3.55 \(\int \frac{(a+b x^2)^2 \cosh (c+d x)}{x^4} \, dx\)

Optimal. Leaf size=133 \[ \frac{1}{6} a^2 d^3 \sinh (c) \text{Chi}(d x)+\frac{1}{6} a^2 d^3 \cosh (c) \text{Shi}(d x)-\frac{a^2 d^2 \cosh (c+d x)}{6 x}-\frac{a^2 d \sinh (c+d x)}{6 x^2}-\frac{a^2 \cosh (c+d x)}{3 x^3}+2 a b d \sinh (c) \text{Chi}(d x)+2 a b d \cosh (c) \text{Shi}(d x)-\frac{2 a b \cosh (c+d x)}{x}+\frac{b^2 \sinh (c+d x)}{d} \]

[Out]

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

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Rubi [A]  time = 0.258916, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {5287, 2637, 3297, 3303, 3298, 3301} \[ \frac{1}{6} a^2 d^3 \sinh (c) \text{Chi}(d x)+\frac{1}{6} a^2 d^3 \cosh (c) \text{Shi}(d x)-\frac{a^2 d^2 \cosh (c+d x)}{6 x}-\frac{a^2 d \sinh (c+d x)}{6 x^2}-\frac{a^2 \cosh (c+d x)}{3 x^3}+2 a b d \sinh (c) \text{Chi}(d x)+2 a b d \cosh (c) \text{Shi}(d x)-\frac{2 a b \cosh (c+d x)}{x}+\frac{b^2 \sinh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

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^4} \, dx &=\int \left (b^2 \cosh (c+d x)+\frac{a^2 \cosh (c+d x)}{x^4}+\frac{2 a b \cosh (c+d x)}{x^2}\right ) \, dx\\ &=a^2 \int \frac{\cosh (c+d x)}{x^4} \, dx+(2 a b) \int \frac{\cosh (c+d x)}{x^2} \, dx+b^2 \int \cosh (c+d x) \, dx\\ &=-\frac{a^2 \cosh (c+d x)}{3 x^3}-\frac{2 a b \cosh (c+d x)}{x}+\frac{b^2 \sinh (c+d x)}{d}+\frac{1}{3} \left (a^2 d\right ) \int \frac{\sinh (c+d x)}{x^3} \, dx+(2 a b d) \int \frac{\sinh (c+d x)}{x} \, dx\\ &=-\frac{a^2 \cosh (c+d x)}{3 x^3}-\frac{2 a b \cosh (c+d x)}{x}+\frac{b^2 \sinh (c+d x)}{d}-\frac{a^2 d \sinh (c+d x)}{6 x^2}+\frac{1}{6} \left (a^2 d^2\right ) \int \frac{\cosh (c+d x)}{x^2} \, 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)}{3 x^3}-\frac{2 a b \cosh (c+d x)}{x}-\frac{a^2 d^2 \cosh (c+d x)}{6 x}+2 a b d \text{Chi}(d x) \sinh (c)+\frac{b^2 \sinh (c+d x)}{d}-\frac{a^2 d \sinh (c+d x)}{6 x^2}+2 a b d \cosh (c) \text{Shi}(d x)+\frac{1}{6} \left (a^2 d^3\right ) \int \frac{\sinh (c+d x)}{x} \, dx\\ &=-\frac{a^2 \cosh (c+d x)}{3 x^3}-\frac{2 a b \cosh (c+d x)}{x}-\frac{a^2 d^2 \cosh (c+d x)}{6 x}+2 a b d \text{Chi}(d x) \sinh (c)+\frac{b^2 \sinh (c+d x)}{d}-\frac{a^2 d \sinh (c+d x)}{6 x^2}+2 a b d \cosh (c) \text{Shi}(d x)+\frac{1}{6} \left (a^2 d^3 \cosh (c)\right ) \int \frac{\sinh (d x)}{x} \, dx+\frac{1}{6} \left (a^2 d^3 \sinh (c)\right ) \int \frac{\cosh (d x)}{x} \, dx\\ &=-\frac{a^2 \cosh (c+d x)}{3 x^3}-\frac{2 a b \cosh (c+d x)}{x}-\frac{a^2 d^2 \cosh (c+d x)}{6 x}+2 a b d \text{Chi}(d x) \sinh (c)+\frac{1}{6} a^2 d^3 \text{Chi}(d x) \sinh (c)+\frac{b^2 \sinh (c+d x)}{d}-\frac{a^2 d \sinh (c+d x)}{6 x^2}+2 a b d \cosh (c) \text{Shi}(d x)+\frac{1}{6} a^2 d^3 \cosh (c) \text{Shi}(d x)\\ \end{align*}

Mathematica [A]  time = 0.401937, size = 114, normalized size = 0.86 \[ \frac{1}{6} \left (-\frac{a^2 d^2 \cosh (c+d x)}{x}-\frac{a^2 d \sinh (c+d x)}{x^2}-\frac{2 a^2 \cosh (c+d x)}{x^3}+a d \sinh (c) \left (a d^2+12 b\right ) \text{Chi}(d x)+a d \cosh (c) \left (a d^2+12 b\right ) \text{Shi}(d x)-\frac{12 a b \cosh (c+d x)}{x}+\frac{6 b^2 \sinh (c+d x)}{d}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*a^2*Cosh[c + d*x])/x^3 - (12*a*b*Cosh[c + d*x])/x - (a^2*d^2*Cosh[c + d*x])/x + a*d*(12*b + a*d^2)*CoshIn
tegral[d*x]*Sinh[c] + (6*b^2*Sinh[c + d*x])/d - (a^2*d*Sinh[c + d*x])/x^2 + a*d*(12*b + a*d^2)*Cosh[c]*SinhInt
egral[d*x])/6

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Maple [A]  time = 0.092, size = 222, normalized size = 1.7 \begin{align*}{\frac{{d}^{3}{a}^{2}{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{12}}-{\frac{{b}^{2}{{\rm e}^{-dx-c}}}{2\,d}}-{\frac{ab{{\rm e}^{-dx-c}}}{x}}+dab{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) -{\frac{{a}^{2}{d}^{2}{{\rm e}^{-dx-c}}}{12\,x}}+{\frac{d{a}^{2}{{\rm e}^{-dx-c}}}{12\,{x}^{2}}}-{\frac{{a}^{2}{{\rm e}^{-dx-c}}}{6\,{x}^{3}}}-{\frac{{{\rm e}^{dx+c}}{a}^{2}}{6\,{x}^{3}}}-{\frac{d{a}^{2}{{\rm e}^{dx+c}}}{12\,{x}^{2}}}-{\frac{{a}^{2}{d}^{2}{{\rm e}^{dx+c}}}{12\,x}}+{\frac{{{\rm e}^{dx+c}}{b}^{2}}{2\,d}}-{\frac{{d}^{3}{a}^{2}{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{12}}-{\frac{ab{{\rm e}^{dx+c}}}{x}}-dab{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2*cosh(d*x+c)/x^4,x)

[Out]

1/12*d^3*a^2*exp(-c)*Ei(1,d*x)-1/2*b^2/d*exp(-d*x-c)-a*b*exp(-d*x-c)/x+d*a*b*exp(-c)*Ei(1,d*x)-1/12*d^2*a^2*ex
p(-d*x-c)/x+1/12*d*a^2*exp(-d*x-c)/x^2-1/6*a^2*exp(-d*x-c)/x^3-1/6*a^2/x^3*exp(d*x+c)-1/12*d*a^2/x^2*exp(d*x+c
)-1/12*d^2*a^2/x*exp(d*x+c)+1/2*b^2/d*exp(d*x+c)-1/12*d^3*a^2*exp(c)*Ei(1,-d*x)-a*b/x*exp(d*x+c)-d*a*b*exp(c)*
Ei(1,-d*x)

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Maxima [A]  time = 1.23669, size = 182, normalized size = 1.37 \begin{align*} \frac{1}{6} \,{\left (a^{2} d^{2} e^{\left (-c\right )} \Gamma \left (-2, d x\right ) - a^{2} d^{2} e^{c} \Gamma \left (-2, -d x\right ) - 6 \, a b{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 6 \, a b{\rm Ei}\left (d x\right ) e^{c} - \frac{3 \,{\left (d x e^{c} - e^{c}\right )} b^{2} e^{\left (d x\right )}}{d^{2}} - \frac{3 \,{\left (d x + 1\right )} b^{2} e^{\left (-d x - c\right )}}{d^{2}}\right )} d + \frac{1}{3} \,{\left (3 \, b^{2} x - \frac{6 \, a b x^{2} + a^{2}}{x^{3}}\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^4,x, algorithm="maxima")

[Out]

1/6*(a^2*d^2*e^(-c)*gamma(-2, d*x) - a^2*d^2*e^c*gamma(-2, -d*x) - 6*a*b*Ei(-d*x)*e^(-c) + 6*a*b*Ei(d*x)*e^c -
3*(d*x*e^c - e^c)*b^2*e^(d*x)/d^2 - 3*(d*x + 1)*b^2*e^(-d*x - c)/d^2)*d + 1/3*(3*b^2*x - (6*a*b*x^2 + a^2)/x^
3)*cosh(d*x + c)

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Fricas [A]  time = 2.04377, size = 381, normalized size = 2.86 \begin{align*} -\frac{2 \,{\left (2 \, a^{2} d +{\left (a^{2} d^{3} + 12 \, a b d\right )} x^{2}\right )} \cosh \left (d x + c\right ) -{\left ({\left (a^{2} d^{4} + 12 \, a b d^{2}\right )} x^{3}{\rm Ei}\left (d x\right ) -{\left (a^{2} d^{4} + 12 \, a b d^{2}\right )} x^{3}{\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) + 2 \,{\left (a^{2} d^{2} x - 6 \, b^{2} x^{3}\right )} \sinh \left (d x + c\right ) -{\left ({\left (a^{2} d^{4} + 12 \, a b d^{2}\right )} x^{3}{\rm Ei}\left (d x\right ) +{\left (a^{2} d^{4} + 12 \, a b d^{2}\right )} x^{3}{\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{12 \, d x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(2*(2*a^2*d + (a^2*d^3 + 12*a*b*d)*x^2)*cosh(d*x + c) - ((a^2*d^4 + 12*a*b*d^2)*x^3*Ei(d*x) - (a^2*d^4 +
12*a*b*d^2)*x^3*Ei(-d*x))*cosh(c) + 2*(a^2*d^2*x - 6*b^2*x^3)*sinh(d*x + c) - ((a^2*d^4 + 12*a*b*d^2)*x^3*Ei(
d*x) + (a^2*d^4 + 12*a*b*d^2)*x^3*Ei(-d*x))*sinh(c))/(d*x^3)

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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^{4}}\, 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**4,x)

[Out]

Integral((a + b*x**2)**2*cosh(c + d*x)/x**4, x)

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Giac [A]  time = 1.1575, size = 319, normalized size = 2.4 \begin{align*} -\frac{a^{2} d^{4} x^{3}{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - a^{2} d^{4} x^{3}{\rm Ei}\left (d x\right ) e^{c} + 12 \, a b d^{2} x^{3}{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - 12 \, a b d^{2} x^{3}{\rm Ei}\left (d x\right ) e^{c} + a^{2} d^{3} x^{2} e^{\left (d x + c\right )} + a^{2} d^{3} x^{2} e^{\left (-d x - c\right )} + a^{2} d^{2} x e^{\left (d x + c\right )} + 12 \, a b d x^{2} e^{\left (d x + c\right )} - 6 \, b^{2} x^{3} e^{\left (d x + c\right )} - a^{2} d^{2} x e^{\left (-d x - c\right )} + 12 \, a b d x^{2} e^{\left (-d x - c\right )} + 6 \, b^{2} x^{3} e^{\left (-d x - c\right )} + 2 \, a^{2} d e^{\left (d x + c\right )} + 2 \, a^{2} d e^{\left (-d x - c\right )}}{12 \, d x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(a^2*d^4*x^3*Ei(-d*x)*e^(-c) - a^2*d^4*x^3*Ei(d*x)*e^c + 12*a*b*d^2*x^3*Ei(-d*x)*e^(-c) - 12*a*b*d^2*x^3
*Ei(d*x)*e^c + a^2*d^3*x^2*e^(d*x + c) + a^2*d^3*x^2*e^(-d*x - c) + a^2*d^2*x*e^(d*x + c) + 12*a*b*d*x^2*e^(d*
x + c) - 6*b^2*x^3*e^(d*x + c) - a^2*d^2*x*e^(-d*x - c) + 12*a*b*d*x^2*e^(-d*x - c) + 6*b^2*x^3*e^(-d*x - c) +
2*a^2*d*e^(d*x + c) + 2*a^2*d*e^(-d*x - c))/(d*x^3)