Optimal. Leaf size=123 \[ \frac{i a (a-b) \sqrt{\frac{b \sinh ^2(x)}{a}+1} \text{EllipticF}\left (i x,\frac{b}{a}\right )}{3 \sqrt{a+b \sinh ^2(x)}}+\frac{1}{3} b \sinh (x) \cosh (x) \sqrt{a+b \sinh ^2(x)}-\frac{2 i (2 a-b) \sqrt{a+b \sinh ^2(x)} E\left (i x\left |\frac{b}{a}\right .\right )}{3 \sqrt{\frac{b \sinh ^2(x)}{a}+1}} \]
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Rubi [A] time = 0.162822, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3180, 3172, 3178, 3177, 3183, 3182} \[ \frac{1}{3} b \sinh (x) \cosh (x) \sqrt{a+b \sinh ^2(x)}+\frac{i a (a-b) \sqrt{\frac{b \sinh ^2(x)}{a}+1} F\left (i x\left |\frac{b}{a}\right .\right )}{3 \sqrt{a+b \sinh ^2(x)}}-\frac{2 i (2 a-b) \sqrt{a+b \sinh ^2(x)} E\left (i x\left |\frac{b}{a}\right .\right )}{3 \sqrt{\frac{b \sinh ^2(x)}{a}+1}} \]
Antiderivative was successfully verified.
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Rule 3180
Rule 3172
Rule 3178
Rule 3177
Rule 3183
Rule 3182
Rubi steps
\begin{align*} \int \left (a+b \sinh ^2(x)\right )^{3/2} \, dx &=\frac{1}{3} b \cosh (x) \sinh (x) \sqrt{a+b \sinh ^2(x)}+\frac{1}{3} \int \frac{a (3 a-b)+2 (2 a-b) b \sinh ^2(x)}{\sqrt{a+b \sinh ^2(x)}} \, dx\\ &=\frac{1}{3} b \cosh (x) \sinh (x) \sqrt{a+b \sinh ^2(x)}-\frac{1}{3} (a (a-b)) \int \frac{1}{\sqrt{a+b \sinh ^2(x)}} \, dx+\frac{1}{3} (2 (2 a-b)) \int \sqrt{a+b \sinh ^2(x)} \, dx\\ &=\frac{1}{3} b \cosh (x) \sinh (x) \sqrt{a+b \sinh ^2(x)}+\frac{\left (2 (2 a-b) \sqrt{a+b \sinh ^2(x)}\right ) \int \sqrt{1+\frac{b \sinh ^2(x)}{a}} \, dx}{3 \sqrt{1+\frac{b \sinh ^2(x)}{a}}}-\frac{\left (a (a-b) \sqrt{1+\frac{b \sinh ^2(x)}{a}}\right ) \int \frac{1}{\sqrt{1+\frac{b \sinh ^2(x)}{a}}} \, dx}{3 \sqrt{a+b \sinh ^2(x)}}\\ &=\frac{1}{3} b \cosh (x) \sinh (x) \sqrt{a+b \sinh ^2(x)}-\frac{2 i (2 a-b) E\left (i x\left |\frac{b}{a}\right .\right ) \sqrt{a+b \sinh ^2(x)}}{3 \sqrt{1+\frac{b \sinh ^2(x)}{a}}}+\frac{i a (a-b) F\left (i x\left |\frac{b}{a}\right .\right ) \sqrt{1+\frac{b \sinh ^2(x)}{a}}}{3 \sqrt{a+b \sinh ^2(x)}}\\ \end{align*}
Mathematica [A] time = 0.378756, size = 132, normalized size = 1.07 \[ \frac{4 i a (a-b) \sqrt{\frac{2 a+b \cosh (2 x)-b}{a}} \text{EllipticF}\left (i x,\frac{b}{a}\right )+\sqrt{2} b \sinh (2 x) (2 a+b \cosh (2 x)-b)-8 i a (2 a-b) \sqrt{\frac{2 a+b \cosh (2 x)-b}{a}} E\left (i x\left |\frac{b}{a}\right .\right )}{12 \sqrt{2 a+b \cosh (2 x)-b}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.07, size = 329, normalized size = 2.7 \begin{align*}{\frac{1}{3\,\cosh \left ( x \right ) } \left ( \sqrt{-{\frac{b}{a}}}{b}^{2}\sinh \left ( x \right ) \left ( \cosh \left ( x \right ) \right ) ^{4}+ \left ( \sqrt{-{\frac{b}{a}}}ab-\sqrt{-{\frac{b}{a}}}{b}^{2} \right ) \left ( \cosh \left ( x \right ) \right ) ^{2}\sinh \left ( x \right ) +3\,{a}^{2}\sqrt{{\frac{b \left ( \cosh \left ( x \right ) \right ) ^{2}}{a}}+{\frac{a-b}{a}}}\sqrt{ \left ( \cosh \left ( x \right ) \right ) ^{2}}{\it EllipticF} \left ( \sinh \left ( x \right ) \sqrt{-{\frac{b}{a}}},\sqrt{{\frac{a}{b}}} \right ) -5\,ab\sqrt{{\frac{b \left ( \cosh \left ( x \right ) \right ) ^{2}}{a}}+{\frac{a-b}{a}}}\sqrt{ \left ( \cosh \left ( x \right ) \right ) ^{2}}{\it EllipticF} \left ( \sinh \left ( x \right ) \sqrt{-{\frac{b}{a}}},\sqrt{{\frac{a}{b}}} \right ) +2\,\sqrt{{\frac{b \left ( \cosh \left ( x \right ) \right ) ^{2}}{a}}+{\frac{a-b}{a}}}\sqrt{ \left ( \cosh \left ( x \right ) \right ) ^{2}}{\it EllipticF} \left ( \sinh \left ( x \right ) \sqrt{-{\frac{b}{a}}},\sqrt{{\frac{a}{b}}} \right ){b}^{2}+4\,ab\sqrt{{\frac{b \left ( \cosh \left ( x \right ) \right ) ^{2}}{a}}+{\frac{a-b}{a}}}\sqrt{ \left ( \cosh \left ( x \right ) \right ) ^{2}}{\it EllipticE} \left ( \sinh \left ( x \right ) \sqrt{-{\frac{b}{a}}},\sqrt{{\frac{a}{b}}} \right ) -2\,\sqrt{{\frac{b \left ( \cosh \left ( x \right ) \right ) ^{2}}{a}}+{\frac{a-b}{a}}}\sqrt{ \left ( \cosh \left ( x \right ) \right ) ^{2}}{\it EllipticE} \left ( \sinh \left ( x \right ) \sqrt{-{\frac{b}{a}}},\sqrt{{\frac{a}{b}}} \right ){b}^{2} \right ){\frac{1}{\sqrt{-{\frac{b}{a}}}}}{\frac{1}{\sqrt{a+b \left ( \sinh \left ( x \right ) \right ) ^{2}}}}} \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{\left (b \sinh \left (x\right )^{2} + 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 ({\left (b \sinh \left (x\right )^{2} + a\right )}^{\frac{3}{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{\left (b \sinh \left (x\right )^{2} + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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