3.49 \(\int \cosh ^4(c+d x) (a+b \text{sech}^2(c+d x)) \, dx\)

Optimal. Leaf size=61 \[ \frac{(3 a+4 b) \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{1}{8} x (3 a+4 b)+\frac{a \sinh (c+d x) \cosh ^3(c+d x)}{4 d} \]

[Out]

((3*a + 4*b)*x)/8 + ((3*a + 4*b)*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) + (a*Cosh[c + d*x]^3*Sinh[c + d*x])/(4*d)

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Rubi [A]  time = 0.0458518, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4045, 2635, 8} \[ \frac{(3 a+4 b) \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{1}{8} x (3 a+4 b)+\frac{a \sinh (c+d x) \cosh ^3(c+d x)}{4 d} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]^4*(a + b*Sech[c + d*x]^2),x]

[Out]

((3*a + 4*b)*x)/8 + ((3*a + 4*b)*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) + (a*Cosh[c + d*x]^3*Sinh[c + d*x])/(4*d)

Rule 4045

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[(A*Cot[e
+ f*x]*(b*Csc[e + f*x])^m)/(f*m), x] + Dist[(C*m + A*(m + 1))/(b^2*m), Int[(b*Csc[e + f*x])^(m + 2), x], x] /
; FreeQ[{b, e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && LeQ[m, -1]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \cosh ^4(c+d x) \left (a+b \text{sech}^2(c+d x)\right ) \, dx &=\frac{a \cosh ^3(c+d x) \sinh (c+d x)}{4 d}+\frac{1}{4} (3 a+4 b) \int \cosh ^2(c+d x) \, dx\\ &=\frac{(3 a+4 b) \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac{a \cosh ^3(c+d x) \sinh (c+d x)}{4 d}+\frac{1}{8} (3 a+4 b) \int 1 \, dx\\ &=\frac{1}{8} (3 a+4 b) x+\frac{(3 a+4 b) \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac{a \cosh ^3(c+d x) \sinh (c+d x)}{4 d}\\ \end{align*}

Mathematica [A]  time = 0.0924414, size = 45, normalized size = 0.74 \[ \frac{4 (3 a+4 b) (c+d x)+8 (a+b) \sinh (2 (c+d x))+a \sinh (4 (c+d x))}{32 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[c + d*x]^4*(a + b*Sech[c + d*x]^2),x]

[Out]

(4*(3*a + 4*b)*(c + d*x) + 8*(a + b)*Sinh[2*(c + d*x)] + a*Sinh[4*(c + d*x)])/(32*d)

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Maple [A]  time = 0.04, size = 66, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ( a \left ( \left ({\frac{ \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}{4}}+{\frac{3\,\cosh \left ( dx+c \right ) }{8}} \right ) \sinh \left ( dx+c \right ) +{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) +b \left ({\frac{\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(d*x+c)^4*(a+b*sech(d*x+c)^2),x)

[Out]

1/d*(a*((1/4*cosh(d*x+c)^3+3/8*cosh(d*x+c))*sinh(d*x+c)+3/8*d*x+3/8*c)+b*(1/2*cosh(d*x+c)*sinh(d*x+c)+1/2*d*x+
1/2*c))

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Maxima [A]  time = 1.13573, size = 131, normalized size = 2.15 \begin{align*} \frac{1}{64} \, a{\left (24 \, x + \frac{e^{\left (4 \, d x + 4 \, c\right )}}{d} + \frac{8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac{8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac{e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} + \frac{1}{8} \, b{\left (4 \, x + \frac{e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac{e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^4*(a+b*sech(d*x+c)^2),x, algorithm="maxima")

[Out]

1/64*a*(24*x + e^(4*d*x + 4*c)/d + 8*e^(2*d*x + 2*c)/d - 8*e^(-2*d*x - 2*c)/d - e^(-4*d*x - 4*c)/d) + 1/8*b*(4
*x + e^(2*d*x + 2*c)/d - e^(-2*d*x - 2*c)/d)

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Fricas [A]  time = 2.06623, size = 163, normalized size = 2.67 \begin{align*} \frac{a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} +{\left (3 \, a + 4 \, b\right )} d x +{\left (a \cosh \left (d x + c\right )^{3} + 4 \,{\left (a + b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^4*(a+b*sech(d*x+c)^2),x, algorithm="fricas")

[Out]

1/8*(a*cosh(d*x + c)*sinh(d*x + c)^3 + (3*a + 4*b)*d*x + (a*cosh(d*x + c)^3 + 4*(a + b)*cosh(d*x + c))*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)**4*(a+b*sech(d*x+c)**2),x)

[Out]

Timed out

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Giac [B]  time = 1.21788, size = 157, normalized size = 2.57 \begin{align*} \frac{8 \,{\left (d x + c\right )}{\left (3 \, a + 4 \, b\right )} + a e^{\left (4 \, d x + 4 \, c\right )} + 8 \, a e^{\left (2 \, d x + 2 \, c\right )} + 8 \, b e^{\left (2 \, d x + 2 \, c\right )} -{\left (18 \, a e^{\left (4 \, d x + 4 \, c\right )} + 24 \, b e^{\left (4 \, d x + 4 \, c\right )} + 8 \, a e^{\left (2 \, d x + 2 \, c\right )} + 8 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{64 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^4*(a+b*sech(d*x+c)^2),x, algorithm="giac")

[Out]

1/64*(8*(d*x + c)*(3*a + 4*b) + a*e^(4*d*x + 4*c) + 8*a*e^(2*d*x + 2*c) + 8*b*e^(2*d*x + 2*c) - (18*a*e^(4*d*x
+ 4*c) + 24*b*e^(4*d*x + 4*c) + 8*a*e^(2*d*x + 2*c) + 8*b*e^(2*d*x + 2*c) + a)*e^(-4*d*x - 4*c))/d