Optimal. Leaf size=44 \[ -\frac {2 \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{d \sqrt {a^2+b^2}} \]
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Rubi [A] time = 0.04, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2660, 618, 204} \[ -\frac {2 \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{d \sqrt {a^2+b^2}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 2660
Rubi steps
\begin {align*} \int \frac {1}{a+b \sinh (c+d x)} \, dx &=-\frac {(2 i) \operatorname {Subst}\left (\int \frac {1}{a-2 i b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{d}\\ &=\frac {(4 i) \operatorname {Subst}\left (\int \frac {1}{-4 \left (a^2+b^2\right )-x^2} \, dx,x,-2 i b+2 a \tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{d}\\ &=-\frac {2 \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} d}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 52, normalized size = 1.18 \[ \frac {2 \tan ^{-1}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )}{d \sqrt {-a^2-b^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.65, size = 162, normalized size = 3.68 \[ \frac {\log \left (\frac {b^{2} \cosh \left (d x + c\right )^{2} + b^{2} \sinh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \left (d x + c\right ) + a b\right )} \sinh \left (d x + c\right ) - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right ) + a\right )}}{b \cosh \left (d x + c\right )^{2} + b \sinh \left (d x + c\right )^{2} + 2 \, a \cosh \left (d x + c\right ) + 2 \, {\left (b \cosh \left (d x + c\right ) + a\right )} \sinh \left (d x + c\right ) - b}\right )}{\sqrt {a^{2} + b^{2}} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 67, normalized size = 1.52 \[ \frac {\log \left (\frac {{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 43, normalized size = 0.98 \[ \frac {2 \arctanh \left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{d \sqrt {a^{2}+b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 67, normalized size = 1.52 \[ \frac {\log \left (\frac {b e^{\left (-d x - c\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-d x - c\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.76, size = 55, normalized size = 1.25 \[ \frac {2\,\mathrm {atan}\left (\frac {a\,d+b\,d\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c}{\sqrt {-a^2\,d^2-b^2\,d^2}}\right )}{\sqrt {-a^2\,d^2-b^2\,d^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.95, size = 187, normalized size = 4.25 \[ \begin {cases} \frac {\log {\left (\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )} \right )}}{b d} & \text {for}\: a = 0 \\\frac {2 i \sqrt {b^{2}}}{b^{2} d \tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )} - i b d \sqrt {b^{2}}} & \text {for}\: a = - \sqrt {- b^{2}} \\- \frac {2 i \sqrt {b^{2}}}{b^{2} d \tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )} + i b d \sqrt {b^{2}}} & \text {for}\: a = \sqrt {- b^{2}} \\\frac {x}{a + b \sinh {\relax (c )}} & \text {for}\: d = 0 \\- \frac {\log {\left (\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )} - \frac {b}{a} - \frac {\sqrt {a^{2} + b^{2}}}{a} \right )}}{d \sqrt {a^{2} + b^{2}}} + \frac {\log {\left (\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )} - \frac {b}{a} + \frac {\sqrt {a^{2} + b^{2}}}{a} \right )}}{d \sqrt {a^{2} + b^{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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