Optimal. Leaf size=43 \[ \frac {2 \sqrt {\sinh (c+d x)}}{b d}-\frac {2 a \log \left (a+b \sqrt {\sinh (c+d x)}\right )}{b^2 d} \]
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Rubi [A] time = 0.05, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3223, 190, 43} \[ \frac {2 \sqrt {\sinh (c+d x)}}{b d}-\frac {2 a \log \left (a+b \sqrt {\sinh (c+d x)}\right )}{b^2 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 190
Rule 3223
Rubi steps
\begin {align*} \int \frac {\cosh (c+d x)}{a+b \sqrt {\sinh (c+d x)}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{a+b \sqrt {x}} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {x}{a+b x} \, dx,x,\sqrt {\sinh (c+d x)}\right )}{d}\\ &=\frac {2 \operatorname {Subst}\left (\int \left (\frac {1}{b}-\frac {a}{b (a+b x)}\right ) \, dx,x,\sqrt {\sinh (c+d x)}\right )}{d}\\ &=-\frac {2 a \log \left (a+b \sqrt {\sinh (c+d x)}\right )}{b^2 d}+\frac {2 \sqrt {\sinh (c+d x)}}{b d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 41, normalized size = 0.95 \[ \frac {2 \left (\frac {\sqrt {\sinh (c+d x)}}{b}-\frac {a \log \left (a+b \sqrt {\sinh (c+d x)}\right )}{b^2}\right )}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 2.31, size = 225, normalized size = 5.23 \[ \frac {a d x + a \log \left (\frac {b^{2} \cosh \left (d x + c\right )^{2} + b^{2} \sinh \left (d x + c\right )^{2} + 2 \, a^{2} \cosh \left (d x + c\right ) - b^{2} + 2 \, {\left (b^{2} \cosh \left (d x + c\right ) + a^{2}\right )} \sinh \left (d x + c\right ) - 4 \, {\left (a b \cosh \left (d x + c\right ) + a b \sinh \left (d x + c\right )\right )} \sqrt {\sinh \left (d x + c\right )}}{b^{2} \cosh \left (d x + c\right )^{2} + b^{2} \sinh \left (d x + c\right )^{2} - 2 \, a^{2} \cosh \left (d x + c\right ) - b^{2} + 2 \, {\left (b^{2} \cosh \left (d x + c\right ) - a^{2}\right )} \sinh \left (d x + c\right )}\right ) - a \log \left (\frac {2 \, {\left (b^{2} \sinh \left (d x + c\right ) - a^{2}\right )}}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 2 \, b \sqrt {\sinh \left (d x + c\right )}}{b^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left (d x + c\right )}{b \sqrt {\sinh \left (d x + c\right )} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 89, normalized size = 2.07 \[ \frac {2 \left (\sqrt {\sinh }\left (d x +c \right )\right )}{b d}+\frac {a \ln \left (b \left (\sqrt {\sinh }\left (d x +c \right )\right )-a \right )}{d \,b^{2}}-\frac {a \ln \left (a +b \left (\sqrt {\sinh }\left (d x +c \right )\right )\right )}{b^{2} d}-\frac {a \ln \left (b^{2} \sinh \left (d x +c \right )-a^{2}\right )}{d \,b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left (d x + c\right )}{b \sqrt {\sinh \left (d x + c\right )} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.01, size = 39, normalized size = 0.91 \[ \frac {2\,\sqrt {\mathrm {sinh}\left (c+d\,x\right )}}{b\,d}-\frac {2\,a\,\ln \left (a+b\,\sqrt {\mathrm {sinh}\left (c+d\,x\right )}\right )}{b^2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.82, size = 68, normalized size = 1.58 \[ \begin {cases} \frac {x \cosh {\relax (c )}}{a} & \text {for}\: b = 0 \wedge d = 0 \\\frac {\sinh {\left (c + d x \right )}}{a d} & \text {for}\: b = 0 \\\frac {x \cosh {\relax (c )}}{a + b \sqrt {\sinh {\relax (c )}}} & \text {for}\: d = 0 \\- \frac {2 a \log {\left (\frac {a}{b} + \sqrt {\sinh {\left (c + d x \right )}} \right )}}{b^{2} d} + \frac {2 \sqrt {\sinh {\left (c + d x \right )}}}{b d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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