\(\int \frac {x \cosh (c+d x)}{a+b x^2} \, dx\) [60]

Optimal result

Integrand size = 17, antiderivative size = 177 \[ \int \frac {x \cosh (c+d x)}{a+b x^2} \, dx=\frac {\cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b}+\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 b}-\frac {\sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b}+\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {5401, 3384, 3379, 3382} \[ \int \frac {x \cosh (c+d x)}{a+b x^2} \, dx=\frac {\cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b}+\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b}-\frac {\sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b}+\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b} \]

[In]

Int[(x*Cosh[c + d*x])/(a + b*x^2),x]

[Out]

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

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 5401

Int[Cosh[(c_.) + (d_.)*(x_)]*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[Cosh[c
+ d*x], x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IntegerQ[m] && IGtQ[n, 0] && (Eq
Q[n, 2] || EqQ[p, -1])

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.20 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.88 \[ \int \frac {x \cosh (c+d x)}{a+b x^2} \, dx=\frac {e^{-c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (e^{2 c+\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+e^{2 c} \operatorname {ExpIntegralEi}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+\operatorname {ExpIntegralEi}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )}{4 b} \]

[In]

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

[Out]

(E^(-c - (I*Sqrt[a]*d)/Sqrt[b])*(E^(2*c + ((2*I)*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[d*(((-I)*Sqrt[a])/Sqrt[b] +
x)] + E^(2*c)*ExpIntegralEi[d*((I*Sqrt[a])/Sqrt[b] + x)] + E^(((2*I)*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[((-I)*
Sqrt[a]*d)/Sqrt[b] - d*x] + ExpIntegralEi[(I*Sqrt[a]*d)/Sqrt[b] - d*x]))/(4*b)

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.13

method	result	size
risch	\(-\frac {{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{4 b}-\frac {{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{4 b}-\frac {{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{4 b}-\frac {{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{4 b}\)	\(200\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.24 \[ \int \frac {x \cosh (c+d x)}{a+b x^2} \, dx=\frac {{\left ({\rm Ei}\left (d x - \sqrt {-\frac {a d^{2}}{b}}\right ) + {\rm Ei}\left (-d x + \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \cosh \left (c + \sqrt {-\frac {a d^{2}}{b}}\right ) + {\left ({\rm Ei}\left (d x + \sqrt {-\frac {a d^{2}}{b}}\right ) + {\rm Ei}\left (-d x - \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \cosh \left (-c + \sqrt {-\frac {a d^{2}}{b}}\right ) + {\left ({\rm Ei}\left (d x - \sqrt {-\frac {a d^{2}}{b}}\right ) - {\rm Ei}\left (-d x + \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \sinh \left (c + \sqrt {-\frac {a d^{2}}{b}}\right ) - {\left ({\rm Ei}\left (d x + \sqrt {-\frac {a d^{2}}{b}}\right ) - {\rm Ei}\left (-d x - \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \sinh \left (-c + \sqrt {-\frac {a d^{2}}{b}}\right )}{4 \, b} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int((x*cosh(c + d*x))/(a + b*x^2),x)

[Out]

int((x*cosh(c + d*x))/(a + b*x^2), x)