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

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Rubi [A]  time = 0.05, antiderivative size = 30, 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 = {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 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]

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]

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*}

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Mathematica [A]  time = 0.02, 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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fricas [A]  time = 0.39, size = 41, normalized size = 1.37 \[ \frac {a \sinh \left (d x + c\right )^{3} + 3 \, {\left (a \cosh \left (d x + c\right )^{2} + 3 \, a + 4 \, b\right )} \sinh \left (d x + c\right )}{12 \, d} \]

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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giac [B]  time = 0.16, size = 72, normalized size = 2.40 \[ \frac {a e^{\left (3 \, d x + 3 \, c\right )} + 9 \, a e^{\left (d x + c\right )} + 12 \, b e^{\left (d x + c\right )} - {\left (9 \, a e^{\left (2 \, d x + 2 \, c\right )} + 12 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{24 \, d} \]

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

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maple [A]  time = 0.43, size = 34, normalized size = 1.13 \[ \frac {a \left (\frac {2}{3}+\frac {\left (\cosh ^{2}\left (d x +c \right )\right )}{3}\right ) \sinh \left (d x +c \right )+b \sinh \left (d x +c \right )}{d} \]

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 = 0.31, size = 85, normalized size = 2.83 \[ \frac {1}{24} \, a {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} - \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + \frac {1}{2} \, b {\left (\frac {e^{\left (d x + c\right )}}{d} - \frac {e^{\left (-d x - c\right )}}{d}\right )} \]

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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mupad [B]  time = 0.09, size = 34, normalized size = 1.13 \[ \frac {3\,a\,\mathrm {sinh}\left (c+d\,x\right )+3\,b\,\mathrm {sinh}\left (c+d\,x\right )+a\,{\mathrm {sinh}\left (c+d\,x\right )}^3}{3\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right ) \cosh ^{3}{\left (c + d x \right )}\, dx \]

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]

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

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