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

Optimal. Leaf size=30 \[ \frac{(a+b) \sinh (c+d x)}{d}+\frac{a \sinh ^3(c+d x)}{3 d} \]

[Out]

((a + b)*Sinh[c + d*x])/d + (a*Sinh[c + d*x]^3)/(3*d)

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Rubi [A]  time = 0.0502306, antiderivative size = 30, 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 = {4044, 3013} \[ \frac{(a+b) \sinh (c+d x)}{d}+\frac{a \sinh ^3(c+d x)}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b)*Sinh[c + d*x])/d + (a*Sinh[c + d*x]^3)/(3*d)

Rule 4044

Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Int[(C + A*Sin[e + f*
x]^2)/Sin[e + f*x]^(m + 2), x] /; FreeQ[{e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && ILtQ[(m + 1)/2, 0]

Rule 3013

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Dist[f^(-1), Subst[I
nt[(1 - x^2)^((m - 1)/2)*(A + C - C*x^2), x], x, Cos[e + f*x]], x] /; FreeQ[{e, f, A, C}, x] && IGtQ[(m + 1)/2
, 0]

Rubi steps

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

Mathematica [A]  time = 0.0173721, size = 50, normalized size = 1.67 \[ \frac{a \sinh ^3(c+d x)}{3 d}+\frac{a \sinh (c+d x)}{d}+\frac{b \sinh (c) \cosh (d x)}{d}+\frac{b \cosh (c) \sinh (d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.037, size = 34, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ( a \left ({\frac{2}{3}}+{\frac{ \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \sinh \left ( dx+c \right ) +b\sinh \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.14902, size = 115, normalized size = 3.83 \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{e^{\left (-d x - c\right )}}{d}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^3*(a+b*sech(d*x+c)^2),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 -
e^(-d*x - c)/d)

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Fricas [A]  time = 2.099, size = 105, normalized size = 3.5 \begin{align*} \frac{a \sinh \left (d x + c\right )^{3} + 3 \,{\left (a \cosh \left (d x + c\right )^{2} + 3 \, a + 4 \, b\right )} \sinh \left (d x + c\right )}{12 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(a*sinh(d*x + c)^3 + 3*(a*cosh(d*x + c)^2 + 3*a + 4*b)*sinh(d*x + c))/d

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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(cosh(d*x+c)**3*(a+b*sech(d*x+c)**2),x)

[Out]

Timed out

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Giac [B]  time = 1.16692, size = 97, normalized size = 3.23 \begin{align*} \frac{a e^{\left (3 \, d x + 3 \, c\right )} + 9 \, a e^{\left (d x + c\right )} + 12 \, b e^{\left (d x + c\right )} -{\left (9 \, a e^{\left (2 \, d x + 2 \, c\right )} + 12 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{24 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(a*e^(3*d*x + 3*c) + 9*a*e^(d*x + c) + 12*b*e^(d*x + c) - (9*a*e^(2*d*x + 2*c) + 12*b*e^(2*d*x + 2*c) + a
)*e^(-3*d*x - 3*c))/d