3.306 \(\int \frac {\cosh ^3(a+b x) \sinh ^2(a+b x)}{x^4} \, dx\)

Optimal. Leaf size=238 \[ -\frac {1}{48} b^3 \sinh (a) \text {Chi}(b x)+\frac {9}{32} b^3 \sinh (3 a) \text {Chi}(3 b x)+\frac {125}{96} b^3 \sinh (5 a) \text {Chi}(5 b x)-\frac {1}{48} b^3 \cosh (a) \text {Shi}(b x)+\frac {9}{32} b^3 \cosh (3 a) \text {Shi}(3 b x)+\frac {125}{96} b^3 \cosh (5 a) \text {Shi}(5 b x)+\frac {b^2 \cosh (a+b x)}{48 x}-\frac {3 b^2 \cosh (3 a+3 b x)}{32 x}-\frac {25 b^2 \cosh (5 a+5 b x)}{96 x}+\frac {\cosh (a+b x)}{24 x^3}-\frac {\cosh (3 a+3 b x)}{48 x^3}-\frac {\cosh (5 a+5 b x)}{48 x^3}+\frac {b \sinh (a+b x)}{48 x^2}-\frac {b \sinh (3 a+3 b x)}{32 x^2}-\frac {5 b \sinh (5 a+5 b x)}{96 x^2} \]

[Out]

1/24*cosh(b*x+a)/x^3+1/48*b^2*cosh(b*x+a)/x-1/48*cosh(3*b*x+3*a)/x^3-3/32*b^2*cosh(3*b*x+3*a)/x-1/48*cosh(5*b*
x+5*a)/x^3-25/96*b^2*cosh(5*b*x+5*a)/x-1/48*b^3*cosh(a)*Shi(b*x)+9/32*b^3*cosh(3*a)*Shi(3*b*x)+125/96*b^3*cosh
(5*a)*Shi(5*b*x)-1/48*b^3*Chi(b*x)*sinh(a)+9/32*b^3*Chi(3*b*x)*sinh(3*a)+125/96*b^3*Chi(5*b*x)*sinh(5*a)+1/48*
b*sinh(b*x+a)/x^2-1/32*b*sinh(3*b*x+3*a)/x^2-5/96*b*sinh(5*b*x+5*a)/x^2

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Rubi [A]  time = 0.44, antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 20, 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} \[ -\frac {1}{48} b^3 \sinh (a) \text {Chi}(b x)+\frac {9}{32} b^3 \sinh (3 a) \text {Chi}(3 b x)+\frac {125}{96} b^3 \sinh (5 a) \text {Chi}(5 b x)-\frac {1}{48} b^3 \cosh (a) \text {Shi}(b x)+\frac {9}{32} b^3 \cosh (3 a) \text {Shi}(3 b x)+\frac {125}{96} b^3 \cosh (5 a) \text {Shi}(5 b x)+\frac {b^2 \cosh (a+b x)}{48 x}-\frac {3 b^2 \cosh (3 a+3 b x)}{32 x}-\frac {25 b^2 \cosh (5 a+5 b x)}{96 x}+\frac {b \sinh (a+b x)}{48 x^2}-\frac {b \sinh (3 a+3 b x)}{32 x^2}-\frac {5 b \sinh (5 a+5 b x)}{96 x^2}+\frac {\cosh (a+b x)}{24 x^3}-\frac {\cosh (3 a+3 b x)}{48 x^3}-\frac {\cosh (5 a+5 b x)}{48 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(Cosh[a + b*x]^3*Sinh[a + b*x]^2)/x^4,x]

[Out]

Cosh[a + b*x]/(24*x^3) + (b^2*Cosh[a + b*x])/(48*x) - Cosh[3*a + 3*b*x]/(48*x^3) - (3*b^2*Cosh[3*a + 3*b*x])/(
32*x) - Cosh[5*a + 5*b*x]/(48*x^3) - (25*b^2*Cosh[5*a + 5*b*x])/(96*x) - (b^3*CoshIntegral[b*x]*Sinh[a])/48 +
(9*b^3*CoshIntegral[3*b*x]*Sinh[3*a])/32 + (125*b^3*CoshIntegral[5*b*x]*Sinh[5*a])/96 + (b*Sinh[a + b*x])/(48*
x^2) - (b*Sinh[3*a + 3*b*x])/(32*x^2) - (5*b*Sinh[5*a + 5*b*x])/(96*x^2) - (b^3*Cosh[a]*SinhIntegral[b*x])/48
+ (9*b^3*Cosh[3*a]*SinhIntegral[3*b*x])/32 + (125*b^3*Cosh[5*a]*SinhIntegral[5*b*x])/96

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 ^3(a+b x) \sinh ^2(a+b x)}{x^4} \, dx &=\int \left (-\frac {\cosh (a+b x)}{8 x^4}+\frac {\cosh (3 a+3 b x)}{16 x^4}+\frac {\cosh (5 a+5 b x)}{16 x^4}\right ) \, dx\\ &=\frac {1}{16} \int \frac {\cosh (3 a+3 b x)}{x^4} \, dx+\frac {1}{16} \int \frac {\cosh (5 a+5 b x)}{x^4} \, dx-\frac {1}{8} \int \frac {\cosh (a+b x)}{x^4} \, dx\\ &=\frac {\cosh (a+b x)}{24 x^3}-\frac {\cosh (3 a+3 b x)}{48 x^3}-\frac {\cosh (5 a+5 b x)}{48 x^3}-\frac {1}{24} b \int \frac {\sinh (a+b x)}{x^3} \, dx+\frac {1}{16} b \int \frac {\sinh (3 a+3 b x)}{x^3} \, dx+\frac {1}{48} (5 b) \int \frac {\sinh (5 a+5 b x)}{x^3} \, dx\\ &=\frac {\cosh (a+b x)}{24 x^3}-\frac {\cosh (3 a+3 b x)}{48 x^3}-\frac {\cosh (5 a+5 b x)}{48 x^3}+\frac {b \sinh (a+b x)}{48 x^2}-\frac {b \sinh (3 a+3 b x)}{32 x^2}-\frac {5 b \sinh (5 a+5 b x)}{96 x^2}-\frac {1}{48} b^2 \int \frac {\cosh (a+b x)}{x^2} \, dx+\frac {1}{32} \left (3 b^2\right ) \int \frac {\cosh (3 a+3 b x)}{x^2} \, dx+\frac {1}{96} \left (25 b^2\right ) \int \frac {\cosh (5 a+5 b x)}{x^2} \, dx\\ &=\frac {\cosh (a+b x)}{24 x^3}+\frac {b^2 \cosh (a+b x)}{48 x}-\frac {\cosh (3 a+3 b x)}{48 x^3}-\frac {3 b^2 \cosh (3 a+3 b x)}{32 x}-\frac {\cosh (5 a+5 b x)}{48 x^3}-\frac {25 b^2 \cosh (5 a+5 b x)}{96 x}+\frac {b \sinh (a+b x)}{48 x^2}-\frac {b \sinh (3 a+3 b x)}{32 x^2}-\frac {5 b \sinh (5 a+5 b x)}{96 x^2}-\frac {1}{48} b^3 \int \frac {\sinh (a+b x)}{x} \, dx+\frac {1}{32} \left (9 b^3\right ) \int \frac {\sinh (3 a+3 b x)}{x} \, dx+\frac {1}{96} \left (125 b^3\right ) \int \frac {\sinh (5 a+5 b x)}{x} \, dx\\ &=\frac {\cosh (a+b x)}{24 x^3}+\frac {b^2 \cosh (a+b x)}{48 x}-\frac {\cosh (3 a+3 b x)}{48 x^3}-\frac {3 b^2 \cosh (3 a+3 b x)}{32 x}-\frac {\cosh (5 a+5 b x)}{48 x^3}-\frac {25 b^2 \cosh (5 a+5 b x)}{96 x}+\frac {b \sinh (a+b x)}{48 x^2}-\frac {b \sinh (3 a+3 b x)}{32 x^2}-\frac {5 b \sinh (5 a+5 b x)}{96 x^2}-\frac {1}{48} \left (b^3 \cosh (a)\right ) \int \frac {\sinh (b x)}{x} \, dx+\frac {1}{32} \left (9 b^3 \cosh (3 a)\right ) \int \frac {\sinh (3 b x)}{x} \, dx+\frac {1}{96} \left (125 b^3 \cosh (5 a)\right ) \int \frac {\sinh (5 b x)}{x} \, dx-\frac {1}{48} \left (b^3 \sinh (a)\right ) \int \frac {\cosh (b x)}{x} \, dx+\frac {1}{32} \left (9 b^3 \sinh (3 a)\right ) \int \frac {\cosh (3 b x)}{x} \, dx+\frac {1}{96} \left (125 b^3 \sinh (5 a)\right ) \int \frac {\cosh (5 b x)}{x} \, dx\\ &=\frac {\cosh (a+b x)}{24 x^3}+\frac {b^2 \cosh (a+b x)}{48 x}-\frac {\cosh (3 a+3 b x)}{48 x^3}-\frac {3 b^2 \cosh (3 a+3 b x)}{32 x}-\frac {\cosh (5 a+5 b x)}{48 x^3}-\frac {25 b^2 \cosh (5 a+5 b x)}{96 x}-\frac {1}{48} b^3 \text {Chi}(b x) \sinh (a)+\frac {9}{32} b^3 \text {Chi}(3 b x) \sinh (3 a)+\frac {125}{96} b^3 \text {Chi}(5 b x) \sinh (5 a)+\frac {b \sinh (a+b x)}{48 x^2}-\frac {b \sinh (3 a+3 b x)}{32 x^2}-\frac {5 b \sinh (5 a+5 b x)}{96 x^2}-\frac {1}{48} b^3 \cosh (a) \text {Shi}(b x)+\frac {9}{32} b^3 \cosh (3 a) \text {Shi}(3 b x)+\frac {125}{96} b^3 \cosh (5 a) \text {Shi}(5 b x)\\ \end {align*}

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Mathematica [A]  time = 0.56, size = 212, normalized size = 0.89 \[ \frac {-2 b^3 x^3 \sinh (a) \text {Chi}(b x)+27 b^3 x^3 \sinh (3 a) \text {Chi}(3 b x)+125 b^3 x^3 \sinh (5 a) \text {Chi}(5 b x)-2 b^3 x^3 \cosh (a) \text {Shi}(b x)+27 b^3 x^3 \cosh (3 a) \text {Shi}(3 b x)+125 b^3 x^3 \cosh (5 a) \text {Shi}(5 b x)+2 b^2 x^2 \cosh (a+b x)-9 b^2 x^2 \cosh (3 (a+b x))-25 b^2 x^2 \cosh (5 (a+b x))+2 b x \sinh (a+b x)-3 b x \sinh (3 (a+b x))-5 b x \sinh (5 (a+b x))+4 \cosh (a+b x)-2 \cosh (3 (a+b x))-2 \cosh (5 (a+b x))}{96 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cosh[a + b*x]^3*Sinh[a + b*x]^2)/x^4,x]

[Out]

(4*Cosh[a + b*x] + 2*b^2*x^2*Cosh[a + b*x] - 2*Cosh[3*(a + b*x)] - 9*b^2*x^2*Cosh[3*(a + b*x)] - 2*Cosh[5*(a +
b*x)] - 25*b^2*x^2*Cosh[5*(a + b*x)] - 2*b^3*x^3*CoshIntegral[b*x]*Sinh[a] + 27*b^3*x^3*CoshIntegral[3*b*x]*S
inh[3*a] + 125*b^3*x^3*CoshIntegral[5*b*x]*Sinh[5*a] + 2*b*x*Sinh[a + b*x] - 3*b*x*Sinh[3*(a + b*x)] - 5*b*x*S
inh[5*(a + b*x)] - 2*b^3*x^3*Cosh[a]*SinhIntegral[b*x] + 27*b^3*x^3*Cosh[3*a]*SinhIntegral[3*b*x] + 125*b^3*x^
3*Cosh[5*a]*SinhIntegral[5*b*x])/(96*x^3)

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fricas [A]  time = 0.58, size = 397, normalized size = 1.67 \[ -\frac {10 \, b x \sinh \left (b x + a\right )^{5} + 2 \, {\left (25 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{5} + 10 \, {\left (25 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} + 2 \, {\left (9 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{3} + 2 \, {\left (50 \, b x \cosh \left (b x + a\right )^{2} + 3 \, b x\right )} \sinh \left (b x + a\right )^{3} + 2 \, {\left (10 \, {\left (25 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{3} + 3 \, {\left (9 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} - 4 \, {\left (b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right ) - 125 \, {\left (b^{3} x^{3} {\rm Ei}\left (5 \, b x\right ) - b^{3} x^{3} {\rm Ei}\left (-5 \, b x\right )\right )} \cosh \left (5 \, a\right ) - 27 \, {\left (b^{3} x^{3} {\rm Ei}\left (3 \, b x\right ) - b^{3} x^{3} {\rm Ei}\left (-3 \, b x\right )\right )} \cosh \left (3 \, a\right ) + 2 \, {\left (b^{3} x^{3} {\rm Ei}\left (b x\right ) - b^{3} x^{3} {\rm Ei}\left (-b x\right )\right )} \cosh \relax (a) + 2 \, {\left (25 \, b x \cosh \left (b x + a\right )^{4} + 9 \, b x \cosh \left (b x + a\right )^{2} - 2 \, b x\right )} \sinh \left (b x + a\right ) - 125 \, {\left (b^{3} x^{3} {\rm Ei}\left (5 \, b x\right ) + b^{3} x^{3} {\rm Ei}\left (-5 \, b x\right )\right )} \sinh \left (5 \, a\right ) - 27 \, {\left (b^{3} x^{3} {\rm Ei}\left (3 \, b x\right ) + b^{3} x^{3} {\rm Ei}\left (-3 \, b x\right )\right )} \sinh \left (3 \, a\right ) + 2 \, {\left (b^{3} x^{3} {\rm Ei}\left (b x\right ) + b^{3} x^{3} {\rm Ei}\left (-b x\right )\right )} \sinh \relax (a)}{192 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^3*sinh(b*x+a)^2/x^4,x, algorithm="fricas")

[Out]

-1/192*(10*b*x*sinh(b*x + a)^5 + 2*(25*b^2*x^2 + 2)*cosh(b*x + a)^5 + 10*(25*b^2*x^2 + 2)*cosh(b*x + a)*sinh(b
*x + a)^4 + 2*(9*b^2*x^2 + 2)*cosh(b*x + a)^3 + 2*(50*b*x*cosh(b*x + a)^2 + 3*b*x)*sinh(b*x + a)^3 + 2*(10*(25
*b^2*x^2 + 2)*cosh(b*x + a)^3 + 3*(9*b^2*x^2 + 2)*cosh(b*x + a))*sinh(b*x + a)^2 - 4*(b^2*x^2 + 2)*cosh(b*x +
a) - 125*(b^3*x^3*Ei(5*b*x) - b^3*x^3*Ei(-5*b*x))*cosh(5*a) - 27*(b^3*x^3*Ei(3*b*x) - b^3*x^3*Ei(-3*b*x))*cosh
(3*a) + 2*(b^3*x^3*Ei(b*x) - b^3*x^3*Ei(-b*x))*cosh(a) + 2*(25*b*x*cosh(b*x + a)^4 + 9*b*x*cosh(b*x + a)^2 - 2
*b*x)*sinh(b*x + a) - 125*(b^3*x^3*Ei(5*b*x) + b^3*x^3*Ei(-5*b*x))*sinh(5*a) - 27*(b^3*x^3*Ei(3*b*x) + b^3*x^3
*Ei(-3*b*x))*sinh(3*a) + 2*(b^3*x^3*Ei(b*x) + b^3*x^3*Ei(-b*x))*sinh(a))/x^3

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giac [A]  time = 0.13, size = 342, normalized size = 1.44 \[ \frac {125 \, b^{3} x^{3} {\rm Ei}\left (5 \, b x\right ) e^{\left (5 \, a\right )} + 27 \, b^{3} x^{3} {\rm Ei}\left (3 \, b x\right ) e^{\left (3 \, a\right )} + 2 \, b^{3} x^{3} {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - 27 \, b^{3} x^{3} {\rm Ei}\left (-3 \, b x\right ) e^{\left (-3 \, a\right )} - 125 \, b^{3} x^{3} {\rm Ei}\left (-5 \, b x\right ) e^{\left (-5 \, a\right )} - 2 \, b^{3} x^{3} {\rm Ei}\left (b x\right ) e^{a} - 25 \, b^{2} x^{2} e^{\left (5 \, b x + 5 \, a\right )} - 9 \, b^{2} x^{2} e^{\left (3 \, b x + 3 \, a\right )} + 2 \, b^{2} x^{2} e^{\left (b x + a\right )} + 2 \, b^{2} x^{2} e^{\left (-b x - a\right )} - 9 \, b^{2} x^{2} e^{\left (-3 \, b x - 3 \, a\right )} - 25 \, b^{2} x^{2} e^{\left (-5 \, b x - 5 \, a\right )} - 5 \, b x e^{\left (5 \, b x + 5 \, a\right )} - 3 \, b x e^{\left (3 \, b x + 3 \, a\right )} + 2 \, b x e^{\left (b x + a\right )} - 2 \, b x e^{\left (-b x - a\right )} + 3 \, b x e^{\left (-3 \, b x - 3 \, a\right )} + 5 \, b x e^{\left (-5 \, b x - 5 \, a\right )} - 2 \, e^{\left (5 \, b x + 5 \, a\right )} - 2 \, e^{\left (3 \, b x + 3 \, a\right )} + 4 \, e^{\left (b x + a\right )} + 4 \, e^{\left (-b x - a\right )} - 2 \, e^{\left (-3 \, b x - 3 \, a\right )} - 2 \, e^{\left (-5 \, b x - 5 \, a\right )}}{192 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^3*sinh(b*x+a)^2/x^4,x, algorithm="giac")

[Out]

1/192*(125*b^3*x^3*Ei(5*b*x)*e^(5*a) + 27*b^3*x^3*Ei(3*b*x)*e^(3*a) + 2*b^3*x^3*Ei(-b*x)*e^(-a) - 27*b^3*x^3*E
i(-3*b*x)*e^(-3*a) - 125*b^3*x^3*Ei(-5*b*x)*e^(-5*a) - 2*b^3*x^3*Ei(b*x)*e^a - 25*b^2*x^2*e^(5*b*x + 5*a) - 9*
b^2*x^2*e^(3*b*x + 3*a) + 2*b^2*x^2*e^(b*x + a) + 2*b^2*x^2*e^(-b*x - a) - 9*b^2*x^2*e^(-3*b*x - 3*a) - 25*b^2
*x^2*e^(-5*b*x - 5*a) - 5*b*x*e^(5*b*x + 5*a) - 3*b*x*e^(3*b*x + 3*a) + 2*b*x*e^(b*x + a) - 2*b*x*e^(-b*x - a)
+ 3*b*x*e^(-3*b*x - 3*a) + 5*b*x*e^(-5*b*x - 5*a) - 2*e^(5*b*x + 5*a) - 2*e^(3*b*x + 3*a) + 4*e^(b*x + a) + 4
*e^(-b*x - a) - 2*e^(-3*b*x - 3*a) - 2*e^(-5*b*x - 5*a))/x^3

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maple [A]  time = 0.55, size = 356, normalized size = 1.50 \[ -\frac {25 b^{2} {\mathrm e}^{-5 b x -5 a}}{192 x}+\frac {5 b \,{\mathrm e}^{-5 b x -5 a}}{192 x^{2}}-\frac {{\mathrm e}^{-5 b x -5 a}}{96 x^{3}}+\frac {125 b^{3} {\mathrm e}^{-5 a} \Ei \left (1, 5 b x \right )}{192}-\frac {3 b^{2} {\mathrm e}^{-3 b x -3 a}}{64 x}+\frac {b \,{\mathrm e}^{-3 b x -3 a}}{64 x^{2}}-\frac {{\mathrm e}^{-3 b x -3 a}}{96 x^{3}}+\frac {9 b^{3} {\mathrm e}^{-3 a} \Ei \left (1, 3 b x \right )}{64}+\frac {b^{2} {\mathrm e}^{-b x -a}}{96 x}-\frac {b \,{\mathrm e}^{-b x -a}}{96 x^{2}}+\frac {{\mathrm e}^{-b x -a}}{48 x^{3}}-\frac {b^{3} {\mathrm e}^{-a} \Ei \left (1, b x \right )}{96}+\frac {{\mathrm e}^{b x +a}}{48 x^{3}}+\frac {b \,{\mathrm e}^{b x +a}}{96 x^{2}}+\frac {b^{2} {\mathrm e}^{b x +a}}{96 x}+\frac {b^{3} {\mathrm e}^{a} \Ei \left (1, -b x \right )}{96}-\frac {{\mathrm e}^{3 b x +3 a}}{96 x^{3}}-\frac {b \,{\mathrm e}^{3 b x +3 a}}{64 x^{2}}-\frac {3 b^{2} {\mathrm e}^{3 b x +3 a}}{64 x}-\frac {9 b^{3} {\mathrm e}^{3 a} \Ei \left (1, -3 b x \right )}{64}-\frac {{\mathrm e}^{5 b x +5 a}}{96 x^{3}}-\frac {5 b \,{\mathrm e}^{5 b x +5 a}}{192 x^{2}}-\frac {25 b^{2} {\mathrm e}^{5 b x +5 a}}{192 x}-\frac {125 b^{3} {\mathrm e}^{5 a} \Ei \left (1, -5 b x \right )}{192} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(b*x+a)^3*sinh(b*x+a)^2/x^4,x)

[Out]

-25/192*b^2*exp(-5*b*x-5*a)/x+5/192*b*exp(-5*b*x-5*a)/x^2-1/96*exp(-5*b*x-5*a)/x^3+125/192*b^3*exp(-5*a)*Ei(1,
5*b*x)-3/64*b^2*exp(-3*b*x-3*a)/x+1/64*b*exp(-3*b*x-3*a)/x^2-1/96*exp(-3*b*x-3*a)/x^3+9/64*b^3*exp(-3*a)*Ei(1,
3*b*x)+1/96*b^2*exp(-b*x-a)/x-1/96*b*exp(-b*x-a)/x^2+1/48*exp(-b*x-a)/x^3-1/96*b^3*exp(-a)*Ei(1,b*x)+1/48/x^3*
exp(b*x+a)+1/96*b/x^2*exp(b*x+a)+1/96*b^2/x*exp(b*x+a)+1/96*b^3*exp(a)*Ei(1,-b*x)-1/96/x^3*exp(3*b*x+3*a)-1/64
*b/x^2*exp(3*b*x+3*a)-3/64*b^2/x*exp(3*b*x+3*a)-9/64*b^3*exp(3*a)*Ei(1,-3*b*x)-1/96/x^3*exp(5*b*x+5*a)-5/192*b
/x^2*exp(5*b*x+5*a)-25/192*b^2/x*exp(5*b*x+5*a)-125/192*b^3*exp(5*a)*Ei(1,-5*b*x)

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maxima [A]  time = 0.45, size = 88, normalized size = 0.37 \[ -\frac {125}{32} \, b^{3} e^{\left (-5 \, a\right )} \Gamma \left (-3, 5 \, b x\right ) - \frac {27}{32} \, b^{3} e^{\left (-3 \, a\right )} \Gamma \left (-3, 3 \, b x\right ) + \frac {1}{16} \, b^{3} e^{\left (-a\right )} \Gamma \left (-3, b x\right ) - \frac {1}{16} \, b^{3} e^{a} \Gamma \left (-3, -b x\right ) + \frac {27}{32} \, b^{3} e^{\left (3 \, a\right )} \Gamma \left (-3, -3 \, b x\right ) + \frac {125}{32} \, b^{3} e^{\left (5 \, a\right )} \Gamma \left (-3, -5 \, b x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^3*sinh(b*x+a)^2/x^4,x, algorithm="maxima")

[Out]

-125/32*b^3*e^(-5*a)*gamma(-3, 5*b*x) - 27/32*b^3*e^(-3*a)*gamma(-3, 3*b*x) + 1/16*b^3*e^(-a)*gamma(-3, b*x) -
1/16*b^3*e^a*gamma(-3, -b*x) + 27/32*b^3*e^(3*a)*gamma(-3, -3*b*x) + 125/32*b^3*e^(5*a)*gamma(-3, -5*b*x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^3\,{\mathrm {sinh}\left (a+b\,x\right )}^2}{x^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cosh(a + b*x)^3*sinh(a + b*x)^2)/x^4,x)

[Out]

int((cosh(a + b*x)^3*sinh(a + b*x)^2)/x^4, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh ^{2}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)**3*sinh(b*x+a)**2/x**4,x)

[Out]

Integral(sinh(a + b*x)**2*cosh(a + b*x)**3/x**4, x)

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