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

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

[Out]

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

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Rubi [A]  time = 0.724394, antiderivative size = 329, normalized size of antiderivative = 1., 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 = {5364, 5288, 5281, 3303, 3298, 3301} \[ \frac{b d \cosh \left (a+b \sqrt [3]{c}\right ) \text{Chi}\left (b \left (\sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}+\frac{(-1)^{2/3} b d \cosh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text{Chi}\left (-b \left ((-1)^{2/3} \sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}-\frac{\sqrt [3]{-1} b d \cosh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text{Chi}\left (b \left (\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}-\frac{b d \sinh \left (a+b \sqrt [3]{c}\right ) \text{Shi}\left (b \left (\sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}-\frac{(-1)^{2/3} b d \sinh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text{Shi}\left (b \left ((-1)^{2/3} \sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}-\frac{\sqrt [3]{-1} b d \sinh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text{Shi}\left (b \left (\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}-\frac{\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*d*Cosh[a + b*c^(1/3)]*CoshIntegral[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)]*CoshIntegral[-(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)]*CoshIntegral[b*((-1)^(1/3)*c^(1/3) + (c + d*x)^(1/3))])/(3*c^(2/3)) - Sinh[a +
b*(c + d*x)^(1/3)]/x - (b*d*Sinh[a + b*c^(1/3)]*SinhIntegral[b*(c^(1/3) - (c + d*x)^(1/3))])/(3*c^(2/3)) - ((-
1)^(2/3)*b*d*Sinh[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*Sinh[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 5364

Int[(x_)^(m_.)*((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(u_)^(n_)])^(p_.), x_Symbol] :> Dist[1/Coefficient[u, x, 1]^(
m + 1), Subst[Int[(x - Coefficient[u, x, 0])^m*(a + b*Sinh[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 5288

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sinh[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[(e^m*(a + b*x
^n)^(p + 1)*Sinh[c + d*x])/(b*n*(p + 1)), x] - Dist[(d*e^m)/(b*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*Cosh[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 5281

Int[Cosh[(c_.) + (d_.)*(x_)]*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[Cosh[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{\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x^2} \, dx &=d \operatorname{Subst}\left (\int \frac{\sinh \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 \sinh (a+b x)}{\left (c-x^3\right )^2} \, dx,x,\sqrt [3]{c+d x}\right )\\ &=-\frac{\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x}-(b d) \operatorname{Subst}\left (\int \frac{\cosh (a+b x)}{c-x^3} \, dx,x,\sqrt [3]{c+d x}\right )\\ &=-\frac{\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x}-(b d) \operatorname{Subst}\left (\int \left (\frac{\cosh (a+b x)}{3 c^{2/3} \left (\sqrt [3]{c}-x\right )}+\frac{\cosh (a+b x)}{3 c^{2/3} \left (\sqrt [3]{c}+\sqrt [3]{-1} x\right )}+\frac{\cosh (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{\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x}-\frac{(b d) \operatorname{Subst}\left (\int \frac{\cosh (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{\cosh (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{\cosh (a+b x)}{\sqrt [3]{c}-(-1)^{2/3} x} \, dx,x,\sqrt [3]{c+d x}\right )}{3 c^{2/3}}\\ &=-\frac{\sinh \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{\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 \cosh \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 \cosh \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{\left (b d \sinh \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 \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]{c}-(-1)^{2/3} x} \, dx,x,\sqrt [3]{c+d x}\right )}{3 c^{2/3}}+\frac{\left (i b d \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 )}{\sqrt [3]{c}+\sqrt [3]{-1} x} \, dx,x,\sqrt [3]{c+d x}\right )}{3 c^{2/3}}\\ &=\frac{b d \cosh \left (a+b \sqrt [3]{c}\right ) \text{Chi}\left (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{Chi}\left (\sqrt [3]{-1} 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{Chi}\left (-(-1)^{2/3} b \sqrt [3]{c}+b \sqrt [3]{c+d x}\right )}{3 c^{2/3}}-\frac{\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x}-\frac{b d \sinh \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 \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 )}{3 c^{2/3}}-\frac{\sqrt [3]{-1} b d \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 )}{3 c^{2/3}}\\ \end{align*}

Mathematica [C]  time = 1.83551, size = 210, normalized size = 0.64 \[ \frac{e^{-a} \left (b d x \text{RootSum}\left [c-\text{$\#$1}^3\& ,\frac{-\sinh (\text{$\#$1} b) \text{Chi}\left (b \left (\sqrt [3]{c+d x}-\text{$\#$1}\right )\right )+\cosh (\text{$\#$1} b) \text{Chi}\left (b \left (\sqrt [3]{c+d x}-\text{$\#$1}\right )\right )+\sinh (\text{$\#$1} b) \text{Shi}\left (b \left (\sqrt [3]{c+d x}-\text{$\#$1}\right )\right )-\cosh (\text{$\#$1} b) \text{Shi}\left (b \left (\sqrt [3]{c+d x}-\text{$\#$1}\right )\right )}{\text{$\#$1}^2}\& \right ]-3 e^{2 a+b \sqrt [3]{c+d x}}+3 e^{-b \sqrt [3]{c+d x}}\right )+b d x \text{RootSum}\left [c-\text{$\#$1}^3\& ,\frac{e^{\text{$\#$1} b+a} \text{ExpIntegralEi}\left (b \left (\sqrt [3]{c+d x}-\text{$\#$1}\right )\right )}{\text{$\#$1}^2}\& \right ]}{6 x} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[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/E^(b*(c + d*x
)^(1/3)) - 3*E^(2*a + 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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Maple [F]  time = 0.015, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}}\sinh \left ( a+b\sqrt [3]{dx+c} \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.50018, size = 2032, normalized size = 6.18 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(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)*(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
) + (-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*sinh((d*x + c)^(1/3)*b + a))/(c*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh{\left (a + b \sqrt [3]{c + d x} \right )}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh \left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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