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

Optimal. Leaf size=44 \[ -\frac{(a-b) \cosh (c+d x)}{d}+\frac{a \cosh ^3(c+d x)}{3 d}+\frac{b \text{sech}(c+d x)}{d} \]

[Out]

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

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Rubi [A]  time = 0.0560561, antiderivative size = 44, 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, 448} \[ -\frac{(a-b) \cosh (c+d x)}{d}+\frac{a \cosh ^3(c+d x)}{3 d}+\frac{b \text{sech}(c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0488596, size = 53, normalized size = 1.2 \[ -\frac{3 a \cosh (c+d x)}{4 d}+\frac{a \cosh (3 (c+d x))}{12 d}+\frac{b \cosh (c+d x)}{d}+\frac{b \text{sech}(c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*a*Cosh[c + d*x])/(4*d) + (b*Cosh[c + d*x])/d + (a*Cosh[3*(c + d*x)])/(12*d) + (b*Sech[c + d*x])/d

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Maple [A]  time = 0.029, size = 55, normalized size = 1.3 \begin{align*}{\frac{1}{d} \left ( a \left ( -{\frac{2}{3}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \cosh \left ( dx+c \right ) +b \left ( -{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{\cosh \left ( dx+c \right ) }}+2\,\cosh \left ( dx+c \right ) \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sech(d*x+c)^2)*sinh(d*x+c)^3,x)

[Out]

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

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Maxima [B]  time = 1.00415, size = 150, normalized size = 3.41 \begin{align*} \frac{1}{24} \, a{\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 )} + \frac{1}{2} \, 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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sech(d*x+c)^2)*sinh(d*x+c)^3,x, algorithm="maxima")

[Out]

1/24*a*(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) + 1/2*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))))

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Fricas [B]  time = 2.52248, size = 220, normalized size = 5. \begin{align*} \frac{a \cosh \left (d x + c\right )^{4} + a \sinh \left (d x + c\right )^{4} - 4 \,{\left (2 \, a - 3 \, b\right )} \cosh \left (d x + c\right )^{2} + 2 \,{\left (3 \, a \cosh \left (d x + c\right )^{2} - 4 \, a + 6 \, b\right )} \sinh \left (d x + c\right )^{2} - 9 \, a + 36 \, b}{24 \, d \cosh \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sech(d*x+c)^2)*sinh(d*x+c)^3,x, algorithm="fricas")

[Out]

1/24*(a*cosh(d*x + c)^4 + a*sinh(d*x + c)^4 - 4*(2*a - 3*b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 - 4*a + 6
*b)*sinh(d*x + c)^2 - 9*a + 36*b)/(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)*sinh(d*x+c)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.18559, size = 132, normalized size = 3. \begin{align*} \frac{2 \, b}{d{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}} + \frac{a d^{2}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 12 \, a d^{2}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 12 \, b d^{2}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}}{24 \, d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sech(d*x+c)^2)*sinh(d*x+c)^3,x, algorithm="giac")

[Out]

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