Optimal. Leaf size=57 \[ \frac {2 (b+c) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b}}+\frac {\log (a+b \cosh (x))}{b} \]
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Rubi [A]
time = 0.10, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {4486, 2738,
214, 2747, 31} \begin {gather*} \frac {2 (b+c) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b}}+\frac {\log (a+b \cosh (x))}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 214
Rule 2738
Rule 2747
Rule 4486
Rubi steps
\begin {align*} \int \frac {b+c+\sinh (x)}{a+b \cosh (x)} \, dx &=\int \left (\frac {b+c}{a+b \cosh (x)}+\frac {\sinh (x)}{a+b \cosh (x)}\right ) \, dx\\ &=(b+c) \int \frac {1}{a+b \cosh (x)} \, dx+\int \frac {\sinh (x)}{a+b \cosh (x)} \, dx\\ &=\frac {\text {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \cosh (x)\right )}{b}+(2 (b+c)) \text {Subst}\left (\int \frac {1}{a+b-(a-b) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )\\ &=\frac {2 (b+c) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b}}+\frac {\log (a+b \cosh (x))}{b}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 56, normalized size = 0.98 \begin {gather*} -\frac {2 (b+c) \text {ArcTan}\left (\frac {(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}+\frac {\log (a+b \cosh (x))}{b} \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. \(103\) vs.
\(2(47)=94\).
time = 0.78, size = 104, normalized size = 1.82
method | result | size |
default | \(\frac {\ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-b \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-a -b \right )-\frac {2 \left (-b^{2}-b c \right ) \arctanh \left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}}{b}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{b}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{b}\) | \(104\) |
risch | \(\frac {x}{b}+\frac {2 x \,a^{2} b}{-a^{2} b^{2}+b^{4}}-\frac {2 x \,b^{3}}{-a^{2} b^{2}+b^{4}}+\frac {\ln \left ({\mathrm e}^{x}-\frac {-a \,b^{2}-a b c +\sqrt {a^{2} b^{4}+2 a^{2} b^{3} c +a^{2} b^{2} c^{2}-b^{6}-2 b^{5} c -b^{4} c^{2}}}{b^{2} \left (b +c \right )}\right ) a^{2}}{\left (a^{2}-b^{2}\right ) b}-\frac {b \ln \left ({\mathrm e}^{x}-\frac {-a \,b^{2}-a b c +\sqrt {a^{2} b^{4}+2 a^{2} b^{3} c +a^{2} b^{2} c^{2}-b^{6}-2 b^{5} c -b^{4} c^{2}}}{b^{2} \left (b +c \right )}\right )}{a^{2}-b^{2}}+\frac {\ln \left ({\mathrm e}^{x}-\frac {-a \,b^{2}-a b c +\sqrt {a^{2} b^{4}+2 a^{2} b^{3} c +a^{2} b^{2} c^{2}-b^{6}-2 b^{5} c -b^{4} c^{2}}}{b^{2} \left (b +c \right )}\right ) \sqrt {a^{2} b^{4}+2 a^{2} b^{3} c +a^{2} b^{2} c^{2}-b^{6}-2 b^{5} c -b^{4} c^{2}}}{\left (a^{2}-b^{2}\right ) b}+\frac {\ln \left ({\mathrm e}^{x}+\frac {a \,b^{2}+a b c +\sqrt {a^{2} b^{4}+2 a^{2} b^{3} c +a^{2} b^{2} c^{2}-b^{6}-2 b^{5} c -b^{4} c^{2}}}{b^{2} \left (b +c \right )}\right ) a^{2}}{\left (a^{2}-b^{2}\right ) b}-\frac {b \ln \left ({\mathrm e}^{x}+\frac {a \,b^{2}+a b c +\sqrt {a^{2} b^{4}+2 a^{2} b^{3} c +a^{2} b^{2} c^{2}-b^{6}-2 b^{5} c -b^{4} c^{2}}}{b^{2} \left (b +c \right )}\right )}{a^{2}-b^{2}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {a \,b^{2}+a b c +\sqrt {a^{2} b^{4}+2 a^{2} b^{3} c +a^{2} b^{2} c^{2}-b^{6}-2 b^{5} c -b^{4} c^{2}}}{b^{2} \left (b +c \right )}\right ) \sqrt {a^{2} b^{4}+2 a^{2} b^{3} c +a^{2} b^{2} c^{2}-b^{6}-2 b^{5} c -b^{4} c^{2}}}{\left (a^{2}-b^{2}\right ) b}\) | \(674\) |
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. 111 vs.
\(2 (47) = 94\).
time = 0.45, size = 289, normalized size = 5.07 \begin {gather*} \left [\frac {\sqrt {a^{2} - b^{2}} {\left (b^{2} + b c\right )} \log \left (\frac {b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + 2 \, a^{2} - b^{2} + 2 \, {\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) - 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) + b}\right ) - {\left (a^{2} - b^{2}\right )} x + {\left (a^{2} - b^{2}\right )} \log \left (\frac {2 \, {\left (b \cosh \left (x\right ) + a\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{a^{2} b - b^{3}}, -\frac {2 \, \sqrt {-a^{2} + b^{2}} {\left (b^{2} + b c\right )} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{a^{2} - b^{2}}\right ) + {\left (a^{2} - b^{2}\right )} x - {\left (a^{2} - b^{2}\right )} \log \left (\frac {2 \, {\left (b \cosh \left (x\right ) + a\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{a^{2} b - b^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 840 vs.
\(2 (49) = 98\).
time = 17.86, size = 840, normalized size = 14.74 \begin {gather*} \begin {cases} \tilde {\infty } \left (2 c \operatorname {atan}{\left (\tanh {\left (\frac {x}{2} \right )} \right )} + x - 2 \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 \right )} + \log {\left (\tanh ^{2}{\left (\frac {x}{2} \right )} + 1 \right )}\right ) & \text {for}\: a = 0 \wedge b = 0 \\- \frac {1}{\tanh {\left (\frac {x}{2} \right )}} - \frac {c}{b \tanh {\left (\frac {x}{2} \right )}} + \frac {x}{b} - \frac {2 \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 \right )}}{b} + \frac {2 \log {\left (\tanh {\left (\frac {x}{2} \right )} \right )}}{b} & \text {for}\: a = - b \\\frac {c x + \cosh {\left (x \right )}}{a} & \text {for}\: b = 0 \\\tanh {\left (\frac {x}{2} \right )} + \frac {c \tanh {\left (\frac {x}{2} \right )}}{b} + \frac {x}{b} - \frac {2 \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 \right )}}{b} & \text {for}\: a = b \\\frac {a x \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}}{a b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} + \frac {a \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} \log {\left (- \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{a b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} + \frac {a \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} \log {\left (\sqrt {\frac {a}{a - b} + \frac {b}{a - b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{a b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} - \frac {2 a \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 \right )}}{a b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} - \frac {b^{2} \log {\left (- \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{a b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} + \frac {b^{2} \log {\left (\sqrt {\frac {a}{a - b} + \frac {b}{a - b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{a b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} - \frac {b c \log {\left (- \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{a b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} + \frac {b c \log {\left (\sqrt {\frac {a}{a - b} + \frac {b}{a - b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{a b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} - \frac {b x \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}}{a b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} - \frac {b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} \log {\left (- \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{a b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} - \frac {b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} \log {\left (\sqrt {\frac {a}{a - b} + \frac {b}{a - b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{a b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} + \frac {2 b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 \right )}}{a b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 60, normalized size = 1.05 \begin {gather*} \frac {2 \, {\left (b + c\right )} \arctan \left (\frac {b e^{x} + a}{\sqrt {-a^{2} + b^{2}}}\right )}{\sqrt {-a^{2} + b^{2}}} - \frac {x}{b} + \frac {\log \left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} + b\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.81, size = 198, normalized size = 3.47 \begin {gather*} \frac {\ln \left (b\,\sqrt {\left (a+b\right )\,\left (a-b\right )}+a^2\,{\mathrm {e}}^x-b^2\,{\mathrm {e}}^x+a\,{\mathrm {e}}^x\,\sqrt {\left (a+b\right )\,\left (a-b\right )}\right )\,\left (b^2\,\sqrt {\left (a+b\right )\,\left (a-b\right )}+a^2-b^2+b\,c\,\sqrt {\left (a+b\right )\,\left (a-b\right )}\right )}{a^2\,b-b^3}-\frac {x}{b}-\frac {\ln \left (b\,\sqrt {\left (a+b\right )\,\left (a-b\right )}-a^2\,{\mathrm {e}}^x+b^2\,{\mathrm {e}}^x+a\,{\mathrm {e}}^x\,\sqrt {\left (a+b\right )\,\left (a-b\right )}\right )\,\left (b^2\,\sqrt {\left (a+b\right )\,\left (a-b\right )}-a^2+b^2+b\,c\,\sqrt {\left (a+b\right )\,\left (a-b\right )}\right )}{a^2\,b-b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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