3.518 \(\int \text {csch}^3(a+b x) \text {sech}^2(a+b x) \, dx\)

Optimal. Leaf size=49 \[ -\frac {3 \text {sech}(a+b x)}{2 b}+\frac {3 \tanh ^{-1}(\cosh (a+b x))}{2 b}-\frac {\text {csch}^2(a+b x) \text {sech}(a+b x)}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2622, 288, 321, 207} \[ -\frac {3 \text {sech}(a+b x)}{2 b}+\frac {3 \tanh ^{-1}(\cosh (a+b x))}{2 b}-\frac {\text {csch}^2(a+b x) \text {sech}(a+b x)}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[Csch[a + b*x]^3*Sech[a + b*x]^2,x]

[Out]

(3*ArcTanh[Cosh[a + b*x]])/(2*b) - (3*Sech[a + b*x])/(2*b) - (Csch[a + b*x]^2*Sech[a + b*x])/(2*b)

Rule 207

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

Rule 288

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 321

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*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 2622

Int[csc[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[1/(f*a^n), Subst[Int
[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n
+ 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 68, normalized size = 1.39 \[ -\frac {\text {csch}^2\left (\frac {1}{2} (a+b x)\right )}{8 b}-\frac {\text {sech}^2\left (\frac {1}{2} (a+b x)\right )}{8 b}-\frac {\text {sech}(a+b x)}{b}-\frac {3 \log \left (\tanh \left (\frac {1}{2} (a+b x)\right )\right )}{2 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[a + b*x]^3*Sech[a + b*x]^2,x]

[Out]

-1/8*Csch[(a + b*x)/2]^2/b - (3*Log[Tanh[(a + b*x)/2]])/(2*b) - Sech[(a + b*x)/2]^2/(8*b) - Sech[a + b*x]/b

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fricas [B]  time = 0.43, size = 709, normalized size = 14.47 \[ -\frac {6 \, \cosh \left (b x + a\right )^{5} + 30 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} + 6 \, \sinh \left (b x + a\right )^{5} + 4 \, {\left (15 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{3} - 4 \, \cosh \left (b x + a\right )^{3} + 12 \, {\left (5 \, \cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} - 3 \, {\left (\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} + {\left (15 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{4} - \cosh \left (b x + a\right )^{4} + 4 \, {\left (5 \, \cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + {\left (15 \, \cosh \left (b x + a\right )^{4} - 6 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{5} - 2 \, \cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + 3 \, {\left (\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} + {\left (15 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{4} - \cosh \left (b x + a\right )^{4} + 4 \, {\left (5 \, \cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + {\left (15 \, \cosh \left (b x + a\right )^{4} - 6 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{5} - 2 \, \cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 6 \, {\left (5 \, \cosh \left (b x + a\right )^{4} - 2 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right ) + 6 \, \cosh \left (b x + a\right )}{2 \, {\left (b \cosh \left (b x + a\right )^{6} + 6 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + b \sinh \left (b x + a\right )^{6} - b \cosh \left (b x + a\right )^{4} + {\left (15 \, b \cosh \left (b x + a\right )^{2} - b\right )} \sinh \left (b x + a\right )^{4} + 4 \, {\left (5 \, b \cosh \left (b x + a\right )^{3} - b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} - b \cosh \left (b x + a\right )^{2} + {\left (15 \, b \cosh \left (b x + a\right )^{4} - 6 \, b \cosh \left (b x + a\right )^{2} - b\right )} \sinh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b \cosh \left (b x + a\right )^{5} - 2 \, b \cosh \left (b x + a\right )^{3} - b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)^3*sech(b*x+a)^2,x, algorithm="fricas")

[Out]

-1/2*(6*cosh(b*x + a)^5 + 30*cosh(b*x + a)*sinh(b*x + a)^4 + 6*sinh(b*x + a)^5 + 4*(15*cosh(b*x + a)^2 - 1)*si
nh(b*x + a)^3 - 4*cosh(b*x + a)^3 + 12*(5*cosh(b*x + a)^3 - cosh(b*x + a))*sinh(b*x + a)^2 - 3*(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 - cosh(b*x +
a)^4 + 4*(5*cosh(b*x + a)^3 - cosh(b*x + a))*sinh(b*x + a)^3 + (15*cosh(b*x + a)^4 - 6*cosh(b*x + a)^2 - 1)*si
nh(b*x + a)^2 - cosh(b*x + a)^2 + 2*(3*cosh(b*x + a)^5 - 2*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)^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 - cosh(b*x + a)^4 + 4*(5*cosh(b*x + a)^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 - cosh(b*x + a)^2 + 2*(3*cosh(b*x + a
)^5 - 2*cosh(b*x + a)^3 - cosh(b*x + a))*sinh(b*x + a) + 1)*log(cosh(b*x + a) + sinh(b*x + a) - 1) + 6*(5*cosh
(b*x + a)^4 - 2*cosh(b*x + a)^2 + 1)*sinh(b*x + a) + 6*cosh(b*x + a))/(b*cosh(b*x + a)^6 + 6*b*cosh(b*x + a)*s
inh(b*x + a)^5 + b*sinh(b*x + a)^6 - b*cosh(b*x + a)^4 + (15*b*cosh(b*x + a)^2 - b)*sinh(b*x + a)^4 + 4*(5*b*c
osh(b*x + a)^3 - b*cosh(b*x + a))*sinh(b*x + a)^3 - b*cosh(b*x + a)^2 + (15*b*cosh(b*x + a)^4 - 6*b*cosh(b*x +
a)^2 - b)*sinh(b*x + a)^2 + 2*(3*b*cosh(b*x + a)^5 - 2*b*cosh(b*x + a)^3 - b*cosh(b*x + a))*sinh(b*x + a) + b
)

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giac [B]  time = 0.14, size = 110, normalized size = 2.24 \[ -\frac {\frac {4 \, {\left (3 \, {\left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}\right )}^{2} - 8\right )}}{{\left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}\right )}^{3} - 4 \, e^{\left (b x + a\right )} - 4 \, e^{\left (-b x - a\right )}} - 3 \, \log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} + 2\right ) + 3 \, \log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} - 2\right )}{4 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)^3*sech(b*x+a)^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.20, size = 43, normalized size = 0.88 \[ \frac {-\frac {1}{2 \sinh \left (b x +a \right )^{2} \cosh \left (b x +a \right )}-\frac {3}{2 \cosh \left (b x +a \right )}+3 \arctanh \left ({\mathrm e}^{b x +a}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(b*x+a)^3*sech(b*x+a)^2,x)

[Out]

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

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maxima [B]  time = 0.31, size = 106, normalized size = 2.16 \[ \frac {3 \, \log \left (e^{\left (-b x - a\right )} + 1\right )}{2 \, b} - \frac {3 \, \log \left (e^{\left (-b x - a\right )} - 1\right )}{2 \, b} + \frac {3 \, e^{\left (-b x - a\right )} - 2 \, e^{\left (-3 \, b x - 3 \, a\right )} + 3 \, e^{\left (-5 \, b x - 5 \, a\right )}}{b {\left (e^{\left (-2 \, b x - 2 \, a\right )} + e^{\left (-4 \, b x - 4 \, a\right )} - e^{\left (-6 \, b x - 6 \, a\right )} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)^3*sech(b*x+a)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.08, size = 111, normalized size = 2.27 \[ \frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^2}}{b}\right )}{\sqrt {-b^2}}-\frac {2\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-2\,{\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}-\frac {{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )}-\frac {2\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(a + b*x)^2*sinh(a + b*x)^3),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {csch}^{3}{\left (a + b x \right )} \operatorname {sech}^{2}{\left (a + b x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)**3*sech(b*x+a)**2,x)

[Out]

Integral(csch(a + b*x)**3*sech(a + b*x)**2, x)

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