3.322 \(\int \frac {\cosh ^2(a+b x) \sinh ^3(a+b x)}{x^2} \, dx\)

Optimal. Leaf size=124 \[ -\frac {1}{8} b \cosh (a) \text {Chi}(b x)-\frac {3}{16} b \cosh (3 a) \text {Chi}(3 b x)+\frac {5}{16} b \cosh (5 a) \text {Chi}(5 b x)-\frac {1}{8} b \sinh (a) \text {Shi}(b x)-\frac {3}{16} b \sinh (3 a) \text {Shi}(3 b x)+\frac {5}{16} b \sinh (5 a) \text {Shi}(5 b x)+\frac {\sinh (a+b x)}{8 x}+\frac {\sinh (3 a+3 b x)}{16 x}-\frac {\sinh (5 a+5 b x)}{16 x} \]

[Out]

-1/8*b*Chi(b*x)*cosh(a)-3/16*b*Chi(3*b*x)*cosh(3*a)+5/16*b*Chi(5*b*x)*cosh(5*a)-1/8*b*Shi(b*x)*sinh(a)-3/16*b*
Shi(3*b*x)*sinh(3*a)+5/16*b*Shi(5*b*x)*sinh(5*a)+1/8*sinh(b*x+a)/x+1/16*sinh(3*b*x+3*a)/x-1/16*sinh(5*b*x+5*a)
/x

________________________________________________________________________________________

Rubi [A]  time = 0.25, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 14, 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} \[ -\frac {1}{8} b \cosh (a) \text {Chi}(b x)-\frac {3}{16} b \cosh (3 a) \text {Chi}(3 b x)+\frac {5}{16} b \cosh (5 a) \text {Chi}(5 b x)-\frac {1}{8} b \sinh (a) \text {Shi}(b x)-\frac {3}{16} b \sinh (3 a) \text {Shi}(3 b x)+\frac {5}{16} b \sinh (5 a) \text {Shi}(5 b x)+\frac {\sinh (a+b x)}{8 x}+\frac {\sinh (3 a+3 b x)}{16 x}-\frac {\sinh (5 a+5 b x)}{16 x} \]

Antiderivative was successfully verified.

[In]

Int[(Cosh[a + b*x]^2*Sinh[a + b*x]^3)/x^2,x]

[Out]

-(b*Cosh[a]*CoshIntegral[b*x])/8 - (3*b*Cosh[3*a]*CoshIntegral[3*b*x])/16 + (5*b*Cosh[5*a]*CoshIntegral[5*b*x]
)/16 + Sinh[a + b*x]/(8*x) + Sinh[3*a + 3*b*x]/(16*x) - Sinh[5*a + 5*b*x]/(16*x) - (b*Sinh[a]*SinhIntegral[b*x
])/8 - (3*b*Sinh[3*a]*SinhIntegral[3*b*x])/16 + (5*b*Sinh[5*a]*SinhIntegral[5*b*x])/16

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 ^3(a+b x)}{x^2} \, dx &=\int \left (-\frac {\sinh (a+b x)}{8 x^2}-\frac {\sinh (3 a+3 b x)}{16 x^2}+\frac {\sinh (5 a+5 b x)}{16 x^2}\right ) \, dx\\ &=-\left (\frac {1}{16} \int \frac {\sinh (3 a+3 b x)}{x^2} \, dx\right )+\frac {1}{16} \int \frac {\sinh (5 a+5 b x)}{x^2} \, dx-\frac {1}{8} \int \frac {\sinh (a+b x)}{x^2} \, dx\\ &=\frac {\sinh (a+b x)}{8 x}+\frac {\sinh (3 a+3 b x)}{16 x}-\frac {\sinh (5 a+5 b x)}{16 x}-\frac {1}{8} b \int \frac {\cosh (a+b x)}{x} \, dx-\frac {1}{16} (3 b) \int \frac {\cosh (3 a+3 b x)}{x} \, dx+\frac {1}{16} (5 b) \int \frac {\cosh (5 a+5 b x)}{x} \, dx\\ &=\frac {\sinh (a+b x)}{8 x}+\frac {\sinh (3 a+3 b x)}{16 x}-\frac {\sinh (5 a+5 b x)}{16 x}-\frac {1}{8} (b \cosh (a)) \int \frac {\cosh (b x)}{x} \, dx-\frac {1}{16} (3 b \cosh (3 a)) \int \frac {\cosh (3 b x)}{x} \, dx+\frac {1}{16} (5 b \cosh (5 a)) \int \frac {\cosh (5 b x)}{x} \, dx-\frac {1}{8} (b \sinh (a)) \int \frac {\sinh (b x)}{x} \, dx-\frac {1}{16} (3 b \sinh (3 a)) \int \frac {\sinh (3 b x)}{x} \, dx+\frac {1}{16} (5 b \sinh (5 a)) \int \frac {\sinh (5 b x)}{x} \, dx\\ &=-\frac {1}{8} b \cosh (a) \text {Chi}(b x)-\frac {3}{16} b \cosh (3 a) \text {Chi}(3 b x)+\frac {5}{16} b \cosh (5 a) \text {Chi}(5 b x)+\frac {\sinh (a+b x)}{8 x}+\frac {\sinh (3 a+3 b x)}{16 x}-\frac {\sinh (5 a+5 b x)}{16 x}-\frac {1}{8} b \sinh (a) \text {Shi}(b x)-\frac {3}{16} b \sinh (3 a) \text {Shi}(3 b x)+\frac {5}{16} b \sinh (5 a) \text {Shi}(5 b x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.27, size = 106, normalized size = 0.85 \[ \frac {-2 b x \cosh (a) \text {Chi}(b x)-3 b x \cosh (3 a) \text {Chi}(3 b x)+5 b x \cosh (5 a) \text {Chi}(5 b x)-2 b x \sinh (a) \text {Shi}(b x)-3 b x \sinh (3 a) \text {Shi}(3 b x)+5 b x \sinh (5 a) \text {Shi}(5 b x)+2 \sinh (a+b x)+\sinh (3 (a+b x))-\sinh (5 (a+b x))}{16 x} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cosh[a + b*x]^2*Sinh[a + b*x]^3)/x^2,x]

[Out]

(-2*b*x*Cosh[a]*CoshIntegral[b*x] - 3*b*x*Cosh[3*a]*CoshIntegral[3*b*x] + 5*b*x*Cosh[5*a]*CoshIntegral[5*b*x]
+ 2*Sinh[a + b*x] + Sinh[3*(a + b*x)] - Sinh[5*(a + b*x)] - 2*b*x*Sinh[a]*SinhIntegral[b*x] - 3*b*x*Sinh[3*a]*
SinhIntegral[3*b*x] + 5*b*x*Sinh[5*a]*SinhIntegral[5*b*x])/(16*x)

________________________________________________________________________________________

fricas [A]  time = 1.01, size = 203, normalized size = 1.64 \[ -\frac {2 \, \sinh \left (b x + a\right )^{5} + 2 \, {\left (10 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{3} - 5 \, {\left (b x {\rm Ei}\left (5 \, b x\right ) + b x {\rm Ei}\left (-5 \, b x\right )\right )} \cosh \left (5 \, a\right ) + 3 \, {\left (b x {\rm Ei}\left (3 \, b x\right ) + b x {\rm Ei}\left (-3 \, b x\right )\right )} \cosh \left (3 \, a\right ) + 2 \, {\left (b x {\rm Ei}\left (b x\right ) + b x {\rm Ei}\left (-b x\right )\right )} \cosh \relax (a) + 2 \, {\left (5 \, \cosh \left (b x + a\right )^{4} - 3 \, \cosh \left (b x + a\right )^{2} - 2\right )} \sinh \left (b x + a\right ) - 5 \, {\left (b x {\rm Ei}\left (5 \, b x\right ) - b x {\rm Ei}\left (-5 \, b x\right )\right )} \sinh \left (5 \, a\right ) + 3 \, {\left (b x {\rm Ei}\left (3 \, b x\right ) - b x {\rm Ei}\left (-3 \, b x\right )\right )} \sinh \left (3 \, a\right ) + 2 \, {\left (b x {\rm Ei}\left (b x\right ) - b x {\rm Ei}\left (-b x\right )\right )} \sinh \relax (a)}{32 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^2*sinh(b*x+a)^3/x^2,x, algorithm="fricas")

[Out]

-1/32*(2*sinh(b*x + a)^5 + 2*(10*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^3 - 5*(b*x*Ei(5*b*x) + b*x*Ei(-5*b*x))*cos
h(5*a) + 3*(b*x*Ei(3*b*x) + b*x*Ei(-3*b*x))*cosh(3*a) + 2*(b*x*Ei(b*x) + b*x*Ei(-b*x))*cosh(a) + 2*(5*cosh(b*x
+ a)^4 - 3*cosh(b*x + a)^2 - 2)*sinh(b*x + a) - 5*(b*x*Ei(5*b*x) - b*x*Ei(-5*b*x))*sinh(5*a) + 3*(b*x*Ei(3*b*
x) - b*x*Ei(-3*b*x))*sinh(3*a) + 2*(b*x*Ei(b*x) - b*x*Ei(-b*x))*sinh(a))/x

________________________________________________________________________________________

giac [A]  time = 0.15, size = 140, normalized size = 1.13 \[ \frac {5 \, b x {\rm Ei}\left (5 \, b x\right ) e^{\left (5 \, a\right )} - 3 \, b x {\rm Ei}\left (3 \, b x\right ) e^{\left (3 \, a\right )} - 2 \, b x {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - 3 \, b x {\rm Ei}\left (-3 \, b x\right ) e^{\left (-3 \, a\right )} + 5 \, b x {\rm Ei}\left (-5 \, b x\right ) e^{\left (-5 \, a\right )} - 2 \, b x {\rm Ei}\left (b x\right ) e^{a} - e^{\left (5 \, b x + 5 \, a\right )} + e^{\left (3 \, b x + 3 \, a\right )} + 2 \, e^{\left (b x + a\right )} - 2 \, e^{\left (-b x - a\right )} - e^{\left (-3 \, b x - 3 \, a\right )} + e^{\left (-5 \, b x - 5 \, a\right )}}{32 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^2*sinh(b*x+a)^3/x^2,x, algorithm="giac")

[Out]

1/32*(5*b*x*Ei(5*b*x)*e^(5*a) - 3*b*x*Ei(3*b*x)*e^(3*a) - 2*b*x*Ei(-b*x)*e^(-a) - 3*b*x*Ei(-3*b*x)*e^(-3*a) +
5*b*x*Ei(-5*b*x)*e^(-5*a) - 2*b*x*Ei(b*x)*e^a - e^(5*b*x + 5*a) + e^(3*b*x + 3*a) + 2*e^(b*x + a) - 2*e^(-b*x
- a) - e^(-3*b*x - 3*a) + e^(-5*b*x - 5*a))/x

________________________________________________________________________________________

maple [A]  time = 0.41, size = 158, normalized size = 1.27 \[ \frac {{\mathrm e}^{-5 b x -5 a}}{32 x}-\frac {5 b \,{\mathrm e}^{-5 a} \Ei \left (1, 5 b x \right )}{32}-\frac {{\mathrm e}^{-3 b x -3 a}}{32 x}+\frac {3 b \,{\mathrm e}^{-3 a} \Ei \left (1, 3 b x \right )}{32}-\frac {{\mathrm e}^{-b x -a}}{16 x}+\frac {b \,{\mathrm e}^{-a} \Ei \left (1, b x \right )}{16}+\frac {{\mathrm e}^{b x +a}}{16 x}+\frac {b \,{\mathrm e}^{a} \Ei \left (1, -b x \right )}{16}+\frac {{\mathrm e}^{3 b x +3 a}}{32 x}+\frac {3 b \,{\mathrm e}^{3 a} \Ei \left (1, -3 b x \right )}{32}-\frac {{\mathrm e}^{5 b x +5 a}}{32 x}-\frac {5 b \,{\mathrm e}^{5 a} \Ei \left (1, -5 b x \right )}{32} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*exp(-5*b*x-5*a)/x-5/32*b*exp(-5*a)*Ei(1,5*b*x)-1/32*exp(-3*b*x-3*a)/x+3/32*b*exp(-3*a)*Ei(1,3*b*x)-1/16*e
xp(-b*x-a)/x+1/16*b*exp(-a)*Ei(1,b*x)+1/16/x*exp(b*x+a)+1/16*b*exp(a)*Ei(1,-b*x)+1/32/x*exp(3*b*x+3*a)+3/32*b*
exp(3*a)*Ei(1,-3*b*x)-1/32/x*exp(5*b*x+5*a)-5/32*b*exp(5*a)*Ei(1,-5*b*x)

________________________________________________________________________________________

maxima [A]  time = 0.47, size = 76, normalized size = 0.61 \[ \frac {5}{32} \, b e^{\left (-5 \, a\right )} \Gamma \left (-1, 5 \, b x\right ) - \frac {3}{32} \, b e^{\left (-3 \, a\right )} \Gamma \left (-1, 3 \, b x\right ) - \frac {1}{16} \, b e^{\left (-a\right )} \Gamma \left (-1, b x\right ) - \frac {1}{16} \, b e^{a} \Gamma \left (-1, -b x\right ) - \frac {3}{32} \, b e^{\left (3 \, a\right )} \Gamma \left (-1, -3 \, b x\right ) + \frac {5}{32} \, b e^{\left (5 \, a\right )} \Gamma \left (-1, -5 \, b x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^2*sinh(b*x+a)^3/x^2,x, algorithm="maxima")

[Out]

5/32*b*e^(-5*a)*gamma(-1, 5*b*x) - 3/32*b*e^(-3*a)*gamma(-1, 3*b*x) - 1/16*b*e^(-a)*gamma(-1, b*x) - 1/16*b*e^
a*gamma(-1, -b*x) - 3/32*b*e^(3*a)*gamma(-1, -3*b*x) + 5/32*b*e^(5*a)*gamma(-1, -5*b*x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^2\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)**2*sinh(b*x+a)**3/x**2,x)

[Out]

Integral(sinh(a + b*x)**3*cosh(a + b*x)**2/x**2, x)

________________________________________________________________________________________