3.3.73 \(\int \frac {\cosh ^3(a+b x) \sinh (a+b x)}{x} \, dx\) [273]

Optimal. Leaf size=53 \[ \frac {1}{4} \text {Chi}(2 b x) \sinh (2 a)+\frac {1}{8} \text {Chi}(4 b x) \sinh (4 a)+\frac {1}{4} \cosh (2 a) \text {Shi}(2 b x)+\frac {1}{8} \cosh (4 a) \text {Shi}(4 b x) \]

[Out]

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

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Rubi [A]
time = 0.11, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5556, 3384, 3379, 3382} \begin {gather*} \frac {1}{4} \sinh (2 a) \text {Chi}(2 b x)+\frac {1}{8} \sinh (4 a) \text {Chi}(4 b x)+\frac {1}{4} \cosh (2 a) \text {Shi}(2 b x)+\frac {1}{8} \cosh (4 a) \text {Shi}(4 b x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Cosh[a + b*x]^3*Sinh[a + b*x])/x,x]

[Out]

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

Rule 3379

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 3382

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 3384

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 5556

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

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Mathematica [A]
time = 0.05, size = 47, normalized size = 0.89 \begin {gather*} \frac {1}{8} (2 \text {Chi}(2 b x) \sinh (2 a)+\text {Chi}(4 b x) \sinh (4 a)+2 \cosh (2 a) \text {Shi}(2 b x)+\cosh (4 a) \text {Shi}(4 b x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Cosh[a + b*x]^3*Sinh[a + b*x])/x,x]

[Out]

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

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Maple [A]
time = 4.75, size = 50, normalized size = 0.94

method	result	size
risch	\(\frac {{\mathrm e}^{-4 a} \expIntegral \left (1, 4 b x \right )}{16}+\frac {{\mathrm e}^{-2 a} \expIntegral \left (1, 2 b x \right )}{8}-\frac {{\mathrm e}^{2 a} \expIntegral \left (1, -2 b x \right )}{8}-\frac {{\mathrm e}^{4 a} \expIntegral \left (1, -4 b x \right )}{16}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(b*x+a)^3*sinh(b*x+a)/x,x,method=_RETURNVERBOSE)

[Out]

1/16*exp(-4*a)*Ei(1,4*b*x)+1/8*exp(-2*a)*Ei(1,2*b*x)-1/8*exp(2*a)*Ei(1,-2*b*x)-1/16*exp(4*a)*Ei(1,-4*b*x)

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Maxima [A]
time = 0.34, size = 45, normalized size = 0.85 \begin {gather*} \frac {1}{16} \, {\rm Ei}\left (4 \, b x\right ) e^{\left (4 \, a\right )} + \frac {1}{8} \, {\rm Ei}\left (2 \, b x\right ) e^{\left (2 \, a\right )} - \frac {1}{8} \, {\rm Ei}\left (-2 \, b x\right ) e^{\left (-2 \, a\right )} - \frac {1}{16} \, {\rm Ei}\left (-4 \, b x\right ) e^{\left (-4 \, a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^3*sinh(b*x+a)/x,x, algorithm="maxima")

[Out]

1/16*Ei(4*b*x)*e^(4*a) + 1/8*Ei(2*b*x)*e^(2*a) - 1/8*Ei(-2*b*x)*e^(-2*a) - 1/16*Ei(-4*b*x)*e^(-4*a)

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Fricas [A]
time = 0.34, size = 73, normalized size = 1.38 \begin {gather*} \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}{8} \, {\left ({\rm Ei}\left (2 \, b x\right ) - {\rm Ei}\left (-2 \, b x\right )\right )} \cosh \left (2 \, 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} \, {\left ({\rm Ei}\left (2 \, b x\right ) + {\rm Ei}\left (-2 \, b x\right )\right )} \sinh \left (2 \, a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^3*sinh(b*x+a)/x,x, algorithm="fricas")

[Out]

1/16*(Ei(4*b*x) - Ei(-4*b*x))*cosh(4*a) + 1/8*(Ei(2*b*x) - Ei(-2*b*x))*cosh(2*a) + 1/16*(Ei(4*b*x) + Ei(-4*b*x
))*sinh(4*a) + 1/8*(Ei(2*b*x) + Ei(-2*b*x))*sinh(2*a)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sinh {\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)**3*sinh(b*x+a)/x,x)

[Out]

Integral(sinh(a + b*x)*cosh(a + b*x)**3/x, x)

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Giac [A]
time = 0.37, size = 45, normalized size = 0.85 \begin {gather*} \frac {1}{16} \, {\rm Ei}\left (4 \, b x\right ) e^{\left (4 \, a\right )} + \frac {1}{8} \, {\rm Ei}\left (2 \, b x\right ) e^{\left (2 \, a\right )} - \frac {1}{8} \, {\rm Ei}\left (-2 \, b x\right ) e^{\left (-2 \, a\right )} - \frac {1}{16} \, {\rm Ei}\left (-4 \, b x\right ) e^{\left (-4 \, a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^3*sinh(b*x+a)/x,x, algorithm="giac")

[Out]

1/16*Ei(4*b*x)*e^(4*a) + 1/8*Ei(2*b*x)*e^(2*a) - 1/8*Ei(-2*b*x)*e^(-2*a) - 1/16*Ei(-4*b*x)*e^(-4*a)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^3\,\mathrm {sinh}\left (a+b\,x\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cosh(a + b*x)^3*sinh(a + b*x))/x,x)

[Out]

int((cosh(a + b*x)^3*sinh(a + b*x))/x, x)

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