3.10 \(\int (a+b \text{sech}^2(c+d x))^2 \sinh ^3(c+d x) \, dx\)

Optimal. Leaf size=72 \[ \frac{a^2 \cosh ^3(c+d x)}{3 d}-\frac{a (a-2 b) \cosh (c+d x)}{d}+\frac{b (2 a-b) \text{sech}(c+d x)}{d}+\frac{b^2 \text{sech}^3(c+d x)}{3 d} \]

[Out]

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

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Rubi [A]  time = 0.0865373, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {4133, 448} \[ \frac{a^2 \cosh ^3(c+d x)}{3 d}-\frac{a (a-2 b) \cosh (c+d x)}{d}+\frac{b (2 a-b) \text{sech}(c+d x)}{d}+\frac{b^2 \text{sech}^3(c+d x)}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*(a - 2*b)*Cosh[c + d*x])/d) + (a^2*Cosh[c + d*x]^3)/(3*d) + ((2*a - b)*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 448

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps

\begin{align*} \int \left (a+b \text{sech}^2(c+d x)\right )^2 \sinh ^3(c+d x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right ) \left (b+a x^2\right )^2}{x^4} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (a (a-2 b)+\frac{b^2}{x^4}+\frac{(2 a-b) b}{x^2}-a^2 x^2\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{a (a-2 b) \cosh (c+d x)}{d}+\frac{a^2 \cosh ^3(c+d x)}{3 d}+\frac{(2 a-b) b \text{sech}(c+d x)}{d}+\frac{b^2 \text{sech}^3(c+d x)}{3 d}\\ \end{align*}

Mathematica [A]  time = 0.509638, size = 83, normalized size = 1.15 \[ \frac{\text{sech}^3(c+d x) \left (-3 \left (11 a^2-64 a b+16 b^2\right ) \cosh (2 (c+d x))+a^2 \cosh (6 (c+d x))-26 a^2-6 a (a-4 b) \cosh (4 (c+d x))+168 a b-16 b^2\right )}{96 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-26*a^2 + 168*a*b - 16*b^2 - 3*(11*a^2 - 64*a*b + 16*b^2)*Cosh[2*(c + d*x)] - 6*a*(a - 4*b)*Cosh[4*(c + d*x)
] + a^2*Cosh[6*(c + d*x)])*Sech[c + d*x]^3)/(96*d)

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Maple [A]  time = 0.045, size = 108, normalized size = 1.5 \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ( -{\frac{2}{3}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \cosh \left ( dx+c \right ) +2\,ab \left ( -{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{\cosh \left ( dx+c \right ) }}+2\,\cosh \left ( dx+c \right ) \right ) +{b}^{2} \left ( -{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{3\, \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}}+{\frac{2\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{3\,\cosh \left ( dx+c \right ) }}-{\frac{2\,\cosh \left ( dx+c \right ) }{3}} \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)^3,x)

[Out]

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

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Maxima [B]  time = 1.04399, size = 359, normalized size = 4.99 \begin{align*} \frac{1}{24} \, a^{2}{\left (\frac{e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac{9 \, e^{\left (d x + c\right )}}{d} - \frac{9 \, e^{\left (-d x - c\right )}}{d} + \frac{e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + a b{\left (\frac{e^{\left (-d x - c\right )}}{d} + \frac{5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 1}{d{\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}\right )} - \frac{2}{3} \, b^{2}{\left (\frac{3 \, e^{\left (-d x - c\right )}}{d{\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}} + \frac{2 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d{\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}} + \frac{3 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d{\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\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)^3,x, algorithm="maxima")

[Out]

1/24*a^2*(e^(3*d*x + 3*c)/d - 9*e^(d*x + c)/d - 9*e^(-d*x - c)/d + e^(-3*d*x - 3*c)/d) + a*b*(e^(-d*x - c)/d +
(5*e^(-2*d*x - 2*c) + 1)/(d*(e^(-d*x - c) + e^(-3*d*x - 3*c)))) - 2/3*b^2*(3*e^(-d*x - c)/(d*(3*e^(-2*d*x - 2
*c) + 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) + 1)) + 2*e^(-3*d*x - 3*c)/(d*(3*e^(-2*d*x - 2*c) + 3*e^(-4*d*x -
4*c) + e^(-6*d*x - 6*c) + 1)) + 3*e^(-5*d*x - 5*c)/(d*(3*e^(-2*d*x - 2*c) + 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6
*c) + 1)))

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Fricas [B]  time = 2.76468, size = 541, normalized size = 7.51 \begin{align*} \frac{a^{2} \cosh \left (d x + c\right )^{6} + a^{2} \sinh \left (d x + c\right )^{6} - 6 \,{\left (a^{2} - 4 \, a b\right )} \cosh \left (d x + c\right )^{4} + 3 \,{\left (5 \, a^{2} \cosh \left (d x + c\right )^{2} - 2 \, a^{2} + 8 \, a b\right )} \sinh \left (d x + c\right )^{4} - 3 \,{\left (11 \, a^{2} - 64 \, a b + 16 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 3 \,{\left (5 \, a^{2} \cosh \left (d x + c\right )^{4} - 12 \,{\left (a^{2} - 4 \, a b\right )} \cosh \left (d x + c\right )^{2} - 11 \, a^{2} + 64 \, a b - 16 \, b^{2}\right )} \sinh \left (d x + c\right )^{2} - 26 \, a^{2} + 168 \, a b - 16 \, b^{2}}{24 \,{\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)^3,x, algorithm="fricas")

[Out]

1/24*(a^2*cosh(d*x + c)^6 + a^2*sinh(d*x + c)^6 - 6*(a^2 - 4*a*b)*cosh(d*x + c)^4 + 3*(5*a^2*cosh(d*x + c)^2 -
2*a^2 + 8*a*b)*sinh(d*x + c)^4 - 3*(11*a^2 - 64*a*b + 16*b^2)*cosh(d*x + c)^2 + 3*(5*a^2*cosh(d*x + c)^4 - 12
*(a^2 - 4*a*b)*cosh(d*x + c)^2 - 11*a^2 + 64*a*b - 16*b^2)*sinh(d*x + c)^2 - 26*a^2 + 168*a*b - 16*b^2)/(d*cos
h(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)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.17876, size = 207, normalized size = 2.88 \begin{align*} \frac{a^{2} d^{2}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 12 \, a^{2} d^{2}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 24 \, a b d^{2}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}}{24 \, d^{3}} + \frac{2 \,{\left (6 \, a b{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 3 \, b^{2}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} + 4 \, 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)^3,x, algorithm="giac")

[Out]

1/24*(a^2*d^2*(e^(d*x + c) + e^(-d*x - c))^3 - 12*a^2*d^2*(e^(d*x + c) + e^(-d*x - c)) + 24*a*b*d^2*(e^(d*x +
c) + e^(-d*x - c)))/d^3 + 2/3*(6*a*b*(e^(d*x + c) + e^(-d*x - c))^2 - 3*b^2*(e^(d*x + c) + e^(-d*x - c))^2 + 4
*b^2)/(d*(e^(d*x + c) + e^(-d*x - c))^3)