3.6 \(\int \text{csch}^2(c+d x) (a+b \text{sech}^2(c+d x)) \, dx\)

Optimal. Leaf size=27 \[ -\frac{(a+b) \coth (c+d x)}{d}-\frac{b \tanh (c+d x)}{d} \]

[Out]

-(((a + b)*Coth[c + d*x])/d) - (b*Tanh[c + d*x])/d

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Rubi [A]  time = 0.0460427, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {4132, 14} \[ -\frac{(a+b) \coth (c+d x)}{d}-\frac{b \tanh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^2*(a + b*Sech[c + d*x]^2),x]

[Out]

-(((a + b)*Coth[c + d*x])/d) - (b*Tanh[c + d*x])/d

Rule 4132

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_)]^(m_), x_Symbol] :> With[{ff = Fr
eeFactors[Tan[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[(x^m*ExpandToSum[a + b*(1 + ff^2*x^2)^(n/2), x]^p)/(
1 + ff^2*x^2)^(m/2 + 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && Integer
Q[n/2]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

Mathematica [A]  time = 0.0696755, size = 37, normalized size = 1.37 \[ -\frac{a \coth (c+d x)}{d}-\frac{b \tanh (c+d x)}{d}-\frac{b \coth (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^2*(a + b*Sech[c + d*x]^2),x]

[Out]

-((a*Coth[c + d*x])/d) - (b*Coth[c + d*x])/d - (b*Tanh[c + d*x])/d

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Maple [A]  time = 0.027, size = 44, normalized size = 1.6 \begin{align*}{\frac{1}{d} \left ( -{\rm coth} \left (dx+c\right )a+b \left ( -{\frac{1}{\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) }}-2\,\tanh \left ( dx+c \right ) \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(-coth(d*x+c)*a+b*(-1/sinh(d*x+c)/cosh(d*x+c)-2*tanh(d*x+c)))

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Maxima [A]  time = 1.03625, size = 53, normalized size = 1.96 \begin{align*} \frac{2 \, a}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} + \frac{4 \, b}{d{\left (e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a/(d*(e^(-2*d*x - 2*c) - 1)) + 4*b/(d*(e^(-4*d*x - 4*c) - 1))

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Fricas [B]  time = 2.41616, size = 246, normalized size = 9.11 \begin{align*} -\frac{4 \,{\left ({\left (a + b\right )} \cosh \left (d x + c\right ) - b \sinh \left (d x + c\right )\right )}}{d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + d \sinh \left (d x + c\right )^{3} - d \cosh \left (d x + c\right ) +{\left (3 \, d \cosh \left (d x + c\right )^{2} + d\right )} \sinh \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*((a + b)*cosh(d*x + c) - b*sinh(d*x + c))/(d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c)*sinh(d*x + c)^2 + d*sinh(d
*x + c)^3 - d*cosh(d*x + c) + (3*d*cosh(d*x + c)^2 + d)*sinh(d*x + c))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{sech}^{2}{\left (c + d x \right )}\right ) \operatorname{csch}^{2}{\left (c + d x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.12958, size = 46, normalized size = 1.7 \begin{align*} -\frac{2 \,{\left (a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b\right )}}{d{\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(a*e^(2*d*x + 2*c) + a + 2*b)/(d*(e^(4*d*x + 4*c) - 1))