Optimal. Leaf size=194 \[ \frac {3 a (b B-c C) \tanh ^{-1}\left (\frac {c-(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2+c^2}}\right )}{\left (a^2-b^2+c^2\right )^{5/2}}-\frac {B c-b C-a C \cosh (x)-a B \sinh (x)}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}-\frac {a (B c-b C)-\left (2 b B c+\left (a^2-2 c^2\right ) C\right ) \cosh (x)-\left (a^2 B+2 b (b B-c C)\right ) \sinh (x)}{2 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))} \]
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Rubi [A]
time = 0.17, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {3235, 3232,
3203, 632, 212} \begin {gather*} \frac {3 a (b B-c C) \tanh ^{-1}\left (\frac {c-(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2+c^2}}\right )}{\left (a^2-b^2+c^2\right )^{5/2}}-\frac {-\cosh (x) \left (C \left (a^2-2 c^2\right )+2 b B c\right )-\sinh (x) \left (a^2 B+2 b (b B-c C)\right )+a (B c-b C)}{2 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))}-\frac {-a B \sinh (x)-a C \cosh (x)-b C+B c}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 3203
Rule 3232
Rule 3235
Rubi steps
\begin {align*} \int \frac {B \cosh (x)+C \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^3} \, dx &=-\frac {B c-b C-a C \cosh (x)-a B \sinh (x)}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}-\frac {\int \frac {2 (b B-c C)-a B \cosh (x)-a C \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^2} \, dx}{2 \left (a^2-b^2+c^2\right )}\\ &=-\frac {B c-b C-a C \cosh (x)-a B \sinh (x)}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}-\frac {a (B c-b C)-\left (2 b B c+\left (a^2-2 c^2\right ) C\right ) \cosh (x)-\left (a^2 B+2 b (b B-c C)\right ) \sinh (x)}{2 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))}-\frac {(3 a (b B-c C)) \int \frac {1}{a+b \cosh (x)+c \sinh (x)} \, dx}{2 \left (a^2-b^2+c^2\right )^2}\\ &=-\frac {B c-b C-a C \cosh (x)-a B \sinh (x)}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}-\frac {a (B c-b C)-\left (2 b B c+\left (a^2-2 c^2\right ) C\right ) \cosh (x)-\left (a^2 B+2 b (b B-c C)\right ) \sinh (x)}{2 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))}-\frac {(3 a (b B-c C)) \text {Subst}\left (\int \frac {1}{a+b+2 c x-(a-b) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{\left (a^2-b^2+c^2\right )^2}\\ &=-\frac {B c-b C-a C \cosh (x)-a B \sinh (x)}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}-\frac {a (B c-b C)-\left (2 b B c+\left (a^2-2 c^2\right ) C\right ) \cosh (x)-\left (a^2 B+2 b (b B-c C)\right ) \sinh (x)}{2 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))}+\frac {(6 a (b B-c C)) \text {Subst}\left (\int \frac {1}{4 \left (a^2-b^2+c^2\right )-x^2} \, dx,x,2 c+2 (-a+b) \tanh \left (\frac {x}{2}\right )\right )}{\left (a^2-b^2+c^2\right )^2}\\ &=\frac {3 a (b B-c C) \tanh ^{-1}\left (\frac {c-(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2+c^2}}\right )}{\left (a^2-b^2+c^2\right )^{5/2}}-\frac {B c-b C-a C \cosh (x)-a B \sinh (x)}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}-\frac {a (B c-b C)-\left (2 b B c+\left (a^2-2 c^2\right ) C\right ) \cosh (x)-\left (a^2 B+2 b (b B-c C)\right ) \sinh (x)}{2 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))}\\ \end {align*}
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Mathematica [A]
time = 0.48, size = 319, normalized size = 1.64 \begin {gather*} -\frac {3 a (b B-c C) \text {ArcTan}\left (\frac {c+(-a+b) \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^2-c^2}}\right )}{\left (-a^2+b^2-c^2\right )^{5/2}}+\frac {-9 a^2 b B c-2 a^4 C+4 a^2 b^2 C-2 b^4 C+5 a^2 c^2 C+4 b^2 c^2 C-2 c^4 C-6 a b c (b B-c C) \cosh (x)+c \left (a^2+2 b^2-2 c^2\right ) (b B-c C) \cosh (2 x)+4 a^3 b B \sinh (x)+2 a b^3 B \sinh (x)-8 a b B c^2 \sinh (x)-4 a^3 c C \sinh (x)-2 a b^2 c C \sinh (x)+8 a c^3 C \sinh (x)+a^2 b^2 B \sinh (2 x)+2 b^4 B \sinh (2 x)-2 b^2 B c^2 \sinh (2 x)-a^2 b c C \sinh (2 x)-2 b^3 c C \sinh (2 x)+2 b c^3 C \sinh (2 x)}{4 b \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(799\) vs.
\(2(181)=362\).
time = 1.70, size = 800, normalized size = 4.12
method | result | size |
default | \(\frac {-\frac {\left (2 B \,a^{4}-3 B \,a^{3} b +2 B \,a^{2} b^{2}+4 B \,a^{2} c^{2}-3 B a \,b^{3}+2 B \,b^{4}-4 B \,b^{2} c^{2}+2 B \,c^{4}+3 C \,a^{3} c -6 C \,a^{2} b c +3 C a \,b^{2} c \right ) \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{\left (a^{4}-2 a^{2} b^{2}+2 a^{2} c^{2}+b^{4}-2 b^{2} c^{2}+c^{4}\right ) \left (a -b \right )}+\frac {\left (2 B \,a^{4} c -9 B \,a^{3} b c +14 B \,a^{2} b^{2} c +4 B \,a^{2} c^{3}-9 B a \,b^{3} c +2 B \,b^{4} c -4 B \,b^{2} c^{3}+2 B \,c^{5}-2 C \,a^{5}+2 C \,a^{4} b +4 C \,a^{3} b^{2}+5 C \,a^{3} c^{2}-4 C \,a^{2} b^{3}-14 C \,a^{2} b \,c^{2}-2 C a \,b^{4}+13 C a \,b^{2} c^{2}-2 C a \,c^{4}+2 C \,b^{5}-4 C \,b^{3} c^{2}+2 C b \,c^{4}\right ) \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{\left (a^{4}-2 a^{2} b^{2}+2 a^{2} c^{2}+b^{4}-2 b^{2} c^{2}+c^{4}\right ) \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (2 B \,a^{5}-3 B \,a^{4} b +B \,a^{3} b^{2}+4 B \,a^{3} c^{2}+B \,a^{2} b^{3}+8 B \,a^{2} b \,c^{2}-3 B a \,b^{4}-8 B a \,b^{2} c^{2}+2 B a \,c^{4}+2 B \,b^{5}-4 B \,b^{3} c^{2}+2 B b \,c^{4}+5 C \,a^{4} c -5 C \,a^{3} b c -5 C \,a^{2} b^{2} c -4 C \,a^{2} c^{3}+5 C a \,b^{3} c +4 C a b \,c^{3}\right ) \tanh \left (\frac {x}{2}\right )}{\left (a^{4}-2 a^{2} b^{2}+2 a^{2} c^{2}+b^{4}-2 b^{2} c^{2}+c^{4}\right ) \left (a^{2}-2 a b +b^{2}\right )}+\frac {a \left (5 B \,a^{2} b c -5 B \,b^{3} c +2 B b \,c^{3}+2 C \,a^{4}-4 C \,a^{2} b^{2}-C \,a^{2} c^{2}+2 C \,b^{4}+C \,b^{2} c^{2}\right )}{\left (a^{4}-2 a^{2} b^{2}+2 a^{2} c^{2}+b^{4}-2 b^{2} c^{2}+c^{4}\right ) \left (a^{2}-2 a b +b^{2}\right )}}{\left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-b \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 c \tanh \left (\frac {x}{2}\right )-a -b \right )^{2}}+\frac {3 a \left (B b -C c \right ) \arctan \left (\frac {2 \left (a -b \right ) \tanh \left (\frac {x}{2}\right )-2 c}{2 \sqrt {-a^{2}+b^{2}-c^{2}}}\right )}{\left (a^{4}-2 a^{2} b^{2}+2 a^{2} c^{2}+b^{4}-2 b^{2} c^{2}+c^{4}\right ) \sqrt {-a^{2}+b^{2}-c^{2}}}\) | \(800\) |
risch | \(-\frac {5 B \,a^{2} b^{2} {\mathrm e}^{2 x}+4 B \,a^{2} c^{2} {\mathrm e}^{2 x}+4 B \,a^{3} b \,{\mathrm e}^{x}+5 B a \,b^{3} {\mathrm e}^{x}+B \,a^{2} b^{2}+C \,a^{2} c^{2}+2 C \,b^{2} c^{2}-3 C a \,b^{2} c \,{\mathrm e}^{3 x}+2 B b \,c^{3}-6 C a b \,c^{2} {\mathrm e}^{3 x}-9 C \,a^{2} b c \,{\mathrm e}^{2 x}+2 C b \,c^{3}+3 B a \,b^{3} {\mathrm e}^{3 x}+2 B \,a^{4} {\mathrm e}^{2 x}+2 B \,b^{4} {\mathrm e}^{2 x}+2 B \,c^{4} {\mathrm e}^{2 x}-2 B \,b^{2} c^{2}+2 C \,b^{4} {\mathrm e}^{2 x}+2 \,{\mathrm e}^{2 x} a^{4} C -B \,a^{2} b c +2 C \,c^{4} {\mathrm e}^{2 x}-2 B \,b^{3} c -5 C a \,b^{2} c \,{\mathrm e}^{x}-5 B a b \,c^{2} {\mathrm e}^{x}+6 B a \,b^{2} c \,{\mathrm e}^{3 x}+3 B a b \,c^{2} {\mathrm e}^{3 x}+9 B \,a^{2} b c \,{\mathrm e}^{2 x}-C \,a^{2} b c -2 C \,b^{3} c +2 B \,b^{4}-4 B \,b^{2} c^{2} {\mathrm e}^{2 x}-2 C \,c^{4}-4 C \,a^{3} c \,{\mathrm e}^{x}+5 C a \,c^{3} {\mathrm e}^{x}-3 C a \,c^{3} {\mathrm e}^{3 x}-4 C \,a^{2} b^{2} {\mathrm e}^{2 x}-5 C \,a^{2} c^{2} {\mathrm e}^{2 x}-4 C \,b^{2} c^{2} {\mathrm e}^{2 x}}{\left (a^{2}-b^{2}+c^{2}\right )^{2} \left (b \,{\mathrm e}^{2 x}+{\mathrm e}^{2 x} c +2 a \,{\mathrm e}^{x}+b -c \right )^{2} \left (b +c \right )}+\frac {3 \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} a +a^{6}-3 a^{4} b^{2}+3 a^{4} c^{2}+3 a^{2} b^{4}-6 a^{2} b^{2} c^{2}+3 a^{2} c^{4}-b^{6}+3 b^{4} c^{2}-3 b^{2} c^{4}+c^{6}}{\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} \left (b +c \right )}\right ) B a b}{2 \left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}}}-\frac {3 \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} a +a^{6}-3 a^{4} b^{2}+3 a^{4} c^{2}+3 a^{2} b^{4}-6 a^{2} b^{2} c^{2}+3 a^{2} c^{4}-b^{6}+3 b^{4} c^{2}-3 b^{2} c^{4}+c^{6}}{\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} \left (b +c \right )}\right ) C a c}{2 \left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}}}-\frac {3 \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} a -a^{6}+3 a^{4} b^{2}-3 a^{4} c^{2}-3 a^{2} b^{4}+6 a^{2} b^{2} c^{2}-3 a^{2} c^{4}+b^{6}-3 b^{4} c^{2}+3 b^{2} c^{4}-c^{6}}{\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} \left (b +c \right )}\right ) B a b}{2 \left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}}}+\frac {3 \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} a -a^{6}+3 a^{4} b^{2}-3 a^{4} c^{2}-3 a^{2} b^{4}+6 a^{2} b^{2} c^{2}-3 a^{2} c^{4}+b^{6}-3 b^{4} c^{2}+3 b^{2} c^{4}-c^{6}}{\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} \left (b +c \right )}\right ) C a c}{2 \left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}}}\) | \(950\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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. 4996 vs.
\(2 (182) = 364\).
time = 0.50, size = 10107, normalized size = 52.10 \begin {gather*} \text {Too large to display} \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 577 vs.
\(2 (182) = 364\).
time = 0.44, size = 577, normalized size = 2.97 \begin {gather*} -\frac {3 \, {\left (B a b - C a c\right )} \arctan \left (\frac {b e^{x} + c e^{x} + a}{\sqrt {-a^{2} + b^{2} - c^{2}}}\right )}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4} + 2 \, a^{2} c^{2} - 2 \, b^{2} c^{2} + c^{4}\right )} \sqrt {-a^{2} + b^{2} - c^{2}}} - \frac {3 \, B a b^{3} e^{\left (3 \, x\right )} + 6 \, B a b^{2} c e^{\left (3 \, x\right )} - 3 \, C a b^{2} c e^{\left (3 \, x\right )} + 3 \, B a b c^{2} e^{\left (3 \, x\right )} - 6 \, C a b c^{2} e^{\left (3 \, x\right )} - 3 \, C a c^{3} e^{\left (3 \, x\right )} + 2 \, B a^{4} e^{\left (2 \, x\right )} + 2 \, C a^{4} e^{\left (2 \, x\right )} + 5 \, B a^{2} b^{2} e^{\left (2 \, x\right )} - 4 \, C a^{2} b^{2} e^{\left (2 \, x\right )} + 2 \, B b^{4} e^{\left (2 \, x\right )} + 2 \, C b^{4} e^{\left (2 \, x\right )} + 9 \, B a^{2} b c e^{\left (2 \, x\right )} - 9 \, C a^{2} b c e^{\left (2 \, x\right )} + 4 \, B a^{2} c^{2} e^{\left (2 \, x\right )} - 5 \, C a^{2} c^{2} e^{\left (2 \, x\right )} - 4 \, B b^{2} c^{2} e^{\left (2 \, x\right )} - 4 \, C b^{2} c^{2} e^{\left (2 \, x\right )} + 2 \, B c^{4} e^{\left (2 \, x\right )} + 2 \, C c^{4} e^{\left (2 \, x\right )} + 4 \, B a^{3} b e^{x} + 5 \, B a b^{3} e^{x} - 4 \, C a^{3} c e^{x} - 5 \, C a b^{2} c e^{x} - 5 \, B a b c^{2} e^{x} + 5 \, C a c^{3} e^{x} + B a^{2} b^{2} + 2 \, B b^{4} - B a^{2} b c - C a^{2} b c - 2 \, B b^{3} c - 2 \, C b^{3} c + C a^{2} c^{2} - 2 \, B b^{2} c^{2} + 2 \, C b^{2} c^{2} + 2 \, B b c^{3} + 2 \, C b c^{3} - 2 \, C c^{4}}{{\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5} + a^{4} c - 2 \, a^{2} b^{2} c + b^{4} c + 2 \, a^{2} b c^{2} - 2 \, b^{3} c^{2} + 2 \, a^{2} c^{3} - 2 \, b^{2} c^{3} + b c^{4} + c^{5}\right )} {\left (b e^{\left (2 \, x\right )} + c e^{\left (2 \, x\right )} + 2 \, a e^{x} + b - c\right )}^{2}} \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 {B\,\mathrm {cosh}\left (x\right )+C\,\mathrm {sinh}\left (x\right )}{{\left (a+b\,\mathrm {cosh}\left (x\right )+c\,\mathrm {sinh}\left (x\right )\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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