Integrand size = 18, antiderivative size = 329 \[ \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x^2} \, dx=\frac {b d \cosh \left (a+b \sqrt [3]{c}\right ) \text {Chi}\left (b \left (\sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}+\frac {(-1)^{2/3} b d \cosh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Chi}\left (-b \left ((-1)^{2/3} \sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}-\frac {\sqrt [3]{-1} b d \cosh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Chi}\left (b \left (\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}-\frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x}-\frac {b d \sinh \left (a+b \sqrt [3]{c}\right ) \text {Shi}\left (b \left (\sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}-\frac {(-1)^{2/3} b d \sinh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Shi}\left (b \left ((-1)^{2/3} \sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}-\frac {\sqrt [3]{-1} b d \sinh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Shi}\left (b \left (\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}} \]
1/3*b*d*Chi(b*(c^(1/3)-(d*x+c)^(1/3)))*cosh(a+b*c^(1/3))/c^(2/3)-1/3*(-1)^ (1/3)*b*d*Chi(b*((-1)^(1/3)*c^(1/3)+(d*x+c)^(1/3)))*cosh(a-(-1)^(1/3)*b*c^ (1/3))/c^(2/3)+1/3*(-1)^(2/3)*b*d*Chi(-b*((-1)^(2/3)*c^(1/3)-(d*x+c)^(1/3) ))*cosh(a+(-1)^(2/3)*b*c^(1/3))/c^(2/3)-1/3*b*d*Shi(b*(c^(1/3)-(d*x+c)^(1/ 3)))*sinh(a+b*c^(1/3))/c^(2/3)-1/3*(-1)^(1/3)*b*d*Shi(b*((-1)^(1/3)*c^(1/3 )+(d*x+c)^(1/3)))*sinh(a-(-1)^(1/3)*b*c^(1/3))/c^(2/3)-1/3*(-1)^(2/3)*b*d* Shi(b*((-1)^(2/3)*c^(1/3)-(d*x+c)^(1/3)))*sinh(a+(-1)^(2/3)*b*c^(1/3))/c^( 2/3)-sinh(a+b*(d*x+c)^(1/3))/x
Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
Time = 2.86 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.64 \[ \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x^2} \, dx=\frac {e^{-a} \left (3 e^{-b \sqrt [3]{c+d x}}-3 e^{2 a+b \sqrt [3]{c+d x}}+b d x \text {RootSum}\left [c-\text {$\#$1}^3\&,\frac {e^{2 a+b \text {$\#$1}} \operatorname {ExpIntegralEi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )}{\text {$\#$1}^2}\&\right ]+b d x \text {RootSum}\left [c-\text {$\#$1}^3\&,\frac {\cosh (b \text {$\#$1}) \text {Chi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )-\text {Chi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right ) \sinh (b \text {$\#$1})-\cosh (b \text {$\#$1}) \text {Shi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )+\sinh (b \text {$\#$1}) \text {Shi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )}{\text {$\#$1}^2}\&\right ]\right )}{6 x} \]
(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)
Time = 0.95 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.02, 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 |
| \(\displaystyle 3 d \left (-\frac {1}{3} b \left (-\frac {\cosh \left (a+b \sqrt [3]{c}\right ) \text {Chi}\left (b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )}{3 c^{2/3}}+\frac {\sqrt [3]{-1} \cosh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Chi}\left (\sqrt [3]{-1} \sqrt [3]{c} b+\sqrt [3]{c+d x} b\right )}{3 c^{2/3}}-\frac {(-1)^{2/3} \cosh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Chi}\left (b \sqrt [3]{c+d x}-(-1)^{2/3} b \sqrt [3]{c}\right )}{3 c^{2/3}}+\frac {\sinh \left (a+b \sqrt [3]{c}\right ) \text {Shi}\left (b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )}{3 c^{2/3}}+\frac {(-1)^{2/3} \sinh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Shi}\left ((-1)^{2/3} b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )}{3 c^{2/3}}+\frac {\sqrt [3]{-1} \sinh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Shi}\left (\sqrt [3]{-1} \sqrt [3]{c} b+\sqrt [3]{c+d x} b\right )}{3 c^{2/3}}\right )-\frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{3 d x}\right )\) |
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)
3.2.2.3.1 Defintions of rubi rules used
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])
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])
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]]
\[\int \frac {\sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )}{x^{2}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 704 vs. \(2 (245) = 490\).
Time = 0.27 (sec) , antiderivative size = 704, normalized size of antiderivative = 2.14 \[ \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x^2} \, dx=\text {Too large to display} \]
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*...
\[ \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x^2} \, dx=\int \frac {\sinh {\left (a + b \sqrt [3]{c + d x} \right )}}{x^{2}}\, dx \]
\[ \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x^2} \, dx=\int { \frac {\sinh \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{x^{2}} \,d x } \]
\[ \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x^2} \, dx=\int { \frac {\sinh \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x^2} \, dx=\int \frac {\mathrm {sinh}\left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )}{x^2} \,d x \]