\(\int \frac {\sinh (a+b \sqrt [3]{c+d x})}{x} \, dx\) [101]

Optimal result

Integrand size = 18, antiderivative size = 235 \[ \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x} \, dx=\text {Chi}\left (b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right ) \sinh \left (a+b \sqrt [3]{c}\right )+\text {Chi}\left (\sqrt [3]{-1} b \sqrt [3]{c}+b \sqrt [3]{c+d x}\right ) \sinh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right )+\text {Chi}\left (-(-1)^{2/3} b \sqrt [3]{c}+b \sqrt [3]{c+d x}\right ) \sinh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right )-\cosh \left (a+b \sqrt [3]{c}\right ) \text {Shi}\left (b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )-\cosh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Shi}\left ((-1)^{2/3} b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )+\cosh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Shi}\left (\sqrt [3]{-1} b \sqrt [3]{c}+b \sqrt [3]{c+d x}\right ) \] Output:

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.

Time = 0.05 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.99 \[ \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x} \, dx=\frac {1}{2} \left (-\text {RootSum}\left [c-\text {$\#$1}^3\&,\cosh (a+b \text {$\#$1}) \text {Chi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )-\text {Chi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right ) \sinh (a+b \text {$\#$1})-\cosh (a+b \text {$\#$1}) \text {Shi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )+\sinh (a+b \text {$\#$1}) \text {Shi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )\&\right ]+\text {RootSum}\left [c-\text {$\#$1}^3\&,\cosh (a+b \text {$\#$1}) \text {Chi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )+\text {Chi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right ) \sinh (a+b \text {$\#$1})+\cosh (a+b \text {$\#$1}) \text {Shi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )+\sinh (a+b \text {$\#$1}) \text {Shi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )\&\right ]\right ) \] Input:

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

Output:

(-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]*CoshInteg 
ral[b*((c + d*x)^(1/3) - #1)] + CoshIntegral[b*((c + d*x)^(1/3) - #1)]*Sin 
h[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
 

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {5887, 25, 7267, 5815, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 5887

\(\displaystyle \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{d x}d(c+d x)\)

\(\Big \downarrow \) 25

\(\displaystyle -\int -\frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{d x}d(c+d x)\)

\(\Big \downarrow \) 7267

\(\displaystyle -3 \int -\frac {(c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{d x}d\sqrt [3]{c+d x}\)

\(\Big \downarrow \) 5815

\(\displaystyle -3 \int \left (\frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{3 \left (-c+\sqrt [3]{c}-d x\right )}+\frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{3 \left (-c-\sqrt [3]{-1} \sqrt [3]{c}-d x\right )}+\frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{3 \left (-c+(-1)^{2/3} \sqrt [3]{c}-d x\right )}\right )d\sqrt [3]{c+d x}\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5815

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sinh[(c_.) + (d_.)*(x_)], x_Sy 
mbol] :> Int[ExpandIntegrand[Sinh[c + d*x], x^m*(a + b*x^n)^p, x], x] /; Fr 
eeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IntegerQ[m] && IGtQ[n, 0] && (EqQ[n, 
2] || EqQ[p, -1])
 

rule 5887

Int[(x_)^(m_.)*((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(u_)^(n_)])^(p_.), x_Symbo 
l] :> Simp[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 7267

Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si 
mp[lst[[2]]*lst[[4]]   Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x 
] /;  !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
 

Maple [F]

\[\int \frac {\sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )}{x}d x\]

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 503 vs. \(2 (183) = 366\).

Time = 0.11 (sec) , antiderivative size = 503, normalized size of antiderivative = 2.14 \[ \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x} \, dx =\text {Too large to display} \] Input:

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

Output:

-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)*(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))
 

Sympy [F]

\[ \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x} \, dx=\int \frac {\sinh {\left (a + b \sqrt [3]{c + d x} \right )}}{x}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x} \, dx=\int { \frac {\sinh \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{x} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x} \, dx=\int { \frac {\sinh \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{x} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x} \, dx=\int \frac {\mathrm {sinh}\left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )}{x} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x} \, dx=\int \frac {\sinh \left (\left (d x +c \right )^{\frac {1}{3}} b +a \right )}{x}d x \] Input:

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

Output:

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