Optimal. Leaf size=435 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt [5]{b}-\sqrt [5]{a} \tanh \left (\frac{x}{2}\right )}{\sqrt{a^{2/5}+b^{2/5}}}\right )}{5 a^{4/5} \sqrt{a^{2/5}+b^{2/5}}}+\frac{2 (-1)^{9/10} \tanh ^{-1}\left (\frac{(-1)^{9/10} \left (\sqrt [5]{a} \tanh \left (\frac{x}{2}\right )+\sqrt [5]{-1} \sqrt [5]{b}\right )}{\sqrt{\sqrt [5]{-1} b^{2/5}-(-1)^{4/5} a^{2/5}}}\right )}{5 a^{4/5} \sqrt{\sqrt [5]{-1} b^{2/5}-(-1)^{4/5} a^{2/5}}}+\frac{2 \sqrt [5]{-1} \tanh ^{-1}\left (\frac{\sqrt [5]{-1} \sqrt [5]{a} \tanh \left (\frac{x}{2}\right )+\sqrt [5]{b}}{\sqrt{(-1)^{2/5} a^{2/5}+b^{2/5}}}\right )}{5 a^{4/5} \sqrt{(-1)^{2/5} a^{2/5}+b^{2/5}}}+\frac{2 (-1)^{9/10} \tanh ^{-1}\left (\frac{(-1)^{3/10} \left ((-1)^{3/5} \sqrt [5]{a} \tanh \left (\frac{x}{2}\right )+\sqrt [5]{b}\right )}{\sqrt{(-1)^{3/5} b^{2/5}-(-1)^{4/5} a^{2/5}}}\right )}{5 a^{4/5} \sqrt{(-1)^{3/5} b^{2/5}-(-1)^{4/5} a^{2/5}}}-\frac{2 (-1)^{9/10} \tanh ^{-1}\left (\frac{-(-1)^{9/10} \sqrt [5]{a} \tanh \left (\frac{x}{2}\right )+i \sqrt [5]{b}}{\sqrt{-(-1)^{4/5} a^{2/5}-b^{2/5}}}\right )}{5 a^{4/5} \sqrt{-(-1)^{4/5} a^{2/5}-b^{2/5}}} \]
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Rubi [A] time = 0.974696, antiderivative size = 435, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3213, 2660, 618, 206, 204} \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt [5]{b}-\sqrt [5]{a} \tanh \left (\frac{x}{2}\right )}{\sqrt{a^{2/5}+b^{2/5}}}\right )}{5 a^{4/5} \sqrt{a^{2/5}+b^{2/5}}}+\frac{2 (-1)^{9/10} \tanh ^{-1}\left (\frac{(-1)^{9/10} \left (\sqrt [5]{a} \tanh \left (\frac{x}{2}\right )+\sqrt [5]{-1} \sqrt [5]{b}\right )}{\sqrt{\sqrt [5]{-1} b^{2/5}-(-1)^{4/5} a^{2/5}}}\right )}{5 a^{4/5} \sqrt{\sqrt [5]{-1} b^{2/5}-(-1)^{4/5} a^{2/5}}}+\frac{2 \sqrt [5]{-1} \tanh ^{-1}\left (\frac{\sqrt [5]{-1} \sqrt [5]{a} \tanh \left (\frac{x}{2}\right )+\sqrt [5]{b}}{\sqrt{(-1)^{2/5} a^{2/5}+b^{2/5}}}\right )}{5 a^{4/5} \sqrt{(-1)^{2/5} a^{2/5}+b^{2/5}}}+\frac{2 (-1)^{9/10} \tanh ^{-1}\left (\frac{(-1)^{3/10} \left ((-1)^{3/5} \sqrt [5]{a} \tanh \left (\frac{x}{2}\right )+\sqrt [5]{b}\right )}{\sqrt{(-1)^{3/5} b^{2/5}-(-1)^{4/5} a^{2/5}}}\right )}{5 a^{4/5} \sqrt{(-1)^{3/5} b^{2/5}-(-1)^{4/5} a^{2/5}}}-\frac{2 (-1)^{9/10} \tanh ^{-1}\left (\frac{-(-1)^{9/10} \sqrt [5]{a} \tanh \left (\frac{x}{2}\right )+i \sqrt [5]{b}}{\sqrt{-(-1)^{4/5} a^{2/5}-b^{2/5}}}\right )}{5 a^{4/5} \sqrt{-(-1)^{4/5} a^{2/5}-b^{2/5}}} \]
Antiderivative was successfully verified.
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Rule 3213
Rule 2660
Rule 618
Rule 206
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{a+b \sinh ^5(x)} \, dx &=\int \left (-\frac{(-1)^{9/10}}{5 a^{4/5} \left (-(-1)^{9/10} \sqrt [5]{a}-i \sqrt [5]{b} \sinh (x)\right )}-\frac{(-1)^{9/10}}{5 a^{4/5} \left (-(-1)^{9/10} \sqrt [5]{a}-\sqrt [10]{-1} \sqrt [5]{b} \sinh (x)\right )}-\frac{(-1)^{9/10}}{5 a^{4/5} \left (-(-1)^{9/10} \sqrt [5]{a}+(-1)^{3/10} \sqrt [5]{b} \sinh (x)\right )}-\frac{(-1)^{9/10}}{5 a^{4/5} \left (-(-1)^{9/10} \sqrt [5]{a}+(-1)^{7/10} \sqrt [5]{b} \sinh (x)\right )}-\frac{(-1)^{9/10}}{5 a^{4/5} \left (-(-1)^{9/10} \sqrt [5]{a}-(-1)^{9/10} \sqrt [5]{b} \sinh (x)\right )}\right ) \, dx\\ &=-\frac{(-1)^{9/10} \int \frac{1}{-(-1)^{9/10} \sqrt [5]{a}-i \sqrt [5]{b} \sinh (x)} \, dx}{5 a^{4/5}}-\frac{(-1)^{9/10} \int \frac{1}{-(-1)^{9/10} \sqrt [5]{a}-\sqrt [10]{-1} \sqrt [5]{b} \sinh (x)} \, dx}{5 a^{4/5}}-\frac{(-1)^{9/10} \int \frac{1}{-(-1)^{9/10} \sqrt [5]{a}+(-1)^{3/10} \sqrt [5]{b} \sinh (x)} \, dx}{5 a^{4/5}}-\frac{(-1)^{9/10} \int \frac{1}{-(-1)^{9/10} \sqrt [5]{a}+(-1)^{7/10} \sqrt [5]{b} \sinh (x)} \, dx}{5 a^{4/5}}-\frac{(-1)^{9/10} \int \frac{1}{-(-1)^{9/10} \sqrt [5]{a}-(-1)^{9/10} \sqrt [5]{b} \sinh (x)} \, dx}{5 a^{4/5}}\\ &=-\frac{\left (2 (-1)^{9/10}\right ) \operatorname{Subst}\left (\int \frac{1}{-(-1)^{9/10} \sqrt [5]{a}-2 i \sqrt [5]{b} x+(-1)^{9/10} \sqrt [5]{a} x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{5 a^{4/5}}-\frac{\left (2 (-1)^{9/10}\right ) \operatorname{Subst}\left (\int \frac{1}{-(-1)^{9/10} \sqrt [5]{a}-2 \sqrt [10]{-1} \sqrt [5]{b} x+(-1)^{9/10} \sqrt [5]{a} x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{5 a^{4/5}}-\frac{\left (2 (-1)^{9/10}\right ) \operatorname{Subst}\left (\int \frac{1}{-(-1)^{9/10} \sqrt [5]{a}+2 (-1)^{3/10} \sqrt [5]{b} x+(-1)^{9/10} \sqrt [5]{a} x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{5 a^{4/5}}-\frac{\left (2 (-1)^{9/10}\right ) \operatorname{Subst}\left (\int \frac{1}{-(-1)^{9/10} \sqrt [5]{a}+2 (-1)^{7/10} \sqrt [5]{b} x+(-1)^{9/10} \sqrt [5]{a} x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{5 a^{4/5}}-\frac{\left (2 (-1)^{9/10}\right ) \operatorname{Subst}\left (\int \frac{1}{-(-1)^{9/10} \sqrt [5]{a}-2 (-1)^{9/10} \sqrt [5]{b} x+(-1)^{9/10} \sqrt [5]{a} x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{5 a^{4/5}}\\ &=\frac{\left (4 (-1)^{9/10}\right ) \operatorname{Subst}\left (\int \frac{1}{-4 (-1)^{4/5} \left (a^{2/5}+b^{2/5}\right )-x^2} \, dx,x,-2 (-1)^{9/10} \sqrt [5]{b}+2 (-1)^{9/10} \sqrt [5]{a} \tanh \left (\frac{x}{2}\right )\right )}{5 a^{4/5}}+\frac{\left (4 (-1)^{9/10}\right ) \operatorname{Subst}\left (\int \frac{1}{-4 (-1)^{2/5} \left ((-1)^{2/5} a^{2/5}+b^{2/5}\right )-x^2} \, dx,x,2 (-1)^{7/10} \sqrt [5]{b}+2 (-1)^{9/10} \sqrt [5]{a} \tanh \left (\frac{x}{2}\right )\right )}{5 a^{4/5}}+\frac{\left (4 (-1)^{9/10}\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left ((-1)^{4/5} a^{2/5}+b^{2/5}\right )-x^2} \, dx,x,-2 i \sqrt [5]{b}+2 (-1)^{9/10} \sqrt [5]{a} \tanh \left (\frac{x}{2}\right )\right )}{5 a^{4/5}}+\frac{\left (4 (-1)^{9/10}\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left ((-1)^{4/5} a^{2/5}-\sqrt [5]{-1} b^{2/5}\right )-x^2} \, dx,x,-2 \sqrt [10]{-1} \sqrt [5]{b}+2 (-1)^{9/10} \sqrt [5]{a} \tanh \left (\frac{x}{2}\right )\right )}{5 a^{4/5}}+\frac{\left (4 (-1)^{9/10}\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left ((-1)^{4/5} a^{2/5}-(-1)^{3/5} b^{2/5}\right )-x^2} \, dx,x,2 (-1)^{3/10} \sqrt [5]{b}+2 (-1)^{9/10} \sqrt [5]{a} \tanh \left (\frac{x}{2}\right )\right )}{5 a^{4/5}}\\ &=-\frac{2 (-1)^{7/10} \tan ^{-1}\left (\frac{i \sqrt [5]{b}+(-1)^{7/10} \sqrt [5]{a} \tanh \left (\frac{x}{2}\right )}{\sqrt{(-1)^{2/5} a^{2/5}+b^{2/5}}}\right )}{5 a^{4/5} \sqrt{(-1)^{2/5} a^{2/5}+b^{2/5}}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt [5]{b}-\sqrt [5]{a} \tanh \left (\frac{x}{2}\right )}{\sqrt{a^{2/5}+b^{2/5}}}\right )}{5 a^{4/5} \sqrt{a^{2/5}+b^{2/5}}}-\frac{2 (-1)^{9/10} \tanh ^{-1}\left (\frac{i \sqrt [5]{b}-(-1)^{9/10} \sqrt [5]{a} \tanh \left (\frac{x}{2}\right )}{\sqrt{-(-1)^{4/5} a^{2/5}-b^{2/5}}}\right )}{5 a^{4/5} \sqrt{-(-1)^{4/5} a^{2/5}-b^{2/5}}}-\frac{2 (-1)^{9/10} \tanh ^{-1}\left (\frac{\sqrt [10]{-1} \sqrt [5]{b}-(-1)^{9/10} \sqrt [5]{a} \tanh \left (\frac{x}{2}\right )}{\sqrt{-(-1)^{4/5} a^{2/5}+\sqrt [5]{-1} b^{2/5}}}\right )}{5 a^{4/5} \sqrt{-(-1)^{4/5} a^{2/5}+\sqrt [5]{-1} b^{2/5}}}+\frac{2 (-1)^{9/10} \tanh ^{-1}\left (\frac{(-1)^{3/10} \sqrt [5]{b}+(-1)^{9/10} \sqrt [5]{a} \tanh \left (\frac{x}{2}\right )}{\sqrt{-(-1)^{4/5} a^{2/5}+(-1)^{3/5} b^{2/5}}}\right )}{5 a^{4/5} \sqrt{-(-1)^{4/5} a^{2/5}+(-1)^{3/5} b^{2/5}}}\\ \end{align*}
Mathematica [C] time = 0.325722, size = 141, normalized size = 0.32 \[ \frac{8}{5} \text{RootSum}\left [32 \text{$\#$1}^5 a+\text{$\#$1}^{10} b-5 \text{$\#$1}^8 b+10 \text{$\#$1}^6 b-10 \text{$\#$1}^4 b+5 \text{$\#$1}^2 b-b\& ,\frac{\text{$\#$1}^3 x+2 \text{$\#$1}^3 \log \left (-\text{$\#$1} \sinh \left (\frac{x}{2}\right )+\text{$\#$1} \cosh \left (\frac{x}{2}\right )-\sinh \left (\frac{x}{2}\right )-\cosh \left (\frac{x}{2}\right )\right )}{16 \text{$\#$1}^3 a+\text{$\#$1}^8 b-4 \text{$\#$1}^6 b+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b}\& \right ] \]
Antiderivative was successfully verified.
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Maple [C] time = 0.032, size = 113, normalized size = 0.3 \begin{align*}{\frac{1}{5}\sum _{{\it \_R}={\it RootOf} \left ( a{{\it \_Z}}^{10}-5\,a{{\it \_Z}}^{8}+10\,a{{\it \_Z}}^{6}-32\,b{{\it \_Z}}^{5}-10\,a{{\it \_Z}}^{4}+5\,a{{\it \_Z}}^{2}-a \right ) }{\frac{-{{\it \_R}}^{8}+4\,{{\it \_R}}^{6}-6\,{{\it \_R}}^{4}+4\,{{\it \_R}}^{2}-1}{{{\it \_R}}^{9}a-4\,{{\it \_R}}^{7}a+6\,{{\it \_R}}^{5}a-16\,{{\it \_R}}^{4}b-4\,{{\it \_R}}^{3}a+{\it \_R}\,a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -{\it \_R} \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{b \sinh \left (x\right )^{5} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{a + b \sinh ^{5}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{b \sinh \left (x\right )^{5} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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