3.93 \(\int \frac {(a+b x^3)^2 \cosh (c+d x)}{x^5} \, dx\)

Optimal. Leaf size=167 \[ \frac {1}{24} a^2 d^4 \cosh (c) \text {Chi}(d x)+\frac {1}{24} a^2 d^4 \sinh (c) \text {Shi}(d x)-\frac {a^2 d^3 \sinh (c+d x)}{24 x}-\frac {a^2 d^2 \cosh (c+d x)}{24 x^2}-\frac {a^2 \cosh (c+d x)}{4 x^4}-\frac {a^2 d \sinh (c+d x)}{12 x^3}+2 a b d \sinh (c) \text {Chi}(d x)+2 a b d \cosh (c) \text {Shi}(d x)-\frac {2 a b \cosh (c+d x)}{x}-\frac {b^2 \cosh (c+d x)}{d^2}+\frac {b^2 x \sinh (c+d x)}{d} \]

[Out]

1/24*a^2*d^4*Chi(d*x)*cosh(c)-b^2*cosh(d*x+c)/d^2-1/4*a^2*cosh(d*x+c)/x^4-1/24*a^2*d^2*cosh(d*x+c)/x^2-2*a*b*c
osh(d*x+c)/x+2*a*b*d*cosh(c)*Shi(d*x)+2*a*b*d*Chi(d*x)*sinh(c)+1/24*a^2*d^4*Shi(d*x)*sinh(c)-1/12*a^2*d*sinh(d
*x+c)/x^3-1/24*a^2*d^3*sinh(d*x+c)/x+b^2*x*sinh(d*x+c)/d

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Rubi [A]  time = 0.31, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {5287, 3297, 3303, 3298, 3301, 3296, 2638} \[ \frac {1}{24} a^2 d^4 \cosh (c) \text {Chi}(d x)+\frac {1}{24} a^2 d^4 \sinh (c) \text {Shi}(d x)-\frac {a^2 d^2 \cosh (c+d x)}{24 x^2}-\frac {a^2 d^3 \sinh (c+d x)}{24 x}-\frac {a^2 d \sinh (c+d x)}{12 x^3}-\frac {a^2 \cosh (c+d x)}{4 x^4}+2 a b d \sinh (c) \text {Chi}(d x)+2 a b d \cosh (c) \text {Shi}(d x)-\frac {2 a b \cosh (c+d x)}{x}-\frac {b^2 \cosh (c+d x)}{d^2}+\frac {b^2 x \sinh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)^2*Cosh[c + d*x])/x^5,x]

[Out]

-((b^2*Cosh[c + d*x])/d^2) - (a^2*Cosh[c + d*x])/(4*x^4) - (a^2*d^2*Cosh[c + d*x])/(24*x^2) - (2*a*b*Cosh[c +
d*x])/x + (a^2*d^4*Cosh[c]*CoshIntegral[d*x])/24 + 2*a*b*d*CoshIntegral[d*x]*Sinh[c] - (a^2*d*Sinh[c + d*x])/(
12*x^3) - (a^2*d^3*Sinh[c + d*x])/(24*x) + (b^2*x*Sinh[c + d*x])/d + 2*a*b*d*Cosh[c]*SinhIntegral[d*x] + (a^2*
d^4*Sinh[c]*SinhIntegral[d*x])/24

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

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 5287

Int[Cosh[(c_.) + (d_.)*(x_)]*((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[Cosh[c + d*x], (e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^5} \, dx &=\int \left (\frac {a^2 \cosh (c+d x)}{x^5}+\frac {2 a b \cosh (c+d x)}{x^2}+b^2 x \cosh (c+d x)\right ) \, dx\\ &=a^2 \int \frac {\cosh (c+d x)}{x^5} \, dx+(2 a b) \int \frac {\cosh (c+d x)}{x^2} \, dx+b^2 \int x \cosh (c+d x) \, dx\\ &=-\frac {a^2 \cosh (c+d x)}{4 x^4}-\frac {2 a b \cosh (c+d x)}{x}+\frac {b^2 x \sinh (c+d x)}{d}-\frac {b^2 \int \sinh (c+d x) \, dx}{d}+\frac {1}{4} \left (a^2 d\right ) \int \frac {\sinh (c+d x)}{x^4} \, dx+(2 a b d) \int \frac {\sinh (c+d x)}{x} \, dx\\ &=-\frac {b^2 \cosh (c+d x)}{d^2}-\frac {a^2 \cosh (c+d x)}{4 x^4}-\frac {2 a b \cosh (c+d x)}{x}-\frac {a^2 d \sinh (c+d x)}{12 x^3}+\frac {b^2 x \sinh (c+d x)}{d}+\frac {1}{12} \left (a^2 d^2\right ) \int \frac {\cosh (c+d x)}{x^3} \, dx+(2 a b d \cosh (c)) \int \frac {\sinh (d x)}{x} \, dx+(2 a b d \sinh (c)) \int \frac {\cosh (d x)}{x} \, dx\\ &=-\frac {b^2 \cosh (c+d x)}{d^2}-\frac {a^2 \cosh (c+d x)}{4 x^4}-\frac {a^2 d^2 \cosh (c+d x)}{24 x^2}-\frac {2 a b \cosh (c+d x)}{x}+2 a b d \text {Chi}(d x) \sinh (c)-\frac {a^2 d \sinh (c+d x)}{12 x^3}+\frac {b^2 x \sinh (c+d x)}{d}+2 a b d \cosh (c) \text {Shi}(d x)+\frac {1}{24} \left (a^2 d^3\right ) \int \frac {\sinh (c+d x)}{x^2} \, dx\\ &=-\frac {b^2 \cosh (c+d x)}{d^2}-\frac {a^2 \cosh (c+d x)}{4 x^4}-\frac {a^2 d^2 \cosh (c+d x)}{24 x^2}-\frac {2 a b \cosh (c+d x)}{x}+2 a b d \text {Chi}(d x) \sinh (c)-\frac {a^2 d \sinh (c+d x)}{12 x^3}-\frac {a^2 d^3 \sinh (c+d x)}{24 x}+\frac {b^2 x \sinh (c+d x)}{d}+2 a b d \cosh (c) \text {Shi}(d x)+\frac {1}{24} \left (a^2 d^4\right ) \int \frac {\cosh (c+d x)}{x} \, dx\\ &=-\frac {b^2 \cosh (c+d x)}{d^2}-\frac {a^2 \cosh (c+d x)}{4 x^4}-\frac {a^2 d^2 \cosh (c+d x)}{24 x^2}-\frac {2 a b \cosh (c+d x)}{x}+2 a b d \text {Chi}(d x) \sinh (c)-\frac {a^2 d \sinh (c+d x)}{12 x^3}-\frac {a^2 d^3 \sinh (c+d x)}{24 x}+\frac {b^2 x \sinh (c+d x)}{d}+2 a b d \cosh (c) \text {Shi}(d x)+\frac {1}{24} \left (a^2 d^4 \cosh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx+\frac {1}{24} \left (a^2 d^4 \sinh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx\\ &=-\frac {b^2 \cosh (c+d x)}{d^2}-\frac {a^2 \cosh (c+d x)}{4 x^4}-\frac {a^2 d^2 \cosh (c+d x)}{24 x^2}-\frac {2 a b \cosh (c+d x)}{x}+\frac {1}{24} a^2 d^4 \cosh (c) \text {Chi}(d x)+2 a b d \text {Chi}(d x) \sinh (c)-\frac {a^2 d \sinh (c+d x)}{12 x^3}-\frac {a^2 d^3 \sinh (c+d x)}{24 x}+\frac {b^2 x \sinh (c+d x)}{d}+2 a b d \cosh (c) \text {Shi}(d x)+\frac {1}{24} a^2 d^4 \sinh (c) \text {Shi}(d x)\\ \end {align*}

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Mathematica [A]  time = 0.58, size = 150, normalized size = 0.90 \[ \frac {1}{24} \left (-\frac {a^2 d^3 \sinh (c+d x)}{x}-\frac {a^2 d^2 \cosh (c+d x)}{x^2}-\frac {6 a^2 \cosh (c+d x)}{x^4}-\frac {2 a^2 d \sinh (c+d x)}{x^3}+a d \text {Chi}(d x) \left (a d^3 \cosh (c)+48 b \sinh (c)\right )+a d \text {Shi}(d x) \left (a d^3 \sinh (c)+48 b \cosh (c)\right )-\frac {48 a b \cosh (c+d x)}{x}-\frac {24 b^2 \cosh (c+d x)}{d^2}+\frac {24 b^2 x \sinh (c+d x)}{d}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^3)^2*Cosh[c + d*x])/x^5,x]

[Out]

((-24*b^2*Cosh[c + d*x])/d^2 - (6*a^2*Cosh[c + d*x])/x^4 - (a^2*d^2*Cosh[c + d*x])/x^2 - (48*a*b*Cosh[c + d*x]
)/x + a*d*CoshIntegral[d*x]*(a*d^3*Cosh[c] + 48*b*Sinh[c]) - (2*a^2*d*Sinh[c + d*x])/x^3 - (a^2*d^3*Sinh[c + d
*x])/x + (24*b^2*x*Sinh[c + d*x])/d + a*d*(48*b*Cosh[c] + a*d^3*Sinh[c])*SinhIntegral[d*x])/24

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fricas [A]  time = 0.40, size = 196, normalized size = 1.17 \[ -\frac {2 \, {\left (a^{2} d^{4} x^{2} + 48 \, a b d^{2} x^{3} + 24 \, b^{2} x^{4} + 6 \, a^{2} d^{2}\right )} \cosh \left (d x + c\right ) - {\left ({\left (a^{2} d^{6} + 48 \, a b d^{3}\right )} x^{4} {\rm Ei}\left (d x\right ) + {\left (a^{2} d^{6} - 48 \, a b d^{3}\right )} x^{4} {\rm Ei}\left (-d x\right )\right )} \cosh \relax (c) + 2 \, {\left (a^{2} d^{5} x^{3} - 24 \, b^{2} d x^{5} + 2 \, a^{2} d^{3} x\right )} \sinh \left (d x + c\right ) - {\left ({\left (a^{2} d^{6} + 48 \, a b d^{3}\right )} x^{4} {\rm Ei}\left (d x\right ) - {\left (a^{2} d^{6} - 48 \, a b d^{3}\right )} x^{4} {\rm Ei}\left (-d x\right )\right )} \sinh \relax (c)}{48 \, d^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2*cosh(d*x+c)/x^5,x, algorithm="fricas")

[Out]

-1/48*(2*(a^2*d^4*x^2 + 48*a*b*d^2*x^3 + 24*b^2*x^4 + 6*a^2*d^2)*cosh(d*x + c) - ((a^2*d^6 + 48*a*b*d^3)*x^4*E
i(d*x) + (a^2*d^6 - 48*a*b*d^3)*x^4*Ei(-d*x))*cosh(c) + 2*(a^2*d^5*x^3 - 24*b^2*d*x^5 + 2*a^2*d^3*x)*sinh(d*x
+ c) - ((a^2*d^6 + 48*a*b*d^3)*x^4*Ei(d*x) - (a^2*d^6 - 48*a*b*d^3)*x^4*Ei(-d*x))*sinh(c))/(d^2*x^4)

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giac [B]  time = 0.14, size = 315, normalized size = 1.89 \[ \frac {a^{2} d^{6} x^{4} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a^{2} d^{6} x^{4} {\rm Ei}\left (d x\right ) e^{c} - a^{2} d^{5} x^{3} e^{\left (d x + c\right )} + a^{2} d^{5} x^{3} e^{\left (-d x - c\right )} - 48 \, a b d^{3} x^{4} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 48 \, a b d^{3} x^{4} {\rm Ei}\left (d x\right ) e^{c} - a^{2} d^{4} x^{2} e^{\left (d x + c\right )} + 24 \, b^{2} d x^{5} e^{\left (d x + c\right )} - a^{2} d^{4} x^{2} e^{\left (-d x - c\right )} - 24 \, b^{2} d x^{5} e^{\left (-d x - c\right )} - 48 \, a b d^{2} x^{3} e^{\left (d x + c\right )} - 48 \, a b d^{2} x^{3} e^{\left (-d x - c\right )} - 2 \, a^{2} d^{3} x e^{\left (d x + c\right )} - 24 \, b^{2} x^{4} e^{\left (d x + c\right )} + 2 \, a^{2} d^{3} x e^{\left (-d x - c\right )} - 24 \, b^{2} x^{4} e^{\left (-d x - c\right )} - 6 \, a^{2} d^{2} e^{\left (d x + c\right )} - 6 \, a^{2} d^{2} e^{\left (-d x - c\right )}}{48 \, d^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2*cosh(d*x+c)/x^5,x, algorithm="giac")

[Out]

1/48*(a^2*d^6*x^4*Ei(-d*x)*e^(-c) + a^2*d^6*x^4*Ei(d*x)*e^c - a^2*d^5*x^3*e^(d*x + c) + a^2*d^5*x^3*e^(-d*x -
c) - 48*a*b*d^3*x^4*Ei(-d*x)*e^(-c) + 48*a*b*d^3*x^4*Ei(d*x)*e^c - a^2*d^4*x^2*e^(d*x + c) + 24*b^2*d*x^5*e^(d
*x + c) - a^2*d^4*x^2*e^(-d*x - c) - 24*b^2*d*x^5*e^(-d*x - c) - 48*a*b*d^2*x^3*e^(d*x + c) - 48*a*b*d^2*x^3*e
^(-d*x - c) - 2*a^2*d^3*x*e^(d*x + c) - 24*b^2*x^4*e^(d*x + c) + 2*a^2*d^3*x*e^(-d*x - c) - 24*b^2*x^4*e^(-d*x
- c) - 6*a^2*d^2*e^(d*x + c) - 6*a^2*d^2*e^(-d*x - c))/(d^2*x^4)

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maple [A]  time = 0.26, size = 292, normalized size = 1.75 \[ -\frac {d^{2} a^{2} {\mathrm e}^{-d x -c}}{48 x^{2}}+\frac {d \,a^{2} {\mathrm e}^{-d x -c}}{24 x^{3}}-\frac {a^{2} {\mathrm e}^{-d x -c}}{8 x^{4}}-\frac {b^{2} {\mathrm e}^{-d x -c} x}{2 d}-\frac {b^{2} {\mathrm e}^{-d x -c}}{2 d^{2}}-\frac {d^{4} a^{2} {\mathrm e}^{-c} \Ei \left (1, d x \right )}{48}+\frac {d^{3} a^{2} {\mathrm e}^{-d x -c}}{48 x}-\frac {a b \,{\mathrm e}^{-d x -c}}{x}+d a b \,{\mathrm e}^{-c} \Ei \left (1, d x \right )+\frac {b^{2} {\mathrm e}^{d x +c} x}{2 d}-\frac {d^{4} a^{2} {\mathrm e}^{c} \Ei \left (1, -d x \right )}{48}-\frac {b^{2} {\mathrm e}^{d x +c}}{2 d^{2}}-\frac {d \,a^{2} {\mathrm e}^{d x +c}}{24 x^{3}}-\frac {d^{2} a^{2} {\mathrm e}^{d x +c}}{48 x^{2}}-\frac {d^{3} a^{2} {\mathrm e}^{d x +c}}{48 x}-\frac {a^{2} {\mathrm e}^{d x +c}}{8 x^{4}}-\frac {a b \,{\mathrm e}^{d x +c}}{x}-d a b \,{\mathrm e}^{c} \Ei \left (1, -d x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^2*cosh(d*x+c)/x^5,x)

[Out]

-1/48*d^2*a^2*exp(-d*x-c)/x^2+1/24*d*a^2*exp(-d*x-c)/x^3-1/8*a^2*exp(-d*x-c)/x^4-1/2/d*b^2*exp(-d*x-c)*x-1/2/d
^2*b^2*exp(-d*x-c)-1/48*d^4*a^2*exp(-c)*Ei(1,d*x)+1/48*d^3*a^2*exp(-d*x-c)/x-a*b*exp(-d*x-c)/x+d*a*b*exp(-c)*E
i(1,d*x)+1/2/d*b^2*exp(d*x+c)*x-1/48*d^4*a^2*exp(c)*Ei(1,-d*x)-1/2/d^2*b^2*exp(d*x+c)-1/24*d*a^2/x^3*exp(d*x+c
)-1/48*d^2*a^2/x^2*exp(d*x+c)-1/48*d^3*a^2/x*exp(d*x+c)-1/8*a^2/x^4*exp(d*x+c)-a*b/x*exp(d*x+c)-d*a*b*exp(c)*E
i(1,-d*x)

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maxima [A]  time = 0.46, size = 154, normalized size = 0.92 \[ \frac {1}{8} \, {\left (a^{2} d^{3} e^{\left (-c\right )} \Gamma \left (-3, d x\right ) + a^{2} d^{3} e^{c} \Gamma \left (-3, -d x\right ) - 8 \, a b {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 8 \, a b {\rm Ei}\left (d x\right ) e^{c} - \frac {2 \, {\left (d^{2} x^{2} e^{c} - 2 \, d x e^{c} + 2 \, e^{c}\right )} b^{2} e^{\left (d x\right )}}{d^{3}} - \frac {2 \, {\left (d^{2} x^{2} + 2 \, d x + 2\right )} b^{2} e^{\left (-d x - c\right )}}{d^{3}}\right )} d + \frac {1}{4} \, {\left (2 \, b^{2} x^{2} - \frac {8 \, a b x^{3} + a^{2}}{x^{4}}\right )} \cosh \left (d x + c\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2*cosh(d*x+c)/x^5,x, algorithm="maxima")

[Out]

1/8*(a^2*d^3*e^(-c)*gamma(-3, d*x) + a^2*d^3*e^c*gamma(-3, -d*x) - 8*a*b*Ei(-d*x)*e^(-c) + 8*a*b*Ei(d*x)*e^c -
2*(d^2*x^2*e^c - 2*d*x*e^c + 2*e^c)*b^2*e^(d*x)/d^3 - 2*(d^2*x^2 + 2*d*x + 2)*b^2*e^(-d*x - c)/d^3)*d + 1/4*(
2*b^2*x^2 - (8*a*b*x^3 + a^2)/x^4)*cosh(d*x + c)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\left (b\,x^3+a\right )}^2}{x^5} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cosh(c + d*x)*(a + b*x^3)^2)/x^5,x)

[Out]

int((cosh(c + d*x)*(a + b*x^3)^2)/x^5, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b x^{3}\right )^{2} \cosh {\left (c + d x \right )}}{x^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**2*cosh(d*x+c)/x**5,x)

[Out]

Integral((a + b*x**3)**2*cosh(c + d*x)/x**5, x)

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