Optimal. Leaf size=329 \[ \frac {b d \cosh \left (a+b \sqrt [3]{c}\right ) \text {Chi}\left (b \left (\sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}+\frac {(-1)^{2/3} b d \cosh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Chi}\left (-b \left ((-1)^{2/3} \sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}-\frac {\sqrt [3]{-1} b d \cosh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Chi}\left (b \left (\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}-\frac {b d \sinh \left (a+b \sqrt [3]{c}\right ) \text {Shi}\left (b \left (\sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}-\frac {(-1)^{2/3} b d \sinh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Shi}\left (b \left ((-1)^{2/3} \sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}-\frac {\sqrt [3]{-1} b d \sinh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Shi}\left (b \left (\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}-\frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x} \]
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Rubi [A] time = 0.72, antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5364, 5288, 5281, 3303, 3298, 3301} \[ \frac {b d \cosh \left (a+b \sqrt [3]{c}\right ) \text {Chi}\left (b \left (\sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}+\frac {(-1)^{2/3} b d \cosh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Chi}\left (-b \left ((-1)^{2/3} \sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}-\frac {\sqrt [3]{-1} b d \cosh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Chi}\left (b \left (\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}-\frac {b d \sinh \left (a+b \sqrt [3]{c}\right ) \text {Shi}\left (b \left (\sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}-\frac {(-1)^{2/3} b d \sinh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Shi}\left (b \left ((-1)^{2/3} \sqrt [3]{c}-\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}-\frac {\sqrt [3]{-1} b d \sinh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Shi}\left (b \left (\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{c+d x}\right )\right )}{3 c^{2/3}}-\frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x} \]
Antiderivative was successfully verified.
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Rule 3298
Rule 3301
Rule 3303
Rule 5281
Rule 5288
Rule 5364
Rubi steps
\begin {align*} \int \frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x^2} \, dx &=d \operatorname {Subst}\left (\int \frac {\sinh \left (a+b \sqrt [3]{x}\right )}{(-c+x)^2} \, dx,x,c+d x\right )\\ &=(3 d) \operatorname {Subst}\left (\int \frac {x^2 \sinh (a+b x)}{\left (c-x^3\right )^2} \, dx,x,\sqrt [3]{c+d x}\right )\\ &=-\frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x}-(b d) \operatorname {Subst}\left (\int \frac {\cosh (a+b x)}{c-x^3} \, dx,x,\sqrt [3]{c+d x}\right )\\ &=-\frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x}-(b d) \operatorname {Subst}\left (\int \left (\frac {\cosh (a+b x)}{3 c^{2/3} \left (\sqrt [3]{c}-x\right )}+\frac {\cosh (a+b x)}{3 c^{2/3} \left (\sqrt [3]{c}+\sqrt [3]{-1} x\right )}+\frac {\cosh (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 {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x}-\frac {(b d) \operatorname {Subst}\left (\int \frac {\cosh (a+b x)}{\sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )}{3 c^{2/3}}-\frac {(b d) \operatorname {Subst}\left (\int \frac {\cosh (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) \operatorname {Subst}\left (\int \frac {\cosh (a+b x)}{\sqrt [3]{c}-(-1)^{2/3} x} \, dx,x,\sqrt [3]{c+d x}\right )}{3 c^{2/3}}\\ &=-\frac {\sinh \left (a+b \sqrt [3]{c+d x}\right )}{x}-\frac {\left (b d \cosh \left (a+b \sqrt [3]{c}\right )\right ) \operatorname {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 \cosh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right )\right ) \operatorname {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 \cosh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right )\right ) \operatorname {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 {\left (b d \sinh \left (a+b \sqrt [3]{c}\right )\right ) \operatorname {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 \sinh \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right )\right ) \operatorname {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 \sinh \left (a+(-1)^{2/3} b \sqrt [3]{c}\right )\right ) \operatorname {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 {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}}\\ \end {align*}
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Mathematica [C] time = 1.85, size = 210, normalized size = 0.64 \[ \frac {e^{-a} \left (b d x \text {RootSum}\left [c-\text {$\#$1}^3\& ,\frac {-\sinh (\text {$\#$1} b) \text {Chi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )+\cosh (\text {$\#$1} b) \text {Chi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )+\sinh (\text {$\#$1} b) \text {Shi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )-\cosh (\text {$\#$1} b) \text {Shi}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )}{\text {$\#$1}^2}\& \right ]-3 e^{2 a+b \sqrt [3]{c+d x}}+3 e^{-b \sqrt [3]{c+d x}}\right )+b d x \text {RootSum}\left [c-\text {$\#$1}^3\& ,\frac {e^{\text {$\#$1} b+a} \text {Ei}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )}{\text {$\#$1}^2}\& \right ]}{6 x} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.64, size = 704, normalized size = 2.14 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {\sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {sinh}\left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh {\left (a + b \sqrt [3]{c + d x} \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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