3.260 \(\int \frac {\cosh (a+b x) (-2+\sinh ^2(a+b x))}{x} \, dx\)

Optimal. Leaf size=47 \[ -\frac {9}{4} \cosh (a) \text {Chi}(b x)+\frac {1}{4} \cosh (3 a) \text {Chi}(3 b x)-\frac {9}{4} \sinh (a) \text {Shi}(b x)+\frac {1}{4} \sinh (3 a) \text {Shi}(3 b x) \]

[Out]

-9/4*Chi(b*x)*cosh(a)+1/4*Chi(3*b*x)*cosh(3*a)-9/4*Shi(b*x)*sinh(a)+1/4*Shi(3*b*x)*sinh(3*a)

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Rubi [A]  time = 0.46, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6742, 3303, 3298, 3301, 5448} \[ -\frac {9}{4} \cosh (a) \text {Chi}(b x)+\frac {1}{4} \cosh (3 a) \text {Chi}(3 b x)-\frac {9}{4} \sinh (a) \text {Shi}(b x)+\frac {1}{4} \sinh (3 a) \text {Shi}(3 b x) \]

Antiderivative was successfully verified.

[In]

Int[(Cosh[a + b*x]*(-2 + Sinh[a + b*x]^2))/x,x]

[Out]

(-9*Cosh[a]*CoshIntegral[b*x])/4 + (Cosh[3*a]*CoshIntegral[3*b*x])/4 - (9*Sinh[a]*SinhIntegral[b*x])/4 + (Sinh
[3*a]*SinhIntegral[3*b*x])/4

Rule 3298

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(I*SinhIntegral[(c*f*fz)
/d + f*fz*x])/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3301

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[(c*f*fz)/d
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 3303

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 5448

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {\cosh (a+b x) \left (-2+\sinh ^2(a+b x)\right )}{x} \, dx &=\int \left (-\frac {2 \cosh (a+b x)}{x}+\frac {\cosh (a+b x) \sinh ^2(a+b x)}{x}\right ) \, dx\\ &=-\left (2 \int \frac {\cosh (a+b x)}{x} \, dx\right )+\int \frac {\cosh (a+b x) \sinh ^2(a+b x)}{x} \, dx\\ &=-\left ((2 \cosh (a)) \int \frac {\cosh (b x)}{x} \, dx\right )-(2 \sinh (a)) \int \frac {\sinh (b x)}{x} \, dx+\int \left (-\frac {\cosh (a+b x)}{4 x}+\frac {\cosh (3 a+3 b x)}{4 x}\right ) \, dx\\ &=-2 \cosh (a) \text {Chi}(b x)-2 \sinh (a) \text {Shi}(b x)-\frac {1}{4} \int \frac {\cosh (a+b x)}{x} \, dx+\frac {1}{4} \int \frac {\cosh (3 a+3 b x)}{x} \, dx\\ &=-2 \cosh (a) \text {Chi}(b x)-2 \sinh (a) \text {Shi}(b x)-\frac {1}{4} \cosh (a) \int \frac {\cosh (b x)}{x} \, dx+\frac {1}{4} \cosh (3 a) \int \frac {\cosh (3 b x)}{x} \, dx-\frac {1}{4} \sinh (a) \int \frac {\sinh (b x)}{x} \, dx+\frac {1}{4} \sinh (3 a) \int \frac {\sinh (3 b x)}{x} \, dx\\ &=-\frac {9}{4} \cosh (a) \text {Chi}(b x)+\frac {1}{4} \cosh (3 a) \text {Chi}(3 b x)-\frac {9}{4} \sinh (a) \text {Shi}(b x)+\frac {1}{4} \sinh (3 a) \text {Shi}(3 b x)\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 41, normalized size = 0.87 \[ \frac {1}{4} (-9 \cosh (a) \text {Chi}(b x)+\cosh (3 a) \text {Chi}(3 b x)-9 \sinh (a) \text {Shi}(b x)+\sinh (3 a) \text {Shi}(3 b x)) \]

Antiderivative was successfully verified.

[In]

Integrate[(Cosh[a + b*x]*(-2 + Sinh[a + b*x]^2))/x,x]

[Out]

(-9*Cosh[a]*CoshIntegral[b*x] + Cosh[3*a]*CoshIntegral[3*b*x] - 9*Sinh[a]*SinhIntegral[b*x] + Sinh[3*a]*SinhIn
tegral[3*b*x])/4

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fricas [A]  time = 1.08, size = 67, normalized size = 1.43 \[ \frac {1}{8} \, {\left ({\rm Ei}\left (3 \, b x\right ) + {\rm Ei}\left (-3 \, b x\right )\right )} \cosh \left (3 \, a\right ) - \frac {9}{8} \, {\left ({\rm Ei}\left (b x\right ) + {\rm Ei}\left (-b x\right )\right )} \cosh \relax (a) + \frac {1}{8} \, {\left ({\rm Ei}\left (3 \, b x\right ) - {\rm Ei}\left (-3 \, b x\right )\right )} \sinh \left (3 \, a\right ) - \frac {9}{8} \, {\left ({\rm Ei}\left (b x\right ) - {\rm Ei}\left (-b x\right )\right )} \sinh \relax (a) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)*(-2+sinh(b*x+a)^2)/x,x, algorithm="fricas")

[Out]

1/8*(Ei(3*b*x) + Ei(-3*b*x))*cosh(3*a) - 9/8*(Ei(b*x) + Ei(-b*x))*cosh(a) + 1/8*(Ei(3*b*x) - Ei(-3*b*x))*sinh(
3*a) - 9/8*(Ei(b*x) - Ei(-b*x))*sinh(a)

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giac [A]  time = 0.34, size = 42, normalized size = 0.89 \[ \frac {1}{8} \, {\rm Ei}\left (3 \, b x\right ) e^{\left (3 \, a\right )} - \frac {9}{8} \, {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + \frac {1}{8} \, {\rm Ei}\left (-3 \, b x\right ) e^{\left (-3 \, a\right )} - \frac {9}{8} \, {\rm Ei}\left (b x\right ) e^{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)*(-2+sinh(b*x+a)^2)/x,x, algorithm="giac")

[Out]

1/8*Ei(3*b*x)*e^(3*a) - 9/8*Ei(-b*x)*e^(-a) + 1/8*Ei(-3*b*x)*e^(-3*a) - 9/8*Ei(b*x)*e^a

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maple [A]  time = 0.21, size = 47, normalized size = 1.00 \[ -\frac {{\mathrm e}^{-3 a} \Ei \left (1, 3 b x \right )}{8}+\frac {9 \,{\mathrm e}^{-a} \Ei \left (1, b x \right )}{8}+\frac {9 \,{\mathrm e}^{a} \Ei \left (1, -b x \right )}{8}-\frac {{\mathrm e}^{3 a} \Ei \left (1, -3 b x \right )}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(b*x+a)*(-2+sinh(b*x+a)^2)/x,x)

[Out]

-1/8*exp(-3*a)*Ei(1,3*b*x)+9/8*exp(-a)*Ei(1,b*x)+9/8*exp(a)*Ei(1,-b*x)-1/8*exp(3*a)*Ei(1,-3*b*x)

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maxima [A]  time = 0.45, size = 42, normalized size = 0.89 \[ \frac {1}{8} \, {\rm Ei}\left (3 \, b x\right ) e^{\left (3 \, a\right )} - \frac {9}{8} \, {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + \frac {1}{8} \, {\rm Ei}\left (-3 \, b x\right ) e^{\left (-3 \, a\right )} - \frac {9}{8} \, {\rm Ei}\left (b x\right ) e^{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)*(-2+sinh(b*x+a)^2)/x,x, algorithm="maxima")

[Out]

1/8*Ei(3*b*x)*e^(3*a) - 9/8*Ei(-b*x)*e^(-a) + 1/8*Ei(-3*b*x)*e^(-3*a) - 9/8*Ei(b*x)*e^a

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {cosh}\left (a+b\,x\right )\,\left ({\mathrm {sinh}\left (a+b\,x\right )}^2-2\right )}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cosh(a + b*x)*(sinh(a + b*x)^2 - 2))/x,x)

[Out]

int((cosh(a + b*x)*(sinh(a + b*x)^2 - 2))/x, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\sinh ^{2}{\left (a + b x \right )} - 2\right ) \cosh {\left (a + b x \right )}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)*(-2+sinh(b*x+a)**2)/x,x)

[Out]

Integral((sinh(a + b*x)**2 - 2)*cosh(a + b*x)/x, x)

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