\(\int \frac {\cosh (c+d x)}{a+b x^2} \, dx\) [61]

Optimal result

Integrand size = 16, antiderivative size = 213 \[ \int \frac {\cosh (c+d x)}{a+b x^2} \, dx=\frac {\cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 \sqrt {-a} \sqrt {b}}-\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 \sqrt {-a} \sqrt {b}}-\frac {\sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 \sqrt {-a} \sqrt {b}}-\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 \sqrt {-a} \sqrt {b}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5389, 3384, 3379, 3382} \[ \int \frac {\cosh (c+d x)}{a+b x^2} \, dx=\frac {\cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 \sqrt {-a} \sqrt {b}}-\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 \sqrt {-a} \sqrt {b}}-\frac {\sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 \sqrt {-a} \sqrt {b}}-\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 \sqrt {-a} \sqrt {b}} \]

[In]

Int[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*Sqrt[-a]*Sqrt[b]) - (Cosh[c - (Sq
rt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*Sqrt[-a]*Sqrt[b]) - (Sinh[c + (Sqrt[-a]*d)/Sqr
t[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*Sqrt[-a]*Sqrt[b]) - (Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhInt
egral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*Sqrt[-a]*Sqrt[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 5389

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

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.32 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.78 \[ \int \frac {\cosh (c+d x)}{a+b x^2} \, dx=-\frac {i e^{-c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (e^{2 c+\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )-e^{2 c} \operatorname {ExpIntegralEi}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )-e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+\operatorname {ExpIntegralEi}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )}{4 \sqrt {a} \sqrt {b}} \]

[In]

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

[Out]

((-1/4*I)*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])*ExpIntegral
Ei[((-I)*Sqrt[a]*d)/Sqrt[b] - d*x] + ExpIntegralEi[(I*Sqrt[a]*d)/Sqrt[b] - d*x]))/(Sqrt[a]*Sqrt[b])

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.00

method	result	size
risch	\(-\frac {{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{4 \sqrt {-a b}}+\frac {{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{4 \sqrt {-a b}}-\frac {{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{4 \sqrt {-a b}}+\frac {{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{4 \sqrt {-a b}}\)	\(212\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 316 vs. \(2 (157) = 314\).

Time = 0.26 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.48 \[ \int \frac {\cosh (c+d x)}{a+b x^2} \, dx=-\frac {{\left (\sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (d x - \sqrt {-\frac {a d^{2}}{b}}\right ) + \sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (-d x + \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \cosh \left (c + \sqrt {-\frac {a d^{2}}{b}}\right ) - {\left (\sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (d x + \sqrt {-\frac {a d^{2}}{b}}\right ) + \sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (-d x - \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \cosh \left (-c + \sqrt {-\frac {a d^{2}}{b}}\right ) + {\left (\sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (d x - \sqrt {-\frac {a d^{2}}{b}}\right ) - \sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (-d x + \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \sinh \left (c + \sqrt {-\frac {a d^{2}}{b}}\right ) + {\left (\sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (d x + \sqrt {-\frac {a d^{2}}{b}}\right ) - \sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (-d x - \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \sinh \left (-c + \sqrt {-\frac {a d^{2}}{b}}\right )}{4 \, a d} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(cosh(c + d*x)/(a + b*x^2),x)

[Out]

int(cosh(c + d*x)/(a + b*x^2), x)