\(\int x \sinh (a+b \sqrt [3]{c+d x}) \, dx\) [99]

Optimal result

Integrand size = 16, antiderivative size = 261 \[ \int x \sinh \left (a+b \sqrt [3]{c+d x}\right ) \, dx=-\frac {6 c \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac {360 \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^2}-\frac {3 c (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac {60 (c+d x) \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac {3 (c+d x)^{5/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}-\frac {360 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^2}+\frac {6 c \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac {180 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^2}-\frac {15 (c+d x)^{4/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2} \] Output:

-6*c*cosh(a+b*(d*x+c)^(1/3))/b^3/d^2+360*(d*x+c)^(1/3)*cosh(a+b*(d*x+c)^(1 
/3))/b^5/d^2-3*c*(d*x+c)^(2/3)*cosh(a+b*(d*x+c)^(1/3))/b/d^2+60*(d*x+c)*co 
sh(a+b*(d*x+c)^(1/3))/b^3/d^2+3*(d*x+c)^(5/3)*cosh(a+b*(d*x+c)^(1/3))/b/d^ 
2-360*sinh(a+b*(d*x+c)^(1/3))/b^6/d^2+6*c*(d*x+c)^(1/3)*sinh(a+b*(d*x+c)^( 
1/3))/b^2/d^2-180*(d*x+c)^(2/3)*sinh(a+b*(d*x+c)^(1/3))/b^4/d^2-15*(d*x+c) 
^(4/3)*sinh(a+b*(d*x+c)^(1/3))/b^2/d^2
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.45 \[ \int x \sinh \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\frac {3 b \left (120 \sqrt [3]{c+d x}+b^4 d x (c+d x)^{2/3}+2 b^2 (9 c+10 d x)\right ) \cosh \left (a+b \sqrt [3]{c+d x}\right )-3 \left (120+60 b^2 (c+d x)^{2/3}+b^4 \sqrt [3]{c+d x} (3 c+5 d x)\right ) \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^2} \] Input:

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

Output:

(3*b*(120*(c + d*x)^(1/3) + b^4*d*x*(c + d*x)^(2/3) + 2*b^2*(9*c + 10*d*x) 
)*Cosh[a + b*(c + d*x)^(1/3)] - 3*(120 + 60*b^2*(c + d*x)^(2/3) + b^4*(c + 
 d*x)^(1/3)*(3*c + 5*d*x))*Sinh[a + b*(c + d*x)^(1/3)])/(b^6*d^2)
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.91, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5887, 25, 7267, 5809, 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.

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

\(\Big \downarrow \) 5887

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 7267

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

\(\Big \downarrow \) 5809

\(\displaystyle -\frac {3 \int \left (c (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )-(c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )\right )d\sqrt [3]{c+d x}}{d^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 \left (\frac {120 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^6}-\frac {120 \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^5}+\frac {60 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^4}-\frac {20 (c+d x) \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3}+\frac {2 c \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3}+\frac {5 (c+d x)^{4/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2}-\frac {2 c \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2}-\frac {(c+d x)^{5/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b}+\frac {c (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d^2}\)

Input:

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

Output:

(-3*((2*c*Cosh[a + b*(c + d*x)^(1/3)])/b^3 - (120*(c + d*x)^(1/3)*Cosh[a + 
 b*(c + d*x)^(1/3)])/b^5 + (c*(c + d*x)^(2/3)*Cosh[a + b*(c + d*x)^(1/3)]) 
/b - (20*(c + d*x)*Cosh[a + b*(c + d*x)^(1/3)])/b^3 - ((c + d*x)^(5/3)*Cos 
h[a + b*(c + d*x)^(1/3)])/b + (120*Sinh[a + b*(c + d*x)^(1/3)])/b^6 - (2*c 
*(c + d*x)^(1/3)*Sinh[a + b*(c + d*x)^(1/3)])/b^2 + (60*(c + d*x)^(2/3)*Si 
nh[a + b*(c + d*x)^(1/3)])/b^4 + (5*(c + d*x)^(4/3)*Sinh[a + b*(c + d*x)^( 
1/3)])/b^2))/d^2
 

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 5809

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*Sinh[(c_.) + (d_.)*(x 
_)], x_Symbol] :> Int[ExpandIntegrand[Sinh[c + d*x], (e*x)^m*(a + b*x^n)^p, 
 x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(658\) vs. \(2(231)=462\).

Time = 0.38 (sec) , antiderivative size = 659, normalized size of antiderivative = 2.52

method	result	size
derivativedivides	\(\frac {-\frac {3 a^{5} \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )}{b^{3}}+\frac {15 a^{4} \left (\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-\sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )}{b^{3}}-\frac {30 a^{3} \left (\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2} \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-2 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+2 \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )}{b^{3}}+\frac {30 a^{2} \left (\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{3} \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-3 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2} \sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+6 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-6 \sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )}{b^{3}}-\frac {15 a \left (\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{4} \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-4 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{3} \sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+12 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2} \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-24 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+24 \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )}{b^{3}}+\frac {3 \left (\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{5} \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-5 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{4} \sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+20 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{3} \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-60 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2} \sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+120 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-120 \sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )}{b^{3}}-3 c \,a^{2} \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+6 c a \left (\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-\sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )-3 c \left (\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2} \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-2 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+2 \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )}{b^{3} d^{2}}\)	\(659\)
default	\(\frac {-\frac {3 a^{5} \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )}{b^{3}}+\frac {15 a^{4} \left (\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-\sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )}{b^{3}}-\frac {30 a^{3} \left (\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2} \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-2 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+2 \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )}{b^{3}}+\frac {30 a^{2} \left (\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{3} \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-3 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2} \sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+6 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-6 \sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )}{b^{3}}-\frac {15 a \left (\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{4} \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-4 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{3} \sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+12 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2} \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-24 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+24 \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )}{b^{3}}+\frac {3 \left (\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{5} \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-5 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{4} \sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+20 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{3} \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-60 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2} \sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+120 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-120 \sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )}{b^{3}}-3 c \,a^{2} \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+6 c a \left (\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-\sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )-3 c \left (\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2} \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-2 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+2 \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )}{b^{3} d^{2}}\)	\(659\)
parts	\(\text {Expression too large to display}\)	\(1211\)

Input:

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

Output:

3/d^2/b^3*(-1/b^3*a^5*cosh(a+b*(d*x+c)^(1/3))+5/b^3*a^4*((a+b*(d*x+c)^(1/3 
))*cosh(a+b*(d*x+c)^(1/3))-sinh(a+b*(d*x+c)^(1/3)))-10/b^3*a^3*((a+b*(d*x+ 
c)^(1/3))^2*cosh(a+b*(d*x+c)^(1/3))-2*(a+b*(d*x+c)^(1/3))*sinh(a+b*(d*x+c) 
^(1/3))+2*cosh(a+b*(d*x+c)^(1/3)))+10/b^3*a^2*((a+b*(d*x+c)^(1/3))^3*cosh( 
a+b*(d*x+c)^(1/3))-3*(a+b*(d*x+c)^(1/3))^2*sinh(a+b*(d*x+c)^(1/3))+6*(a+b* 
(d*x+c)^(1/3))*cosh(a+b*(d*x+c)^(1/3))-6*sinh(a+b*(d*x+c)^(1/3)))-5/b^3*a* 
((a+b*(d*x+c)^(1/3))^4*cosh(a+b*(d*x+c)^(1/3))-4*(a+b*(d*x+c)^(1/3))^3*sin 
h(a+b*(d*x+c)^(1/3))+12*(a+b*(d*x+c)^(1/3))^2*cosh(a+b*(d*x+c)^(1/3))-24*( 
a+b*(d*x+c)^(1/3))*sinh(a+b*(d*x+c)^(1/3))+24*cosh(a+b*(d*x+c)^(1/3)))+1/b 
^3*((a+b*(d*x+c)^(1/3))^5*cosh(a+b*(d*x+c)^(1/3))-5*(a+b*(d*x+c)^(1/3))^4* 
sinh(a+b*(d*x+c)^(1/3))+20*(a+b*(d*x+c)^(1/3))^3*cosh(a+b*(d*x+c)^(1/3))-6 
0*(a+b*(d*x+c)^(1/3))^2*sinh(a+b*(d*x+c)^(1/3))+120*(a+b*(d*x+c)^(1/3))*co 
sh(a+b*(d*x+c)^(1/3))-120*sinh(a+b*(d*x+c)^(1/3)))-c*a^2*cosh(a+b*(d*x+c)^ 
(1/3))+2*c*a*((a+b*(d*x+c)^(1/3))*cosh(a+b*(d*x+c)^(1/3))-sinh(a+b*(d*x+c) 
^(1/3)))-c*((a+b*(d*x+c)^(1/3))^2*cosh(a+b*(d*x+c)^(1/3))-2*(a+b*(d*x+c)^( 
1/3))*sinh(a+b*(d*x+c)^(1/3))+2*cosh(a+b*(d*x+c)^(1/3))))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.42 \[ \int x \sinh \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\frac {3 \, {\left ({\left ({\left (d x + c\right )}^{\frac {2}{3}} b^{5} d x + 20 \, b^{3} d x + 18 \, b^{3} c + 120 \, {\left (d x + c\right )}^{\frac {1}{3}} b\right )} \cosh \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - {\left (60 \, {\left (d x + c\right )}^{\frac {2}{3}} b^{2} + {\left (5 \, b^{4} d x + 3 \, b^{4} c\right )} {\left (d x + c\right )}^{\frac {1}{3}} + 120\right )} \sinh \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )}}{b^{6} d^{2}} \] Input:

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

Output:

3*(((d*x + c)^(2/3)*b^5*d*x + 20*b^3*d*x + 18*b^3*c + 120*(d*x + c)^(1/3)* 
b)*cosh((d*x + c)^(1/3)*b + a) - (60*(d*x + c)^(2/3)*b^2 + (5*b^4*d*x + 3* 
b^4*c)*(d*x + c)^(1/3) + 120)*sinh((d*x + c)^(1/3)*b + a))/(b^6*d^2)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.42 \[ \int x \sinh \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\frac {2 \, d^{2} x^{2} \sinh \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - {\left (\frac {c^{2} e^{\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}}{b} - \frac {c^{2} e^{\left (-{\left (d x + c\right )}^{\frac {1}{3}} b - a\right )}}{b} - \frac {2 \, {\left ({\left (d x + c\right )} b^{3} e^{a} - 3 \, {\left (d x + c\right )}^{\frac {2}{3}} b^{2} e^{a} + 6 \, {\left (d x + c\right )}^{\frac {1}{3}} b e^{a} - 6 \, e^{a}\right )} c e^{\left ({\left (d x + c\right )}^{\frac {1}{3}} b\right )}}{b^{4}} + \frac {2 \, {\left ({\left (d x + c\right )} b^{3} + 3 \, {\left (d x + c\right )}^{\frac {2}{3}} b^{2} + 6 \, {\left (d x + c\right )}^{\frac {1}{3}} b + 6\right )} c e^{\left (-{\left (d x + c\right )}^{\frac {1}{3}} b - a\right )}}{b^{4}} + \frac {{\left ({\left (d x + c\right )}^{2} b^{6} e^{a} - 6 \, {\left (d x + c\right )}^{\frac {5}{3}} b^{5} e^{a} + 30 \, {\left (d x + c\right )}^{\frac {4}{3}} b^{4} e^{a} - 120 \, {\left (d x + c\right )} b^{3} e^{a} + 360 \, {\left (d x + c\right )}^{\frac {2}{3}} b^{2} e^{a} - 720 \, {\left (d x + c\right )}^{\frac {1}{3}} b e^{a} + 720 \, e^{a}\right )} e^{\left ({\left (d x + c\right )}^{\frac {1}{3}} b\right )}}{b^{7}} - \frac {{\left ({\left (d x + c\right )}^{2} b^{6} + 6 \, {\left (d x + c\right )}^{\frac {5}{3}} b^{5} + 30 \, {\left (d x + c\right )}^{\frac {4}{3}} b^{4} + 120 \, {\left (d x + c\right )} b^{3} + 360 \, {\left (d x + c\right )}^{\frac {2}{3}} b^{2} + 720 \, {\left (d x + c\right )}^{\frac {1}{3}} b + 720\right )} e^{\left (-{\left (d x + c\right )}^{\frac {1}{3}} b - a\right )}}{b^{7}}\right )} b}{4 \, d^{2}} \] Input:

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

Output:

1/4*(2*d^2*x^2*sinh((d*x + c)^(1/3)*b + a) - (c^2*e^((d*x + c)^(1/3)*b + a 
)/b - c^2*e^(-(d*x + c)^(1/3)*b - a)/b - 2*((d*x + c)*b^3*e^a - 3*(d*x + c 
)^(2/3)*b^2*e^a + 6*(d*x + c)^(1/3)*b*e^a - 6*e^a)*c*e^((d*x + c)^(1/3)*b) 
/b^4 + 2*((d*x + c)*b^3 + 3*(d*x + c)^(2/3)*b^2 + 6*(d*x + c)^(1/3)*b + 6) 
*c*e^(-(d*x + c)^(1/3)*b - a)/b^4 + ((d*x + c)^2*b^6*e^a - 6*(d*x + c)^(5/ 
3)*b^5*e^a + 30*(d*x + c)^(4/3)*b^4*e^a - 120*(d*x + c)*b^3*e^a + 360*(d*x 
 + c)^(2/3)*b^2*e^a - 720*(d*x + c)^(1/3)*b*e^a + 720*e^a)*e^((d*x + c)^(1 
/3)*b)/b^7 - ((d*x + c)^2*b^6 + 6*(d*x + c)^(5/3)*b^5 + 30*(d*x + c)^(4/3) 
*b^4 + 120*(d*x + c)*b^3 + 360*(d*x + c)^(2/3)*b^2 + 720*(d*x + c)^(1/3)*b 
 + 720)*e^(-(d*x + c)^(1/3)*b - a)/b^7)*b)/d^2
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 706 vs. \(2 (231) = 462\).

Time = 0.14 (sec) , antiderivative size = 706, normalized size of antiderivative = 2.70 \[ \int x \sinh \left (a+b \sqrt [3]{c+d x}\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

-3/2*((((d*x + c)^(1/3)*b + a)^2*b^3*c - 2*((d*x + c)^(1/3)*b + a)*a*b^3*c 
 + a^2*b^3*c - ((d*x + c)^(1/3)*b + a)^5 + 5*((d*x + c)^(1/3)*b + a)^4*a - 
 10*((d*x + c)^(1/3)*b + a)^3*a^2 + 10*((d*x + c)^(1/3)*b + a)^2*a^3 - 5*( 
(d*x + c)^(1/3)*b + a)*a^4 + a^5 - 2*((d*x + c)^(1/3)*b + a)*b^3*c + 2*a*b 
^3*c + 5*((d*x + c)^(1/3)*b + a)^4 - 20*((d*x + c)^(1/3)*b + a)^3*a + 30*( 
(d*x + c)^(1/3)*b + a)^2*a^2 - 20*((d*x + c)^(1/3)*b + a)*a^3 + 5*a^4 + 2* 
b^3*c - 20*((d*x + c)^(1/3)*b + a)^3 + 60*((d*x + c)^(1/3)*b + a)^2*a - 60 
*((d*x + c)^(1/3)*b + a)*a^2 + 20*a^3 + 60*((d*x + c)^(1/3)*b + a)^2 - 120 
*((d*x + c)^(1/3)*b + a)*a + 60*a^2 - 120*(d*x + c)^(1/3)*b + 120)*e^((d*x 
 + c)^(1/3)*b + a)/(b^5*d) + (((d*x + c)^(1/3)*b + a)^2*b^3*c - 2*((d*x + 
c)^(1/3)*b + a)*a*b^3*c + a^2*b^3*c - ((d*x + c)^(1/3)*b + a)^5 + 5*((d*x 
+ c)^(1/3)*b + a)^4*a - 10*((d*x + c)^(1/3)*b + a)^3*a^2 + 10*((d*x + c)^( 
1/3)*b + a)^2*a^3 - 5*((d*x + c)^(1/3)*b + a)*a^4 + a^5 + 2*((d*x + c)^(1/ 
3)*b + a)*b^3*c - 2*a*b^3*c - 5*((d*x + c)^(1/3)*b + a)^4 + 20*((d*x + c)^ 
(1/3)*b + a)^3*a - 30*((d*x + c)^(1/3)*b + a)^2*a^2 + 20*((d*x + c)^(1/3)* 
b + a)*a^3 - 5*a^4 + 2*b^3*c - 20*((d*x + c)^(1/3)*b + a)^3 + 60*((d*x + c 
)^(1/3)*b + a)^2*a - 60*((d*x + c)^(1/3)*b + a)*a^2 + 20*a^3 - 60*((d*x + 
c)^(1/3)*b + a)^2 + 120*((d*x + c)^(1/3)*b + a)*a - 60*a^2 - 120*(d*x + c) 
^(1/3)*b - 120)*e^(-(d*x + c)^(1/3)*b - a)/(b^5*d))/(b*d)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.70 \[ \int x \sinh \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\frac {3 \left (d x +c \right )^{\frac {2}{3}} \cosh \left (\left (d x +c \right )^{\frac {1}{3}} b +a \right ) b^{5} d x +360 \left (d x +c \right )^{\frac {1}{3}} \cosh \left (\left (d x +c \right )^{\frac {1}{3}} b +a \right ) b +54 \cosh \left (\left (d x +c \right )^{\frac {1}{3}} b +a \right ) b^{3} c +60 \cosh \left (\left (d x +c \right )^{\frac {1}{3}} b +a \right ) b^{3} d x -180 \left (d x +c \right )^{\frac {2}{3}} \sinh \left (\left (d x +c \right )^{\frac {1}{3}} b +a \right ) b^{2}-9 \left (d x +c \right )^{\frac {1}{3}} \sinh \left (\left (d x +c \right )^{\frac {1}{3}} b +a \right ) b^{4} c -15 \left (d x +c \right )^{\frac {1}{3}} \sinh \left (\left (d x +c \right )^{\frac {1}{3}} b +a \right ) b^{4} d x -360 \sinh \left (\left (d x +c \right )^{\frac {1}{3}} b +a \right )}{b^{6} d^{2}} \] Input:

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

Output:

(3*((c + d*x)**(2/3)*cosh((c + d*x)**(1/3)*b + a)*b**5*d*x + 120*(c + d*x) 
**(1/3)*cosh((c + d*x)**(1/3)*b + a)*b + 18*cosh((c + d*x)**(1/3)*b + a)*b 
**3*c + 20*cosh((c + d*x)**(1/3)*b + a)*b**3*d*x - 60*(c + d*x)**(2/3)*sin 
h((c + d*x)**(1/3)*b + a)*b**2 - 3*(c + d*x)**(1/3)*sinh((c + d*x)**(1/3)* 
b + a)*b**4*c - 5*(c + d*x)**(1/3)*sinh((c + d*x)**(1/3)*b + a)*b**4*d*x - 
 120*sinh((c + d*x)**(1/3)*b + a)))/(b**6*d**2)