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

Optimal. Leaf size=52 \[ \frac {1}{2} b \sinh (4 a) \text {Chi}(4 b x)+\frac {1}{2} b \cosh (4 a) \text {Shi}(4 b x)-\frac {\cosh (4 a+4 b x)}{8 x}+\frac {1}{8 x} \]

[Out]

1/8/x-1/8*cosh(4*b*x+4*a)/x+1/2*b*cosh(4*a)*Shi(4*b*x)+1/2*b*Chi(4*b*x)*sinh(4*a)

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Rubi [A]  time = 0.11, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5448, 3297, 3303, 3298, 3301} \[ \frac {1}{2} b \sinh (4 a) \text {Chi}(4 b x)+\frac {1}{2} b \cosh (4 a) \text {Shi}(4 b x)-\frac {\cosh (4 a+4 b x)}{8 x}+\frac {1}{8 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

1/(8*x) - Cosh[4*a + 4*b*x]/(8*x) + (b*CoshIntegral[4*b*x]*Sinh[4*a])/2 + (b*Cosh[4*a]*SinhIntegral[4*b*x])/2

Rule 3297

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x])/(d*(
m + 1)), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]

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]

Rubi steps

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

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Mathematica [A]  time = 0.09, size = 45, normalized size = 0.87 \[ \frac {4 b x \sinh (4 a) \text {Chi}(4 b x)+4 b x \cosh (4 a) \text {Shi}(4 b x)-\cosh (4 (a+b x))+1}{8 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1 - Cosh[4*(a + b*x)] + 4*b*x*CoshIntegral[4*b*x]*Sinh[4*a] + 4*b*x*Cosh[4*a]*SinhIntegral[4*b*x])/(8*x)

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fricas [A]  time = 0.63, size = 88, normalized size = 1.69 \[ -\frac {\cosh \left (b x + a\right )^{4} + 6 \, \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{4} - 2 \, {\left (b x {\rm Ei}\left (4 \, b x\right ) - b x {\rm Ei}\left (-4 \, b x\right )\right )} \cosh \left (4 \, a\right ) - 2 \, {\left (b x {\rm Ei}\left (4 \, b x\right ) + b x {\rm Ei}\left (-4 \, b x\right )\right )} \sinh \left (4 \, a\right ) - 1}{8 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(cosh(b*x + a)^4 + 6*cosh(b*x + a)^2*sinh(b*x + a)^2 + sinh(b*x + a)^4 - 2*(b*x*Ei(4*b*x) - b*x*Ei(-4*b*x
))*cosh(4*a) - 2*(b*x*Ei(4*b*x) + b*x*Ei(-4*b*x))*sinh(4*a) - 1)/x

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giac [A]  time = 0.15, size = 55, normalized size = 1.06 \[ \frac {4 \, b x {\rm Ei}\left (4 \, b x\right ) e^{\left (4 \, a\right )} - 4 \, b x {\rm Ei}\left (-4 \, b x\right ) e^{\left (-4 \, a\right )} - e^{\left (4 \, b x + 4 \, a\right )} - e^{\left (-4 \, b x - 4 \, a\right )} + 2}{16 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(4*b*x*Ei(4*b*x)*e^(4*a) - 4*b*x*Ei(-4*b*x)*e^(-4*a) - e^(4*b*x + 4*a) - e^(-4*b*x - 4*a) + 2)/x

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maple [A]  time = 0.35, size = 61, normalized size = 1.17 \[ \frac {1}{8 x}-\frac {{\mathrm e}^{-4 b x -4 a}}{16 x}+\frac {b \,{\mathrm e}^{-4 a} \Ei \left (1, 4 b x \right )}{4}-\frac {{\mathrm e}^{4 b x +4 a}}{16 x}-\frac {b \,{\mathrm e}^{4 a} \Ei \left (1, -4 b x \right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8/x-1/16*exp(-4*b*x-4*a)/x+1/4*b*exp(-4*a)*Ei(1,4*b*x)-1/16/x*exp(4*b*x+4*a)-1/4*b*exp(4*a)*Ei(1,-4*b*x)

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maxima [A]  time = 0.60, size = 32, normalized size = 0.62 \[ -\frac {1}{4} \, b e^{\left (-4 \, a\right )} \Gamma \left (-1, 4 \, b x\right ) + \frac {1}{4} \, b e^{\left (4 \, a\right )} \Gamma \left (-1, -4 \, b x\right ) + \frac {1}{8 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*b*e^(-4*a)*gamma(-1, 4*b*x) + 1/4*b*e^(4*a)*gamma(-1, -4*b*x) + 1/8/x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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