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

Optimal. Leaf size=213 \[ \frac {\cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 \sqrt {-a} \sqrt {b}}-\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 \sqrt {-a} \sqrt {b}}-\frac {\sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 \sqrt {-a} \sqrt {b}}-\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 \sqrt {-a} \sqrt {b}} \]

[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)

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Rubi [A]  time = 0.26, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5281, 3303, 3298, 3301} \[ \frac {\cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 \sqrt {-a} \sqrt {b}}-\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 \sqrt {-a} \sqrt {b}}-\frac {\sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 \sqrt {-a} \sqrt {b}}-\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 \sqrt {-a} \sqrt {b}} \]

Antiderivative was successfully verified.

[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 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 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 5281

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

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Mathematica [C]  time = 0.21, size = 180, normalized size = 0.85 \[ \frac {i \left (\cosh \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (i d x-\frac {\sqrt {a} d}{\sqrt {b}}\right )-\cosh \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )+i \left (\sinh \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )+\sinh \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )\right )\right )}{2 \sqrt {a} \sqrt {b}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((I/2)*(Cosh[c - (I*Sqrt[a]*d)/Sqrt[b]]*CosIntegral[-((Sqrt[a]*d)/Sqrt[b]) + I*d*x] - Cosh[c + (I*Sqrt[a]*d)/S
qrt[b]]*CosIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x] + I*(Sinh[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[a]*d)/
Sqrt[b] - I*d*x] + Sinh[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x])))/(Sqrt[a]*Sqrt[b
])

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fricas [B]  time = 0.73, size = 316, normalized size = 1.48 \[ -\frac {{\left (\sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (d x - \sqrt {-\frac {a d^{2}}{b}}\right ) + \sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (-d x + \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \cosh \left (c + \sqrt {-\frac {a d^{2}}{b}}\right ) - {\left (\sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (d x + \sqrt {-\frac {a d^{2}}{b}}\right ) + \sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (-d x - \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \cosh \left (-c + \sqrt {-\frac {a d^{2}}{b}}\right ) + {\left (\sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (d x - \sqrt {-\frac {a d^{2}}{b}}\right ) - \sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (-d x + \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \sinh \left (c + \sqrt {-\frac {a d^{2}}{b}}\right ) + {\left (\sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (d x + \sqrt {-\frac {a d^{2}}{b}}\right ) - \sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (-d x - \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \sinh \left (-c + \sqrt {-\frac {a d^{2}}{b}}\right )}{4 \, a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left (d x + c\right )}{b x^{2} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.10, size = 212, normalized size = 1.00 \[ -\frac {{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \Ei \left (1, -\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{4 \sqrt {-a b}}+\frac {{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \Ei \left (1, \frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{4 \sqrt {-a b}}-\frac {{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \Ei \left (1, \frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{4 \sqrt {-a b}}+\frac {{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \Ei \left (1, -\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{4 \sqrt {-a b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left (d x + c\right )}{b x^{2} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {cosh}\left (c+d\,x\right )}{b\,x^2+a} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(c + d*x)/(a + b*x^2),x)

[Out]

int(cosh(c + d*x)/(a + b*x^2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh {\left (c + d x \right )}}{a + b x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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