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

Optimal. Leaf size=26 \[ \frac {a x^2}{2}+\frac {b \tan ^{-1}\left (\sinh \left (c+d x^2\right )\right )}{2 d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {14, 5436, 3770} \[ \frac {a x^2}{2}+\frac {b \tan ^{-1}\left (\sinh \left (c+d x^2\right )\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*Sech[c + d*x^2]),x]

[Out]

(a*x^2)/2 + (b*ArcTan[Sinh[c + d*x^2]])/(2*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 3770

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

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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 26, normalized size = 1.00 \[ \frac {a x^2}{2}+\frac {b \tan ^{-1}\left (\sinh \left (c+d x^2\right )\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.42, size = 33, normalized size = 1.27 \[ \frac {a d x^{2} + 2 \, b \arctan \left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right )\right )}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.12, size = 28, normalized size = 1.08 \[ \frac {{\left (d x^{2} + c\right )} a}{2 \, d} + \frac {b \arctan \left (e^{\left (d x^{2} + c\right )}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.10, size = 30, normalized size = 1.15 \[ \frac {a \,x^{2}}{2}+\frac {b \arctan \left (\sinh \left (d \,x^{2}+c \right )\right )}{2 d}+\frac {a c}{2 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.31, size = 22, normalized size = 0.85 \[ \frac {1}{2} \, a x^{2} + \frac {b \arctan \left (\sinh \left (d x^{2} + c\right )\right )}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.07, size = 42, normalized size = 1.62 \[ \frac {a\,x^2}{2}+\frac {\mathrm {atan}\left (\frac {b\,{\mathrm {e}}^{d\,x^2}\,{\mathrm {e}}^c\,\sqrt {d^2}}{d\,\sqrt {b^2}}\right )\,\sqrt {b^2}}{\sqrt {d^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b/cosh(c + d*x^2)),x)

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________