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

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

[Out]

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

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Rubi [A]  time = 0.518501, antiderivative size = 232, normalized size of antiderivative = 1., 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 = {5364, 5292, 3303, 3298, 3301} \[ \sinh \left (a+b \sqrt [3]{c}\right ) \text{Chi}\left (b \left (\sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )+\sinh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text{Chi}\left (b \left (\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{c+d x}\right )\right )+\sinh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text{Chi}\left (-b \left ((-1)^{2/3} \sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )-\cosh \left (a+b \sqrt [3]{c}\right ) \text{Shi}\left (b \left (\sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )-\cosh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text{Shi}\left (b \left ((-1)^{2/3} \sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )+\cosh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text{Shi}\left (b \left (\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{c+d x}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b*(c + d*x)^(1/3)]/x,x]

[Out]

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

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 5292

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

Mathematica [C]  time = 0.0679446, size = 233, normalized size = 1. \[ \frac{1}{2} \left (\text{RootSum}\left [c-\text{$\#$1}^3\& ,\sinh (\text{$\#$1} b+a) \text{Chi}\left (b \left (\sqrt [3]{c+d x}-\text{$\#$1}\right )\right )+\cosh (\text{$\#$1} b+a) \text{Chi}\left (b \left (\sqrt [3]{c+d x}-\text{$\#$1}\right )\right )+\sinh (\text{$\#$1} b+a) \text{Shi}\left (b \left (\sqrt [3]{c+d x}-\text{$\#$1}\right )\right )+\cosh (\text{$\#$1} b+a) \text{Shi}\left (b \left (\sqrt [3]{c+d x}-\text{$\#$1}\right )\right )\& \right ]-\text{RootSum}\left [c-\text{$\#$1}^3\& ,-\sinh (\text{$\#$1} b+a) \text{Chi}\left (b \left (\sqrt [3]{c+d x}-\text{$\#$1}\right )\right )+\cosh (\text{$\#$1} b+a) \text{Chi}\left (b \left (\sqrt [3]{c+d x}-\text{$\#$1}\right )\right )+\sinh (\text{$\#$1} b+a) \text{Shi}\left (b \left (\sqrt [3]{c+d x}-\text{$\#$1}\right )\right )-\cosh (\text{$\#$1} b+a) \text{Shi}\left (b \left (\sqrt [3]{c+d x}-\text{$\#$1}\right )\right )\& \right ]\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[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)] + Co
shIntegral[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)] & ])/2

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Maple [F]  time = 0.016, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x}\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,x)

[Out]

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

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Maxima [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}\,{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,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.42165, size = 1539, normalized size = 6.63 \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,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*
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*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)*(sqr
t(-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))

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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}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sinh(a + b*(c + d*x)**(1/3))/x, 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}\,{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,x, algorithm="giac")

[Out]

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