Optimal. Leaf size=156 \[ -\frac {2 (c \cosh (x)+b \sinh (x))}{\left (a^2-b^2+c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}-\frac {2 i E\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}{\left (a^2-b^2+c^2\right ) \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}} \]
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Rubi [A]
time = 0.08, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3207, 3198,
2732} \begin {gather*} -\frac {2 (b \sinh (x)+c \cosh (x))}{\left (a^2-b^2+c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}-\frac {2 i \sqrt {a+b \cosh (x)+c \sinh (x)} E\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right )}{\left (a^2-b^2+c^2\right ) \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2732
Rule 3198
Rule 3207
Rubi steps
\begin {align*} \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{3/2}} \, dx &=-\frac {2 (c \cosh (x)+b \sinh (x))}{\left (a^2-b^2+c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}+\frac {\int \sqrt {a+b \cosh (x)+c \sinh (x)} \, dx}{a^2-b^2+c^2}\\ &=-\frac {2 (c \cosh (x)+b \sinh (x))}{\left (a^2-b^2+c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}+\frac {\sqrt {a+b \cosh (x)+c \sinh (x)} \int \sqrt {\frac {a}{a+\sqrt {b^2-c^2}}+\frac {\sqrt {b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}{a+\sqrt {b^2-c^2}}} \, dx}{\left (a^2-b^2+c^2\right ) \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}\\ &=-\frac {2 (c \cosh (x)+b \sinh (x))}{\left (a^2-b^2+c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}-\frac {2 i E\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}{\left (a^2-b^2+c^2\right ) \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in
optimal.
time = 5.00, size = 806, normalized size = 5.17 \begin {gather*} \frac {2 \left (a b+\left (b^2-c^2\right ) \cosh (x)\right )-\frac {2 b^3 (a+b \cosh (x)+c \sinh (x))}{b^2-c^2}+\frac {2 a F_1\left (\frac {1}{2};\frac {1}{2},\frac {1}{2};\frac {3}{2};\frac {a+b \cosh (x)+c \sinh (x)}{a+i \sqrt {1-\frac {b^2}{c^2}} c},\frac {a+b \cosh (x)+c \sinh (x)}{a-i \sqrt {1-\frac {b^2}{c^2}} c}\right ) \text {sech}\left (x+\tanh ^{-1}\left (\frac {b}{c}\right )\right ) (a+b \cosh (x)+c \sinh (x)) \sqrt {-\frac {-i \sqrt {1-\frac {b^2}{c^2}} c+b \cosh (x)+c \sinh (x)}{a+i \sqrt {1-\frac {b^2}{c^2}} c}} \sqrt {-\frac {i \sqrt {1-\frac {b^2}{c^2}} c+b \cosh (x)+c \sinh (x)}{a-i \sqrt {1-\frac {b^2}{c^2}} c}}}{\sqrt {1-\frac {b^2}{c^2}}}+\frac {b c \sinh \left (x+\tanh ^{-1}\left (\frac {c}{b}\right )\right )}{\sqrt {1-\frac {c^2}{b^2}}}-\frac {b c F_1\left (-\frac {1}{2};-\frac {1}{2},-\frac {1}{2};\frac {1}{2};\frac {a+b \cosh (x)+c \sinh (x)}{a+b \sqrt {1-\frac {c^2}{b^2}}},\frac {a+b \cosh (x)+c \sinh (x)}{a-b \sqrt {1-\frac {c^2}{b^2}}}\right ) \sinh \left (x+\tanh ^{-1}\left (\frac {c}{b}\right )\right )}{\sqrt {1-\frac {c^2}{b^2}} \sqrt {\frac {b \sqrt {1-\frac {c^2}{b^2}}-b \cosh (x)-c \sinh (x)}{a+b \sqrt {1-\frac {c^2}{b^2}}}} \sqrt {\frac {b \sqrt {1-\frac {c^2}{b^2}}+b \cosh (x)+c \sinh (x)}{-a+b \sqrt {1-\frac {c^2}{b^2}}}}}+\frac {c^2 \left (\frac {2 b^2 (a+b \cosh (x)+c \sinh (x))}{b^2-c^2}-\frac {c \sinh \left (x+\tanh ^{-1}\left (\frac {c}{b}\right )\right )}{\sqrt {1-\frac {c^2}{b^2}}}+\frac {c F_1\left (-\frac {1}{2};-\frac {1}{2},-\frac {1}{2};\frac {1}{2};\frac {a+b \cosh (x)+c \sinh (x)}{a+b \sqrt {1-\frac {c^2}{b^2}}},\frac {a+b \cosh (x)+c \sinh (x)}{a-b \sqrt {1-\frac {c^2}{b^2}}}\right ) \sinh \left (x+\tanh ^{-1}\left (\frac {c}{b}\right )\right )}{\sqrt {1-\frac {c^2}{b^2}} \sqrt {\frac {b \sqrt {1-\frac {c^2}{b^2}}-b \cosh (x)-c \sinh (x)}{a+b \sqrt {1-\frac {c^2}{b^2}}}} \sqrt {\frac {b \sqrt {1-\frac {c^2}{b^2}}+b \cosh (x)+c \sinh (x)}{-a+b \sqrt {1-\frac {c^2}{b^2}}}}}\right )}{b}}{c \left (a^2-b^2+c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1431\) vs.
\(2(177)=354\).
time = 3.38, size = 1432, normalized size = 9.18
method | result | size |
default | \(\text {Expression too large to display}\) | \(1432\) |
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 [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.22, size = 798, normalized size = 5.12 \begin {gather*} \frac {2 \, {\left ({\left (2 \, \sqrt {2} a^{2} \cosh \left (x\right ) + \sqrt {2} {\left (a b + a c\right )} \cosh \left (x\right )^{2} + \sqrt {2} {\left (a b + a c\right )} \sinh \left (x\right )^{2} + 2 \, {\left (\sqrt {2} a^{2} + \sqrt {2} {\left (a b + a c\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + \sqrt {2} {\left (a b - a c\right )}\right )} \sqrt {b + c} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2} + 3 \, c^{2}\right )}}{3 \, {\left (b^{2} + 2 \, b c + c^{2}\right )}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2} + 9 \, a c^{2}\right )}}{27 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )}}, \frac {3 \, {\left (b + c\right )} \cosh \left (x\right ) + 3 \, {\left (b + c\right )} \sinh \left (x\right ) + 2 \, a}{3 \, {\left (b + c\right )}}\right ) - 3 \, {\left (\sqrt {2} {\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{2} + \sqrt {2} {\left (b^{2} + 2 \, b c + c^{2}\right )} \sinh \left (x\right )^{2} + 2 \, \sqrt {2} {\left (a b + a c\right )} \cosh \left (x\right ) + 2 \, {\left (\sqrt {2} {\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right ) + \sqrt {2} {\left (a b + a c\right )}\right )} \sinh \left (x\right ) + \sqrt {2} {\left (b^{2} - c^{2}\right )}\right )} \sqrt {b + c} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2} + 3 \, c^{2}\right )}}{3 \, {\left (b^{2} + 2 \, b c + c^{2}\right )}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2} + 9 \, a c^{2}\right )}}{27 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2} + 3 \, c^{2}\right )}}{3 \, {\left (b^{2} + 2 \, b c + c^{2}\right )}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2} + 9 \, a c^{2}\right )}}{27 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )}}, \frac {3 \, {\left (b + c\right )} \cosh \left (x\right ) + 3 \, {\left (b + c\right )} \sinh \left (x\right ) + 2 \, a}{3 \, {\left (b + c\right )}}\right )\right ) - 6 \, {\left ({\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{2} + {\left (b^{2} + 2 \, b c + c^{2}\right )} \sinh \left (x\right )^{2} + {\left (a b + a c\right )} \cosh \left (x\right ) + {\left (a b + a c + 2 \, {\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {b \cosh \left (x\right ) + c \sinh \left (x\right ) + a}\right )}}{3 \, {\left (a^{2} b^{2} - b^{4} - c^{4} - {\left (a^{2} - 2 \, b^{2}\right )} c^{2} + {\left (a^{2} b^{2} - b^{4} + a^{2} c^{2} + 2 \, b c^{3} + c^{4} + 2 \, {\left (a^{2} b - b^{3}\right )} c\right )} \cosh \left (x\right )^{2} + {\left (a^{2} b^{2} - b^{4} + a^{2} c^{2} + 2 \, b c^{3} + c^{4} + 2 \, {\left (a^{2} b - b^{3}\right )} c\right )} \sinh \left (x\right )^{2} + 2 \, {\left (a^{3} b - a b^{3} + a b c^{2} + a c^{3} + {\left (a^{3} - a b^{2}\right )} c\right )} \cosh \left (x\right ) + 2 \, {\left (a^{3} b - a b^{3} + a b c^{2} + a c^{3} + {\left (a^{3} - a b^{2}\right )} c + {\left (a^{2} b^{2} - b^{4} + a^{2} c^{2} + 2 \, b c^{3} + c^{4} + 2 \, {\left (a^{2} b - b^{3}\right )} c\right )} \cosh \left (x\right )\right )} \sinh \left (x\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 \frac {1}{\left (a + b \cosh {\left (x \right )} + c \sinh {\left (x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 \frac {1}{{\left (a+b\,\mathrm {cosh}\left (x\right )+c\,\mathrm {sinh}\left (x\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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