Optimal. Leaf size=90 \[ -\frac {(a-b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a-b}}\right )}{f}+\frac {(a-b) \sqrt {a+b \sinh ^2(e+f x)}}{f}+\frac {\left (a+b \sinh ^2(e+f x)\right )^{3/2}}{3 f} \]
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Rubi [A]
time = 0.06, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3273, 52, 65,
214} \begin {gather*} \frac {(a-b) \sqrt {a+b \sinh ^2(e+f x)}}{f}+\frac {\left (a+b \sinh ^2(e+f x)\right )^{3/2}}{3 f}-\frac {(a-b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a-b}}\right )}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 214
Rule 3273
Rubi steps
\begin {align*} \int \left (a+b \sinh ^2(e+f x)\right )^{3/2} \tanh (e+f x) \, dx &=\frac {\text {Subst}\left (\int \frac {(a+b x)^{3/2}}{1+x} \, dx,x,\sinh ^2(e+f x)\right )}{2 f}\\ &=\frac {\left (a+b \sinh ^2(e+f x)\right )^{3/2}}{3 f}+\frac {(a-b) \text {Subst}\left (\int \frac {\sqrt {a+b x}}{1+x} \, dx,x,\sinh ^2(e+f x)\right )}{2 f}\\ &=\frac {(a-b) \sqrt {a+b \sinh ^2(e+f x)}}{f}+\frac {\left (a+b \sinh ^2(e+f x)\right )^{3/2}}{3 f}+\frac {(a-b)^2 \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x}} \, dx,x,\sinh ^2(e+f x)\right )}{2 f}\\ &=\frac {(a-b) \sqrt {a+b \sinh ^2(e+f x)}}{f}+\frac {\left (a+b \sinh ^2(e+f x)\right )^{3/2}}{3 f}+\frac {(a-b)^2 \text {Subst}\left (\int \frac {1}{1-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^2(e+f x)}\right )}{b f}\\ &=-\frac {(a-b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a-b}}\right )}{f}+\frac {(a-b) \sqrt {a+b \sinh ^2(e+f x)}}{f}+\frac {\left (a+b \sinh ^2(e+f x)\right )^{3/2}}{3 f}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 86, normalized size = 0.96 \begin {gather*} \frac {-3 (a-b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a-b+b \cosh ^2(e+f x)}}{\sqrt {a-b}}\right )+\left (4 a-4 b+b \cosh ^2(e+f x)\right ) \sqrt {a-b+b \cosh ^2(e+f x)}}{3 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.88, size = 69, normalized size = 0.77
method | result | size |
default | \(\frac {\mathit {`\,int/indef0`\,}\left (\frac {\sinh \left (f x +e \right ) \left (b^{2} \left (\sinh ^{4}\left (f x +e \right )\right )+2 a b \left (\sinh ^{2}\left (f x +e \right )\right )+a^{2}\right )}{\cosh \left (f x +e \right )^{2} \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}}, \sinh \left (f x +e \right )\right )}{f}\) | \(69\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 428 vs.
\(2 (78) = 156\).
time = 0.86, size = 1052, normalized size = 11.69 \begin {gather*} \left [-\frac {12 \, {\left ({\left (a - b\right )} \cosh \left (f x + e\right )^{3} + 3 \, {\left (a - b\right )} \cosh \left (f x + e\right )^{2} \sinh \left (f x + e\right ) + 3 \, {\left (a - b\right )} \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{2} + {\left (a - b\right )} \sinh \left (f x + e\right )^{3}\right )} \sqrt {a - b} \log \left (\frac {b \cosh \left (f x + e\right )^{4} + 4 \, b \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + b \sinh \left (f x + e\right )^{4} + 2 \, {\left (4 \, a - 3 \, b\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (3 \, b \cosh \left (f x + e\right )^{2} + 4 \, a - 3 \, b\right )} \sinh \left (f x + e\right )^{2} + 4 \, \sqrt {2} \sqrt {a - b} \sqrt {\frac {b \cosh \left (f x + e\right )^{2} + b \sinh \left (f x + e\right )^{2} + 2 \, a - b}{\cosh \left (f x + e\right )^{2} - 2 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + \sinh \left (f x + e\right )^{2}}} {\left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right )} + 4 \, {\left (b \cosh \left (f x + e\right )^{3} + {\left (4 \, a - 3 \, b\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right ) + b}{\cosh \left (f x + e\right )^{4} + 4 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + \sinh \left (f x + e\right )^{4} + 2 \, {\left (3 \, \cosh \left (f x + e\right )^{2} + 1\right )} \sinh \left (f x + e\right )^{2} + 2 \, \cosh \left (f x + e\right )^{2} + 4 \, {\left (\cosh \left (f x + e\right )^{3} + \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right ) + 1}\right ) - \sqrt {2} {\left (b \cosh \left (f x + e\right )^{4} + 4 \, b \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + b \sinh \left (f x + e\right )^{4} + 2 \, {\left (8 \, a - 7 \, b\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (3 \, b \cosh \left (f x + e\right )^{2} + 8 \, a - 7 \, b\right )} \sinh \left (f x + e\right )^{2} + 4 \, {\left (b \cosh \left (f x + e\right )^{3} + {\left (8 \, a - 7 \, b\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right ) + b\right )} \sqrt {\frac {b \cosh \left (f x + e\right )^{2} + b \sinh \left (f x + e\right )^{2} + 2 \, a - b}{\cosh \left (f x + e\right )^{2} - 2 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + \sinh \left (f x + e\right )^{2}}}}{24 \, {\left (f \cosh \left (f x + e\right )^{3} + 3 \, f \cosh \left (f x + e\right )^{2} \sinh \left (f x + e\right ) + 3 \, f \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{2} + f \sinh \left (f x + e\right )^{3}\right )}}, -\frac {24 \, {\left ({\left (a - b\right )} \cosh \left (f x + e\right )^{3} + 3 \, {\left (a - b\right )} \cosh \left (f x + e\right )^{2} \sinh \left (f x + e\right ) + 3 \, {\left (a - b\right )} \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{2} + {\left (a - b\right )} \sinh \left (f x + e\right )^{3}\right )} \sqrt {-a + b} \arctan \left (-\frac {\sqrt {2} \sqrt {-a + b} \sqrt {\frac {b \cosh \left (f x + e\right )^{2} + b \sinh \left (f x + e\right )^{2} + 2 \, a - b}{\cosh \left (f x + e\right )^{2} - 2 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + \sinh \left (f x + e\right )^{2}}}}{2 \, {\left ({\left (a - b\right )} \cosh \left (f x + e\right ) + {\left (a - b\right )} \sinh \left (f x + e\right )\right )}}\right ) - \sqrt {2} {\left (b \cosh \left (f x + e\right )^{4} + 4 \, b \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + b \sinh \left (f x + e\right )^{4} + 2 \, {\left (8 \, a - 7 \, b\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (3 \, b \cosh \left (f x + e\right )^{2} + 8 \, a - 7 \, b\right )} \sinh \left (f x + e\right )^{2} + 4 \, {\left (b \cosh \left (f x + e\right )^{3} + {\left (8 \, a - 7 \, b\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right ) + b\right )} \sqrt {\frac {b \cosh \left (f x + e\right )^{2} + b \sinh \left (f x + e\right )^{2} + 2 \, a - b}{\cosh \left (f x + e\right )^{2} - 2 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + \sinh \left (f x + e\right )^{2}}}}{24 \, {\left (f \cosh \left (f x + e\right )^{3} + 3 \, f \cosh \left (f x + e\right )^{2} \sinh \left (f x + e\right ) + 3 \, f \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{2} + f \sinh \left (f x + e\right )^{3}\right )}}\right ] \end {gather*}
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 \begin {gather*} \int \left (a + b \sinh ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}} \tanh {\left (e + f x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \mathrm {tanh}\left (e+f\,x\right )\,{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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