Integrand size = 18, antiderivative size = 232 \[ \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x} \, dx=\text {Chi}\left (b \left (\sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right ) \sinh \left (a+b \sqrt [3]{c}\right )+\text {Chi}\left (b \left (\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{c+d x}\right )\right ) \sinh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right )+\text {Chi}\left (-b \left ((-1)^{2/3} \sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right ) \sinh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right )-\cosh \left (a+b \sqrt [3]{c}\right ) \text {Shi}\left (b \left (\sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )-\cosh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Shi}\left (b \left ((-1)^{2/3} \sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )+\cosh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Shi}\left (b \left (\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{c+d x}\right )\right ) \]
-cosh(a+b*c^(1/3))*Shi(b*(c^(1/3)-(d*x+c)^(1/3)))-cosh(a+(-1)^(2/3)*b*c^(1 /3))*Shi(b*((-1)^(2/3)*c^(1/3)-(d*x+c)^(1/3)))+cosh(a-(-1)^(1/3)*b*c^(1/3) )*Shi(b*((-1)^(1/3)*c^(1/3)+(d*x+c)^(1/3)))+Chi(b*(c^(1/3)-(d*x+c)^(1/3))) *sinh(a+b*c^(1/3))+Chi(b*((-1)^(1/3)*c^(1/3)+(d*x+c)^(1/3)))*sinh(a-(-1)^( 1/3)*b*c^(1/3))+Chi(-b*((-1)^(2/3)*c^(1/3)-(d*x+c)^(1/3)))*sinh(a+(-1)^(2/ 3)*b*c^(1/3))
Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
Time = 0.06 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.00 \[ \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x} \, dx=\frac {1}{2} \left (-\text {RootSum}\left [c-\text {$\#$1}^3\&,\cosh (a+b \text {$\#$1}) \text {Chi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )-\text {Chi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right ) \sinh (a+b \text {$\#$1})-\cosh (a+b \text {$\#$1}) \text {Shi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )+\sinh (a+b \text {$\#$1}) \text {Shi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )\&\right ]+\text {RootSum}\left [c-\text {$\#$1}^3\&,\cosh (a+b \text {$\#$1}) \text {Chi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )+\text {Chi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right ) \sinh (a+b \text {$\#$1})+\cosh (a+b \text {$\#$1}) \text {Shi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )+\sinh (a+b \text {$\#$1}) \text {Shi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )\&\right ]\right ) \]
(-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
Time = 0.82 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.09, 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 |
| \(\displaystyle -3 \left (-\frac {1}{3} \sinh \left (a+b \sqrt [3]{c}\right ) \text {Chi}\left (b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )-\frac {1}{3} \sinh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Chi}\left (\sqrt [3]{-1} \sqrt [3]{c} b+\sqrt [3]{c+d x} b\right )-\frac {1}{3} \sinh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Chi}\left (b \sqrt [3]{c+d x}-(-1)^{2/3} b \sqrt [3]{c}\right )+\frac {1}{3} \cosh \left (a+b \sqrt [3]{c}\right ) \text {Shi}\left (b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )+\frac {1}{3} \cosh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Shi}\left ((-1)^{2/3} b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )-\frac {1}{3} \cosh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Shi}\left (\sqrt [3]{-1} \sqrt [3]{c} b+\sqrt [3]{c+d x} b\right )\right )\) |
-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)
3.2.1.3.1 Defintions of rubi rules used
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])
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]
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]]
\[\int \frac {\sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )}{x}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 503 vs. \(2 (182) = 364\).
Time = 0.26 (sec) , antiderivative size = 503, normalized size of antiderivative = 2.17 \[ \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x} \, dx=-\frac {1}{2} \, {\rm Ei}\left (-{\left (d x + c\right )}^{\frac {1}{3}} b - \frac {1}{2} \, \left (b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )}\right ) \cosh \left (\frac {1}{2} \, \left (b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} - a\right ) + \frac {1}{2} \, {\rm Ei}\left ({\left (d x + c\right )}^{\frac {1}{3}} b - \frac {1}{2} \, \left (-b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )}\right ) \cosh \left (\frac {1}{2} \, \left (-b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} + a\right ) - \frac {1}{2} \, {\rm Ei}\left (-{\left (d x + c\right )}^{\frac {1}{3}} b + \frac {1}{2} \, \left (b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )}\right ) \cosh \left (\frac {1}{2} \, \left (b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} + a\right ) + \frac {1}{2} \, {\rm Ei}\left ({\left (d x + c\right )}^{\frac {1}{3}} b + \frac {1}{2} \, \left (-b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )}\right ) \cosh \left (\frac {1}{2} \, \left (-b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} - a\right ) - \frac {1}{2} \, {\rm Ei}\left (-{\left (d x + c\right )}^{\frac {1}{3}} b + \left (b^{3} c\right )^{\frac {1}{3}}\right ) \cosh \left (a + \left (b^{3} c\right )^{\frac {1}{3}}\right ) + \frac {1}{2} \, {\rm Ei}\left ({\left (d x + c\right )}^{\frac {1}{3}} b + \left (-b^{3} c\right )^{\frac {1}{3}}\right ) \cosh \left (-a + \left (-b^{3} c\right )^{\frac {1}{3}}\right ) - \frac {1}{2} \, {\rm Ei}\left (-{\left (d x + c\right )}^{\frac {1}{3}} b - \frac {1}{2} \, \left (b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )}\right ) \sinh \left (\frac {1}{2} \, \left (b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} - a\right ) + \frac {1}{2} \, {\rm Ei}\left ({\left (d x + c\right )}^{\frac {1}{3}} b - \frac {1}{2} \, \left (-b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )}\right ) \sinh \left (\frac {1}{2} \, \left (-b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} + a\right ) + \frac {1}{2} \, {\rm Ei}\left (-{\left (d x + c\right )}^{\frac {1}{3}} b + \frac {1}{2} \, \left (b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )}\right ) \sinh \left (\frac {1}{2} \, \left (b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} + a\right ) - \frac {1}{2} \, {\rm Ei}\left ({\left (d x + c\right )}^{\frac {1}{3}} b + \frac {1}{2} \, \left (-b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )}\right ) \sinh \left (\frac {1}{2} \, \left (-b^{3} c\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} - a\right ) + \frac {1}{2} \, {\rm Ei}\left (-{\left (d x + c\right )}^{\frac {1}{3}} b + \left (b^{3} c\right )^{\frac {1}{3}}\right ) \sinh \left (a + \left (b^{3} c\right )^{\frac {1}{3}}\right ) - \frac {1}{2} \, {\rm Ei}\left ({\left (d x + c\right )}^{\frac {1}{3}} b + \left (-b^{3} c\right )^{\frac {1}{3}}\right ) \sinh \left (-a + \left (-b^{3} c\right )^{\frac {1}{3}}\right ) \]
-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))
\[ \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x} \, dx=\int \frac {\sinh {\left (a + b \sqrt [3]{c + d x} \right )}}{x}\, dx \]
\[ \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x} \, dx=\int { \frac {\sinh \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{x} \,d x } \]
\[ \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x} \, dx=\int { \frac {\sinh \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{x} \,d x } \]
Timed out. \[ \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x} \, dx=\int \frac {\mathrm {sinh}\left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )}{x} \,d x \]