### 3.294 $$\int \frac{\cosh ^2(a+b x) \sinh ^2(a+b x)}{x} \, dx$$

Optimal. Leaf size=33 $\frac{1}{8} \cosh (4 a) \text{Chi}(4 b x)+\frac{1}{8} \sinh (4 a) \text{Shi}(4 b x)-\frac{\log (x)}{8}$

[Out]

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

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Rubi [A]  time = 0.0848538, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.2, Rules used = {5448, 3303, 3298, 3301} $\frac{1}{8} \cosh (4 a) \text{Chi}(4 b x)+\frac{1}{8} \sinh (4 a) \text{Shi}(4 b x)-\frac{\log (x)}{8}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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 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 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]

Rubi steps

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

Mathematica [A]  time = 0.0974682, size = 32, normalized size = 0.97 $\frac{1}{8} (\cosh (4 a) \text{Chi}(4 b x)+\sinh (4 a) \text{Shi}(4 b x)-\log (2 b x))$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.049, size = 30, normalized size = 0.9 \begin{align*} -{\frac{\ln \left ( x \right ) }{8}}-{\frac{{{\rm e}^{-4\,a}}{\it Ei} \left ( 1,4\,bx \right ) }{16}}-{\frac{{{\rm e}^{4\,a}}{\it Ei} \left ( 1,-4\,bx \right ) }{16}} \end{align*}

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

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Maxima [A]  time = 1.26553, size = 36, normalized size = 1.09 \begin{align*} \frac{1}{16} \,{\rm Ei}\left (4 \, b x\right ) e^{\left (4 \, a\right )} + \frac{1}{16} \,{\rm Ei}\left (-4 \, b x\right ) e^{\left (-4 \, a\right )} - \frac{1}{8} \, \log \left (x\right ) \end{align*}

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]

1/16*Ei(4*b*x)*e^(4*a) + 1/16*Ei(-4*b*x)*e^(-4*a) - 1/8*log(x)

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Fricas [A]  time = 1.83623, size = 130, normalized size = 3.94 \begin{align*} \frac{1}{16} \,{\left ({\rm Ei}\left (4 \, b x\right ) +{\rm Ei}\left (-4 \, b x\right )\right )} \cosh \left (4 \, a\right ) + \frac{1}{16} \,{\left ({\rm Ei}\left (4 \, b x\right ) -{\rm Ei}\left (-4 \, b x\right )\right )} \sinh \left (4 \, a\right ) - \frac{1}{8} \, \log \left (x\right ) \end{align*}

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]

1/16*(Ei(4*b*x) + Ei(-4*b*x))*cosh(4*a) + 1/16*(Ei(4*b*x) - Ei(-4*b*x))*sinh(4*a) - 1/8*log(x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{x}\, dx \end{align*}

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*cosh(a + b*x)**2/x, x)

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Giac [A]  time = 1.15044, size = 36, normalized size = 1.09 \begin{align*} \frac{1}{16} \,{\rm Ei}\left (4 \, b x\right ) e^{\left (4 \, a\right )} + \frac{1}{16} \,{\rm Ei}\left (-4 \, b x\right ) e^{\left (-4 \, a\right )} - \frac{1}{8} \, \log \left (x\right ) \end{align*}

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]

1/16*Ei(4*b*x)*e^(4*a) + 1/16*Ei(-4*b*x)*e^(-4*a) - 1/8*log(x)