3.62 \(\int \frac{\cosh (a+b \sqrt{c+d x})}{x} \, dx\)

Optimal. Leaf size=124 \[ \cosh \left (a+b \sqrt{c}\right ) \text{Chi}\left (b \left (\sqrt{c}-\sqrt{c+d x}\right )\right )+\cosh \left (a-b \sqrt{c}\right ) \text{Chi}\left (b \left (\sqrt{c}+\sqrt{c+d x}\right )\right )-\sinh \left (a+b \sqrt{c}\right ) \text{Shi}\left (b \left (\sqrt{c}-\sqrt{c+d x}\right )\right )+\sinh \left (a-b \sqrt{c}\right ) \text{Shi}\left (b \left (\sqrt{c}+\sqrt{c+d x}\right )\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.289778, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {5365, 5293, 3303, 3298, 3301} \[ \cosh \left (a+b \sqrt{c}\right ) \text{Chi}\left (b \left (\sqrt{c}-\sqrt{c+d x}\right )\right )+\cosh \left (a-b \sqrt{c}\right ) \text{Chi}\left (b \left (\sqrt{c}+\sqrt{c+d x}\right )\right )-\sinh \left (a+b \sqrt{c}\right ) \text{Shi}\left (b \left (\sqrt{c}-\sqrt{c+d x}\right )\right )+\sinh \left (a-b \sqrt{c}\right ) \text{Shi}\left (b \left (\sqrt{c}+\sqrt{c+d x}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*Sqrt[c + d*x]]/x,x]

[Out]

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

Rule 5365

Int[((a_.) + Cosh[(c_.) + (d_.)*(u_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/Coefficient[u, x, 1]^(
m + 1), Subst[Int[(x - Coefficient[u, x, 0])^m*(a + b*Cosh[c + d*x^n])^p, x], x, u], x] /; FreeQ[{a, b, c, d,
n, p}, x] && LinearQ[u, x] && NeQ[u, x] && IntegerQ[m]

Rule 5293

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])

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 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]

Rubi steps

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

Mathematica [A]  time = 0.492219, size = 127, normalized size = 1.02 \[ \frac{1}{2} e^{-a-b \sqrt{c}} \left (e^{2 \left (a+b \sqrt{c}\right )} \text{ExpIntegralEi}\left (b \left (\sqrt{c+d x}-\sqrt{c}\right )\right )+e^{2 a} \text{ExpIntegralEi}\left (b \left (\sqrt{c+d x}+\sqrt{c}\right )\right )+\text{ExpIntegralEi}\left (b \left (\sqrt{c}-\sqrt{c+d x}\right )\right )+e^{2 b \sqrt{c}} \text{ExpIntegralEi}\left (-b \left (\sqrt{c+d x}+\sqrt{c}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + b*Sqrt[c + d*x]]/x,x]

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.02, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x}\cosh \left ( a+b\sqrt{dx+c} \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a+b*(d*x+c)^(1/2))/x,x)

[Out]

int(cosh(a+b*(d*x+c)^(1/2))/x,x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh \left (\sqrt{d x + c} b + a\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(a+b*(d*x+c)^(1/2))/x,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 1.84215, size = 547, normalized size = 4.41 \begin{align*} \frac{1}{2} \,{\left ({\rm Ei}\left (\sqrt{d x + c} b - \sqrt{b^{2} c}\right ) +{\rm Ei}\left (-\sqrt{d x + c} b + \sqrt{b^{2} c}\right )\right )} \cosh \left (a + \sqrt{b^{2} c}\right ) + \frac{1}{2} \,{\left ({\rm Ei}\left (\sqrt{d x + c} b + \sqrt{b^{2} c}\right ) +{\rm Ei}\left (-\sqrt{d x + c} b - \sqrt{b^{2} c}\right )\right )} \cosh \left (-a + \sqrt{b^{2} c}\right ) + \frac{1}{2} \,{\left ({\rm Ei}\left (\sqrt{d x + c} b - \sqrt{b^{2} c}\right ) -{\rm Ei}\left (-\sqrt{d x + c} b + \sqrt{b^{2} c}\right )\right )} \sinh \left (a + \sqrt{b^{2} c}\right ) - \frac{1}{2} \,{\left ({\rm Ei}\left (\sqrt{d x + c} b + \sqrt{b^{2} c}\right ) -{\rm Ei}\left (-\sqrt{d x + c} b - \sqrt{b^{2} c}\right )\right )} \sinh \left (-a + \sqrt{b^{2} c}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(a+b*(d*x+c)^(1/2))/x,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh{\left (a + b \sqrt{c + d x} \right )}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(a+b*(d*x+c)**(1/2))/x,x)

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh \left (\sqrt{d x + c} b + a\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(a+b*(d*x+c)^(1/2))/x,x, algorithm="giac")

[Out]

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