\(\int \frac {(a+b x^2)^2 \cosh (c+d x)}{x^2} \, dx\) [53]

Optimal result

Integrand size = 19, antiderivative size = 95 \[ \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^2} \, dx=-\frac {a^2 \cosh (c+d x)}{x}-\frac {2 b^2 x \cosh (c+d x)}{d^2}+a^2 d \text {Chi}(d x) \sinh (c)+\frac {2 b^2 \sinh (c+d x)}{d^3}+\frac {2 a b \sinh (c+d x)}{d}+\frac {b^2 x^2 \sinh (c+d x)}{d}+a^2 d \cosh (c) \text {Shi}(d x) \]

[Out]

-a^2*cosh(d*x+c)/x-2*b^2*x*cosh(d*x+c)/d^2+a^2*d*cosh(c)*Shi(d*x)+a^2*d*Chi(d*x)*sinh(c)+2*b^2*sinh(d*x+c)/d^3
+2*a*b*sinh(d*x+c)/d+b^2*x^2*sinh(d*x+c)/d

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {5395, 2717, 3378, 3384, 3379, 3382, 3377} \[ \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^2} \, dx=a^2 d \sinh (c) \text {Chi}(d x)+a^2 d \cosh (c) \text {Shi}(d x)-\frac {a^2 \cosh (c+d x)}{x}+\frac {2 a b \sinh (c+d x)}{d}+\frac {2 b^2 \sinh (c+d x)}{d^3}-\frac {2 b^2 x \cosh (c+d x)}{d^2}+\frac {b^2 x^2 \sinh (c+d x)}{d} \]

[In]

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

[Out]

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

Rule 2717

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

Rule 3377

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 3378

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 3379

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 3382

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 3384

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 5395

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

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^2} \, dx=-\frac {a^2 \cosh (c+d x)}{x}-\frac {2 b^2 x \cosh (c+d x)}{d^2}+a^2 d \text {Chi}(d x) \sinh (c)+\frac {2 b^2 \sinh (c+d x)}{d^3}+\frac {2 a b \sinh (c+d x)}{d}+\frac {b^2 x^2 \sinh (c+d x)}{d}+a^2 d \cosh (c) \text {Shi}(d x) \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(199\) vs. \(2(95)=190\).

Time = 0.26 (sec) , antiderivative size = 200, normalized size of antiderivative = 2.11

method	result	size
risch	\(-\frac {{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right ) a^{2} d^{4} x -{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) a^{2} d^{4} x +d^{2} {\mathrm e}^{-d x -c} b^{2} x^{3}-d^{2} {\mathrm e}^{d x +c} b^{2} x^{3}+{\mathrm e}^{-d x -c} a^{2} d^{3}+2 \,{\mathrm e}^{-d x -c} a b \,d^{2} x +2 d \,{\mathrm e}^{-d x -c} b^{2} x^{2}+{\mathrm e}^{d x +c} a^{2} d^{3}-2 \,{\mathrm e}^{d x +c} a b \,d^{2} x +2 d \,{\mathrm e}^{d x +c} b^{2} x^{2}+2 \,{\mathrm e}^{-d x -c} b^{2} x -2 \,{\mathrm e}^{d x +c} b^{2} x}{2 d^{3} x}\)	\(200\)
meijerg	\(\frac {4 i b^{2} \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {i x d \cosh \left (d x \right )}{2 \sqrt {\pi }}-\frac {i \left (\frac {3 x^{2} d^{2}}{2}+3\right ) \sinh \left (d x \right )}{6 \sqrt {\pi }}\right )}{d^{3}}+\frac {4 b^{2} \sinh \left (c \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\left (\frac {x^{2} d^{2}}{2}+1\right ) \cosh \left (d x \right )}{2 \sqrt {\pi }}-\frac {d x \sinh \left (d x \right )}{2 \sqrt {\pi }}\right )}{d^{3}}+\frac {2 a b \cosh \left (c \right ) \sinh \left (d x \right )}{d}-\frac {2 b a \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cosh \left (d x \right )}{\sqrt {\pi }}\right )}{d}+\frac {i a^{2} \cosh \left (c \right ) \sqrt {\pi }\, d \left (\frac {4 i \cosh \left (d x \right )}{d x \sqrt {\pi }}-\frac {4 i \operatorname {Shi}\left (d x \right )}{\sqrt {\pi }}\right )}{4}+\frac {a^{2} \sinh \left (c \right ) \sqrt {\pi }\, d \left (\frac {4 \gamma -4+4 \ln \left (x \right )+4 \ln \left (i d \right )}{\sqrt {\pi }}+\frac {4}{\sqrt {\pi }}-\frac {4 \sinh \left (d x \right )}{\sqrt {\pi }\, x d}+\frac {4 \,\operatorname {Chi}\left (d x \right )-4 \ln \left (d x \right )-4 \gamma }{\sqrt {\pi }}\right )}{4}\)	\(247\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.34 \[ \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^2} \, dx=-\frac {2 \, {\left (a^{2} d^{3} + 2 \, b^{2} d x^{2}\right )} \cosh \left (d x + c\right ) - {\left (a^{2} d^{4} x {\rm Ei}\left (d x\right ) - a^{2} d^{4} x {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) - 2 \, {\left (b^{2} d^{2} x^{3} + 2 \, {\left (a b d^{2} + b^{2}\right )} x\right )} \sinh \left (d x + c\right ) - {\left (a^{2} d^{4} x {\rm Ei}\left (d x\right ) + a^{2} d^{4} x {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{2 \, d^{3} x} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^2} \, dx=\int \frac {\left (a + b x^{2}\right )^{2} \cosh {\left (c + d x \right )}}{x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.88 \[ \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^2} \, dx=-\frac {1}{6} \, {\left (3 \, a^{2} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - 3 \, a^{2} {\rm Ei}\left (d x\right ) e^{c} + \frac {6 \, {\left (d x e^{c} - e^{c}\right )} a b e^{\left (d x\right )}}{d^{2}} + \frac {6 \, {\left (d x + 1\right )} a b e^{\left (-d x - c\right )}}{d^{2}} + \frac {{\left (d^{3} x^{3} e^{c} - 3 \, d^{2} x^{2} e^{c} + 6 \, d x e^{c} - 6 \, e^{c}\right )} b^{2} e^{\left (d x\right )}}{d^{4}} + \frac {{\left (d^{3} x^{3} + 3 \, d^{2} x^{2} + 6 \, d x + 6\right )} b^{2} e^{\left (-d x - c\right )}}{d^{4}}\right )} d + \frac {1}{3} \, {\left (b^{2} x^{3} + 6 \, a b x - \frac {3 \, a^{2}}{x}\right )} \cosh \left (d x + c\right ) \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (95) = 190\).

Time = 0.27 (sec) , antiderivative size = 197, normalized size of antiderivative = 2.07 \[ \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^2} \, dx=-\frac {a^{2} d^{4} x {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - a^{2} d^{4} x {\rm Ei}\left (d x\right ) e^{c} - b^{2} d^{2} x^{3} e^{\left (d x + c\right )} + b^{2} d^{2} x^{3} e^{\left (-d x - c\right )} + a^{2} d^{3} e^{\left (d x + c\right )} - 2 \, a b d^{2} x e^{\left (d x + c\right )} + 2 \, b^{2} d x^{2} e^{\left (d x + c\right )} + a^{2} d^{3} e^{\left (-d x - c\right )} + 2 \, a b d^{2} x e^{\left (-d x - c\right )} + 2 \, b^{2} d x^{2} e^{\left (-d x - c\right )} - 2 \, b^{2} x e^{\left (d x + c\right )} + 2 \, b^{2} x e^{\left (-d x - c\right )}}{2 \, d^{3} x} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^2} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\left (b\,x^2+a\right )}^2}{x^2} \,d x \]

[In]

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

[Out]

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