3.68 \(\int \frac {\cosh (a+b \sqrt [3]{c+d x})}{x^2} \, dx\)

Optimal. Leaf size=329 \[ \frac {b d \sinh \left (a+b \sqrt [3]{c}\right ) \text {Chi}\left (b \left (\sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}-\frac {\sqrt [3]{-1} b d \sinh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Chi}\left (b \left (\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}+\frac {(-1)^{2/3} b d \sinh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Chi}\left (-b \left ((-1)^{2/3} \sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}-\frac {b d \cosh \left (a+b \sqrt [3]{c}\right ) \text {Shi}\left (b \left (\sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}-\frac {(-1)^{2/3} b d \cosh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Shi}\left (b \left ((-1)^{2/3} \sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}-\frac {\sqrt [3]{-1} b d \cosh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Shi}\left (b \left (\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}-\frac {\cosh \left (a+b \sqrt [3]{c+d x}\right )}{x} \]

[Out]

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

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Rubi [A]  time = 0.69, antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps used = 14, 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} \[ \frac {b d \sinh \left (a+b \sqrt [3]{c}\right ) \text {Chi}\left (b \left (\sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}-\frac {\sqrt [3]{-1} b d \sinh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Chi}\left (b \left (\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}+\frac {(-1)^{2/3} b d \sinh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Chi}\left (-b \left ((-1)^{2/3} \sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}-\frac {b d \cosh \left (a+b \sqrt [3]{c}\right ) \text {Shi}\left (b \left (\sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}-\frac {(-1)^{2/3} b d \cosh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Shi}\left (b \left ((-1)^{2/3} \sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}-\frac {\sqrt [3]{-1} b d \cosh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Shi}\left (b \left (\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}-\frac {\cosh \left (a+b \sqrt [3]{c+d x}\right )}{x} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*(c + d*x)^(1/3)]/x^2,x]

[Out]

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

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

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Mathematica [C]  time = 0.55, size = 211, normalized size = 0.64 \[ \frac {e^{-a} \left (-b d x \text {RootSum}\left [c-\text {$\#$1}^3\& ,\frac {-\sinh (\text {$\#$1} b) \text {Chi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )+\cosh (\text {$\#$1} b) \text {Chi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )+\sinh (\text {$\#$1} b) \text {Shi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )-\cosh (\text {$\#$1} b) \text {Shi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )}{\text {$\#$1}^2}\& \right ]-3 e^{-b \sqrt [3]{c+d x}} \left (e^{2 \left (a+b \sqrt [3]{c+d x}\right )}+1\right )\right )+b d x \text {RootSum}\left [c-\text {$\#$1}^3\& ,\frac {e^{\text {$\#$1} b+a} \text {Ei}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )}{\text {$\#$1}^2}\& \right ]}{6 x} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + b*(c + d*x)^(1/3)]/x^2,x]

[Out]

(b*d*x*RootSum[c - #1^3 & , (E^(a + b*#1)*ExpIntegralEi[b*((c + d*x)^(1/3) - #1)])/#1^2 & ] + ((-3*(1 + E^(2*(
a + b*(c + d*x)^(1/3)))))/E^(b*(c + d*x)^(1/3)) - b*d*x*RootSum[c - #1^3 & , (Cosh[b*#1]*CoshIntegral[b*((c +
d*x)^(1/3) - #1)] - CoshIntegral[b*((c + d*x)^(1/3) - #1)]*Sinh[b*#1] - Cosh[b*#1]*SinhIntegral[b*((c + d*x)^(
1/3) - #1)] + Sinh[b*#1]*SinhIntegral[b*((c + d*x)^(1/3) - #1)])/#1^2 & ])/E^a)/(6*x)

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fricas [B]  time = 0.52, size = 706, normalized size = 2.15 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(a+b*(d*x+c)^(1/3))/x^2,x, algorithm="fricas")

[Out]

-1/12*(2*(b^3*c)^(1/3)*d*x*Ei(-(d*x + c)^(1/3)*b + (b^3*c)^(1/3))*cosh(a + (b^3*c)^(1/3)) + 2*(-b^3*c)^(1/3)*d
*x*Ei((d*x + c)^(1/3)*b + (-b^3*c)^(1/3))*cosh(-a + (-b^3*c)^(1/3)) - 2*(b^3*c)^(1/3)*d*x*Ei(-(d*x + c)^(1/3)*
b + (b^3*c)^(1/3))*sinh(a + (b^3*c)^(1/3)) - 2*(-b^3*c)^(1/3)*d*x*Ei((d*x + c)^(1/3)*b + (-b^3*c)^(1/3))*sinh(
-a + (-b^3*c)^(1/3)) - (b^3*c)^(1/3)*(sqrt(-3)*d*x + d*x)*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) - (-b^3*c)^(1/3)*(sqrt(-3)*d*x + d*x)*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) + (b^3*c)^(1/3)*(sqrt(-3)*d*x -
d*x)*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) + (
-b^3*c)^(1/3)*(sqrt(-3)*d*x - d*x)*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) - (b^3*c)^(1/3)*(sqrt(-3)*d*x + d*x)*Ei(-(d*x + c)^(1/3)*b - 1/2*(b^3*c)^(1/3)*(sqr
t(-3) + 1))*sinh(1/2*(b^3*c)^(1/3)*(sqrt(-3) + 1) - a) - (-b^3*c)^(1/3)*(sqrt(-3)*d*x + d*x)*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) - (b^3*c)^(1/3)*(sqrt(-3)
*d*x - d*x)*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) - (-b^3*c)^(1/3)*(sqrt(-3)*d*x - d*x)*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) + 12*c*cosh((d*x + c)^(1/3)*b + a))/(c*x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(a+b*(d*x+c)^(1/3))/x^2,x, algorithm="giac")

[Out]

integrate(cosh((d*x + c)^(1/3)*b + a)/x^2, x)

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maple [F]  time = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a+b*(d*x+c)^(1/3))/x^2,x)

[Out]

int(cosh(a+b*(d*x+c)^(1/3))/x^2,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(a+b*(d*x+c)^(1/3))/x^2,x, algorithm="maxima")

[Out]

integrate(cosh((d*x + c)^(1/3)*b + a)/x^2, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {cosh}\left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*(c + d*x)^(1/3))/x^2,x)

[Out]

int(cosh(a + b*(c + d*x)^(1/3))/x^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh {\left (a + b \sqrt [3]{c + d x} \right )}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(a+b*(d*x+c)**(1/3))/x**2,x)

[Out]

Integral(cosh(a + b*(c + d*x)**(1/3))/x**2, x)

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