3.67 \(\int \frac {\cosh (a+b \sqrt [3]{c+d x})}{x} \, dx\)

Optimal. Leaf size=232 \[ \cosh \left (a+b \sqrt [3]{c}\right ) \text {Chi}\left (b \left (\sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )+\cosh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Chi}\left (-b \left ((-1)^{2/3} \sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )+\cosh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Chi}\left (b \left (\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{c+d x}\right )\right )-\sinh \left (a+b \sqrt [3]{c}\right ) \text {Shi}\left (b \left (\sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )-\sinh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Shi}\left (b \left ((-1)^{2/3} \sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )+\sinh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Shi}\left (b \left (\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{c+d x}\right )\right ) \]

[Out]

Chi(b*(c^(1/3)-(d*x+c)^(1/3)))*cosh(a+b*c^(1/3))+Chi(b*((-1)^(1/3)*c^(1/3)+(d*x+c)^(1/3)))*cosh(a-(-1)^(1/3)*b
*c^(1/3))+Chi(-b*((-1)^(2/3)*c^(1/3)-(d*x+c)^(1/3)))*cosh(a+(-1)^(2/3)*b*c^(1/3))-Shi(b*(c^(1/3)-(d*x+c)^(1/3)
))*sinh(a+b*c^(1/3))+Shi(b*((-1)^(1/3)*c^(1/3)+(d*x+c)^(1/3)))*sinh(a-(-1)^(1/3)*b*c^(1/3))-Shi(b*((-1)^(2/3)*
c^(1/3)-(d*x+c)^(1/3)))*sinh(a+(-1)^(2/3)*b*c^(1/3))

________________________________________________________________________________________

Rubi [A]  time = 0.52, antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 13, 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} \[ \cosh \left (a+b \sqrt [3]{c}\right ) \text {Chi}\left (b \left (\sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )+\cosh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Chi}\left (-b \left ((-1)^{2/3} \sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )+\cosh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Chi}\left (b \left (\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{c+d x}\right )\right )-\sinh \left (a+b \sqrt [3]{c}\right ) \text {Shi}\left (b \left (\sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )-\sinh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Shi}\left (b \left ((-1)^{2/3} \sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )+\sinh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Shi}\left (b \left (\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{c+d x}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*(c + d*x)^(1/3)]/x,x]

[Out]

Cosh[a + b*c^(1/3)]*CoshIntegral[b*(c^(1/3) - (c + d*x)^(1/3))] + Cosh[a + (-1)^(2/3)*b*c^(1/3)]*CoshIntegral[
-(b*((-1)^(2/3)*c^(1/3) - (c + d*x)^(1/3)))] + Cosh[a - (-1)^(1/3)*b*c^(1/3)]*CoshIntegral[b*((-1)^(1/3)*c^(1/
3) + (c + d*x)^(1/3))] - Sinh[a + b*c^(1/3)]*SinhIntegral[b*(c^(1/3) - (c + d*x)^(1/3))] - Sinh[a + (-1)^(2/3)
*b*c^(1/3)]*SinhIntegral[b*((-1)^(2/3)*c^(1/3) - (c + d*x)^(1/3))] + Sinh[a - (-1)^(1/3)*b*c^(1/3)]*SinhIntegr
al[b*((-1)^(1/3)*c^(1/3) + (c + d*x)^(1/3))]

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 [3]{c+d x}\right )}{x} \, dx &=\operatorname {Subst}\left (\int \frac {\cosh \left (a+b \sqrt [3]{x}\right )}{-c+x} \, dx,x,c+d x\right )\\ &=3 \operatorname {Subst}\left (\int \frac {x^2 \cosh (a+b x)}{-c+x^3} \, dx,x,\sqrt [3]{c+d x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (-\frac {\cosh (a+b x)}{3 \left (\sqrt [3]{c}-x\right )}-\frac {\cosh (a+b x)}{3 \left (-\sqrt [3]{-1} \sqrt [3]{c}-x\right )}-\frac {\cosh (a+b x)}{3 \left ((-1)^{2/3} \sqrt [3]{c}-x\right )}\right ) \, dx,x,\sqrt [3]{c+d x}\right )\\ &=-\operatorname {Subst}\left (\int \frac {\cosh (a+b x)}{\sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )-\operatorname {Subst}\left (\int \frac {\cosh (a+b x)}{-\sqrt [3]{-1} \sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )-\operatorname {Subst}\left (\int \frac {\cosh (a+b x)}{(-1)^{2/3} \sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )\\ &=-\left (\cosh \left (a+b \sqrt [3]{c}\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (b \sqrt [3]{c}-b x\right )}{\sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )\right )-\cosh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \operatorname {Subst}\left (\int \frac {\cos \left ((-1)^{5/6} b \sqrt [3]{c}+i b x\right )}{-\sqrt [3]{-1} \sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )-\cosh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\sqrt [6]{-1} b \sqrt [3]{c}+i b x\right )}{(-1)^{2/3} \sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )+\sinh \left (a+b \sqrt [3]{c}\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (b \sqrt [3]{c}-b x\right )}{\sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )+\left (i \sinh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left ((-1)^{5/6} b \sqrt [3]{c}+i b x\right )}{-\sqrt [3]{-1} \sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )+\left (i \sinh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\sqrt [6]{-1} b \sqrt [3]{c}+i b x\right )}{(-1)^{2/3} \sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )\\ &=\cosh \left (a+b \sqrt [3]{c}\right ) \text {Chi}\left (b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )+\cosh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Chi}\left (\sqrt [3]{-1} b \sqrt [3]{c}+b \sqrt [3]{c+d x}\right )+\cosh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Chi}\left (-(-1)^{2/3} b \sqrt [3]{c}+b \sqrt [3]{c+d x}\right )-\sinh \left (a+b \sqrt [3]{c}\right ) \text {Shi}\left (b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )-\sinh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Shi}\left ((-1)^{2/3} b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )+\sinh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Shi}\left (\sqrt [3]{-1} b \sqrt [3]{c}+b \sqrt [3]{c+d x}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.07, size = 231, normalized size = 1.00 \[ \frac {1}{2} \left (\text {RootSum}\left [c-\text {$\#$1}^3\& ,-\sinh (\text {$\#$1} b+a) \text {Chi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )+\cosh (\text {$\#$1} b+a) \text {Chi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )+\sinh (\text {$\#$1} b+a) \text {Shi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )-\cosh (\text {$\#$1} b+a) \text {Shi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )\& \right ]+\text {RootSum}\left [c-\text {$\#$1}^3\& ,\sinh (\text {$\#$1} b+a) \text {Chi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )+\cosh (\text {$\#$1} b+a) \text {Chi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )+\sinh (\text {$\#$1} b+a) \text {Shi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )+\cosh (\text {$\#$1} b+a) \text {Shi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )\& \right ]\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + b*(c + d*x)^(1/3)]/x,x]

[Out]

(RootSum[c - #1^3 & , Cosh[a + b*#1]*CoshIntegral[b*((c + d*x)^(1/3) - #1)] - CoshIntegral[b*((c + d*x)^(1/3)
- #1)]*Sinh[a + b*#1] - Cosh[a + b*#1]*SinhIntegral[b*((c + d*x)^(1/3) - #1)] + Sinh[a + b*#1]*SinhIntegral[b*
((c + d*x)^(1/3) - #1)] & ] + RootSum[c - #1^3 & , Cosh[a + b*#1]*CoshIntegral[b*((c + d*x)^(1/3) - #1)] + Cos
hIntegral[b*((c + d*x)^(1/3) - #1)]*Sinh[a + b*#1] + Cosh[a + b*#1]*SinhIntegral[b*((c + d*x)^(1/3) - #1)] + S
inh[a + b*#1]*SinhIntegral[b*((c + d*x)^(1/3) - #1)] & ])/2

________________________________________________________________________________________

fricas [B]  time = 0.77, size = 503, normalized size = 2.17 \[ \frac {1}{2} \, {\rm Ei}\left (-{\left (d x + c\right )}^{\frac {1}{3}} b - \frac {1}{2} \, \left (b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )}\right ) \cosh \left (\frac {1}{2} \, \left (b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} - a\right ) + \frac {1}{2} \, {\rm Ei}\left ({\left (d x + c\right )}^{\frac {1}{3}} b - \frac {1}{2} \, \left (-b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )}\right ) \cosh \left (\frac {1}{2} \, \left (-b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} + a\right ) + \frac {1}{2} \, {\rm Ei}\left (-{\left (d x + c\right )}^{\frac {1}{3}} b + \frac {1}{2} \, \left (b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )}\right ) \cosh \left (\frac {1}{2} \, \left (b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} + a\right ) + \frac {1}{2} \, {\rm Ei}\left ({\left (d x + c\right )}^{\frac {1}{3}} b + \frac {1}{2} \, \left (-b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )}\right ) \cosh \left (\frac {1}{2} \, \left (-b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} - a\right ) + \frac {1}{2} \, {\rm Ei}\left (-{\left (d x + c\right )}^{\frac {1}{3}} b + \left (b^{3} c\right )^{\frac {1}{3}}\right ) \cosh \left (a + \left (b^{3} c\right )^{\frac {1}{3}}\right ) + \frac {1}{2} \, {\rm Ei}\left ({\left (d x + c\right )}^{\frac {1}{3}} b + \left (-b^{3} c\right )^{\frac {1}{3}}\right ) \cosh \left (-a + \left (-b^{3} c\right )^{\frac {1}{3}}\right ) + \frac {1}{2} \, {\rm Ei}\left (-{\left (d x + c\right )}^{\frac {1}{3}} b - \frac {1}{2} \, \left (b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )}\right ) \sinh \left (\frac {1}{2} \, \left (b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} - a\right ) + \frac {1}{2} \, {\rm Ei}\left ({\left (d x + c\right )}^{\frac {1}{3}} b - \frac {1}{2} \, \left (-b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )}\right ) \sinh \left (\frac {1}{2} \, \left (-b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} + a\right ) - \frac {1}{2} \, {\rm Ei}\left (-{\left (d x + c\right )}^{\frac {1}{3}} b + \frac {1}{2} \, \left (b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )}\right ) \sinh \left (\frac {1}{2} \, \left (b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} + a\right ) - \frac {1}{2} \, {\rm Ei}\left ({\left (d x + c\right )}^{\frac {1}{3}} b + \frac {1}{2} \, \left (-b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )}\right ) \sinh \left (\frac {1}{2} \, \left (-b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} - a\right ) - \frac {1}{2} \, {\rm Ei}\left (-{\left (d x + c\right )}^{\frac {1}{3}} b + \left (b^{3} c\right )^{\frac {1}{3}}\right ) \sinh \left (a + \left (b^{3} c\right )^{\frac {1}{3}}\right ) - \frac {1}{2} \, {\rm Ei}\left ({\left (d x + c\right )}^{\frac {1}{3}} b + \left (-b^{3} c\right )^{\frac {1}{3}}\right ) \sinh \left (-a + \left (-b^{3} c\right )^{\frac {1}{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(a+b*(d*x+c)^(1/3))/x,x, algorithm="fricas")

[Out]

1/2*Ei(-(d*x + c)^(1/3)*b - 1/2*(b^3*c)^(1/3)*(sqrt(-3) + 1))*cosh(1/2*(b^3*c)^(1/3)*(sqrt(-3) + 1) - a) + 1/2
*Ei((d*x + c)^(1/3)*b - 1/2*(-b^3*c)^(1/3)*(sqrt(-3) + 1))*cosh(1/2*(-b^3*c)^(1/3)*(sqrt(-3) + 1) + a) + 1/2*E
i(-(d*x + c)^(1/3)*b + 1/2*(b^3*c)^(1/3)*(sqrt(-3) - 1))*cosh(1/2*(b^3*c)^(1/3)*(sqrt(-3) - 1) + a) + 1/2*Ei((
d*x + c)^(1/3)*b + 1/2*(-b^3*c)^(1/3)*(sqrt(-3) - 1))*cosh(1/2*(-b^3*c)^(1/3)*(sqrt(-3) - 1) - a) + 1/2*Ei(-(d
*x + c)^(1/3)*b + (b^3*c)^(1/3))*cosh(a + (b^3*c)^(1/3)) + 1/2*Ei((d*x + c)^(1/3)*b + (-b^3*c)^(1/3))*cosh(-a
+ (-b^3*c)^(1/3)) + 1/2*Ei(-(d*x + c)^(1/3)*b - 1/2*(b^3*c)^(1/3)*(sqrt(-3) + 1))*sinh(1/2*(b^3*c)^(1/3)*(sqrt
(-3) + 1) - a) + 1/2*Ei((d*x + c)^(1/3)*b - 1/2*(-b^3*c)^(1/3)*(sqrt(-3) + 1))*sinh(1/2*(-b^3*c)^(1/3)*(sqrt(-
3) + 1) + a) - 1/2*Ei(-(d*x + c)^(1/3)*b + 1/2*(b^3*c)^(1/3)*(sqrt(-3) - 1))*sinh(1/2*(b^3*c)^(1/3)*(sqrt(-3)
- 1) + a) - 1/2*Ei((d*x + c)^(1/3)*b + 1/2*(-b^3*c)^(1/3)*(sqrt(-3) - 1))*sinh(1/2*(-b^3*c)^(1/3)*(sqrt(-3) -
1) - a) - 1/2*Ei(-(d*x + c)^(1/3)*b + (b^3*c)^(1/3))*sinh(a + (b^3*c)^(1/3)) - 1/2*Ei((d*x + c)^(1/3)*b + (-b^
3*c)^(1/3))*sinh(-a + (-b^3*c)^(1/3))

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(a+b*(d*x+c)^(1/3))/x,x, algorithm="giac")

[Out]

integrate(cosh((d*x + c)^(1/3)*b + a)/x, x)

________________________________________________________________________________________

maple [F]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a+b*(d*x+c)^(1/3))/x,x)

[Out]

int(cosh(a+b*(d*x+c)^(1/3))/x,x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(a+b*(d*x+c)^(1/3))/x,x, algorithm="maxima")

[Out]

integrate(cosh((d*x + c)^(1/3)*b + a)/x, x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {cosh}\left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*(c + d*x)^(1/3))/x,x)

[Out]

int(cosh(a + b*(c + d*x)^(1/3))/x, x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh {\left (a + b \sqrt [3]{c + d x} \right )}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(a+b*(d*x+c)**(1/3))/x,x)

[Out]

Integral(cosh(a + b*(c + d*x)**(1/3))/x, x)

________________________________________________________________________________________