Optimal. Leaf size=233 \[ -\frac {\left (2 a^2 A+A b^2-3 a b B-A c^2+3 a c C\right ) \tanh ^{-1}\left (\frac {c-(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2+c^2}}\right )}{\left (a^2-b^2+c^2\right )^{5/2}}-\frac {B c-b C+(A c-a C) \cosh (x)+(A b-a B) \sinh (x)}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}-\frac {a (B c-b C)+\left (3 a A c-a^2 C-2 c (b B-c C)\right ) \cosh (x)+\left (3 a A b-a^2 B-2 b (b B-c C)\right ) \sinh (x)}{2 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))} \]
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Rubi [A]
time = 0.35, antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3235, 3232,
3203, 632, 212} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {c-(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2+c^2}}\right ) \left (2 a^2 A-3 a b B+3 a c C+A b^2-A c^2\right )}{\left (a^2-b^2+c^2\right )^{5/2}}-\frac {\sinh (x) \left (a^2 (-B)+3 a A b-2 b (b B-c C)\right )+\cosh (x) \left (a^2 (-C)+3 a A c-2 c (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 {\sinh (x) (A b-a B)+\cosh (x) (A c-a C)-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 {A+B \cosh (x)+C \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^3} \, dx &=-\frac {B c-b C+(A c-a C) \cosh (x)+(A b-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 (a A-b B+c C)+(A b-a B) \cosh (x)+(A c-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-a C) \cosh (x)+(A b-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 (3 a A c-a^2 C-2 c (b B-c C)\right ) \cosh (x)+\left (3 a A b-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 {\left (2 a^2 A+A b^2-3 a b B-A c^2+3 a c C\right ) \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-a C) \cosh (x)+(A b-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 (3 a A c-a^2 C-2 c (b B-c C)\right ) \cosh (x)+\left (3 a A b-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 {\left (2 a^2 A+A b^2-3 a b B-A c^2+3 a c C\right ) \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-a C) \cosh (x)+(A b-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 (3 a A c-a^2 C-2 c (b B-c C)\right ) \cosh (x)+\left (3 a A b-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 {\left (2 \left (2 a^2 A+A b^2-3 a b B-A c^2+3 a c C\right )\right ) \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 {\left (2 a^2 A+A b^2-3 a b B-A c^2+3 a c C\right ) \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-a C) \cosh (x)+(A b-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 (3 a A c-a^2 C-2 c (b B-c C)\right ) \cosh (x)+\left (3 a A b-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 = 0.69, size = 465, normalized size = 2.00 \begin {gather*} \frac {\left (2 a^2 A+A b^2-3 a b B-A c^2+3 a c C\right ) \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 {6 a^3 A c+3 a A b^2 c-9 a^2 b B c-3 a A c^3-2 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+2 b c \left (2 a^2 A+A b^2-3 a b B-A c^2+3 a c C\right ) \cosh (x)+c \left (3 a A \left (-b^2+c^2\right )+a^2 (b B-c C)+2 \left (b^2-c^2\right ) (b B-c C)\right ) \cosh (2 x)-8 a^2 A b^2 \sinh (x)+2 A b^4 \sinh (x)+4 a^3 b B \sinh (x)+2 a b^3 B \sinh (x)+12 a^2 A c^2 \sinh (x)-2 A b^2 c^2 \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)-3 a A b^3 \sinh (2 x)+a^2 b^2 B \sinh (2 x)+2 b^4 B \sinh (2 x)+3 a A b c^2 \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. \(1083\) vs.
\(2(228)=456\).
time = 1.80, size = 1084, normalized size = 4.65
method | result | size |
default | \(-\frac {2 \left (-\frac {\left (4 A \,a^{3} b -7 A \,a^{2} b^{2}+5 A \,a^{2} c^{2}+2 A a \,b^{3}-2 A a b \,c^{2}+A \,b^{4}-3 A \,b^{2} c^{2}+2 A \,c^{4}-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 )}{2 \left (a -b \right ) \left (a^{4}-2 a^{2} b^{2}+2 a^{2} c^{2}+b^{4}-2 b^{2} c^{2}+c^{4}\right )}-\frac {\left (4 A \,a^{4} c -12 A \,a^{3} b c +13 A \,a^{2} b^{2} c -7 A \,a^{2} c^{3}-6 A a \,b^{3} c +6 A a b \,c^{3}+A \,b^{4} c +A \,b^{2} c^{3}-2 A \,c^{5}+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 )}{2 \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 (4 A \,a^{4} b -5 A \,a^{3} b^{2}+11 A \,a^{3} c^{2}-3 A \,a^{2} b^{3}-3 A \,a^{2} b \,c^{2}+5 A a \,b^{4}-7 A a \,b^{2} c^{2}+2 A a \,c^{4}-A \,b^{5}-A \,b^{3} c^{2}+2 A b \,c^{4}-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 )}{2 \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 {4 A \,a^{4} c -3 A \,a^{2} b^{2} c +A \,a^{2} c^{3}-A \,b^{4} c +A \,b^{2} c^{3}-5 B \,a^{3} b c +5 B a \,b^{3} c -2 B a b \,c^{3}-2 C \,a^{5}+4 C \,a^{3} b^{2}+C \,a^{3} c^{2}-2 C a \,b^{4}-C a \,b^{2} c^{2}}{2 \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 )}\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 {\left (2 a^{2} A +A \,b^{2}-A \,c^{2}-3 B a b +3 C a 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}}}\) | \(1084\) |
risch | \(\text {Expression too large to display}\) | \(1966\) |
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. 6850 vs.
\(2 (223) = 446\).
time = 0.56, size = 13813, normalized size = 59.28 \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. 819 vs.
\(2 (223) = 446\).
time = 0.43, size = 819, normalized size = 3.52 \begin {gather*} \frac {{\left (2 \, A a^{2} - 3 \, B a b + A b^{2} + 3 \, C a c - A c^{2}\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 {2 \, A a^{2} b^{2} e^{\left (3 \, x\right )} - 3 \, B a b^{3} e^{\left (3 \, x\right )} + A b^{4} e^{\left (3 \, x\right )} + 4 \, A a^{2} b c 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 )} + 2 \, A b^{3} c e^{\left (3 \, x\right )} + 2 \, A a^{2} c^{2} 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 \, A b c^{3} e^{\left (3 \, x\right )} - A c^{4} e^{\left (3 \, x\right )} - 2 \, B a^{4} e^{\left (2 \, x\right )} - 2 \, C a^{4} e^{\left (2 \, x\right )} + 6 \, A a^{3} b 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 )} + 3 \, A a b^{3} e^{\left (2 \, x\right )} - 2 \, B b^{4} e^{\left (2 \, x\right )} - 2 \, C b^{4} e^{\left (2 \, x\right )} + 6 \, A a^{3} c 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 )} + 3 \, A a b^{2} 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 )} - 3 \, A a b 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 )} - 3 \, A a c^{3} 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} + 10 \, A a^{2} b^{2} e^{x} - 5 \, B a b^{3} e^{x} - A b^{4} e^{x} + 4 \, C a^{3} c e^{x} + 5 \, C a b^{2} c e^{x} - 10 \, A a^{2} c^{2} e^{x} + 5 \, B a b c^{2} e^{x} + 2 \, A b^{2} c^{2} e^{x} - 5 \, C a c^{3} e^{x} - A c^{4} e^{x} - B a^{2} b^{2} + 3 \, A a b^{3} - 2 \, B b^{4} + B a^{2} b c + C a^{2} b c - 3 \, A a b^{2} c + 2 \, B b^{3} c + 2 \, C b^{3} c - C a^{2} c^{2} - 3 \, A a b c^{2} + 2 \, B b^{2} c^{2} - 2 \, C b^{2} c^{2} + 3 \, A a c^{3} - 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 = -0.00 \begin {gather*} \int \frac {A+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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