Integrand size = 18, antiderivative size = 329 \[ \int \frac {\cosh \left (a+b \sqrt [3]{c+d x}\right )}{x^2} \, dx=-\frac {\cosh \left (a+b \sqrt [3]{c+d x}\right )}{x}+\frac {b d \text {Chi}\left (b \left (\sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right ) \sinh \left (a+b \sqrt [3]{c}\right )}{3 c^{2/3}}-\frac {\sqrt [3]{-1} b d \text {Chi}\left (b \left (\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{c+d x}\right )\right ) \sinh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right )}{3 c^{2/3}}+\frac {(-1)^{2/3} b d \text {Chi}\left (-b \left ((-1)^{2/3} \sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right ) \sinh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right )}{3 c^{2/3}}-\frac {b d \cosh \left (a+b \sqrt [3]{c}\right ) \text {Shi}\left (b \left (\sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}-\frac {(-1)^{2/3} b d \cosh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Shi}\left (b \left ((-1)^{2/3} \sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}-\frac {\sqrt [3]{-1} b d \cosh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Shi}\left (b \left (\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}} \]
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Time = 0.56 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5473, 5397, 5388, 3384, 3379, 3382} \[ \int \frac {\cosh \left (a+b \sqrt [3]{c+d x}\right )}{x^2} \, dx=\frac {b d \sinh \left (a+b \sqrt [3]{c}\right ) \text {Chi}\left (b \left (\sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}-\frac {\sqrt [3]{-1} b d \sinh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Chi}\left (b \left (\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}+\frac {(-1)^{2/3} b d \sinh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Chi}\left (-b \left ((-1)^{2/3} \sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}-\frac {b d \cosh \left (a+b \sqrt [3]{c}\right ) \text {Shi}\left (b \left (\sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}-\frac {(-1)^{2/3} b d \cosh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Shi}\left (b \left ((-1)^{2/3} \sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}-\frac {\sqrt [3]{-1} b d \cosh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Shi}\left (b \left (\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}-\frac {\cosh \left (a+b \sqrt [3]{c+d x}\right )}{x} \]
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Rule 3379
Rule 3382
Rule 3384
Rule 5388
Rule 5397
Rule 5473
Rubi steps \begin{align*} \text {integral}& = d \text {Subst}\left (\int \frac {\cosh \left (a+b \sqrt [3]{x}\right )}{(-c+x)^2} \, dx,x,c+d x\right ) \\ & = (3 d) \text {Subst}\left (\int \frac {x^2 \cosh (a+b x)}{\left (c-x^3\right )^2} \, dx,x,\sqrt [3]{c+d x}\right ) \\ & = -\frac {\cosh \left (a+b \sqrt [3]{c+d x}\right )}{x}-(b d) \text {Subst}\left (\int \frac {\sinh (a+b x)}{c-x^3} \, dx,x,\sqrt [3]{c+d x}\right ) \\ & = -\frac {\cosh \left (a+b \sqrt [3]{c+d x}\right )}{x}-(b d) \text {Subst}\left (\int \left (\frac {\sinh (a+b x)}{3 c^{2/3} \left (\sqrt [3]{c}-x\right )}+\frac {\sinh (a+b x)}{3 c^{2/3} \left (\sqrt [3]{c}+\sqrt [3]{-1} x\right )}+\frac {\sinh (a+b x)}{3 c^{2/3} \left (\sqrt [3]{c}-(-1)^{2/3} x\right )}\right ) \, dx,x,\sqrt [3]{c+d x}\right ) \\ & = -\frac {\cosh \left (a+b \sqrt [3]{c+d x}\right )}{x}-\frac {(b d) \text {Subst}\left (\int \frac {\sinh (a+b x)}{\sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )}{3 c^{2/3}}-\frac {(b d) \text {Subst}\left (\int \frac {\sinh (a+b x)}{\sqrt [3]{c}+\sqrt [3]{-1} x} \, dx,x,\sqrt [3]{c+d x}\right )}{3 c^{2/3}}-\frac {(b d) \text {Subst}\left (\int \frac {\sinh (a+b x)}{\sqrt [3]{c}-(-1)^{2/3} x} \, dx,x,\sqrt [3]{c+d x}\right )}{3 c^{2/3}} \\ & = -\frac {\cosh \left (a+b \sqrt [3]{c+d x}\right )}{x}+\frac {\left (b d \cosh \left (a+b \sqrt [3]{c}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (b \sqrt [3]{c}-b x\right )}{\sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )}{3 c^{2/3}}+\frac {\left (i b d \cosh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right )\right ) \text {Subst}\left (\int \frac {\sin \left ((-1)^{5/6} b \sqrt [3]{c}+i b x\right )}{\sqrt [3]{c}-(-1)^{2/3} x} \, dx,x,\sqrt [3]{c+d x}\right )}{3 c^{2/3}}+\frac {\left (i b d \cosh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\sqrt [6]{-1} b \sqrt [3]{c}+i b x\right )}{\sqrt [3]{c}+\sqrt [3]{-1} x} \, dx,x,\sqrt [3]{c+d x}\right )}{3 c^{2/3}}-\frac {\left (b d \sinh \left (a+b \sqrt [3]{c}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (b \sqrt [3]{c}-b x\right )}{\sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )}{3 c^{2/3}}-\frac {\left (b d \sinh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right )\right ) \text {Subst}\left (\int \frac {\cos \left ((-1)^{5/6} b \sqrt [3]{c}+i b x\right )}{\sqrt [3]{c}-(-1)^{2/3} x} \, dx,x,\sqrt [3]{c+d x}\right )}{3 c^{2/3}}-\frac {\left (b d \sinh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\sqrt [6]{-1} b \sqrt [3]{c}+i b x\right )}{\sqrt [3]{c}+\sqrt [3]{-1} x} \, dx,x,\sqrt [3]{c+d x}\right )}{3 c^{2/3}} \\ & = -\frac {\cosh \left (a+b \sqrt [3]{c+d x}\right )}{x}+\frac {b d \text {Chi}\left (b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right ) \sinh \left (a+b \sqrt [3]{c}\right )}{3 c^{2/3}}-\frac {\sqrt [3]{-1} b d \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 )}{3 c^{2/3}}+\frac {(-1)^{2/3} b d \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 )}{3 c^{2/3}}-\frac {b d \cosh \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 \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 )}{3 c^{2/3}}-\frac {\sqrt [3]{-1} b d \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 )}{3 c^{2/3}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
Time = 0.36 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.64 \[ \int \frac {\cosh \left (a+b \sqrt [3]{c+d x}\right )}{x^2} \, dx=\frac {e^{-a} \left (-3 e^{-b \sqrt [3]{c+d x}} \left (1+e^{2 \left (a+b \sqrt [3]{c+d x}\right )}\right )+b d x \text {RootSum}\left [c-\text {$\#$1}^3\&,\frac {e^{2 a+b \text {$\#$1}} \operatorname {ExpIntegralEi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )}{\text {$\#$1}^2}\&\right ]-b d x \text {RootSum}\left [c-\text {$\#$1}^3\&,\frac {\cosh (b \text {$\#$1}) \text {Chi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )-\text {Chi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right ) \sinh (b \text {$\#$1})-\cosh (b \text {$\#$1}) \text {Shi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )+\sinh (b \text {$\#$1}) \text {Shi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )}{\text {$\#$1}^2}\&\right ]\right )}{6 x} \]
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\[\int \frac {\cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )}{x^{2}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 706 vs. \(2 (245) = 490\).
Time = 0.29 (sec) , antiderivative size = 706, normalized size of antiderivative = 2.15 \[ \int \frac {\cosh \left (a+b \sqrt [3]{c+d x}\right )}{x^2} \, dx=\text {Too large to display} \]
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\[ \int \frac {\cosh \left (a+b \sqrt [3]{c+d x}\right )}{x^2} \, dx=\int \frac {\cosh {\left (a + b \sqrt [3]{c + d x} \right )}}{x^{2}}\, dx \]
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\[ \int \frac {\cosh \left (a+b \sqrt [3]{c+d x}\right )}{x^2} \, dx=\int { \frac {\cosh \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{x^{2}} \,d x } \]
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\[ \int \frac {\cosh \left (a+b \sqrt [3]{c+d x}\right )}{x^2} \, dx=\int { \frac {\cosh \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\cosh \left (a+b \sqrt [3]{c+d x}\right )}{x^2} \, dx=\int \frac {\mathrm {cosh}\left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )}{x^2} \,d x \]
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