x) \text {csch}(a+b x) \, dx &=-\frac {\coth ^3(a+b x) \text {csch}(a+b x)}{4 b}+\frac {3}{4} \int \coth ^2(a+b x) \text {csch}(a+b x) \, dx\\ &=-\frac {3 \coth (a+b x) \text {csch}(a+b x)}{8 b}-\frac {\coth ^3(a+b x) \text {csch}(a+b x)}{4 b}+\frac {3}{8} \int \text {csch}(a+b x) \, dx\\ &=-\frac {3 \tanh ^{-1}(\cosh (a+b x))}{8 b}-\frac {3 \coth (a+b x) \text {csch}(a+b x)}{8 b}-\frac {\coth ^3(a+b x) \text {csch}(a+b x)}{4 b}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 95, normalized size = 1.73 \[ -\frac {\text {csch}^4\left (\frac {1}{2} (a+b x)\right )}{64 b}-\frac {5 \text {csch}^2\left (\frac {1}{2} (a+b x)\right )}{32 b}+\frac {\text {sech}^4\left (\frac {1}{2} (a+b x)\right )}{64 b}-\frac {5 \text {sech}^2\left (\frac {1}{2} (a+b x)\right )}{32 b}+\frac {3 \log \left (\tanh \left (\frac {1}{2} (a+b x)\right )\right )}{8 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*Csch[(a + b*x)/2]^2)/(32*b) - Csch[(a + b*x)/2]^4/(64*b) + (3*Log[Tanh[(a + b*x)/2]])/(8*b) - (5*Sech[(a +
b*x)/2]^2)/(32*b) + Sech[(a + b*x)/2]^4/(64*b)

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fricas [B]  time = 0.53, size = 1114, normalized size = 20.25 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(10*cosh(b*x + a)^7 + 70*cosh(b*x + a)*sinh(b*x + a)^6 + 10*sinh(b*x + a)^7 + 6*(35*cosh(b*x + a)^2 + 1)*
sinh(b*x + a)^5 + 6*cosh(b*x + a)^5 + 10*(35*cosh(b*x + a)^3 + 3*cosh(b*x + a))*sinh(b*x + a)^4 + 2*(175*cosh(
b*x + a)^4 + 30*cosh(b*x + a)^2 + 3)*sinh(b*x + a)^3 + 6*cosh(b*x + a)^3 + 6*(35*cosh(b*x + a)^5 + 10*cosh(b*x
+ a)^3 + 3*cosh(b*x + a))*sinh(b*x + a)^2 + 3*(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*cosh(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(cosh(b*x + a) + sinh(b*x + a) + 1) - 3*(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*cosh(b*x + a)^4 + 8*(7*cosh(b*x + a)^5 - 10*cosh(b*x + a)^3 + 3*cosh(b*x + a))*si
nh(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(
cosh(b*x + a) + sinh(b*x + a) - 1) + 2*(35*cosh(b*x + a)^6 + 15*cosh(b*x + a)^4 + 9*cosh(b*x + a)^2 + 5)*sinh(
b*x + a) + 10*cosh(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))*si
nh(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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giac [A]  time = 0.16, size = 86, normalized size = 1.56 \[ -\frac {\frac {2 \, {\left (5 \, e^{\left (7 \, b x + 7 \, a\right )} + 3 \, e^{\left (5 \, b x + 5 \, a\right )} + 3 \, e^{\left (3 \, b x + 3 \, a\right )} + 5 \, e^{\left (b x + a\right )}\right )}}{{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{4}} + 3 \, \log \left (e^{\left (b x + a\right )} + 1\right ) - 3 \, \log \left ({\left | e^{\left (b x + a\right )} - 1 \right |}\right )}{8 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(2*(5*e^(7*b*x + 7*a) + 3*e^(5*b*x + 5*a) + 3*e^(3*b*x + 3*a) + 5*e^(b*x + a))/(e^(2*b*x + 2*a) - 1)^4 +
3*log(e^(b*x + a) + 1) - 3*log(abs(e^(b*x + a) - 1)))/b

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maple [A]  time = 0.34, size = 74, normalized size = 1.35 \[ \frac {-\frac {\cosh ^{3}\left (b x +a \right )}{\sinh \left (b x +a \right )^{4}}+\frac {\cosh \left (b x +a \right )}{\sinh \left (b x +a \right )^{4}}+\left (-\frac {\mathrm {csch}\left (b x +a \right )^{3}}{4}+\frac {3 \,\mathrm {csch}\left (b x +a \right )}{8}\right ) \coth \left (b x +a \right )-\frac {3 \arctanh \left ({\mathrm e}^{b x +a}\right )}{4}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b*(-1/sinh(b*x+a)^4*cosh(b*x+a)^3+1/sinh(b*x+a)^4*cosh(b*x+a)+(-1/4*csch(b*x+a)^3+3/8*csch(b*x+a))*coth(b*x+
a)-3/4*arctanh(exp(b*x+a)))

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maxima [B]  time = 0.30, size = 133, normalized size = 2.42 \[ -\frac {3 \, \log \left (e^{\left (-b x - a\right )} + 1\right )}{8 \, b} + \frac {3 \, \log \left (e^{\left (-b x - a\right )} - 1\right )}{8 \, b} + \frac {5 \, e^{\left (-b x - a\right )} + 3 \, e^{\left (-3 \, b x - 3 \, a\right )} + 3 \, e^{\left (-5 \, b x - 5 \, a\right )} + 5 \, e^{\left (-7 \, b x - 7 \, a\right )}}{4 \, b {\left (4 \, e^{\left (-2 \, b x - 2 \, a\right )} - 6 \, e^{\left (-4 \, b x - 4 \, a\right )} + 4 \, e^{\left (-6 \, b x - 6 \, a\right )} - e^{\left (-8 \, b x - 8 \, a\right )} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/8*log(e^(-b*x - a) + 1)/b + 3/8*log(e^(-b*x - a) - 1)/b + 1/4*(5*e^(-b*x - a) + 3*e^(-3*b*x - 3*a) + 3*e^(-
5*b*x - 5*a) + 5*e^(-7*b*x - 7*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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mupad [B]  time = 1.41, size = 190, normalized size = 3.45 \[ -\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^2}}{b}\right )}{4\,\sqrt {-b^2}}-\frac {9\,{\mathrm {e}}^{a+b\,x}}{2\,b\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-2\,{\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}-\frac {6\,{\mathrm {e}}^{a+b\,x}}{b\,\left (3\,{\mathrm {e}}^{2\,a+2\,b\,x}-3\,{\mathrm {e}}^{4\,a+4\,b\,x}+{\mathrm {e}}^{6\,a+6\,b\,x}-1\right )}-\frac {4\,{\mathrm {e}}^{a+b\,x}}{b\,\left (6\,{\mathrm {e}}^{4\,a+4\,b\,x}-4\,{\mathrm {e}}^{2\,a+2\,b\,x}-4\,{\mathrm {e}}^{6\,a+6\,b\,x}+{\mathrm {e}}^{8\,a+8\,b\,x}+1\right )}-\frac {5\,{\mathrm {e}}^{a+b\,x}}{4\,b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(a + b*x)^4/sinh(a + b*x),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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