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

Optimal. Leaf size=44 \[ \frac{a^2 x^2}{2}+\frac{a b \tan ^{-1}\left (\sinh \left (c+d x^2\right )\right )}{d}+\frac{b^2 \tanh \left (c+d x^2\right )}{2 d} \]

[Out]

(a^2*x^2)/2 + (a*b*ArcTan[Sinh[c + d*x^2]])/d + (b^2*Tanh[c + d*x^2])/(2*d)

________________________________________________________________________________________

Rubi [A]  time = 0.0545031, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {5436, 3773, 3770, 3767, 8} \[ \frac{a^2 x^2}{2}+\frac{a b \tan ^{-1}\left (\sinh \left (c+d x^2\right )\right )}{d}+\frac{b^2 \tanh \left (c+d x^2\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*Sech[c + d*x^2])^2,x]

[Out]

(a^2*x^2)/2 + (a*b*ArcTan[Sinh[c + d*x^2]])/d + (b^2*Tanh[c + d*x^2])/(2*d)

Rule 5436

Int[(x_)^(m_.)*((a_.) + (b_.)*Sech[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simpli
fy[(m + 1)/n] - 1)*(a + b*Sech[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplif
y[(m + 1)/n], 0] && IntegerQ[p]

Rule 3773

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^2, x_Symbol] :> Simp[a^2*x, x] + (Dist[2*a*b, Int[Csc[c + d*x], x],
x] + Dist[b^2, Int[Csc[c + d*x]^2, x], x]) /; FreeQ[{a, b, c, d}, x]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int x \left (a+b \text{sech}\left (c+d x^2\right )\right )^2 \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int (a+b \text{sech}(c+d x))^2 \, dx,x,x^2\right )\\ &=\frac{a^2 x^2}{2}+(a b) \operatorname{Subst}\left (\int \text{sech}(c+d x) \, dx,x,x^2\right )+\frac{1}{2} b^2 \operatorname{Subst}\left (\int \text{sech}^2(c+d x) \, dx,x,x^2\right )\\ &=\frac{a^2 x^2}{2}+\frac{a b \tan ^{-1}\left (\sinh \left (c+d x^2\right )\right )}{d}+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int 1 \, dx,x,-i \tanh \left (c+d x^2\right )\right )}{2 d}\\ &=\frac{a^2 x^2}{2}+\frac{a b \tan ^{-1}\left (\sinh \left (c+d x^2\right )\right )}{d}+\frac{b^2 \tanh \left (c+d x^2\right )}{2 d}\\ \end{align*}

Mathematica [A]  time = 0.0910586, size = 44, normalized size = 1. \[ \frac{a \left (a \left (c+d x^2\right )+2 b \tan ^{-1}\left (\sinh \left (c+d x^2\right )\right )\right )+b^2 \tanh \left (c+d x^2\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(a*(c + d*x^2) + 2*b*ArcTan[Sinh[c + d*x^2]]) + b^2*Tanh[c + d*x^2])/(2*d)

________________________________________________________________________________________

Maple [A]  time = 0.086, size = 51, normalized size = 1.2 \begin{align*}{\frac{{a}^{2}{x}^{2}}{2}}+{\frac{{b}^{2}\tanh \left ( d{x}^{2}+c \right ) }{2\,d}}+2\,{\frac{ba\arctan \left ({{\rm e}^{d{x}^{2}+c}} \right ) }{d}}+{\frac{{a}^{2}c}{2\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^2*x^2+1/2*b^2*tanh(d*x^2+c)/d+2/d*b*a*arctan(exp(d*x^2+c))+1/2/d*a^2*c

________________________________________________________________________________________

Maxima [A]  time = 1.16694, size = 62, normalized size = 1.41 \begin{align*} \frac{1}{2} \, a^{2} x^{2} + \frac{a b \arctan \left (\sinh \left (d x^{2} + c\right )\right )}{d} + \frac{b^{2}}{d{\left (e^{\left (-2 \, d x^{2} - 2 \, c\right )} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 2.17978, size = 482, normalized size = 10.95 \begin{align*} \frac{a^{2} d x^{2} \cosh \left (d x^{2} + c\right )^{2} + 2 \, a^{2} d x^{2} \cosh \left (d x^{2} + c\right ) \sinh \left (d x^{2} + c\right ) + a^{2} d x^{2} \sinh \left (d x^{2} + c\right )^{2} + a^{2} d x^{2} - 2 \, b^{2} + 4 \,{\left (a b \cosh \left (d x^{2} + c\right )^{2} + 2 \, a b \cosh \left (d x^{2} + c\right ) \sinh \left (d x^{2} + c\right ) + a b \sinh \left (d x^{2} + c\right )^{2} + a b\right )} \arctan \left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right )\right )}{2 \,{\left (d \cosh \left (d x^{2} + c\right )^{2} + 2 \, d \cosh \left (d x^{2} + c\right ) \sinh \left (d x^{2} + c\right ) + d \sinh \left (d x^{2} + c\right )^{2} + d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a^2*d*x^2*cosh(d*x^2 + c)^2 + 2*a^2*d*x^2*cosh(d*x^2 + c)*sinh(d*x^2 + c) + a^2*d*x^2*sinh(d*x^2 + c)^2 +
a^2*d*x^2 - 2*b^2 + 4*(a*b*cosh(d*x^2 + c)^2 + 2*a*b*cosh(d*x^2 + c)*sinh(d*x^2 + c) + a*b*sinh(d*x^2 + c)^2
+ a*b)*arctan(cosh(d*x^2 + c) + sinh(d*x^2 + c)))/(d*cosh(d*x^2 + c)^2 + 2*d*cosh(d*x^2 + c)*sinh(d*x^2 + c) +
d*sinh(d*x^2 + c)^2 + d)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.17188, size = 74, normalized size = 1.68 \begin{align*} \frac{{\left (d x^{2} + c\right )} a^{2}}{2 \, d} + \frac{2 \, a b \arctan \left (e^{\left (d x^{2} + c\right )}\right )}{d} - \frac{b^{2}}{d{\left (e^{\left (2 \, d x^{2} + 2 \, c\right )} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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