Optimal. Leaf size=220 \[ -\frac {a \left (2 a^2+3 b^2-3 c^2\right ) \tanh ^{-1}\left (\frac {c-(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2+c^2}}\right )}{\left (a^2-b^2+c^2\right )^{7/2}}-\frac {b \sinh (x) \left (11 a^2+4 b^2-4 c^2\right )+c \cosh (x) \left (11 a^2+4 b^2-4 c^2\right )}{6 \left (a^2-b^2+c^2\right )^3 (a+b \cosh (x)+c \sinh (x))}-\frac {5 (a b \sinh (x)+a c \cosh (x))}{6 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))^2}-\frac {b \sinh (x)+c \cosh (x)}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^3} \]
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Rubi [A] time = 0.30, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3129, 3156, 3153, 3124, 618, 206} \[ -\frac {a \left (2 a^2+3 b^2-3 c^2\right ) \tanh ^{-1}\left (\frac {c-(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2+c^2}}\right )}{\left (a^2-b^2+c^2\right )^{7/2}}-\frac {b \sinh (x) \left (11 a^2+4 b^2-4 c^2\right )+c \cosh (x) \left (11 a^2+4 b^2-4 c^2\right )}{6 \left (a^2-b^2+c^2\right )^3 (a+b \cosh (x)+c \sinh (x))}-\frac {5 (a b \sinh (x)+a c \cosh (x))}{6 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))^2}-\frac {b \sinh (x)+c \cosh (x)}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^3} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 3124
Rule 3129
Rule 3153
Rule 3156
Rubi steps
\begin {align*} \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^4} \, dx &=-\frac {c \cosh (x)+b \sinh (x)}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^3}-\frac {\int \frac {-3 a+2 b \cosh (x)+2 c \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^3} \, dx}{3 \left (a^2-b^2+c^2\right )}\\ &=-\frac {c \cosh (x)+b \sinh (x)}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^3}-\frac {5 (a c \cosh (x)+a b \sinh (x))}{6 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))^2}+\frac {\int \frac {2 \left (3 a^2+2 b^2-2 c^2\right )-5 a b \cosh (x)-5 a c \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^2} \, dx}{6 \left (a^2-b^2+c^2\right )^2}\\ &=-\frac {c \cosh (x)+b \sinh (x)}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^3}-\frac {5 (a c \cosh (x)+a b \sinh (x))}{6 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))^2}-\frac {c \left (11 a^2+4 b^2-4 c^2\right ) \cosh (x)+b \left (11 a^2+4 b^2-4 c^2\right ) \sinh (x)}{6 \left (a^2-b^2+c^2\right )^3 (a+b \cosh (x)+c \sinh (x))}+\frac {\left (a \left (2 a^2+3 b^2-3 c^2\right )\right ) \int \frac {1}{a+b \cosh (x)+c \sinh (x)} \, dx}{2 \left (a^2-b^2+c^2\right )^3}\\ &=-\frac {c \cosh (x)+b \sinh (x)}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^3}-\frac {5 (a c \cosh (x)+a b \sinh (x))}{6 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))^2}-\frac {c \left (11 a^2+4 b^2-4 c^2\right ) \cosh (x)+b \left (11 a^2+4 b^2-4 c^2\right ) \sinh (x)}{6 \left (a^2-b^2+c^2\right )^3 (a+b \cosh (x)+c \sinh (x))}+\frac {\left (a \left (2 a^2+3 b^2-3 c^2\right )\right ) \operatorname {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 )^3}\\ &=-\frac {c \cosh (x)+b \sinh (x)}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^3}-\frac {5 (a c \cosh (x)+a b \sinh (x))}{6 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))^2}-\frac {c \left (11 a^2+4 b^2-4 c^2\right ) \cosh (x)+b \left (11 a^2+4 b^2-4 c^2\right ) \sinh (x)}{6 \left (a^2-b^2+c^2\right )^3 (a+b \cosh (x)+c \sinh (x))}-\frac {\left (2 a \left (2 a^2+3 b^2-3 c^2\right )\right ) \operatorname {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 )^3}\\ &=-\frac {a \left (2 a^2+3 b^2-3 c^2\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 )^{7/2}}-\frac {c \cosh (x)+b \sinh (x)}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^3}-\frac {5 (a c \cosh (x)+a b \sinh (x))}{6 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))^2}-\frac {c \left (11 a^2+4 b^2-4 c^2\right ) \cosh (x)+b \left (11 a^2+4 b^2-4 c^2\right ) \sinh (x)}{6 \left (a^2-b^2+c^2\right )^3 (a+b \cosh (x)+c \sinh (x))}\\ \end {align*}
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Mathematica [B] time = 0.96, size = 488, normalized size = 2.22 \[ -\frac {a \left (2 a^2+3 b^2-3 c^2\right ) \tan ^{-1}\left (\frac {(b-a) \tanh \left (\frac {x}{2}\right )+c}{\sqrt {-a^2+b^2-c^2}}\right )}{\left (-a^2+b^2-c^2\right )^{7/2}}-\frac {-44 a^5 c+72 a^4 b^2 \sinh (x)-132 a^4 c^2 \sinh (x)+54 a^3 b^3 \sinh (2 x)-82 a^3 b^2 c-78 a^3 b c^2 \sinh (2 x)+82 a^3 c^3-9 a^2 b^4 \sinh (x)+11 a^2 b^4 \sinh (3 x)+22 a^2 b^3 c \cosh (3 x)-72 a^2 b^2 c^2 \sinh (x)-30 a^2 b c \cosh (x) \left (2 a^2+3 b^2-3 c^2\right )-6 a c \cosh (2 x) \left (a^2 \left (11 c^2-7 b^2\right )+2 \left (b^4+b^2 c^2-2 c^4\right )\right )-22 a^2 b c^3 \cosh (3 x)+81 a^2 c^4 \sinh (x)-11 a^2 c^4 \sinh (3 x)+6 a b^5 \sinh (2 x)-24 a b^4 c-48 a b^3 c^2 \sinh (2 x)+48 a b^2 c^3+42 a b c^4 \sinh (2 x)-24 a c^5+12 b^6 \sinh (x)+4 b^6 \sinh (3 x)+8 b^5 c \cosh (3 x)-36 b^4 c^2 \sinh (x)-4 b^4 c^2 \sinh (3 x)-16 b^3 c^3 \cosh (3 x)+36 b^2 c^4 \sinh (x)-4 b^2 c^4 \sinh (3 x)+8 b c^5 \cosh (3 x)-12 c^6 \sinh (x)+4 c^6 \sinh (3 x)}{24 b \left (a^2-b^2+c^2\right )^3 (a+b \cosh (x)+c \sinh (x))^3} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 717, normalized size = 3.26 \[ \frac {{\left (2 \, a^{3} + 3 \, a b^{2} - 3 \, a c^{2}\right )} \arctan \left (\frac {b e^{x} + c e^{x} + a}{\sqrt {-a^{2} + b^{2} - c^{2}}}\right )}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6} + 3 \, a^{4} c^{2} - 6 \, a^{2} b^{2} c^{2} + 3 \, b^{4} c^{2} + 3 \, a^{2} c^{4} - 3 \, b^{2} c^{4} + c^{6}\right )} \sqrt {-a^{2} + b^{2} - c^{2}}} + \frac {6 \, a^{3} b^{2} e^{\left (5 \, x\right )} + 9 \, a b^{4} e^{\left (5 \, x\right )} + 12 \, a^{3} b c e^{\left (5 \, x\right )} + 18 \, a b^{3} c e^{\left (5 \, x\right )} + 6 \, a^{3} c^{2} e^{\left (5 \, x\right )} - 18 \, a b c^{3} e^{\left (5 \, x\right )} - 9 \, a c^{4} e^{\left (5 \, x\right )} + 30 \, a^{4} b e^{\left (4 \, x\right )} + 45 \, a^{2} b^{3} e^{\left (4 \, x\right )} + 30 \, a^{4} c e^{\left (4 \, x\right )} + 45 \, a^{2} b^{2} c e^{\left (4 \, x\right )} - 45 \, a^{2} b c^{2} e^{\left (4 \, x\right )} - 45 \, a^{2} c^{3} e^{\left (4 \, x\right )} + 44 \, a^{5} e^{\left (3 \, x\right )} + 82 \, a^{3} b^{2} e^{\left (3 \, x\right )} + 24 \, a b^{4} e^{\left (3 \, x\right )} - 82 \, a^{3} c^{2} e^{\left (3 \, x\right )} - 48 \, a b^{2} c^{2} e^{\left (3 \, x\right )} + 24 \, a c^{4} e^{\left (3 \, x\right )} + 102 \, a^{4} b e^{\left (2 \, x\right )} + 36 \, a^{2} b^{3} e^{\left (2 \, x\right )} + 12 \, b^{5} e^{\left (2 \, x\right )} - 102 \, a^{4} c e^{\left (2 \, x\right )} - 36 \, a^{2} b^{2} c e^{\left (2 \, x\right )} - 12 \, b^{4} c e^{\left (2 \, x\right )} - 36 \, a^{2} b c^{2} e^{\left (2 \, x\right )} - 24 \, b^{3} c^{2} e^{\left (2 \, x\right )} + 36 \, a^{2} c^{3} e^{\left (2 \, x\right )} + 24 \, b^{2} c^{3} e^{\left (2 \, x\right )} + 12 \, b c^{4} e^{\left (2 \, x\right )} - 12 \, c^{5} e^{\left (2 \, x\right )} + 60 \, a^{3} b^{2} e^{x} + 15 \, a b^{4} e^{x} - 120 \, a^{3} b c e^{x} - 30 \, a b^{3} c e^{x} + 60 \, a^{3} c^{2} e^{x} + 30 \, a b c^{3} e^{x} - 15 \, a c^{4} e^{x} + 11 \, a^{2} b^{3} + 4 \, b^{5} - 33 \, a^{2} b^{2} c - 12 \, b^{4} c + 33 \, a^{2} b c^{2} + 8 \, b^{3} c^{2} - 11 \, a^{2} c^{3} + 8 \, b^{2} c^{3} - 12 \, b c^{4} + 4 \, c^{5}}{3 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6} + 3 \, a^{4} c^{2} - 6 \, a^{2} b^{2} c^{2} + 3 \, b^{4} c^{2} + 3 \, a^{2} c^{4} - 3 \, b^{2} c^{4} + c^{6}\right )} {\left (b e^{\left (2 \, x\right )} + c e^{\left (2 \, x\right )} + 2 \, a e^{x} + b - c\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.36, size = 1842, normalized size = 8.37 \[ \text {result too large to display} \]
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 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {1}{{\left (a+b\,\mathrm {cosh}\relax (x)+c\,\mathrm {sinh}\relax (x)\right )}^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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