Optimal. Leaf size=251 \[ -\frac{2 i (A b-a B) \sqrt{\frac{a+b \sinh (x)}{a-i b}} \text{EllipticF}\left (\frac{\pi }{4}-\frac{i x}{2},\frac{2 b}{b+i a}\right )}{3 b \left (a^2+b^2\right ) \sqrt{a+b \sinh (x)}}-\frac{2 \cosh (x) \left (a^2 (-B)+4 a A b+3 b^2 B\right )}{3 \left (a^2+b^2\right )^2 \sqrt{a+b \sinh (x)}}-\frac{2 \cosh (x) (A b-a B)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^{3/2}}+\frac{2 i \left (a^2 (-B)+4 a A b+3 b^2 B\right ) \sqrt{a+b \sinh (x)} E\left (\frac{\pi }{4}-\frac{i x}{2}|\frac{2 b}{i a+b}\right )}{3 b \left (a^2+b^2\right )^2 \sqrt{\frac{a+b \sinh (x)}{a-i b}}} \]
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Rubi [A] time = 0.334651, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{2 \cosh (x) \left (a^2 (-B)+4 a A b+3 b^2 B\right )}{3 \left (a^2+b^2\right )^2 \sqrt{a+b \sinh (x)}}-\frac{2 \cosh (x) (A b-a B)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^{3/2}}-\frac{2 i (A b-a B) \sqrt{\frac{a+b \sinh (x)}{a-i b}} F\left (\frac{\pi }{4}-\frac{i x}{2}|\frac{2 b}{i a+b}\right )}{3 b \left (a^2+b^2\right ) \sqrt{a+b \sinh (x)}}+\frac{2 i \left (a^2 (-B)+4 a A b+3 b^2 B\right ) \sqrt{a+b \sinh (x)} E\left (\frac{\pi }{4}-\frac{i x}{2}|\frac{2 b}{i a+b}\right )}{3 b \left (a^2+b^2\right )^2 \sqrt{\frac{a+b \sinh (x)}{a-i b}}} \]
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 \sinh (x)}{(a+b \sinh (x))^{5/2}} \, dx &=-\frac{2 (A b-a B) \cosh (x)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^{3/2}}-\frac{2 \int \frac{-\frac{3}{2} (a A+b B)+\frac{1}{2} (A b-a B) \sinh (x)}{(a+b \sinh (x))^{3/2}} \, dx}{3 \left (a^2+b^2\right )}\\ &=-\frac{2 (A b-a B) \cosh (x)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^{3/2}}-\frac{2 \left (4 a A b-a^2 B+3 b^2 B\right ) \cosh (x)}{3 \left (a^2+b^2\right )^2 \sqrt{a+b \sinh (x)}}+\frac{4 \int \frac{\frac{1}{4} \left (3 a^2 A-A b^2+4 a b B\right )+\frac{1}{4} \left (4 a A b-a^2 B+3 b^2 B\right ) \sinh (x)}{\sqrt{a+b \sinh (x)}} \, dx}{3 \left (a^2+b^2\right )^2}\\ &=-\frac{2 (A b-a B) \cosh (x)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^{3/2}}-\frac{2 \left (4 a A b-a^2 B+3 b^2 B\right ) \cosh (x)}{3 \left (a^2+b^2\right )^2 \sqrt{a+b \sinh (x)}}-\frac{(A b-a B) \int \frac{1}{\sqrt{a+b \sinh (x)}} \, dx}{3 b \left (a^2+b^2\right )}+\frac{\left (4 a A b-a^2 B+3 b^2 B\right ) \int \sqrt{a+b \sinh (x)} \, dx}{3 b \left (a^2+b^2\right )^2}\\ &=-\frac{2 (A b-a B) \cosh (x)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^{3/2}}-\frac{2 \left (4 a A b-a^2 B+3 b^2 B\right ) \cosh (x)}{3 \left (a^2+b^2\right )^2 \sqrt{a+b \sinh (x)}}+\frac{\left (\left (4 a A b-a^2 B+3 b^2 B\right ) \sqrt{a+b \sinh (x)}\right ) \int \sqrt{\frac{a}{a-i b}+\frac{b \sinh (x)}{a-i b}} \, dx}{3 b \left (a^2+b^2\right )^2 \sqrt{\frac{a+b \sinh (x)}{a-i b}}}-\frac{\left ((A b-a B) \sqrt{\frac{a+b \sinh (x)}{a-i b}}\right ) \int \frac{1}{\sqrt{\frac{a}{a-i b}+\frac{b \sinh (x)}{a-i b}}} \, dx}{3 b \left (a^2+b^2\right ) \sqrt{a+b \sinh (x)}}\\ &=-\frac{2 (A b-a B) \cosh (x)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^{3/2}}-\frac{2 \left (4 a A b-a^2 B+3 b^2 B\right ) \cosh (x)}{3 \left (a^2+b^2\right )^2 \sqrt{a+b \sinh (x)}}+\frac{2 i \left (4 a A b-a^2 B+3 b^2 B\right ) E\left (\frac{\pi }{4}-\frac{i x}{2}|\frac{2 b}{i a+b}\right ) \sqrt{a+b \sinh (x)}}{3 b \left (a^2+b^2\right )^2 \sqrt{\frac{a+b \sinh (x)}{a-i b}}}-\frac{2 i (A b-a B) F\left (\frac{\pi }{4}-\frac{i x}{2}|\frac{2 b}{i a+b}\right ) \sqrt{\frac{a+b \sinh (x)}{a-i b}}}{3 b \left (a^2+b^2\right ) \sqrt{a+b \sinh (x)}}\\ \end{align*}
Mathematica [A] time = 0.776822, size = 236, normalized size = 0.94 \[ \frac{2 i \left (\sqrt{\frac{a+b \sinh (x)}{a-i b}} (a+b \sinh (x)) \left (b \left (3 a^2 A+4 a b B-A b^2\right ) \text{EllipticF}\left (\frac{1}{4} (\pi -2 i x),-\frac{2 i b}{a-i b}\right )+\left (a^2 (-B)+4 a A b+3 b^2 B\right ) \left ((a-i b) E\left (\frac{1}{4} (\pi -2 i x)|-\frac{2 i b}{a-i b}\right )-a \text{EllipticF}\left (\frac{1}{4} (\pi -2 i x),-\frac{2 i b}{a-i b}\right )\right )\right )+i b \cosh (x) \left (-\left (a^2 B-4 a A b-3 b^2 B\right ) (a+b \sinh (x))-\left (a^2+b^2\right ) (a B-A b)\right )\right )}{3 b \left (a^2+b^2\right )^2 (a+b \sinh (x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.239, size = 806, 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 \sinh \left (x\right ) + A}{{\left (b \sinh \left (x\right ) + a\right )}^{\frac{5}{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 \sinh \left (x\right ) + A\right )} \sqrt{b \sinh \left (x\right ) + a}}{b^{3} \sinh \left (x\right )^{3} + 3 \, a b^{2} \sinh \left (x\right )^{2} + 3 \, a^{2} b \sinh \left (x\right ) + a^{3}}, 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 \sinh \left (x\right ) + A}{{\left (b \sinh \left (x\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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