Optimal. Leaf size=32 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} d} \]
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Rubi [A] time = 0.04, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3190, 205} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} d} \]
Antiderivative was successfully verified.
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Rule 205
Rule 3190
Rubi steps
\begin {align*} \int \frac {\cosh (c+d x)}{a+b \sinh ^2(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 32, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.59, size = 459, normalized size = 14.34 \[ \left [-\frac {\sqrt {-a b} \log \left (\frac {b \cosh \left (d x + c\right )^{4} + 4 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b \sinh \left (d x + c\right )^{4} - 2 \, {\left (2 \, a + b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, b \cosh \left (d x + c\right )^{2} - 2 \, a - b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (b \cosh \left (d x + c\right )^{3} - {\left (2 \, a + b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - 4 \, {\left (\cosh \left (d x + c\right )^{3} + 3 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + \sinh \left (d x + c\right )^{3} + {\left (3 \, \cosh \left (d x + c\right )^{2} - 1\right )} \sinh \left (d x + c\right ) - \cosh \left (d x + c\right )\right )} \sqrt {-a b} + b}{b \cosh \left (d x + c\right )^{4} + 4 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b \sinh \left (d x + c\right )^{4} + 2 \, {\left (2 \, a - b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, b \cosh \left (d x + c\right )^{2} + 2 \, a - b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (b \cosh \left (d x + c\right )^{3} + {\left (2 \, a - b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + b}\right )}{2 \, a b d}, \frac {\sqrt {a b} \arctan \left (\frac {\sqrt {a b} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}}{2 \, a}\right ) + \sqrt {a b} \arctan \left (\frac {{\left (b \cosh \left (d x + c\right )^{3} + 3 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + b \sinh \left (d x + c\right )^{3} + {\left (4 \, a - b\right )} \cosh \left (d x + c\right ) + {\left (3 \, b \cosh \left (d x + c\right )^{2} + 4 \, a - b\right )} \sinh \left (d x + c\right )\right )} \sqrt {a b}}{2 \, a b}\right )}{a b d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 24, normalized size = 0.75 \[ \frac {\arctan \left (\frac {\sinh \left (d x +c \right ) b}{\sqrt {a b}}\right )}{d \sqrt {a b}} \]
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 \[ \int \frac {\cosh \left (d x + c\right )}{b \sinh \left (d x + c\right )^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.87, size = 23, normalized size = 0.72 \[ \frac {\mathrm {atan}\left (\frac {b\,\mathrm {sinh}\left (c+d\,x\right )}{\sqrt {a\,b}}\right )}{d\,\sqrt {a\,b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.50, size = 128, normalized size = 4.00 \[ \begin {cases} \frac {\tilde {\infty } x \cosh {\relax (c )}}{\sinh ^{2}{\relax (c )}} & \text {for}\: a = 0 \wedge b = 0 \wedge d = 0 \\\frac {\sinh {\left (c + d x \right )}}{a d} & \text {for}\: b = 0 \\\frac {x \cosh {\relax (c )}}{a + b \sinh ^{2}{\relax (c )}} & \text {for}\: d = 0 \\- \frac {1}{b d \sinh {\left (c + d x \right )}} & \text {for}\: a = 0 \\- \frac {i \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sinh {\left (c + d x \right )} \right )}}{2 \sqrt {a} b d \sqrt {\frac {1}{b}}} + \frac {i \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sinh {\left (c + d x \right )} \right )}}{2 \sqrt {a} b d \sqrt {\frac {1}{b}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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