Optimal. Leaf size=120 \[ \frac {A \text {ArcTan}\left (\frac {c \cosh (x)+b \sinh (x)}{\sqrt {b^2-c^2}}\right )}{2 \left (b^2-c^2\right )^{3/2}}+\frac {B c+A c \cosh (x)+A b \sinh (x)}{2 \left (b^2-c^2\right ) (b \cosh (x)+c \sinh (x))^2}+\frac {b B c \cosh (x)+b^2 B \sinh (x)}{\left (b^2-c^2\right )^2 (b \cosh (x)+c \sinh (x))} \]
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Rubi [A]
time = 0.09, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3237, 3232,
3153, 212} \begin {gather*} \frac {A \text {ArcTan}\left (\frac {b \sinh (x)+c \cosh (x)}{\sqrt {b^2-c^2}}\right )}{2 \left (b^2-c^2\right )^{3/2}}+\frac {A b \sinh (x)+A c \cosh (x)+B c}{2 \left (b^2-c^2\right ) (b \cosh (x)+c \sinh (x))^2}+\frac {b^2 B \sinh (x)+b B c \cosh (x)}{\left (b^2-c^2\right )^2 (b \cosh (x)+c \sinh (x))} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 3153
Rule 3232
Rule 3237
Rubi steps
\begin {align*} \int \frac {A+B \cosh (x)}{(b \cosh (x)+c \sinh (x))^3} \, dx &=\frac {B c+A c \cosh (x)+A b \sinh (x)}{2 \left (b^2-c^2\right ) (b \cosh (x)+c \sinh (x))^2}+\frac {\int \frac {2 b B+A b \cosh (x)+A c \sinh (x)}{(b \cosh (x)+c \sinh (x))^2} \, dx}{2 \left (b^2-c^2\right )}\\ &=\frac {B c+A c \cosh (x)+A b \sinh (x)}{2 \left (b^2-c^2\right ) (b \cosh (x)+c \sinh (x))^2}+\frac {b B c \cosh (x)+b^2 B \sinh (x)}{\left (b^2-c^2\right )^2 (b \cosh (x)+c \sinh (x))}+\frac {A \int \frac {1}{b \cosh (x)+c \sinh (x)} \, dx}{2 \left (b^2-c^2\right )}\\ &=\frac {B c+A c \cosh (x)+A b \sinh (x)}{2 \left (b^2-c^2\right ) (b \cosh (x)+c \sinh (x))^2}+\frac {b B c \cosh (x)+b^2 B \sinh (x)}{\left (b^2-c^2\right )^2 (b \cosh (x)+c \sinh (x))}+\frac {(i A) \text {Subst}\left (\int \frac {1}{b^2-c^2-x^2} \, dx,x,-i c \cosh (x)-i b \sinh (x)\right )}{2 \left (b^2-c^2\right )}\\ &=\frac {A \tan ^{-1}\left (\frac {c \cosh (x)+b \sinh (x)}{\sqrt {b^2-c^2}}\right )}{2 \left (b^2-c^2\right )^{3/2}}+\frac {B c+A c \cosh (x)+A b \sinh (x)}{2 \left (b^2-c^2\right ) (b \cosh (x)+c \sinh (x))^2}+\frac {b B c \cosh (x)+b^2 B \sinh (x)}{\left (b^2-c^2\right )^2 (b \cosh (x)+c \sinh (x))}\\ \end {align*}
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Mathematica [A]
time = 0.72, size = 134, normalized size = 1.12 \begin {gather*} \frac {1}{2} \left (\frac {2 A \text {ArcTan}\left (\frac {c+b \tanh \left (\frac {x}{2}\right )}{\sqrt {b-c} \sqrt {b+c}}\right )}{(b-c)^{3/2} (b+c)^{3/2}}+\frac {A c+2 b B \sinh (x)}{b (b-c) (b+c) (b \cosh (x)+c \sinh (x))}+\frac {b B c+A \left (b^2-c^2\right ) \sinh (x)}{b (b-c) (b+c) (b \cosh (x)+c \sinh (x))^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.42, size = 214, normalized size = 1.78
method | result | size |
risch | \(\frac {A \,b^{2} {\mathrm e}^{3 x}+2 A b c \,{\mathrm e}^{3 x}+A \,c^{2} {\mathrm e}^{3 x}-2 B \,b^{2} {\mathrm e}^{2 x}+2 B \,c^{2} {\mathrm e}^{2 x}-A \,b^{2} {\mathrm e}^{x}+A \,c^{2} {\mathrm e}^{x}-2 B \,b^{2}+2 b B c}{\left (b -c \right ) \left (b \,{\mathrm e}^{2 x}+{\mathrm e}^{2 x} c +b -c \right )^{2} \left (b^{2}+2 b c +c^{2}\right )}-\frac {A \ln \left ({\mathrm e}^{x}-\frac {b -c}{\sqrt {-b^{2}+c^{2}}}\right )}{2 \sqrt {-b^{2}+c^{2}}\, \left (b +c \right ) \left (b -c \right )}+\frac {A \ln \left ({\mathrm e}^{x}+\frac {b -c}{\sqrt {-b^{2}+c^{2}}}\right )}{2 \sqrt {-b^{2}+c^{2}}\, \left (b +c \right ) \left (b -c \right )}\) | \(211\) |
default | \(\frac {-\frac {\left (A \,b^{2}-2 A \,c^{2}-2 B \,b^{2}+2 B \,c^{2}\right ) \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{\left (b^{2}-c^{2}\right ) b}+\frac {c \left (A \,b^{2}+2 A \,c^{2}+2 B \,b^{2}-2 B \,c^{2}\right ) \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{\left (b^{2}-c^{2}\right ) b^{2}}+\frac {\left (A \,b^{2}+2 A \,c^{2}+2 B \,b^{2}-2 B \,c^{2}\right ) \tanh \left (\frac {x}{2}\right )}{\left (b^{2}-c^{2}\right ) b}+\frac {2 A c}{2 b^{2}-2 c^{2}}}{\left (b \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+2 c \tanh \left (\frac {x}{2}\right )+b \right )^{2}}+\frac {A \arctan \left (\frac {2 b \tanh \left (\frac {x}{2}\right )+2 c}{2 \sqrt {b^{2}-c^{2}}}\right )}{\left (b^{2}-c^{2}\right )^{\frac {3}{2}}}\) | \(214\) |
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. 898 vs.
\(2 (112) = 224\).
time = 0.42, size = 1855, normalized size = 15.46 \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 [A]
time = 0.41, size = 152, normalized size = 1.27 \begin {gather*} \frac {A \arctan \left (\frac {b e^{x} + c e^{x}}{\sqrt {b^{2} - c^{2}}}\right )}{{\left (b^{2} - c^{2}\right )}^{\frac {3}{2}}} + \frac {A b^{2} e^{\left (3 \, x\right )} + 2 \, A b c e^{\left (3 \, x\right )} + A c^{2} e^{\left (3 \, x\right )} - 2 \, B b^{2} e^{\left (2 \, x\right )} + 2 \, B c^{2} e^{\left (2 \, x\right )} - A b^{2} e^{x} + A c^{2} e^{x} - 2 \, B b^{2} + 2 \, B b c}{{\left (b^{3} + b^{2} c - b c^{2} - c^{3}\right )} {\left (b e^{\left (2 \, x\right )} + c e^{\left (2 \, x\right )} + b - c\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.67, size = 216, normalized size = 1.80 \begin {gather*} \frac {\mathrm {atan}\left (\frac {A\,{\mathrm {e}}^x\,\sqrt {b^6-3\,b^4\,c^2+3\,b^2\,c^4-c^6}}{b^3\,\sqrt {A^2}+c^3\,\sqrt {A^2}-b\,c^2\,\sqrt {A^2}-b^2\,c\,\sqrt {A^2}}\right )\,\sqrt {A^2}}{\sqrt {b^6-3\,b^4\,c^2+3\,b^2\,c^4-c^6}}-\frac {\frac {B}{{\left (b+c\right )}^2}-\frac {A\,{\mathrm {e}}^x}{\left (b+c\right )\,\left (b-c\right )}}{b-c+{\mathrm {e}}^{2\,x}\,\left (b+c\right )}-\frac {\frac {B}{b+c}+\frac {2\,A\,{\mathrm {e}}^x}{b+c}+\frac {B\,{\mathrm {e}}^{2\,x}}{b+c}}{{\mathrm {e}}^{4\,x}\,{\left (b+c\right )}^2+{\left (b-c\right )}^2+2\,{\mathrm {e}}^{2\,x}\,\left (b+c\right )\,\left (b-c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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