Optimal. Leaf size=49 \[ \frac {2 \log \left (a+b \sqrt {\sinh (c+d x)}\right )}{b^2 d}+\frac {2 a}{b^2 d \left (a+b \sqrt {\sinh (c+d x)}\right )} \]
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Rubi [A]
time = 0.04, antiderivative size = 49, 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 = {3302, 196, 45}
\begin {gather*} \frac {2 a}{b^2 d \left (a+b \sqrt {\sinh (c+d x)}\right )}+\frac {2 \log \left (a+b \sqrt {\sinh (c+d x)}\right )}{b^2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 196
Rule 3302
Rubi steps
\begin {align*} \int \frac {\cosh (c+d x)}{\left (a+b \sqrt {\sinh (c+d x)}\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (a+b \sqrt {x}\right )^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {2 \text {Subst}\left (\int \frac {x}{(a+b x)^2} \, dx,x,\sqrt {\sinh (c+d x)}\right )}{d}\\ &=\frac {2 \text {Subst}\left (\int \left (-\frac {a}{b (a+b x)^2}+\frac {1}{b (a+b x)}\right ) \, dx,x,\sqrt {\sinh (c+d x)}\right )}{d}\\ &=\frac {2 \log \left (a+b \sqrt {\sinh (c+d x)}\right )}{b^2 d}+\frac {2 a}{b^2 d \left (a+b \sqrt {\sinh (c+d x)}\right )}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 42, normalized size = 0.86 \begin {gather*} \frac {2 \left (\log \left (a+b \sqrt {\sinh (c+d x)}\right )+\frac {a}{a+b \sqrt {\sinh (c+d x)}}\right )}{b^2 d} \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. \(126\) vs.
\(2(45)=90\).
time = 1.05, size = 127, normalized size = 2.59
method | result | size |
derivativedivides | \(\frac {\frac {2 a^{2}}{\left (-b^{2} \sinh \left (d x +c \right )+a^{2}\right ) b^{2}}+\frac {\ln \left (-b^{2} \sinh \left (d x +c \right )+a^{2}\right )}{b^{2}}+\frac {a}{b^{2} \left (a +b \left (\sqrt {\sinh }\left (d x +c \right )\right )\right )}+\frac {\ln \left (a +b \left (\sqrt {\sinh }\left (d x +c \right )\right )\right )}{b^{2}}-\frac {a}{b^{2} \left (-b \left (\sqrt {\sinh }\left (d x +c \right )\right )+a \right )}-\frac {\ln \left (-b \left (\sqrt {\sinh }\left (d x +c \right )\right )+a \right )}{b^{2}}}{d}\) | \(127\) |
default | \(\frac {\frac {2 a^{2}}{\left (-b^{2} \sinh \left (d x +c \right )+a^{2}\right ) b^{2}}+\frac {\ln \left (-b^{2} \sinh \left (d x +c \right )+a^{2}\right )}{b^{2}}+\frac {a}{b^{2} \left (a +b \left (\sqrt {\sinh }\left (d x +c \right )\right )\right )}+\frac {\ln \left (a +b \left (\sqrt {\sinh }\left (d x +c \right )\right )\right )}{b^{2}}-\frac {a}{b^{2} \left (-b \left (\sqrt {\sinh }\left (d x +c \right )\right )+a \right )}-\frac {\ln \left (-b \left (\sqrt {\sinh }\left (d x +c \right )\right )+a \right )}{b^{2}}}{d}\) | \(127\) |
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 \begin {gather*} \text {Failed to integrate} \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. 564 vs.
\(2 (45) = 90\).
time = 0.51, size = 564, normalized size = 11.51 \begin {gather*} \frac {b^{2} d x + b^{2} c - {\left (b^{2} d x + b^{2} c\right )} \cosh \left (d x + c\right )^{2} - {\left (b^{2} d x + b^{2} c\right )} \sinh \left (d x + c\right )^{2} + 2 \, {\left (a^{2} d x + a^{2} c - 2 \, a^{2}\right )} \cosh \left (d x + c\right ) + {\left (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 )} \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 ) + {\left (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 )} \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 \, {\left (a^{2} d x + a^{2} c - 2 \, a^{2} - {\left (b^{2} d x + b^{2} c\right )} \cosh \left (d x + c\right )\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^{4} d \cosh \left (d x + c\right )^{2} + b^{4} d \sinh \left (d x + c\right )^{2} - 2 \, a^{2} b^{2} d \cosh \left (d x + c\right ) - b^{4} d + 2 \, {\left (b^{4} d \cosh \left (d x + c\right ) - a^{2} b^{2} d\right )} \sinh \left (d x + c\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. 151 vs.
\(2 (42) = 84\).
time = 4.46, size = 151, normalized size = 3.08 \begin {gather*} \begin {cases} \frac {x \cosh {\left (c \right )}}{a^{2}} & \text {for}\: b = 0 \wedge d = 0 \\\frac {\sinh {\left (c + d x \right )}}{a^{2} d} & \text {for}\: b = 0 \\\frac {x \cosh {\left (c \right )}}{\left (a + b \sqrt {\sinh {\left (c \right )}}\right )^{2}} & \text {for}\: d = 0 \\\frac {2 a \log {\left (\frac {a}{b} + \sqrt {\sinh {\left (c + d x \right )}} \right )}}{a b^{2} d + b^{3} d \sqrt {\sinh {\left (c + d x \right )}}} + \frac {2 a}{a b^{2} d + b^{3} d \sqrt {\sinh {\left (c + d x \right )}}} + \frac {2 b \log {\left (\frac {a}{b} + \sqrt {\sinh {\left (c + d x \right )}} \right )} \sqrt {\sinh {\left (c + d x \right )}}}{a b^{2} d + b^{3} d \sqrt {\sinh {\left (c + d x \right )}}} & \text {otherwise} \end {cases} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.41, size = 45, normalized size = 0.92 \begin {gather*} \frac {2\,a}{b^2\,\left (a\,d+b\,d\,\sqrt {\mathrm {sinh}\left (c+d\,x\right )}\right )}+\frac {2\,\ln \left (a+b\,\sqrt {\mathrm {sinh}\left (c+d\,x\right )}\right )}{b^2\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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