Optimal. Leaf size=104 \[ -\frac {a x \left (2 a^2-3 b^2\right )}{2 b^4}+\frac {\sinh (x) \left (2 \left (a^2-b^2\right )-a b \cosh (x)\right )}{2 b^3}+\frac {2 (a-b)^{3/2} (a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{b^4}+\frac {\sinh ^3(x)}{3 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.24, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2695, 2865, 2735, 2659, 208} \[ -\frac {a x \left (2 a^2-3 b^2\right )}{2 b^4}+\frac {\sinh (x) \left (2 \left (a^2-b^2\right )-a b \cosh (x)\right )}{2 b^3}+\frac {2 (a-b)^{3/2} (a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{b^4}+\frac {\sinh ^3(x)}{3 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 208
Rule 2659
Rule 2695
Rule 2735
Rule 2865
Rubi steps
\begin {align*} \int \frac {\sinh ^4(x)}{a+b \cosh (x)} \, dx &=\frac {\sinh ^3(x)}{3 b}+\frac {\int \frac {(-b-a \cosh (x)) \sinh ^2(x)}{a+b \cosh (x)} \, dx}{b}\\ &=\frac {\left (2 \left (a^2-b^2\right )-a b \cosh (x)\right ) \sinh (x)}{2 b^3}+\frac {\sinh ^3(x)}{3 b}-\frac {\int \frac {b \left (a^2-2 b^2\right )+a \left (2 a^2-3 b^2\right ) \cosh (x)}{a+b \cosh (x)} \, dx}{2 b^3}\\ &=-\frac {a \left (2 a^2-3 b^2\right ) x}{2 b^4}+\frac {\left (2 \left (a^2-b^2\right )-a b \cosh (x)\right ) \sinh (x)}{2 b^3}+\frac {\sinh ^3(x)}{3 b}+\frac {\left (a^2-b^2\right )^2 \int \frac {1}{a+b \cosh (x)} \, dx}{b^4}\\ &=-\frac {a \left (2 a^2-3 b^2\right ) x}{2 b^4}+\frac {\left (2 \left (a^2-b^2\right )-a b \cosh (x)\right ) \sinh (x)}{2 b^3}+\frac {\sinh ^3(x)}{3 b}+\frac {\left (2 \left (a^2-b^2\right )^2\right ) \operatorname {Subst}\left (\int \frac {1}{a+b-(a-b) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{b^4}\\ &=-\frac {a \left (2 a^2-3 b^2\right ) x}{2 b^4}+\frac {2 (a-b)^{3/2} (a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{b^4}+\frac {\left (2 \left (a^2-b^2\right )-a b \cosh (x)\right ) \sinh (x)}{2 b^3}+\frac {\sinh ^3(x)}{3 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.19, size = 95, normalized size = 0.91 \[ \frac {-12 a^3 x-24 \left (b^2-a^2\right )^{3/2} \tan ^{-1}\left (\frac {(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {b^2-a^2}}\right )+12 a^2 b \sinh (x)+18 a b^2 x-3 a b^2 \sinh (2 x)-15 b^3 \sinh (x)+b^3 \sinh (3 x)}{12 b^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 2.78, size = 1099, normalized size = 10.57 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.12, size = 146, normalized size = 1.40 \[ \frac {b^{2} e^{\left (3 \, x\right )} - 3 \, a b e^{\left (2 \, x\right )} + 12 \, a^{2} e^{x} - 15 \, b^{2} e^{x}}{24 \, b^{3}} - \frac {{\left (2 \, a^{3} - 3 \, a b^{2}\right )} x}{2 \, b^{4}} + \frac {{\left (3 \, a b^{2} e^{x} - b^{3} - 3 \, {\left (4 \, a^{2} b - 5 \, b^{3}\right )} e^{\left (2 \, x\right )}\right )} e^{\left (-3 \, x\right )}}{24 \, b^{4}} + \frac {2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \arctan \left (\frac {b e^{x} + a}{\sqrt {-a^{2} + b^{2}}}\right )}{\sqrt {-a^{2} + b^{2}} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.08, size = 338, normalized size = 3.25 \[ -\frac {1}{3 b \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {a}{2 b^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {1}{2 b \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {a^{2}}{b^{3} \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {a}{2 b^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {1}{b \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {a^{3} \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{b^{4}}-\frac {3 a \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 b^{2}}-\frac {1}{3 b \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {a}{2 b^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {1}{2 b \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {a^{2}}{b^{3} \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {a}{2 b^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {1}{b \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {a^{3} \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{b^{4}}+\frac {3 a \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2 b^{2}}+\frac {2 \arctanh \left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right ) a^{4}}{b^{4} \sqrt {\left (a +b \right ) \left (a -b \right )}}-\frac {4 \arctanh \left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right ) a^{2}}{b^{2} \sqrt {\left (a +b \right ) \left (a -b \right )}}+\frac {2 \arctanh \left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.31, size = 222, normalized size = 2.13 \[ \frac {{\mathrm {e}}^{3\,x}}{24\,b}-\frac {{\mathrm {e}}^{-3\,x}}{24\,b}+\frac {x\,\left (3\,a\,b^2-2\,a^3\right )}{2\,b^4}+\frac {{\mathrm {e}}^x\,\left (4\,a^2-5\,b^2\right )}{8\,b^3}+\frac {a\,{\mathrm {e}}^{-2\,x}}{8\,b^2}-\frac {a\,{\mathrm {e}}^{2\,x}}{8\,b^2}-\frac {{\mathrm {e}}^{-x}\,\left (4\,a^2-5\,b^2\right )}{8\,b^3}+\frac {\ln \left (-\frac {2\,{\mathrm {e}}^x\,\left (a^4-2\,a^2\,b^2+b^4\right )}{b^5}-\frac {2\,{\left (a+b\right )}^{3/2}\,\left (b+a\,{\mathrm {e}}^x\right )\,{\left (a-b\right )}^{3/2}}{b^5}\right )\,{\left (a+b\right )}^{3/2}\,{\left (a-b\right )}^{3/2}}{b^4}-\frac {\ln \left (\frac {2\,{\left (a+b\right )}^{3/2}\,\left (b+a\,{\mathrm {e}}^x\right )\,{\left (a-b\right )}^{3/2}}{b^5}-\frac {2\,{\mathrm {e}}^x\,\left (a^4-2\,a^2\,b^2+b^4\right )}{b^5}\right )\,{\left (a+b\right )}^{3/2}\,{\left (a-b\right )}^{3/2}}{b^4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________