Optimal. Leaf size=195 \[ \frac{a \left (a^2+2 b^2\right ) \sinh (x)}{b^3 \left (a^2-b^2\right )}+\frac{\left (2 a^2+b^2\right ) \cosh (x)}{b^4-a^2 b^2}+\frac{2 a^2 \left (a+b \tanh \left (\frac{x}{2}\right )\right )}{\left (a^2-b^2\right )^2 \left (a \tanh ^2\left (\frac{x}{2}\right )+a+2 b \tanh \left (\frac{x}{2}\right )\right )}-\frac{a^3}{b^3 (a+b)^2 \left (1-\tanh \left (\frac{x}{2}\right )\right )}+\frac{a^3}{b^3 (a-b)^2 \left (\tanh \left (\frac{x}{2}\right )+1\right )}+\frac{3 a^2 b \tan ^{-1}\left (\frac{a \sinh (x)+b \cosh (x)}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}} \]
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Rubi [A] time = 1.22499, antiderivative size = 301, normalized size of antiderivative = 1.54, number of steps used = 16, number of rules used = 10, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {4401, 2637, 2638, 6742, 638, 618, 204, 3100, 3074, 206} \[ \frac{3 a^3 \sinh (x)}{b^3 \left (a^2-b^2\right )}-\frac{3 a^2 \cosh (x)}{b^2 \left (a^2-b^2\right )}+\frac{2 a^2 \left (a+b \tanh \left (\frac{x}{2}\right )\right )}{\left (a^2-b^2\right )^2 \left (a \tanh ^2\left (\frac{x}{2}\right )+a+2 b \tanh \left (\frac{x}{2}\right )\right )}-\frac{a^3}{b^3 (a+b)^2 \left (1-\tanh \left (\frac{x}{2}\right )\right )}+\frac{a^3}{b^3 (a-b)^2 \left (\tanh \left (\frac{x}{2}\right )+1\right )}+\frac{2 a^2 \left (3 a^2-b^2\right ) \tan ^{-1}\left (\frac{a \tanh \left (\frac{x}{2}\right )+b}{\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{5/2}}+\frac{2 a^2 b \tan ^{-1}\left (\frac{a \tanh \left (\frac{x}{2}\right )+b}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}-\frac{3 a^2 \tan ^{-1}\left (\frac{a \sinh (x)+b \cosh (x)}{\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2}}-\frac{2 a \sinh (x)}{b^3}+\frac{\cosh (x)}{b^2} \]
Antiderivative was successfully verified.
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Rule 4401
Rule 2637
Rule 2638
Rule 6742
Rule 638
Rule 618
Rule 204
Rule 3100
Rule 3074
Rule 206
Rubi steps
\begin{align*} \int \frac{\sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx &=i \int \left (\frac{2 i a \cosh (x)}{b^3}-\frac{i \sinh (x)}{b^2}-\frac{i a^3 \cosh ^3(x)}{b^3 (i a \cosh (x)+i b \sinh (x))^2}-\frac{3 i a^2 \cosh ^2(x)}{b^3 (a \cosh (x)+b \sinh (x))}\right ) \, dx\\ &=-\frac{(2 a) \int \cosh (x) \, dx}{b^3}+\frac{\left (3 a^2\right ) \int \frac{\cosh ^2(x)}{a \cosh (x)+b \sinh (x)} \, dx}{b^3}+\frac{a^3 \int \frac{\cosh ^3(x)}{(i a \cosh (x)+i b \sinh (x))^2} \, dx}{b^3}+\frac{\int \sinh (x) \, dx}{b^2}\\ &=\frac{\cosh (x)}{b^2}-\frac{3 a^2 \cosh (x)}{b^2 \left (a^2-b^2\right )}-\frac{2 a \sinh (x)}{b^3}+\frac{\left (2 a^3\right ) \operatorname{Subst}\left (\int \frac{\left (-1-x^2\right )^3}{\left (1-x^2\right )^2 \left (a+2 b x+a x^2\right )^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{b^3}+\frac{\left (3 a^3\right ) \int \cosh (x) \, dx}{b^3 \left (a^2-b^2\right )}-\frac{\left (3 a^2\right ) \int \frac{1}{a \cosh (x)+b \sinh (x)} \, dx}{b \left (a^2-b^2\right )}\\ &=\frac{\cosh (x)}{b^2}-\frac{3 a^2 \cosh (x)}{b^2 \left (a^2-b^2\right )}-\frac{2 a \sinh (x)}{b^3}+\frac{3 a^3 \sinh (x)}{b^3 \left (a^2-b^2\right )}+\frac{\left (2 a^3\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{2 (a+b)^2 (-1+x)^2}-\frac{1}{2 (a-b)^2 (1+x)^2}+\frac{2 b^3 x}{a \left (-a^2+b^2\right ) \left (a+2 b x+a x^2\right )^2}+\frac{3 a^2 b^2-b^4}{a \left (a^2-b^2\right )^2 \left (a+2 b x+a x^2\right )}\right ) \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{b^3}-\frac{\left (3 i a^2\right ) \operatorname{Subst}\left (\int \frac{1}{a^2-b^2-x^2} \, dx,x,-i b \cosh (x)-i a \sinh (x)\right )}{b \left (a^2-b^2\right )}\\ &=-\frac{3 a^2 \tan ^{-1}\left (\frac{b \cosh (x)+a \sinh (x)}{\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2}}+\frac{\cosh (x)}{b^2}-\frac{3 a^2 \cosh (x)}{b^2 \left (a^2-b^2\right )}-\frac{2 a \sinh (x)}{b^3}+\frac{3 a^3 \sinh (x)}{b^3 \left (a^2-b^2\right )}-\frac{a^3}{b^3 (a+b)^2 \left (1-\tanh \left (\frac{x}{2}\right )\right )}+\frac{a^3}{(a-b)^2 b^3 \left (1+\tanh \left (\frac{x}{2}\right )\right )}-\frac{\left (4 a^2\right ) \operatorname{Subst}\left (\int \frac{x}{\left (a+2 b x+a x^2\right )^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{a^2-b^2}+\frac{\left (2 a^2 \left (3 a^2-b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+2 b x+a x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{b \left (a^2-b^2\right )^2}\\ &=-\frac{3 a^2 \tan ^{-1}\left (\frac{b \cosh (x)+a \sinh (x)}{\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2}}+\frac{\cosh (x)}{b^2}-\frac{3 a^2 \cosh (x)}{b^2 \left (a^2-b^2\right )}-\frac{2 a \sinh (x)}{b^3}+\frac{3 a^3 \sinh (x)}{b^3 \left (a^2-b^2\right )}-\frac{a^3}{b^3 (a+b)^2 \left (1-\tanh \left (\frac{x}{2}\right )\right )}+\frac{a^3}{(a-b)^2 b^3 \left (1+\tanh \left (\frac{x}{2}\right )\right )}+\frac{2 a^2 \left (a+b \tanh \left (\frac{x}{2}\right )\right )}{\left (a^2-b^2\right )^2 \left (a+2 b \tanh \left (\frac{x}{2}\right )+a \tanh ^2\left (\frac{x}{2}\right )\right )}+\frac{\left (2 a^2 b\right ) \operatorname{Subst}\left (\int \frac{1}{a+2 b x+a x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{\left (a^2-b^2\right )^2}-\frac{\left (4 a^2 \left (3 a^2-b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tanh \left (\frac{x}{2}\right )\right )}{b \left (a^2-b^2\right )^2}\\ &=-\frac{3 a^2 \tan ^{-1}\left (\frac{b \cosh (x)+a \sinh (x)}{\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2}}+\frac{2 a^2 \left (3 a^2-b^2\right ) \tan ^{-1}\left (\frac{b+a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{5/2}}+\frac{\cosh (x)}{b^2}-\frac{3 a^2 \cosh (x)}{b^2 \left (a^2-b^2\right )}-\frac{2 a \sinh (x)}{b^3}+\frac{3 a^3 \sinh (x)}{b^3 \left (a^2-b^2\right )}-\frac{a^3}{b^3 (a+b)^2 \left (1-\tanh \left (\frac{x}{2}\right )\right )}+\frac{a^3}{(a-b)^2 b^3 \left (1+\tanh \left (\frac{x}{2}\right )\right )}+\frac{2 a^2 \left (a+b \tanh \left (\frac{x}{2}\right )\right )}{\left (a^2-b^2\right )^2 \left (a+2 b \tanh \left (\frac{x}{2}\right )+a \tanh ^2\left (\frac{x}{2}\right )\right )}-\frac{\left (4 a^2 b\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tanh \left (\frac{x}{2}\right )\right )}{\left (a^2-b^2\right )^2}\\ &=-\frac{3 a^2 \tan ^{-1}\left (\frac{b \cosh (x)+a \sinh (x)}{\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2}}+\frac{2 a^2 b \tan ^{-1}\left (\frac{b+a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+\frac{2 a^2 \left (3 a^2-b^2\right ) \tan ^{-1}\left (\frac{b+a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{5/2}}+\frac{\cosh (x)}{b^2}-\frac{3 a^2 \cosh (x)}{b^2 \left (a^2-b^2\right )}-\frac{2 a \sinh (x)}{b^3}+\frac{3 a^3 \sinh (x)}{b^3 \left (a^2-b^2\right )}-\frac{a^3}{b^3 (a+b)^2 \left (1-\tanh \left (\frac{x}{2}\right )\right )}+\frac{a^3}{(a-b)^2 b^3 \left (1+\tanh \left (\frac{x}{2}\right )\right )}+\frac{2 a^2 \left (a+b \tanh \left (\frac{x}{2}\right )\right )}{\left (a^2-b^2\right )^2 \left (a+2 b \tanh \left (\frac{x}{2}\right )+a \tanh ^2\left (\frac{x}{2}\right )\right )}\\ \end{align*}
Mathematica [A] time = 0.4389, size = 205, normalized size = 1.05 \[ \frac{a \sqrt{a-b} \left (a^2 b+a^3+a b^2+b^3\right ) \cosh ^2(x)+a \left (a^2 \sqrt{a-b} (a+b)-2 b^2 \sqrt{a-b} (a+b) \sinh ^2(x)+6 a b^2 \sqrt{a+b} \sinh (x) \tan ^{-1}\left (\frac{a \tanh \left (\frac{x}{2}\right )+b}{\sqrt{a-b} \sqrt{a+b}}\right )\right )-b \cosh (x) \left ((a-b)^{3/2} (a+b)^2 \sinh (x)-6 a^3 \sqrt{a+b} \tan ^{-1}\left (\frac{a \tanh \left (\frac{x}{2}\right )+b}{\sqrt{a-b} \sqrt{a+b}}\right )\right )}{(a-b)^{5/2} (a+b)^3 (a \cosh (x)+b \sinh (x))} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.072, size = 164, normalized size = 0.8 \begin{align*}{\frac{1}{ \left ( a-b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}+2\,{\frac{{a}^{2}\tanh \left ( x/2 \right ) b}{ \left ( a-b \right ) ^{2} \left ( a+b \right ) ^{2} \left ( a+2\,\tanh \left ( x/2 \right ) b+a \left ( \tanh \left ( x/2 \right ) \right ) ^{2} \right ) }}+2\,{\frac{{a}^{3}}{ \left ( a-b \right ) ^{2} \left ( a+b \right ) ^{2} \left ( a+2\,\tanh \left ( x/2 \right ) b+a \left ( \tanh \left ( x/2 \right ) \right ) ^{2} \right ) }}+6\,{\frac{{a}^{2}b}{ \left ( a-b \right ) ^{2} \left ( a+b \right ) ^{2}\sqrt{{a}^{2}-{b}^{2}}}\arctan \left ( 1/2\,{\frac{2\,a\tanh \left ( x/2 \right ) +2\,b}{\sqrt{{a}^{2}-{b}^{2}}}} \right ) }-{\frac{1}{ \left ( a+b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.0956, size = 3664, normalized size = 18.79 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15241, size = 235, normalized size = 1.21 \begin{align*} \frac{6 \, a^{2} b \arctan \left (\frac{a e^{x} + b e^{x}}{\sqrt{a^{2} - b^{2}}}\right )}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt{a^{2} - b^{2}}} + \frac{e^{x}}{2 \,{\left (a^{2} + 2 \, a b + b^{2}\right )}} + \frac{5 \, a^{3} e^{\left (2 \, x\right )} + 3 \, a^{2} b e^{\left (2 \, x\right )} + 3 \, a b^{2} e^{\left (2 \, x\right )} + b^{3} e^{\left (2 \, x\right )} + a^{3} + a^{2} b - a b^{2} - b^{3}}{2 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )}{\left (a e^{\left (3 \, x\right )} + b e^{\left (3 \, x\right )} + a e^{x} - b e^{x}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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