\(\int \frac {\cosh (a+b \sqrt {c+d x})}{x^2} \, dx\) [63]

Optimal result

Integrand size = 18, antiderivative size = 182 \[ \int \frac {\cosh \left (a+b \sqrt {c+d x}\right )}{x^2} \, dx=-\frac {\cosh \left (a+b \sqrt {c+d x}\right )}{x}-\frac {b d \text {Chi}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right ) \sinh \left (a-b \sqrt {c}\right )}{2 \sqrt {c}}+\frac {b d \text {Chi}\left (b \left (\sqrt {c}-\sqrt {c+d x}\right )\right ) \sinh \left (a+b \sqrt {c}\right )}{2 \sqrt {c}}-\frac {b d \cosh \left (a+b \sqrt {c}\right ) \text {Shi}\left (b \left (\sqrt {c}-\sqrt {c+d x}\right )\right )}{2 \sqrt {c}}-\frac {b d \cosh \left (a-b \sqrt {c}\right ) \text {Shi}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )}{2 \sqrt {c}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5473, 5397, 5388, 3384, 3379, 3382} \[ \int \frac {\cosh \left (a+b \sqrt {c+d x}\right )}{x^2} \, dx=-\frac {b d \sinh \left (a-b \sqrt {c}\right ) \text {Chi}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )}{2 \sqrt {c}}+\frac {b d \sinh \left (a+b \sqrt {c}\right ) \text {Chi}\left (b \left (\sqrt {c}-\sqrt {c+d x}\right )\right )}{2 \sqrt {c}}-\frac {b d \cosh \left (a+b \sqrt {c}\right ) \text {Shi}\left (b \left (\sqrt {c}-\sqrt {c+d x}\right )\right )}{2 \sqrt {c}}-\frac {b d \cosh \left (a-b \sqrt {c}\right ) \text {Shi}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )}{2 \sqrt {c}}-\frac {\cosh \left (a+b \sqrt {c+d x}\right )}{x} \]

[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 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 5388

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 5397

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 5473

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*} \text {integral}& = d \text {Subst}\left (\int \frac {\cosh \left (a+b \sqrt {x}\right )}{(-c+x)^2} \, dx,x,c+d x\right ) \\ & = (2 d) \text {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) \text {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) \text {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) \text {Subst}\left (\int \frac {\sinh (a+b x)}{\sqrt {c}-x} \, dx,x,\sqrt {c+d x}\right )}{2 \sqrt {c}}-\frac {(b d) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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] (verified)

Time = 1.03 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.12 \[ \int \frac {\cosh \left (a+b \sqrt {c+d x}\right )}{x^2} \, dx=-\frac {e^{-a} \left (2 \sqrt {c} e^{-b \sqrt {c+d x}}+2 \sqrt {c} e^{2 a+b \sqrt {c+d x}}+b d e^{-b \sqrt {c}} x \operatorname {ExpIntegralEi}\left (b \left (\sqrt {c}-\sqrt {c+d x}\right )\right )-b d e^{2 a+b \sqrt {c}} x \operatorname {ExpIntegralEi}\left (b \left (-\sqrt {c}+\sqrt {c+d x}\right )\right )-b d e^{b \sqrt {c}} x \operatorname {ExpIntegralEi}\left (-b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )+b d e^{2 a-b \sqrt {c}} x \operatorname {ExpIntegralEi}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )\right )}{4 \sqrt {c} x} \]

[In]

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

[Out]

-1/4*((2*Sqrt[c])/E^(b*Sqrt[c + d*x]) + 2*Sqrt[c]*E^(2*a + b*Sqrt[c + d*x]) + (b*d*x*ExpIntegralEi[b*(Sqrt[c]
- Sqrt[c + d*x])])/E^(b*Sqrt[c]) - b*d*E^(2*a + b*Sqrt[c])*x*ExpIntegralEi[b*(-Sqrt[c] + Sqrt[c + d*x])] - b*d
*E^(b*Sqrt[c])*x*ExpIntegralEi[-(b*(Sqrt[c] + Sqrt[c + d*x]))] + b*d*E^(2*a - b*Sqrt[c])*x*ExpIntegralEi[b*(Sq
rt[c] + Sqrt[c + d*x])])/(Sqrt[c]*E^a*x)

Maple [F]

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

[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)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 317 vs. \(2 (142) = 284\).

Time = 0.26 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.74 \[ \int \frac {\cosh \left (a+b \sqrt {c+d x}\right )}{x^2} \, dx=-\frac {4 \, c \cosh \left (\sqrt {d x + c} b + a\right ) - {\left (\sqrt {b^{2} c} d x {\rm Ei}\left (\sqrt {d x + c} b - \sqrt {b^{2} c}\right ) - \sqrt {b^{2} c} d x {\rm Ei}\left (-\sqrt {d x + c} b + \sqrt {b^{2} c}\right )\right )} \cosh \left (a + \sqrt {b^{2} c}\right ) + {\left (\sqrt {b^{2} c} d x {\rm Ei}\left (\sqrt {d x + c} b + \sqrt {b^{2} c}\right ) - \sqrt {b^{2} c} d x {\rm Ei}\left (-\sqrt {d x + c} b - \sqrt {b^{2} c}\right )\right )} \cosh \left (-a + \sqrt {b^{2} c}\right ) - {\left (\sqrt {b^{2} c} d x {\rm Ei}\left (\sqrt {d x + c} b - \sqrt {b^{2} c}\right ) + \sqrt {b^{2} c} d x {\rm Ei}\left (-\sqrt {d x + c} b + \sqrt {b^{2} c}\right )\right )} \sinh \left (a + \sqrt {b^{2} c}\right ) - {\left (\sqrt {b^{2} c} d x {\rm Ei}\left (\sqrt {d x + c} b + \sqrt {b^{2} c}\right ) + \sqrt {b^{2} c} d x {\rm Ei}\left (-\sqrt {d x + c} b - \sqrt {b^{2} c}\right )\right )} \sinh \left (-a + \sqrt {b^{2} c}\right )}{4 \, c x} \]

[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)

Sympy [F]

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

[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)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[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)

Mupad [F(-1)]

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

[In]

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

[Out]

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