Optimal. Leaf size=294 \[ \frac {16}{15} (a c \cosh (x)+a b \sinh (x)) \sqrt {a+b \cosh (x)+c \sinh (x)}+\frac {2}{5} (c \cosh (x)+b \sinh (x)) (a+b \cosh (x)+c \sinh (x))^{3/2}-\frac {2 i \left (23 a^2+9 b^2-9 c^2\right ) E\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}{15 \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}+\frac {16 i a \left (a^2-b^2+c^2\right ) F\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right ) \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}{15 \sqrt {a+b \cosh (x)+c \sinh (x)}} \]
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Rubi [A]
time = 0.36, antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3199, 3225,
3228, 3198, 2732, 3206, 2740} \begin {gather*} \frac {16 i a \left (a^2-b^2+c^2\right ) \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}} F\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 )}{15 \sqrt {a+b \cosh (x)+c \sinh (x)}}-\frac {2 i \left (23 a^2+9 b^2-9 c^2\right ) \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 )}{15 \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}+\frac {2}{5} (b \sinh (x)+c \cosh (x)) (a+b \cosh (x)+c \sinh (x))^{3/2}+\frac {16}{15} (a b \sinh (x)+a c \cosh (x)) \sqrt {a+b \cosh (x)+c \sinh (x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2732
Rule 2740
Rule 3198
Rule 3199
Rule 3206
Rule 3225
Rule 3228
Rubi steps
\begin {align*} \int (a+b \cosh (x)+c \sinh (x))^{5/2} \, dx &=\frac {2}{5} (c \cosh (x)+b \sinh (x)) (a+b \cosh (x)+c \sinh (x))^{3/2}+\frac {2}{5} \int \sqrt {a+b \cosh (x)+c \sinh (x)} \left (\frac {1}{2} \left (5 a^2+3 b^2-3 c^2\right )+4 a b \cosh (x)+4 a c \sinh (x)\right ) \, dx\\ &=\frac {16}{15} (a c \cosh (x)+a b \sinh (x)) \sqrt {a+b \cosh (x)+c \sinh (x)}+\frac {2}{5} (c \cosh (x)+b \sinh (x)) (a+b \cosh (x)+c \sinh (x))^{3/2}+\frac {4 \int \frac {\frac {1}{4} a^2 \left (15 a^2+17 b^2-17 c^2\right )+\frac {1}{4} a b \left (23 a^2+9 b^2-9 c^2\right ) \cosh (x)+\frac {1}{4} a c \left (23 a^2+9 b^2-9 c^2\right ) \sinh (x)}{\sqrt {a+b \cosh (x)+c \sinh (x)}} \, dx}{15 a}\\ &=\frac {16}{15} (a c \cosh (x)+a b \sinh (x)) \sqrt {a+b \cosh (x)+c \sinh (x)}+\frac {2}{5} (c \cosh (x)+b \sinh (x)) (a+b \cosh (x)+c \sinh (x))^{3/2}+\frac {1}{15} \left (23 a^2+9 b^2-9 c^2\right ) \int \sqrt {a+b \cosh (x)+c \sinh (x)} \, dx-\frac {1}{15} \left (8 a \left (a^2-b^2+c^2\right )\right ) \int \frac {1}{\sqrt {a+b \cosh (x)+c \sinh (x)}} \, dx\\ &=\frac {16}{15} (a c \cosh (x)+a b \sinh (x)) \sqrt {a+b \cosh (x)+c \sinh (x)}+\frac {2}{5} (c \cosh (x)+b \sinh (x)) (a+b \cosh (x)+c \sinh (x))^{3/2}+\frac {\left (\left (23 a^2+9 b^2-9 c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}\right ) \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}{15 \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}-\frac {\left (8 a \left (a^2-b^2+c^2\right ) \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}\right ) \int \frac {1}{\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}{15 \sqrt {a+b \cosh (x)+c \sinh (x)}}\\ &=\frac {16}{15} (a c \cosh (x)+a b \sinh (x)) \sqrt {a+b \cosh (x)+c \sinh (x)}+\frac {2}{5} (c \cosh (x)+b \sinh (x)) (a+b \cosh (x)+c \sinh (x))^{3/2}-\frac {2 i \left (23 a^2+9 b^2-9 c^2\right ) 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)}}{15 \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}+\frac {16 i a \left (a^2-b^2+c^2\right ) F\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 {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}{15 \sqrt {a+b \cosh (x)+c \sinh (x)}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in
optimal.
time = 6.25, size = 3775, normalized size = 12.84 \begin {gather*} \text {Result too large to show} \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. \(898\) vs.
\(2(328)=656\).
time = 2.87, size = 899, normalized size = 3.06
method | result | size |
default | \(-\frac {\sqrt {\left (b -c \right ) \left (b +c \right )}\, \left (\frac {\left (\cosh ^{3}\left (x \right )\right ) b^{2}}{3}-\frac {\left (\cosh ^{3}\left (x \right )\right ) c^{2}}{3}+3 a^{2} \cosh \left (x \right )-b^{2} \cosh \left (x \right )+c^{2} \cosh \left (x \right )\right )}{\sqrt {\frac {-\sinh \left (x \right ) b^{2}+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}}{\sqrt {b^{2}-c^{2}}}}}+\frac {\sqrt {\frac {\left (-\sinh \left (x \right ) b^{2}+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}\right ) \left (\sinh ^{2}\left (x \right )\right )}{\sqrt {b^{2}-c^{2}}}}\, \left (\frac {a \cosh \left (x \right ) \sqrt {b^{2}-c^{2}}\, \sqrt {\frac {\left (-\sinh \left (x \right ) b^{2}+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}\right ) \left (\sinh ^{2}\left (x \right )\right )}{\sqrt {b^{2}-c^{2}}}}\, b^{2}}{-2 \sinh \left (x \right ) b^{2}+2 \sinh \left (x \right ) c^{2}+2 a \sqrt {b^{2}-c^{2}}}-\frac {a \cosh \left (x \right ) \sqrt {b^{2}-c^{2}}\, \sqrt {\frac {\left (-\sinh \left (x \right ) b^{2}+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}\right ) \left (\sinh ^{2}\left (x \right )\right )}{\sqrt {b^{2}-c^{2}}}}\, c^{2}}{2 \left (-\sinh \left (x \right ) b^{2}+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}\right )}-\frac {a \ln \left (\frac {\left (-\sinh \left (x \right ) b^{2}+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}\right ) \cosh \left (x \right )}{\sqrt {b^{2}-c^{2}}\, \sqrt {\frac {-\sinh \left (x \right ) b^{2}+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}}{\sqrt {b^{2}-c^{2}}}}}+\sqrt {\frac {\left (-\sinh \left (x \right ) b^{2}+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}\right ) \left (\sinh ^{2}\left (x \right )\right )}{\sqrt {b^{2}-c^{2}}}}\right ) b^{2}}{2 \sqrt {\frac {-\sinh \left (x \right ) b^{2}+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}}{\sqrt {b^{2}-c^{2}}}}}+\frac {a \ln \left (\frac {\left (-\sinh \left (x \right ) b^{2}+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}\right ) \cosh \left (x \right )}{\sqrt {b^{2}-c^{2}}\, \sqrt {\frac {-\sinh \left (x \right ) b^{2}+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}}{\sqrt {b^{2}-c^{2}}}}}+\sqrt {\frac {\left (-\sinh \left (x \right ) b^{2}+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}\right ) \left (\sinh ^{2}\left (x \right )\right )}{\sqrt {b^{2}-c^{2}}}}\right ) c^{2}}{2 \sqrt {\frac {-\sinh \left (x \right ) b^{2}+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}}{\sqrt {b^{2}-c^{2}}}}}+\frac {a^{3} \ln \left (\frac {\left (-\sinh \left (x \right ) b^{2}+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}\right ) \cosh \left (x \right )}{\sqrt {b^{2}-c^{2}}\, \sqrt {\frac {-\sinh \left (x \right ) b^{2}+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}}{\sqrt {b^{2}-c^{2}}}}}+\sqrt {\frac {\left (-\sinh \left (x \right ) b^{2}+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}\right ) \left (\sinh ^{2}\left (x \right )\right )}{\sqrt {b^{2}-c^{2}}}}\right )}{\sqrt {\frac {-\sinh \left (x \right ) b^{2}+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}}{\sqrt {b^{2}-c^{2}}}}}\right )}{\sinh \left (x \right ) \sqrt {\frac {-\sinh \left (x \right ) b^{2}+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}}{\sqrt {b^{2}-c^{2}}}}}\) | \(899\) |
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 = 0.14, size = 928, normalized size = 3.16 \begin {gather*} -\frac {4 \, {\left (\sqrt {2} {\left (a^{3} - 33 \, a b^{2} + 33 \, a c^{2}\right )} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} {\left (a^{3} - 33 \, a b^{2} + 33 \, a c^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} {\left (a^{3} - 33 \, a b^{2} + 33 \, a c^{2}\right )} \sinh \left (x\right )^{2}\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 ) + 12 \, {\left (\sqrt {2} {\left (23 \, a^{2} b + 9 \, b^{3} - 9 \, b c^{2} - 9 \, c^{3} + {\left (23 \, a^{2} + 9 \, b^{2}\right )} c\right )} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} {\left (23 \, a^{2} b + 9 \, b^{3} - 9 \, b c^{2} - 9 \, c^{3} + {\left (23 \, a^{2} + 9 \, b^{2}\right )} c\right )} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} {\left (23 \, a^{2} b + 9 \, b^{3} - 9 \, b c^{2} - 9 \, c^{3} + {\left (23 \, a^{2} + 9 \, b^{2}\right )} c\right )} \sinh \left (x\right )^{2}\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 ) - 3 \, {\left (3 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )} \cosh \left (x\right )^{4} + 3 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )} \sinh \left (x\right )^{4} + 22 \, {\left (a b^{2} + 2 \, a b c + a c^{2}\right )} \cosh \left (x\right )^{3} + 2 \, {\left (11 \, a b^{2} + 22 \, a b c + 11 \, a c^{2} + 6 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} - 3 \, b^{3} + 3 \, b^{2} c + 3 \, b c^{2} - 3 \, c^{3} - 4 \, {\left (23 \, a^{2} b + 9 \, b^{3} - 9 \, b c^{2} - 9 \, c^{3} + {\left (23 \, a^{2} + 9 \, b^{2}\right )} c\right )} \cosh \left (x\right )^{2} - 2 \, {\left (46 \, a^{2} b + 18 \, b^{3} - 18 \, b c^{2} - 18 \, c^{3} - 9 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (23 \, a^{2} + 9 \, b^{2}\right )} c - 33 \, {\left (a b^{2} + 2 \, a b c + a c^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} - 22 \, {\left (a b^{2} - a c^{2}\right )} \cosh \left (x\right ) + 2 \, {\left (6 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )} \cosh \left (x\right )^{3} - 11 \, a b^{2} + 11 \, a c^{2} + 33 \, {\left (a b^{2} + 2 \, a b c + a c^{2}\right )} \cosh \left (x\right )^{2} - 4 \, {\left (23 \, a^{2} b + 9 \, b^{3} - 9 \, b c^{2} - 9 \, c^{3} + {\left (23 \, a^{2} + 9 \, b^{2}\right )} c\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {b \cosh \left (x\right ) + c \sinh \left (x\right ) + a}}{90 \, {\left ({\left (b + c\right )} \cosh \left (x\right )^{2} + 2 \, {\left (b + c\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (b + c\right )} \sinh \left (x\right )^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = -0.00 \begin {gather*} \int {\left (a+b\,\mathrm {cosh}\left (x\right )+c\,\mathrm {sinh}\left (x\right )\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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