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(d*x+c)/d-b*tanh(d*x+c)/d

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Rubi [A]  time = 0.05, antiderivative size = 27, normalized size of antiderivative = 1.00, 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 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]]

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]

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*}

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Mathematica [A]  time = 0.07, 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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fricas [B]  time = 0.40, size = 91, normalized size = 3.37 \[ -\frac {4 \, {\left ({\left (a + b\right )} \cosh \left (d x + c\right ) - b \sinh \left (d x + c\right )\right )}}{d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + d \sinh \left (d x + c\right )^{3} - d \cosh \left (d x + c\right ) + {\left (3 \, d \cosh \left (d x + c\right )^{2} + d\right )} \sinh \left (d x + c\right )} \]

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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giac [A]  time = 0.14, size = 34, normalized size = 1.26 \[ -\frac {2 \, {\left (a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b\right )}}{d {\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )}} \]

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))

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maple [A]  time = 0.38, size = 44, normalized size = 1.63 \[ \frac {-\coth \left (d x +c \right ) a +b \left (-\frac {1}{\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}-2 \tanh \left (d x +c \right )\right )}{d} \]

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 = 0.33, size = 39, normalized size = 1.44 \[ \frac {2 \, a}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} + \frac {4 \, b}{d {\left (e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}} \]

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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mupad [B]  time = 1.38, size = 34, normalized size = 1.26 \[ -\frac {2\,\left (a+2\,b+a\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/cosh(c + d*x)^2)/sinh(c + d*x)^2,x)

[Out]

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

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

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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