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

Optimal. Leaf size=110 \[ a^2 \cosh (c) \text{Chi}(d x)+a^2 \sinh (c) \text{Shi}(d x)-\frac{2 a b \cosh (c+d x)}{d^2}+\frac{2 a b x \sinh (c+d x)}{d}-\frac{3 b^2 x^2 \cosh (c+d x)}{d^2}+\frac{6 b^2 x \sinh (c+d x)}{d^3}-\frac{6 b^2 \cosh (c+d x)}{d^4}+\frac{b^2 x^3 \sinh (c+d x)}{d} \]

[Out]

(-6*b^2*Cosh[c + d*x])/d^4 - (2*a*b*Cosh[c + d*x])/d^2 - (3*b^2*x^2*Cosh[c + d*x])/d^2 + a^2*Cosh[c]*CoshInteg
ral[d*x] + (6*b^2*x*Sinh[c + d*x])/d^3 + (2*a*b*x*Sinh[c + d*x])/d + (b^2*x^3*Sinh[c + d*x])/d + a^2*Sinh[c]*S
inhIntegral[d*x]

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Rubi [A]  time = 0.188808, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {5287, 3303, 3298, 3301, 3296, 2638} \[ a^2 \cosh (c) \text{Chi}(d x)+a^2 \sinh (c) \text{Shi}(d x)-\frac{2 a b \cosh (c+d x)}{d^2}+\frac{2 a b x \sinh (c+d x)}{d}-\frac{3 b^2 x^2 \cosh (c+d x)}{d^2}+\frac{6 b^2 x \sinh (c+d x)}{d^3}-\frac{6 b^2 \cosh (c+d x)}{d^4}+\frac{b^2 x^3 \sinh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*b^2*Cosh[c + d*x])/d^4 - (2*a*b*Cosh[c + d*x])/d^2 - (3*b^2*x^2*Cosh[c + d*x])/d^2 + a^2*Cosh[c]*CoshInteg
ral[d*x] + (6*b^2*x*Sinh[c + d*x])/d^3 + (2*a*b*x*Sinh[c + d*x])/d + (b^2*x^3*Sinh[c + d*x])/d + a^2*Sinh[c]*S
inhIntegral[d*x]

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 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]

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 2638

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

Rubi steps

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

Mathematica [A]  time = 0.360715, size = 82, normalized size = 0.75 \[ a^2 \cosh (c) \text{Chi}(d x)+a^2 \sinh (c) \text{Shi}(d x)+\frac{b x \left (2 a d^2+b \left (d^2 x^2+6\right )\right ) \sinh (c+d x)}{d^3}-\frac{b \left (2 a d^2+3 b \left (d^2 x^2+2\right )\right ) \cosh (c+d x)}{d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.05, size = 226, normalized size = 2.1 \begin{align*} -{\frac{{a}^{2}{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{2}}-{\frac{ab{{\rm e}^{-dx-c}}}{{d}^{2}}}-3\,{\frac{{{\rm e}^{dx+c}}{b}^{2}}{{d}^{4}}}-{\frac{{a}^{2}{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{2}}-3\,{\frac{{b}^{2}{{\rm e}^{-dx-c}}}{{d}^{4}}}+{\frac{ab{{\rm e}^{dx+c}}x}{d}}-{\frac{ab{{\rm e}^{dx+c}}}{{d}^{2}}}+{\frac{{{\rm e}^{dx+c}}{b}^{2}{x}^{3}}{2\,d}}-{\frac{3\,{{\rm e}^{dx+c}}{b}^{2}{x}^{2}}{2\,{d}^{2}}}+3\,{\frac{{{\rm e}^{dx+c}}{b}^{2}x}{{d}^{3}}}-{\frac{ab{{\rm e}^{-dx-c}}x}{d}}-{\frac{{b}^{2}{{\rm e}^{-dx-c}}{x}^{3}}{2\,d}}-{\frac{3\,{b}^{2}{{\rm e}^{-dx-c}}{x}^{2}}{2\,{d}^{2}}}-3\,{\frac{{b}^{2}{{\rm e}^{-dx-c}}x}{{d}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.18744, size = 317, normalized size = 2.88 \begin{align*} -\frac{1}{8} \,{\left (4 \, a b{\left (\frac{{\left (d^{2} x^{2} e^{c} - 2 \, d x e^{c} + 2 \, e^{c}\right )} e^{\left (d x\right )}}{d^{3}} + \frac{{\left (d^{2} x^{2} + 2 \, d x + 2\right )} e^{\left (-d x - c\right )}}{d^{3}}\right )} + b^{2}{\left (\frac{{\left (d^{4} x^{4} e^{c} - 4 \, d^{3} x^{3} e^{c} + 12 \, d^{2} x^{2} e^{c} - 24 \, d x e^{c} + 24 \, e^{c}\right )} e^{\left (d x\right )}}{d^{5}} + \frac{{\left (d^{4} x^{4} + 4 \, d^{3} x^{3} + 12 \, d^{2} x^{2} + 24 \, d x + 24\right )} e^{\left (-d x - c\right )}}{d^{5}}\right )} + \frac{4 \, a^{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 )} a^{2}}{d}\right )} d + \frac{1}{4} \,{\left (b^{2} x^{4} + 4 \, a b x^{2} + 2 \, a^{2} \log \left (x^{2}\right )\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,x, algorithm="maxima")

[Out]

-1/8*(4*a*b*((d^2*x^2*e^c - 2*d*x*e^c + 2*e^c)*e^(d*x)/d^3 + (d^2*x^2 + 2*d*x + 2)*e^(-d*x - c)/d^3) + b^2*((d
^4*x^4*e^c - 4*d^3*x^3*e^c + 12*d^2*x^2*e^c - 24*d*x*e^c + 24*e^c)*e^(d*x)/d^5 + (d^4*x^4 + 4*d^3*x^3 + 12*d^2
*x^2 + 24*d*x + 24)*e^(-d*x - c)/d^5) + 4*a^2*cosh(d*x + c)*log(x^2)/d - 4*(Ei(-d*x)*e^(-c) + Ei(d*x)*e^c)*a^2
/d)*d + 1/4*(b^2*x^4 + 4*a*b*x^2 + 2*a^2*log(x^2))*cosh(d*x + c)

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Fricas [A]  time = 2.47665, size = 292, normalized size = 2.65 \begin{align*} -\frac{2 \,{\left (3 \, b^{2} d^{2} x^{2} + 2 \, a b d^{2} + 6 \, b^{2}\right )} \cosh \left (d x + c\right ) -{\left (a^{2} d^{4}{\rm Ei}\left (d x\right ) + a^{2} d^{4}{\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) - 2 \,{\left (b^{2} d^{3} x^{3} + 2 \,{\left (a b d^{3} + 3 \, b^{2} d\right )} x\right )} \sinh \left (d x + c\right ) -{\left (a^{2} d^{4}{\rm Ei}\left (d x\right ) - a^{2} d^{4}{\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{2 \, d^{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,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 5.73347, size = 121, normalized size = 1.1 \begin{align*} a^{2} \sinh{\left (c \right )} \operatorname{Shi}{\left (d x \right )} + a^{2} \cosh{\left (c \right )} \operatorname{Chi}\left (d x\right ) + 2 a b \left (\begin{cases} \frac{x \sinh{\left (c + d x \right )}}{d} - \frac{\cosh{\left (c + d x \right )}}{d^{2}} & \text{for}\: d \neq 0 \\\frac{x^{2} \cosh{\left (c \right )}}{2} & \text{otherwise} \end{cases}\right ) + b^{2} \left (\begin{cases} \frac{x^{3} \sinh{\left (c + d x \right )}}{d} - \frac{3 x^{2} \cosh{\left (c + d x \right )}}{d^{2}} + \frac{6 x \sinh{\left (c + d x \right )}}{d^{3}} - \frac{6 \cosh{\left (c + d x \right )}}{d^{4}} & \text{for}\: d \neq 0 \\\frac{x^{4} \cosh{\left (c \right )}}{4} & \text{otherwise} \end{cases}\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,x)

[Out]

a**2*sinh(c)*Shi(d*x) + a**2*cosh(c)*Chi(d*x) + 2*a*b*Piecewise((x*sinh(c + d*x)/d - cosh(c + d*x)/d**2, Ne(d,
0)), (x**2*cosh(c)/2, True)) + b**2*Piecewise((x**3*sinh(c + d*x)/d - 3*x**2*cosh(c + d*x)/d**2 + 6*x*sinh(c
+ d*x)/d**3 - 6*cosh(c + d*x)/d**4, Ne(d, 0)), (x**4*cosh(c)/4, True))

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Giac [B]  time = 1.20143, size = 300, normalized size = 2.73 \begin{align*} \frac{b^{2} d^{3} x^{3} e^{\left (d x + c\right )} - b^{2} d^{3} x^{3} e^{\left (-d x - c\right )} + a^{2} d^{4}{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a^{2} d^{4}{\rm Ei}\left (d x\right ) e^{c} + 2 \, a b d^{3} x e^{\left (d x + c\right )} - 3 \, b^{2} d^{2} x^{2} e^{\left (d x + c\right )} - 2 \, a b d^{3} x e^{\left (-d x - c\right )} - 3 \, b^{2} d^{2} x^{2} e^{\left (-d x - c\right )} - 2 \, a b d^{2} e^{\left (d x + c\right )} + 6 \, b^{2} d x e^{\left (d x + c\right )} - 2 \, a b d^{2} e^{\left (-d x - c\right )} - 6 \, b^{2} d x e^{\left (-d x - c\right )} - 6 \, b^{2} e^{\left (d x + c\right )} - 6 \, b^{2} e^{\left (-d x - c\right )}}{2 \, d^{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,x, algorithm="giac")

[Out]

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