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

Optimal. Leaf size=182 \[ -\frac{b d \sinh \left (a-b \sqrt{c}\right ) \text{Chi}\left (b \left (\sqrt{c}+\sqrt{c+d x}\right )\right )}{2 \sqrt{c}}+\frac{b d \sinh \left (a+b \sqrt{c}\right ) \text{Chi}\left (b \left (\sqrt{c}-\sqrt{c+d x}\right )\right )}{2 \sqrt{c}}-\frac{b d \cosh \left (a+b \sqrt{c}\right ) \text{Shi}\left (b \left (\sqrt{c}-\sqrt{c+d x}\right )\right )}{2 \sqrt{c}}-\frac{b d \cosh \left (a-b \sqrt{c}\right ) \text{Shi}\left (b \left (\sqrt{c}+\sqrt{c+d x}\right )\right )}{2 \sqrt{c}}-\frac{\cosh \left (a+b \sqrt{c+d x}\right )}{x} \]

[Out]

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

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Rubi [A]  time = 0.382418, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5365, 5289, 5280, 3303, 3298, 3301} \[ -\frac{b d \sinh \left (a-b \sqrt{c}\right ) \text{Chi}\left (b \left (\sqrt{c}+\sqrt{c+d x}\right )\right )}{2 \sqrt{c}}+\frac{b d \sinh \left (a+b \sqrt{c}\right ) \text{Chi}\left (b \left (\sqrt{c}-\sqrt{c+d x}\right )\right )}{2 \sqrt{c}}-\frac{b d \cosh \left (a+b \sqrt{c}\right ) \text{Shi}\left (b \left (\sqrt{c}-\sqrt{c+d x}\right )\right )}{2 \sqrt{c}}-\frac{b d \cosh \left (a-b \sqrt{c}\right ) \text{Shi}\left (b \left (\sqrt{c}+\sqrt{c+d x}\right )\right )}{2 \sqrt{c}}-\frac{\cosh \left (a+b \sqrt{c+d x}\right )}{x} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*Sqrt[c + d*x]]/x^2,x]

[Out]

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

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 5289

Int[Cosh[(c_.) + (d_.)*(x_)]*((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(e^m*(a + b*x
^n)^(p + 1)*Cosh[c + d*x])/(b*n*(p + 1)), x] - Dist[(d*e^m)/(b*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*Sinh[c + d*
x], x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IntegerQ[p] && EqQ[m - n + 1, 0] && LtQ[p, -1] && (IntegerQ[n
] || GtQ[e, 0])

Rule 5280

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*Sinh[(c_.) + (d_.)*(x_)], x_Symbol] :> Int[ExpandIntegrand[Sinh[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])

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

Mathematica [A]  time = 1.75491, size = 199, normalized size = 1.09 \[ \frac{e^{-a} \left (-b d x e^{-b \sqrt{c}} \text{ExpIntegralEi}\left (b \left (\sqrt{c}-\sqrt{c+d x}\right )\right )+b d x e^{b \sqrt{c}} \text{ExpIntegralEi}\left (-b \left (\sqrt{c+d x}+\sqrt{c}\right )\right )-2 \sqrt{c} e^{-b \sqrt{c+d x}}\right )+e^a \left (b d x e^{b \sqrt{c}} \text{ExpIntegralEi}\left (b \left (\sqrt{c+d x}-\sqrt{c}\right )\right )-b d x e^{-b \sqrt{c}} \text{ExpIntegralEi}\left (b \left (\sqrt{c+d x}+\sqrt{c}\right )\right )-2 \sqrt{c} e^{b \sqrt{c+d x}}\right )}{4 \sqrt{c} x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.021, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}}\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^2,x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.90036, size = 759, normalized size = 4.17 \begin{align*} -\frac{4 \, c \cosh \left (\sqrt{d x + c} b + a\right ) -{\left (\sqrt{b^{2} c} d x{\rm Ei}\left (\sqrt{d x + c} b - \sqrt{b^{2} c}\right ) - \sqrt{b^{2} c} d x{\rm Ei}\left (-\sqrt{d x + c} b + \sqrt{b^{2} c}\right )\right )} \cosh \left (a + \sqrt{b^{2} c}\right ) +{\left (\sqrt{b^{2} c} d x{\rm Ei}\left (\sqrt{d x + c} b + \sqrt{b^{2} c}\right ) - \sqrt{b^{2} c} d x{\rm Ei}\left (-\sqrt{d x + c} b - \sqrt{b^{2} c}\right )\right )} \cosh \left (-a + \sqrt{b^{2} c}\right ) -{\left (\sqrt{b^{2} c} d x{\rm Ei}\left (\sqrt{d x + c} b - \sqrt{b^{2} c}\right ) + \sqrt{b^{2} c} d x{\rm Ei}\left (-\sqrt{d x + c} b + \sqrt{b^{2} c}\right )\right )} \sinh \left (a + \sqrt{b^{2} c}\right ) -{\left (\sqrt{b^{2} c} d x{\rm Ei}\left (\sqrt{d x + c} b + \sqrt{b^{2} c}\right ) + \sqrt{b^{2} c} d x{\rm Ei}\left (-\sqrt{d x + c} b - \sqrt{b^{2} c}\right )\right )} \sinh \left (-a + \sqrt{b^{2} c}\right )}{4 \, c x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh{\left (a + b \sqrt{c + d x} \right )}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh \left (\sqrt{d x + c} b + a\right )}{x^{2}}\,{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^2,x, algorithm="giac")

[Out]

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