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

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 45, 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 = {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 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]

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
]

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 = 0.17, 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)

________________________________________________________________________________________

fricas [B]  time = 0.40, size = 133, normalized size = 2.96 \[ \frac {3 \, a^{2} \cosh \left (d x + c\right )^{4} + 3 \, a^{2} \sinh \left (d x + c\right )^{4} + 12 \, {\left (a^{2} - 2 \, a b\right )} \cosh \left (d x + c\right )^{2} + 6 \, {\left (3 \, a^{2} \cosh \left (d x + c\right )^{2} + 2 \, a^{2} - 4 \, a b\right )} \sinh \left (d x + c\right )^{2} + 9 \, a^{2} - 24 \, a b - 8 \, b^{2}}{6 \, {\left (d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 3 \, d \cosh \left (d x + c\right )\right )}} \]

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))

________________________________________________________________________________________

giac [A]  time = 0.14, size = 75, normalized size = 1.67 \[ \frac {3 \, a^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - \frac {8 \, {\left (3 \, a b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} + 2 \, b^{2}\right )}}{{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3}}}{6 \, d} \]

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]

1/6*(3*a^2*(e^(d*x + c) + e^(-d*x - c)) - 8*(3*a*b*(e^(d*x + c) + e^(-d*x - c))^2 + 2*b^2)/(e^(d*x + c) + e^(-
d*x - c))^3)/d

________________________________________________________________________________________

maple [A]  time = 0.10, size = 43, normalized size = 0.96 \[ -\frac {\frac {b^{2} \mathrm {sech}\left (d x +c \right )^{3}}{3}+2 a b \,\mathrm {sech}\left (d x +c \right )-\frac {a^{2}}{\mathrm {sech}\left (d x +c \right )}}{d} \]

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))

________________________________________________________________________________________

maxima [A]  time = 0.32, size = 65, normalized size = 1.44 \[ \frac {a^{2} \cosh \left (d x + c\right )}{d} - \frac {4 \, a b}{d {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}} - \frac {8 \, b^{2}}{3 \, d {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3}} \]

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)

________________________________________________________________________________________

mupad [B]  time = 1.47, size = 45, normalized size = 1.00 \[ \frac {a^2\,\mathrm {cosh}\left (c+d\,x\right )}{d}-\frac {\frac {b^2}{3}+2\,a\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^2}{d\,{\mathrm {cosh}\left (c+d\,x\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

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]

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

________________________________________________________________________________________