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

Optimal. Leaf size=45 \[ \frac{a^2 \cosh (c+d x)}{d}-\frac{2 a b \text{sech}(c+d x)}{d}-\frac{b^2 \text{sech}^3(c+d x)}{3 d} \]

[Out]

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

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Rubi [A]  time = 0.0457674, antiderivative size = 45, 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 = {4133, 270} \[ \frac{a^2 \cosh (c+d x)}{d}-\frac{2 a b \text{sech}(c+d x)}{d}-\frac{b^2 \text{sech}^3(c+d x)}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4133

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

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 0.170857, size = 59, normalized size = 1.31 \[ \frac{\text{sech}^3(c+d x) \left (3 a^2 \cosh (4 (c+d x))+9 a^2+12 a (a-2 b) \cosh (2 (c+d x))-24 a b-8 b^2\right )}{24 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((9*a^2 - 24*a*b - 8*b^2 + 12*a*(a - 2*b)*Cosh[2*(c + d*x)] + 3*a^2*Cosh[4*(c + d*x)])*Sech[c + d*x]^3)/(24*d)

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Maple [A]  time = 0.018, size = 43, normalized size = 1. \begin{align*} -{\frac{1}{d} \left ({\frac{{b}^{2} \left ({\rm sech} \left (dx+c\right ) \right ) ^{3}}{3}}+2\,ab{\rm sech} \left (dx+c\right )-{\frac{{a}^{2}}{{\rm sech} \left (dx+c\right )}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/d*(1/3*b^2*sech(d*x+c)^3+2*a*b*sech(d*x+c)-a^2/sech(d*x+c))

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Maxima [A]  time = 1.05052, size = 88, normalized size = 1.96 \begin{align*} \frac{a^{2} \cosh \left (d x + c\right )}{d} - \frac{4 \, a b}{d{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}} - \frac{8 \, b^{2}}{3 \, d{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*cosh(d*x + c)/d - 4*a*b/(d*(e^(d*x + c) + e^(-d*x - c))) - 8/3*b^2/(d*(e^(d*x + c) + e^(-d*x - c))^3)

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Fricas [B]  time = 2.53017, size = 336, normalized size = 7.47 \begin{align*} \frac{3 \, a^{2} \cosh \left (d x + c\right )^{4} + 3 \, a^{2} \sinh \left (d x + c\right )^{4} + 12 \,{\left (a^{2} - 2 \, a b\right )} \cosh \left (d x + c\right )^{2} + 6 \,{\left (3 \, a^{2} \cosh \left (d x + c\right )^{2} + 2 \, a^{2} - 4 \, a b\right )} \sinh \left (d x + c\right )^{2} + 9 \, a^{2} - 24 \, a b - 8 \, b^{2}}{6 \,{\left (d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 3 \, d \cosh \left (d x + c\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(3*a^2*cosh(d*x + c)^4 + 3*a^2*sinh(d*x + c)^4 + 12*(a^2 - 2*a*b)*cosh(d*x + c)^2 + 6*(3*a^2*cosh(d*x + c)
^2 + 2*a^2 - 4*a*b)*sinh(d*x + c)^2 + 9*a^2 - 24*a*b - 8*b^2)/(d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c)*sinh(d*x
+ c)^2 + 3*d*cosh(d*x + c))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.20174, size = 103, normalized size = 2.29 \begin{align*} \frac{a^{2}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}}{2 \, d} - \frac{4 \,{\left (3 \, a b{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} + 2 \, b^{2}\right )}}{3 \, d{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^2*(e^(d*x + c) + e^(-d*x - c))/d - 4/3*(3*a*b*(e^(d*x + c) + e^(-d*x - c))^2 + 2*b^2)/(d*(e^(d*x + c) +
e^(-d*x - c))^3)