\(\int \frac {(a+b x^2)^2 \cosh (c+d x)}{x^3} \, dx\) [54]

Optimal result

Integrand size = 19, antiderivative size = 114 \[ \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^3} \, dx=-\frac {b^2 \cosh (c+d x)}{d^2}-\frac {a^2 \cosh (c+d x)}{2 x^2}+2 a b \cosh (c) \text {Chi}(d x)+\frac {1}{2} a^2 d^2 \cosh (c) \text {Chi}(d x)-\frac {a^2 d \sinh (c+d x)}{2 x}+\frac {b^2 x \sinh (c+d x)}{d}+2 a b \sinh (c) \text {Shi}(d x)+\frac {1}{2} a^2 d^2 \sinh (c) \text {Shi}(d x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {5395, 3378, 3384, 3379, 3382, 3377, 2718} \[ \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^3} \, dx=\frac {1}{2} a^2 d^2 \cosh (c) \text {Chi}(d x)+\frac {1}{2} a^2 d^2 \sinh (c) \text {Shi}(d x)-\frac {a^2 \cosh (c+d x)}{2 x^2}-\frac {a^2 d \sinh (c+d x)}{2 x}+2 a b \cosh (c) \text {Chi}(d x)+2 a b \sinh (c) \text {Shi}(d x)-\frac {b^2 \cosh (c+d x)}{d^2}+\frac {b^2 x \sinh (c+d x)}{d} \]

[In]

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

[Out]

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

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[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 (\frac {a^2 \cosh (c+d x)}{x^3}+\frac {2 a b \cosh (c+d x)}{x}+b^2 x \cosh (c+d x)\right ) \, dx \\ & = a^2 \int \frac {\cosh (c+d x)}{x^3} \, dx+(2 a b) \int \frac {\cosh (c+d x)}{x} \, dx+b^2 \int x \cosh (c+d x) \, dx \\ & = -\frac {a^2 \cosh (c+d x)}{2 x^2}+\frac {b^2 x \sinh (c+d x)}{d}-\frac {b^2 \int \sinh (c+d x) \, dx}{d}+\frac {1}{2} \left (a^2 d\right ) \int \frac {\sinh (c+d x)}{x^2} \, dx+(2 a b \cosh (c)) \int \frac {\cosh (d x)}{x} \, dx+(2 a b \sinh (c)) \int \frac {\sinh (d x)}{x} \, dx \\ & = -\frac {b^2 \cosh (c+d x)}{d^2}-\frac {a^2 \cosh (c+d x)}{2 x^2}+2 a b \cosh (c) \text {Chi}(d x)-\frac {a^2 d \sinh (c+d x)}{2 x}+\frac {b^2 x \sinh (c+d x)}{d}+2 a b \sinh (c) \text {Shi}(d x)+\frac {1}{2} \left (a^2 d^2\right ) \int \frac {\cosh (c+d x)}{x} \, dx \\ & = -\frac {b^2 \cosh (c+d x)}{d^2}-\frac {a^2 \cosh (c+d x)}{2 x^2}+2 a b \cosh (c) \text {Chi}(d x)-\frac {a^2 d \sinh (c+d x)}{2 x}+\frac {b^2 x \sinh (c+d x)}{d}+2 a b \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 {b^2 \cosh (c+d x)}{d^2}-\frac {a^2 \cosh (c+d x)}{2 x^2}+2 a b \cosh (c) \text {Chi}(d x)+\frac {1}{2} a^2 d^2 \cosh (c) \text {Chi}(d x)-\frac {a^2 d \sinh (c+d x)}{2 x}+\frac {b^2 x \sinh (c+d x)}{d}+2 a b \sinh (c) \text {Shi}(d x)+\frac {1}{2} a^2 d^2 \sinh (c) \text {Shi}(d x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^3} \, dx=\frac {1}{2} \left (-\frac {2 b^2 \cosh (c+d x)}{d^2}-\frac {a^2 \cosh (c+d x)}{x^2}+a \left (4 b+a d^2\right ) \cosh (c) \text {Chi}(d x)-\frac {a^2 d \sinh (c+d x)}{x}+\frac {2 b^2 x \sinh (c+d x)}{d}+a \left (4 b+a d^2\right ) \sinh (c) \text {Shi}(d x)\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.83

method	result	size
risch	\(-\frac {{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right ) a^{2} d^{4} x^{2}+{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) a^{2} d^{4} x^{2}+4 \,{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right ) a b \,d^{2} x^{2}+4 \,{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) a b \,d^{2} x^{2}-{\mathrm e}^{-d x -c} a^{2} d^{3} x +2 \,{\mathrm e}^{-d x -c} b^{2} d \,x^{3}+{\mathrm e}^{d x +c} a^{2} d^{3} x -2 \,{\mathrm e}^{d x +c} b^{2} d \,x^{3}+d^{2} {\mathrm e}^{-d x -c} a^{2}+2 \,{\mathrm e}^{-d x -c} b^{2} x^{2}+d^{2} {\mathrm e}^{d x +c} a^{2}+2 \,{\mathrm e}^{d x +c} b^{2} x^{2}}{4 d^{2} x^{2}}\)	\(209\)
meijerg	\(-\frac {2 b^{2} \cosh \left (c \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\cosh \left (d x \right )}{2 \sqrt {\pi }}-\frac {d x \sinh \left (d x \right )}{2 \sqrt {\pi }}\right )}{d^{2}}+\frac {b^{2} \sinh \left (c \right ) \left (\cosh \left (d x \right ) x d -\sinh \left (d x \right )\right )}{d^{2}}+a b \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {2 \gamma +2 \ln \left (x \right )+2 \ln \left (i d \right )}{\sqrt {\pi }}+\frac {2 \,\operatorname {Chi}\left (d x \right )-2 \ln \left (d x \right )-2 \gamma }{\sqrt {\pi }}\right )+2 a b \,\operatorname {Shi}\left (d x \right ) \sinh \left (c \right )-\frac {a^{2} \cosh \left (c \right ) \sqrt {\pi }\, d^{2} \left (\frac {4}{\sqrt {\pi }\, x^{2} d^{2}}-\frac {2 \left (2 \gamma -3+2 \ln \left (x \right )+2 \ln \left (i d \right )\right )}{\sqrt {\pi }}-\frac {4 \left (\frac {9 x^{2} d^{2}}{2}+3\right )}{3 \sqrt {\pi }\, x^{2} d^{2}}+\frac {4 \cosh \left (d x \right )}{\sqrt {\pi }\, x^{2} d^{2}}+\frac {4 \sinh \left (d x \right )}{\sqrt {\pi }\, x d}-\frac {4 \left (\operatorname {Chi}\left (d x \right )-\ln \left (d x \right )-\gamma \right )}{\sqrt {\pi }}\right )}{8}+\frac {i a^{2} \sinh \left (c \right ) \sqrt {\pi }\, d^{2} \left (\frac {4 i \cosh \left (d x \right )}{d x \sqrt {\pi }}+\frac {4 i \sinh \left (d x \right )}{x^{2} d^{2} \sqrt {\pi }}-\frac {4 i \operatorname {Shi}\left (d x \right )}{\sqrt {\pi }}\right )}{8}\)	\(294\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.44 \[ \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^3} \, dx=-\frac {2 \, {\left (a^{2} d^{2} + 2 \, b^{2} x^{2}\right )} \cosh \left (d x + c\right ) - {\left ({\left (a^{2} d^{4} + 4 \, a b d^{2}\right )} x^{2} {\rm Ei}\left (d x\right ) + {\left (a^{2} d^{4} + 4 \, a b d^{2}\right )} x^{2} {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) + 2 \, {\left (a^{2} d^{3} x - 2 \, b^{2} d x^{3}\right )} \sinh \left (d x + c\right ) - {\left ({\left (a^{2} d^{4} + 4 \, a b d^{2}\right )} x^{2} {\rm Ei}\left (d x\right ) - {\left (a^{2} d^{4} + 4 \, a b d^{2}\right )} x^{2} {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{4 \, d^{2} x^{2}} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^3} \, dx=\int \frac {\left (a + b x^{2}\right )^{2} \cosh {\left (c + d x \right )}}{x^{3}}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.45 \[ \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^3} \, dx=\frac {1}{4} \, {\left ({\left (d e^{\left (-c\right )} \Gamma \left (-1, d x\right ) + d e^{c} \Gamma \left (-1, -d x\right )\right )} a^{2} - b^{2} {\left (\frac {{\left (d^{2} x^{2} e^{c} - 2 \, d x e^{c} + 2 \, e^{c}\right )} e^{\left (d x\right )}}{d^{3}} + \frac {{\left (d^{2} x^{2} + 2 \, d x + 2\right )} e^{\left (-d x - c\right )}}{d^{3}}\right )} - \frac {4 \, a b \cosh \left (d x + c\right ) \log \left (x^{2}\right )}{d} + \frac {4 \, {\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + {\rm Ei}\left (d x\right ) e^{c}\right )} a b}{d}\right )} d + \frac {1}{2} \, {\left (b^{2} x^{2} + 2 \, a b \log \left (x^{2}\right ) - \frac {a^{2}}{x^{2}}\right )} \cosh \left (d x + c\right ) \]

[In]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.81 \[ \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^3} \, dx=\frac {a^{2} d^{4} x^{2} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a^{2} d^{4} x^{2} {\rm Ei}\left (d x\right ) e^{c} + 4 \, a b d^{2} x^{2} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 4 \, a b d^{2} x^{2} {\rm Ei}\left (d x\right ) e^{c} - a^{2} d^{3} x e^{\left (d x + c\right )} + 2 \, b^{2} d x^{3} e^{\left (d x + c\right )} + a^{2} d^{3} x e^{\left (-d x - c\right )} - 2 \, b^{2} d x^{3} e^{\left (-d x - c\right )} - a^{2} d^{2} e^{\left (d x + c\right )} - 2 \, b^{2} x^{2} e^{\left (d x + c\right )} - a^{2} d^{2} e^{\left (-d x - c\right )} - 2 \, b^{2} x^{2} e^{\left (-d x - c\right )}}{4 \, d^{2} x^{2}} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^3} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\left (b\,x^2+a\right )}^2}{x^3} \,d x \]

[In]

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

[Out]

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