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

Optimal. Leaf size=121 \[ \frac {1}{2} a^2 d^2 \cosh (c) \text {Chi}(d x)+\frac {1}{2} a^2 d^2 \sinh (c) \text {Shi}(d x)-\frac {a^2 \cosh (c+d x)}{2 x^2}-\frac {a^2 d \sinh (c+d x)}{2 x}+2 a b d \sinh (c) \text {Chi}(d x)+2 a b d \cosh (c) \text {Shi}(d x)-\frac {2 a b \cosh (c+d x)}{x}+b^2 \cosh (c) \text {Chi}(d x)+b^2 \sinh (c) \text {Shi}(d x) \]

[Out]

b^2*Chi(d*x)*cosh(c)+1/2*a^2*d^2*Chi(d*x)*cosh(c)-1/2*a^2*cosh(d*x+c)/x^2-2*a*b*cosh(d*x+c)/x+2*a*b*d*cosh(c)*
Shi(d*x)+2*a*b*d*Chi(d*x)*sinh(c)+b^2*Shi(d*x)*sinh(c)+1/2*a^2*d^2*Shi(d*x)*sinh(c)-1/2*a^2*d*sinh(d*x+c)/x

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Rubi [A]  time = 0.34, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {6742, 3297, 3303, 3298, 3301} \[ \frac {1}{2} a^2 d^2 \cosh (c) \text {Chi}(d x)+\frac {1}{2} a^2 d^2 \sinh (c) \text {Shi}(d x)-\frac {a^2 \cosh (c+d x)}{2 x^2}-\frac {a^2 d \sinh (c+d x)}{2 x}+2 a b d \sinh (c) \text {Chi}(d x)+2 a b d \cosh (c) \text {Shi}(d x)-\frac {2 a b \cosh (c+d x)}{x}+b^2 \cosh (c) \text {Chi}(d x)+b^2 \sinh (c) \text {Shi}(d x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^2*Cosh[c + d*x])/(2*x^2) - (2*a*b*Cosh[c + d*x])/x + b^2*Cosh[c]*CoshIntegral[d*x] + (a^2*d^2*Cosh[c]*Cosh
Integral[d*x])/2 + 2*a*b*d*CoshIntegral[d*x]*Sinh[c] - (a^2*d*Sinh[c + d*x])/(2*x) + 2*a*b*d*Cosh[c]*SinhInteg
ral[d*x] + b^2*Sinh[c]*SinhIntegral[d*x] + (a^2*d^2*Sinh[c]*SinhIntegral[d*x])/2

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 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

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

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Mathematica [A]  time = 0.40, size = 93, normalized size = 0.77 \[ \frac {1}{2} \left (\text {Chi}(d x) \left (\cosh (c) \left (a^2 d^2+2 b^2\right )+4 a b d \sinh (c)\right )+\text {Shi}(d x) \left (\sinh (c) \left (a^2 d^2+2 b^2\right )+4 a b d \cosh (c)\right )-\frac {a ((a+4 b x) \cosh (c+d x)+a d x \sinh (c+d x))}{x^2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.63, size = 153, normalized size = 1.26 \[ -\frac {2 \, a^{2} d x \sinh \left (d x + c\right ) + 2 \, {\left (4 \, a b x + a^{2}\right )} \cosh \left (d x + c\right ) - {\left ({\left (a^{2} d^{2} + 4 \, a b d + 2 \, b^{2}\right )} x^{2} {\rm Ei}\left (d x\right ) + {\left (a^{2} d^{2} - 4 \, a b d + 2 \, b^{2}\right )} x^{2} {\rm Ei}\left (-d x\right )\right )} \cosh \relax (c) - {\left ({\left (a^{2} d^{2} + 4 \, a b d + 2 \, b^{2}\right )} x^{2} {\rm Ei}\left (d x\right ) - {\left (a^{2} d^{2} - 4 \, a b d + 2 \, b^{2}\right )} x^{2} {\rm Ei}\left (-d x\right )\right )} \sinh \relax (c)}{4 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*a^2*d*x*sinh(d*x + c) + 2*(4*a*b*x + a^2)*cosh(d*x + c) - ((a^2*d^2 + 4*a*b*d + 2*b^2)*x^2*Ei(d*x) + (
a^2*d^2 - 4*a*b*d + 2*b^2)*x^2*Ei(-d*x))*cosh(c) - ((a^2*d^2 + 4*a*b*d + 2*b^2)*x^2*Ei(d*x) - (a^2*d^2 - 4*a*b
*d + 2*b^2)*x^2*Ei(-d*x))*sinh(c))/x^2

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giac [A]  time = 0.12, size = 181, normalized size = 1.50 \[ \frac {a^{2} d^{2} x^{2} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a^{2} d^{2} x^{2} {\rm Ei}\left (d x\right ) e^{c} - 4 \, a b d x^{2} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 4 \, a b d x^{2} {\rm Ei}\left (d x\right ) e^{c} + 2 \, b^{2} x^{2} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 2 \, b^{2} x^{2} {\rm Ei}\left (d x\right ) e^{c} - a^{2} d x e^{\left (d x + c\right )} + a^{2} d x e^{\left (-d x - c\right )} - 4 \, a b x e^{\left (d x + c\right )} - 4 \, a b x e^{\left (-d x - c\right )} - a^{2} e^{\left (d x + c\right )} - a^{2} e^{\left (-d x - c\right )}}{4 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(a^2*d^2*x^2*Ei(-d*x)*e^(-c) + a^2*d^2*x^2*Ei(d*x)*e^c - 4*a*b*d*x^2*Ei(-d*x)*e^(-c) + 4*a*b*d*x^2*Ei(d*x)
*e^c + 2*b^2*x^2*Ei(-d*x)*e^(-c) + 2*b^2*x^2*Ei(d*x)*e^c - a^2*d*x*e^(d*x + c) + a^2*d*x*e^(-d*x - c) - 4*a*b*
x*e^(d*x + c) - 4*a*b*x*e^(-d*x - c) - a^2*e^(d*x + c) - a^2*e^(-d*x - c))/x^2

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maple [A]  time = 0.11, size = 181, normalized size = 1.50 \[ -\frac {b^{2} {\mathrm e}^{-c} \Ei \left (1, d x \right )}{2}-\frac {a^{2} {\mathrm e}^{-d x -c}}{4 x^{2}}+\frac {d \,a^{2} {\mathrm e}^{-d x -c}}{4 x}-\frac {a b \,{\mathrm e}^{-d x -c}}{x}+d a b \,{\mathrm e}^{-c} \Ei \left (1, d x \right )-\frac {d^{2} a^{2} {\mathrm e}^{-c} \Ei \left (1, d x \right )}{4}-\frac {b^{2} {\mathrm e}^{c} \Ei \left (1, -d x \right )}{2}-\frac {a^{2} {\mathrm e}^{d x +c}}{4 x^{2}}-\frac {d \,a^{2} {\mathrm e}^{d x +c}}{4 x}-\frac {a b \,{\mathrm e}^{d x +c}}{x}-d a b \,{\mathrm e}^{c} \Ei \left (1, -d x \right )-\frac {d^{2} a^{2} {\mathrm e}^{c} \Ei \left (1, -d x \right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b^2*exp(-c)*Ei(1,d*x)-1/4*a^2*exp(-d*x-c)/x^2+1/4*d*a^2*exp(-d*x-c)/x-a*b*exp(-d*x-c)/x+d*a*b*exp(-c)*Ei(
1,d*x)-1/4*d^2*a^2*exp(-c)*Ei(1,d*x)-1/2*b^2*exp(c)*Ei(1,-d*x)-1/4*a^2/x^2*exp(d*x+c)-1/4*d*a^2/x*exp(d*x+c)-a
*b/x*exp(d*x+c)-d*a*b*exp(c)*Ei(1,-d*x)-1/4*d^2*a^2*exp(c)*Ei(1,-d*x)

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maxima [A]  time = 0.44, size = 126, normalized size = 1.04 \[ \frac {1}{4} \, {\left ({\left (d e^{\left (-c\right )} \Gamma \left (-1, d x\right ) + d e^{c} \Gamma \left (-1, -d x\right )\right )} a^{2} - 4 \, {\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - {\rm Ei}\left (d x\right ) e^{c}\right )} a b - \frac {4 \, b^{2} \cosh \left (d x + c\right ) \log \relax (x)}{d} + \frac {2 \, {\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + {\rm Ei}\left (d x\right ) e^{c}\right )} b^{2}}{d}\right )} d + \frac {1}{2} \, {\left (2 \, b^{2} \log \relax (x) - \frac {4 \, a b x + a^{2}}{x^{2}}\right )} \cosh \left (d x + c\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*((d*e^(-c)*gamma(-1, d*x) + d*e^c*gamma(-1, -d*x))*a^2 - 4*(Ei(-d*x)*e^(-c) - Ei(d*x)*e^c)*a*b - 4*b^2*cos
h(d*x + c)*log(x)/d + 2*(Ei(-d*x)*e^(-c) + Ei(d*x)*e^c)*b^2/d)*d + 1/2*(2*b^2*log(x) - (4*a*b*x + a^2)/x^2)*co
sh(d*x + c)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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