Optimal. Leaf size=152 \[ -\frac{2 i B \sqrt{\frac{a+b \cosh (x)}{a+b}} \text{EllipticF}\left (\frac{i x}{2},\frac{2 b}{a+b}\right )}{b \sqrt{a+b \cosh (x)}}-\frac{2 \sinh (x) (A b-a B)}{\left (a^2-b^2\right ) \sqrt{a+b \cosh (x)}}-\frac{2 i (A b-a B) \sqrt{a+b \cosh (x)} E\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{b \left (a^2-b^2\right ) \sqrt{\frac{a+b \cosh (x)}{a+b}}} \]
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Rubi [A] time = 0.204889, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {2754, 2752, 2663, 2661, 2655, 2653} \[ -\frac{2 \sinh (x) (A b-a B)}{\left (a^2-b^2\right ) \sqrt{a+b \cosh (x)}}-\frac{2 i (A b-a B) \sqrt{a+b \cosh (x)} E\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{b \left (a^2-b^2\right ) \sqrt{\frac{a+b \cosh (x)}{a+b}}}-\frac{2 i B \sqrt{\frac{a+b \cosh (x)}{a+b}} F\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{b \sqrt{a+b \cosh (x)}} \]
Antiderivative was successfully verified.
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Rule 2754
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \frac{A+B \cosh (x)}{(a+b \cosh (x))^{3/2}} \, dx &=-\frac{2 (A b-a B) \sinh (x)}{\left (a^2-b^2\right ) \sqrt{a+b \cosh (x)}}-\frac{2 \int \frac{\frac{1}{2} (-a A+b B)-\frac{1}{2} (A b-a B) \cosh (x)}{\sqrt{a+b \cosh (x)}} \, dx}{a^2-b^2}\\ &=-\frac{2 (A b-a B) \sinh (x)}{\left (a^2-b^2\right ) \sqrt{a+b \cosh (x)}}+\frac{B \int \frac{1}{\sqrt{a+b \cosh (x)}} \, dx}{b}+\frac{(A b-a B) \int \sqrt{a+b \cosh (x)} \, dx}{b \left (a^2-b^2\right )}\\ &=-\frac{2 (A b-a B) \sinh (x)}{\left (a^2-b^2\right ) \sqrt{a+b \cosh (x)}}+\frac{\left ((A b-a B) \sqrt{a+b \cosh (x)}\right ) \int \sqrt{\frac{a}{a+b}+\frac{b \cosh (x)}{a+b}} \, dx}{b \left (a^2-b^2\right ) \sqrt{\frac{a+b \cosh (x)}{a+b}}}+\frac{\left (B \sqrt{\frac{a+b \cosh (x)}{a+b}}\right ) \int \frac{1}{\sqrt{\frac{a}{a+b}+\frac{b \cosh (x)}{a+b}}} \, dx}{b \sqrt{a+b \cosh (x)}}\\ &=-\frac{2 i (A b-a B) \sqrt{a+b \cosh (x)} E\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{b \left (a^2-b^2\right ) \sqrt{\frac{a+b \cosh (x)}{a+b}}}-\frac{2 i B \sqrt{\frac{a+b \cosh (x)}{a+b}} F\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{b \sqrt{a+b \cosh (x)}}-\frac{2 (A b-a B) \sinh (x)}{\left (a^2-b^2\right ) \sqrt{a+b \cosh (x)}}\\ \end{align*}
Mathematica [A] time = 0.356721, size = 133, normalized size = 0.88 \[ \frac{-2 i B \left (a^2-b^2\right ) \sqrt{\frac{a+b \cosh (x)}{a+b}} \text{EllipticF}\left (\frac{i x}{2},\frac{2 b}{a+b}\right )+2 b \sinh (x) (a B-A b)+2 i (a+b) (a B-A b) \sqrt{\frac{a+b \cosh (x)}{a+b}} E\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{b (a-b) (a+b) \sqrt{a+b \cosh (x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.197, size = 483, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B \cosh \left (x\right ) + A}{{\left (b \cosh \left (x\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (B \cosh \left (x\right ) + A\right )} \sqrt{b \cosh \left (x\right ) + a}}{b^{2} \cosh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B \cosh \left (x\right ) + A}{{\left (b \cosh \left (x\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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