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]

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

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Rubi [A]  time = 0.29, antiderivative size = 124, normalized size of antiderivative = 1.00, 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 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 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 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]

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*}

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Mathematica [A]  time = 0.53, size = 127, normalized size = 1.02 \[ \frac {1}{2} e^{-a-b \sqrt {c}} \left (e^{2 \left (a+b \sqrt {c}\right )} \text {Ei}\left (b \left (\sqrt {c+d x}-\sqrt {c}\right )\right )+e^{2 a} \text {Ei}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )+\text {Ei}\left (b \left (\sqrt {c}-\sqrt {c+d x}\right )\right )+e^{2 b \sqrt {c}} \text {Ei}\left (-b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )\right ) \]

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

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fricas [B]  time = 0.92, size = 217, normalized size = 1.75 \[ \frac {1}{2} \, {\left ({\rm Ei}\left (\sqrt {d x + c} b - \sqrt {b^{2} c}\right ) + {\rm Ei}\left (-\sqrt {d x + c} b + \sqrt {b^{2} c}\right )\right )} \cosh \left (a + \sqrt {b^{2} c}\right ) + \frac {1}{2} \, {\left ({\rm Ei}\left (\sqrt {d x + c} b + \sqrt {b^{2} c}\right ) + {\rm Ei}\left (-\sqrt {d x + c} b - \sqrt {b^{2} c}\right )\right )} \cosh \left (-a + \sqrt {b^{2} c}\right ) + \frac {1}{2} \, {\left ({\rm Ei}\left (\sqrt {d x + c} b - \sqrt {b^{2} c}\right ) - {\rm Ei}\left (-\sqrt {d x + c} b + \sqrt {b^{2} c}\right )\right )} \sinh \left (a + \sqrt {b^{2} c}\right ) - \frac {1}{2} \, {\left ({\rm Ei}\left (\sqrt {d x + c} b + \sqrt {b^{2} c}\right ) - {\rm Ei}\left (-\sqrt {d x + c} b - \sqrt {b^{2} c}\right )\right )} \sinh \left (-a + \sqrt {b^{2} c}\right ) \]

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))

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

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)

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maple [F]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left (a +b \sqrt {d x +c}\right )}{x}\, dx \]

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)

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

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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {cosh}\left (a+b\,\sqrt {c+d\,x}\right )}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*(c + d*x)^(1/2))/x,x)

[Out]

int(cosh(a + b*(c + d*x)^(1/2))/x, x)

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

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)

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