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*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)

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Rubi [A]  time = 0.0570593, antiderivative size = 55, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0389819, 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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Maple [A]  time = 0.015, size = 74, normalized size = 1.4 \begin{align*}{\frac{1}{b} \left ( -{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{3}}{ \left ( \sinh \left ( bx+a \right ) \right ) ^{4}}}+{\frac{\cosh \left ( bx+a \right ) }{ \left ( \sinh \left ( bx+a \right ) \right ) ^{4}}}+ \left ( -{\frac{ \left ({\rm csch} \left (bx+a\right ) \right ) ^{3}}{4}}+{\frac{3\,{\rm csch} \left (bx+a\right )}{8}} \right ){\rm coth} \left (bx+a\right )-{\frac{3\,{\it Artanh} \left ({{\rm e}^{bx+a}} \right ) }{4}} \right ) } \end{align*}

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 = 1.06964, size = 180, normalized size = 3.27 \begin{align*} -\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 )}} \end{align*}

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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Fricas [B]  time = 1.94066, size = 3081, normalized size = 56.02 \begin{align*} \text{result too large to display} \end{align*}

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

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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Giac [A]  time = 1.31105, size = 116, normalized size = 2.11 \begin{align*} -\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} \end{align*}

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