\(\int \frac {\sinh (a+b \sqrt [3]{c+d x})}{x^2} \, dx\) [102]

Optimal result

Integrand size = 18, antiderivative size = 332 \[ \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x^2} \, dx=\frac {b d \cosh \left (a+b \sqrt [3]{c}\right ) \text {Chi}\left (b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )}{3 c^{2/3}}-\frac {\sqrt [3]{-1} b d \cosh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Chi}\left (\sqrt [3]{-1} b \sqrt [3]{c}+b \sqrt [3]{c+d x}\right )}{3 c^{2/3}}+\frac {(-1)^{2/3} b d \cosh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Chi}\left (-(-1)^{2/3} b \sqrt [3]{c}+b \sqrt [3]{c+d x}\right )}{3 c^{2/3}}-\frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x}-\frac {b d \sinh \left (a+b \sqrt [3]{c}\right ) \text {Shi}\left (b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )}{3 c^{2/3}}-\frac {(-1)^{2/3} b d \sinh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Shi}\left ((-1)^{2/3} b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )}{3 c^{2/3}}-\frac {\sqrt [3]{-1} b d \sinh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Shi}\left (\sqrt [3]{-1} b \sqrt [3]{c}+b \sqrt [3]{c+d x}\right )}{3 c^{2/3}} \] Output:

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

Mathematica [C] (verified)

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

Time = 2.58 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.63 \[ \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x^2} \, dx=\frac {e^{-a} \left (3 e^{-b \sqrt [3]{c+d x}}-3 e^{2 a+b \sqrt [3]{c+d x}}+b d x \text {RootSum}\left [c-\text {$\#$1}^3\&,\frac {e^{2 a+b \text {$\#$1}} \operatorname {ExpIntegralEi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )}{\text {$\#$1}^2}\&\right ]+b d x \text {RootSum}\left [c-\text {$\#$1}^3\&,\frac {\cosh (b \text {$\#$1}) \text {Chi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )-\text {Chi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right ) \sinh (b \text {$\#$1})-\cosh (b \text {$\#$1}) \text {Shi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )+\sinh (b \text {$\#$1}) \text {Shi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )}{\text {$\#$1}^2}\&\right ]\right )}{6 x} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {5887, 7267, 5811, 5804, 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.

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

\(\Big \downarrow \) 5887

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

\(\Big \downarrow \) 7267

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

\(\Big \downarrow \) 5811

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

\(\Big \downarrow \) 5804

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

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

rule 5804

Int[Cosh[(c_.) + (d_.)*(x_)]*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> In 
t[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 5811

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] - Simp[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 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]

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

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 704 vs. \(2 (246) = 492\).

Time = 0.12 (sec) , antiderivative size = 704, normalized size of antiderivative = 2.12 \[ \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x^2} \, dx=\text {Too large to display} \] Input:

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

Output:

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)*(sqr 
t(-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)*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) - (-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)*(s 
qrt(-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)*(sqr 
t(-3) - 1))*sinh(1/2*(-b^3*c)^(1/3)*(sqrt(-3) - 1) - a) - 12*c*sinh((d*...
 

Sympy [F]

\[ \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x^2} \, dx=\int \frac {\sinh {\left (a + b \sqrt [3]{c + d x} \right )}}{x^{2}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x^2} \, dx=\int { \frac {\sinh \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{x^{2}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x^2} \, dx=\int { \frac {\sinh \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{x^{2}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x^2} \, dx=\int \frac {\mathrm {sinh}\left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )}{x^2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x^2} \, dx=\int \frac {\sinh \left (\left (d x +c \right )^{\frac {1}{3}} b +a \right )}{x^{2}}d x \] Input:

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

Output:

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