3.54 \(\int \text{csch}^2(c+d x) (a+b \tanh ^3(c+d x)) \, dx\)

Optimal. Leaf size=29 \[ \frac{b \tanh ^2(c+d x)}{2 d}-\frac{a \coth (c+d x)}{d} \]

[Out]

-((a*Coth[c + d*x])/d) + (b*Tanh[c + d*x]^2)/(2*d)

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Rubi [A]  time = 0.0341582, antiderivative size = 29, 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 = {3663, 14} \[ \frac{b \tanh ^2(c+d x)}{2 d}-\frac{a \coth (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^2*(a + b*Tanh[c + d*x]^3),x]

[Out]

-((a*Coth[c + d*x])/d) + (b*Tanh[c + d*x]^2)/(2*d)

Rule 3663

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(c*ff^(m + 1))/f, Subst[Int[(x^m*(a + b*(ff*x)^n)^p)/(c^2 + ff^2*x^2
)^(m/2 + 1), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[m/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 \tanh ^3(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a+b x^3}{x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a}{x^2}+b x\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac{a \coth (c+d x)}{d}+\frac{b \tanh ^2(c+d x)}{2 d}\\ \end{align*}

Mathematica [A]  time = 0.0245542, size = 29, normalized size = 1. \[ -\frac{a \coth (c+d x)}{d}-\frac{b \text{sech}^2(c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^2*(a + b*Tanh[c + d*x]^3),x]

[Out]

-((a*Coth[c + d*x])/d) - (b*Sech[c + d*x]^2)/(2*d)

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Maple [A]  time = 0.05, size = 34, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ( -{\rm coth} \left (dx+c\right )a+{\frac{b \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{2\, \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^2*(a+b*tanh(d*x+c)^3),x)

[Out]

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

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Maxima [A]  time = 1.04275, size = 59, normalized size = 2.03 \begin{align*} \frac{2 \, a}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} - \frac{2 \, b}{d{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.44496, size = 375, normalized size = 12.93 \begin{align*} -\frac{2 \,{\left ({\left (2 \, a + b\right )} \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) +{\left (2 \, a + b\right )} \sinh \left (d x + c\right )^{2} + 2 \, a - b\right )}}{d \cosh \left (d x + c\right )^{4} + 6 \, d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{2} + 4 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + d \sinh \left (d x + c\right )^{4} + 4 \,{\left (d \cosh \left (d x + c\right )^{3} + d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*((2*a + b)*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + (2*a + b)*sinh(d*x + c)^2 + 2*a - b)/(d*cosh
(d*x + c)^4 + 6*d*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*d*cosh(d*x + c)*sinh(d*x + c)^3 + d*sinh(d*x + c)^4 + 4*
(d*cosh(d*x + c)^3 + d*cosh(d*x + c))*sinh(d*x + c) - d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \tanh ^{3}{\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*tanh(d*x+c)**3),x)

[Out]

Integral((a + b*tanh(c + d*x)**3)*csch(c + d*x)**2, x)

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Giac [A]  time = 1.2117, size = 61, normalized size = 2.1 \begin{align*} -\frac{2 \,{\left (\frac{a}{e^{\left (2 \, d x + 2 \, c\right )} - 1} + \frac{b e^{\left (2 \, d x + 2 \, c\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{2}}\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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