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

Optimal. Leaf size=175 \[ \frac{1}{24} a^2 d^4 \cosh (c) \text{Chi}(d x)+\frac{1}{24} a^2 d^4 \sinh (c) \text{Shi}(d x)-\frac{a^2 d^2 \cosh (c+d x)}{24 x^2}-\frac{a^2 d^3 \sinh (c+d x)}{24 x}-\frac{a^2 d \sinh (c+d x)}{12 x^3}-\frac{a^2 \cosh (c+d x)}{4 x^4}+a b d^2 \cosh (c) \text{Chi}(d x)+a b d^2 \sinh (c) \text{Shi}(d x)-\frac{a b \cosh (c+d x)}{x^2}-\frac{a b d \sinh (c+d x)}{x}+b^2 \cosh (c) \text{Chi}(d x)+b^2 \sinh (c) \text{Shi}(d x) \]

[Out]

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

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Rubi [A]  time = 0.35956, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {5287, 3297, 3303, 3298, 3301} \[ \frac{1}{24} a^2 d^4 \cosh (c) \text{Chi}(d x)+\frac{1}{24} a^2 d^4 \sinh (c) \text{Shi}(d x)-\frac{a^2 d^2 \cosh (c+d x)}{24 x^2}-\frac{a^2 d^3 \sinh (c+d x)}{24 x}-\frac{a^2 d \sinh (c+d x)}{12 x^3}-\frac{a^2 \cosh (c+d x)}{4 x^4}+a b d^2 \cosh (c) \text{Chi}(d x)+a b d^2 \sinh (c) \text{Shi}(d x)-\frac{a b \cosh (c+d x)}{x^2}-\frac{a b d \sinh (c+d x)}{x}+b^2 \cosh (c) \text{Chi}(d x)+b^2 \sinh (c) \text{Shi}(d x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

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 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 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]

Rubi steps

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

Mathematica [A]  time = 0.438824, size = 124, normalized size = 0.71 \[ \frac{x^4 \cosh (c) \left (a^2 d^4+24 a b d^2+24 b^2\right ) \text{Chi}(d x)+x^4 \sinh (c) \left (a^2 d^4+24 a b d^2+24 b^2\right ) \text{Shi}(d x)-a \left (d x \left (a d^2 x^2+2 a+24 b x^2\right ) \sinh (c+d x)+\left (a d^2 x^2+6 a+24 b x^2\right ) \cosh (c+d x)\right )}{24 x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.116, size = 291, normalized size = 1.7 \begin{align*} -{\frac{{d}^{4}{a}^{2}{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{48}}+{\frac{bda{{\rm e}^{-dx-c}}}{2\,x}}-{\frac{ab{{\rm e}^{-dx-c}}}{2\,{x}^{2}}}-{\frac{{d}^{2}ab{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{2}}-{\frac{{b}^{2}{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{2}}+{\frac{{a}^{2}{d}^{3}{{\rm e}^{-dx-c}}}{48\,x}}-{\frac{{a}^{2}{d}^{2}{{\rm e}^{-dx-c}}}{48\,{x}^{2}}}+{\frac{d{a}^{2}{{\rm e}^{-dx-c}}}{24\,{x}^{3}}}-{\frac{{a}^{2}{{\rm e}^{-dx-c}}}{8\,{x}^{4}}}-{\frac{{{\rm e}^{dx+c}}{a}^{2}}{8\,{x}^{4}}}-{\frac{d{a}^{2}{{\rm e}^{dx+c}}}{24\,{x}^{3}}}-{\frac{{a}^{2}{d}^{2}{{\rm e}^{dx+c}}}{48\,{x}^{2}}}-{\frac{{a}^{2}{d}^{3}{{\rm e}^{dx+c}}}{48\,x}}-{\frac{{d}^{4}{a}^{2}{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{48}}-{\frac{ab{{\rm e}^{dx+c}}}{2\,{x}^{2}}}-{\frac{bda{{\rm e}^{dx+c}}}{2\,x}}-{\frac{{d}^{2}ab{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{2}}-{\frac{{b}^{2}{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.27655, size = 188, normalized size = 1.07 \begin{align*} \frac{1}{8} \,{\left ({\left (d^{3} e^{\left (-c\right )} \Gamma \left (-3, d x\right ) + d^{3} e^{c} \Gamma \left (-3, -d x\right )\right )} a^{2} + 4 \,{\left (d e^{\left (-c\right )} \Gamma \left (-1, d x\right ) + d e^{c} \Gamma \left (-1, -d x\right )\right )} a b - \frac{4 \, b^{2} \cosh \left (d x + c\right ) \log \left (x^{2}\right )}{d} + \frac{4 \,{\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}{4} \,{\left (2 \, b^{2} \log \left (x^{2}\right ) - \frac{4 \, a b x^{2} + a^{2}}{x^{4}}\right )} \cosh \left (d x + c\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.9739, size = 439, normalized size = 2.51 \begin{align*} -\frac{2 \,{\left ({\left (a^{2} d^{2} + 24 \, a b\right )} x^{2} + 6 \, a^{2}\right )} \cosh \left (d x + c\right ) -{\left ({\left (a^{2} d^{4} + 24 \, a b d^{2} + 24 \, b^{2}\right )} x^{4}{\rm Ei}\left (d x\right ) +{\left (a^{2} d^{4} + 24 \, a b d^{2} + 24 \, b^{2}\right )} x^{4}{\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) + 2 \,{\left (2 \, a^{2} d x +{\left (a^{2} d^{3} + 24 \, a b d\right )} x^{3}\right )} \sinh \left (d x + c\right ) -{\left ({\left (a^{2} d^{4} + 24 \, a b d^{2} + 24 \, b^{2}\right )} x^{4}{\rm Ei}\left (d x\right ) -{\left (a^{2} d^{4} + 24 \, a b d^{2} + 24 \, b^{2}\right )} x^{4}{\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{48 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x^{2}\right )^{2} \cosh{\left (c + d x \right )}}{x^{5}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.17539, size = 397, normalized size = 2.27 \begin{align*} \frac{a^{2} d^{4} x^{4}{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a^{2} d^{4} x^{4}{\rm Ei}\left (d x\right ) e^{c} + 24 \, a b d^{2} x^{4}{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 24 \, a b d^{2} x^{4}{\rm Ei}\left (d x\right ) e^{c} - a^{2} d^{3} x^{3} e^{\left (d x + c\right )} + a^{2} d^{3} x^{3} e^{\left (-d x - c\right )} + 24 \, b^{2} x^{4}{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 24 \, b^{2} x^{4}{\rm Ei}\left (d x\right ) e^{c} - a^{2} d^{2} x^{2} e^{\left (d x + c\right )} - 24 \, a b d x^{3} e^{\left (d x + c\right )} - a^{2} d^{2} x^{2} e^{\left (-d x - c\right )} + 24 \, a b d x^{3} e^{\left (-d x - c\right )} - 2 \, a^{2} d x e^{\left (d x + c\right )} - 24 \, a b x^{2} e^{\left (d x + c\right )} + 2 \, a^{2} d x e^{\left (-d x - c\right )} - 24 \, a b x^{2} e^{\left (-d x - c\right )} - 6 \, a^{2} e^{\left (d x + c\right )} - 6 \, a^{2} e^{\left (-d x - c\right )}}{48 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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