Optimal. Leaf size=211 \[ -\frac{3 \tanh ^{-1}\left (\frac{\sqrt [4]{b^2-c^2} \sinh \left (x+i \tan ^{-1}(b,-i c)\right )}{\sqrt{2} \sqrt{-\sqrt{b^2-c^2}+\sqrt{b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}}\right )}{16 \sqrt{2} \left (b^2-c^2\right )^{5/4}}+\frac{3 (b \sinh (x)+c \cosh (x))}{16 \left (b^2-c^2\right ) \left (-\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}}-\frac{b \sinh (x)+c \cosh (x)}{4 \sqrt{b^2-c^2} \left (-\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{5/2}} \]
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Rubi [A] time = 0.166733, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3116, 3115, 2649, 204} \[ -\frac{3 \tanh ^{-1}\left (\frac{\sqrt [4]{b^2-c^2} \sinh \left (x+i \tan ^{-1}(b,-i c)\right )}{\sqrt{2} \sqrt{-\sqrt{b^2-c^2}+\sqrt{b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}}\right )}{16 \sqrt{2} \left (b^2-c^2\right )^{5/4}}+\frac{3 (b \sinh (x)+c \cosh (x))}{16 \left (b^2-c^2\right ) \left (-\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}}-\frac{b \sinh (x)+c \cosh (x)}{4 \sqrt{b^2-c^2} \left (-\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3116
Rule 3115
Rule 2649
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{\left (-\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{5/2}} \, dx &=-\frac{c \cosh (x)+b \sinh (x)}{4 \sqrt{b^2-c^2} \left (-\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{5/2}}-\frac{3 \int \frac{1}{\left (-\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}} \, dx}{8 \sqrt{b^2-c^2}}\\ &=-\frac{c \cosh (x)+b \sinh (x)}{4 \sqrt{b^2-c^2} \left (-\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{5/2}}+\frac{3 (c \cosh (x)+b \sinh (x))}{16 \left (b^2-c^2\right ) \left (-\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}}+\frac{3 \int \frac{1}{\sqrt{-\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)}} \, dx}{32 \left (b^2-c^2\right )}\\ &=-\frac{c \cosh (x)+b \sinh (x)}{4 \sqrt{b^2-c^2} \left (-\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{5/2}}+\frac{3 (c \cosh (x)+b \sinh (x))}{16 \left (b^2-c^2\right ) \left (-\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}}+\frac{3 \int \frac{1}{\sqrt{-\sqrt{b^2-c^2}+\sqrt{b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}} \, dx}{32 \left (b^2-c^2\right )}\\ &=-\frac{c \cosh (x)+b \sinh (x)}{4 \sqrt{b^2-c^2} \left (-\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{5/2}}+\frac{3 (c \cosh (x)+b \sinh (x))}{16 \left (b^2-c^2\right ) \left (-\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}}+\frac{(3 i) \operatorname{Subst}\left (\int \frac{1}{-2 \sqrt{b^2-c^2}-x^2} \, dx,x,-\frac{i \sqrt{b^2-c^2} \sinh \left (x+i \tan ^{-1}(b,-i c)\right )}{\sqrt{-\sqrt{b^2-c^2}+\sqrt{b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}}\right )}{16 \left (b^2-c^2\right )}\\ &=-\frac{3 \tanh ^{-1}\left (\frac{\sqrt [4]{b^2-c^2} \sinh \left (x+i \tan ^{-1}(b,-i c)\right )}{\sqrt{2} \sqrt{-\sqrt{b^2-c^2}+\sqrt{b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}}\right )}{16 \sqrt{2} \left (b^2-c^2\right )^{5/4}}-\frac{c \cosh (x)+b \sinh (x)}{4 \sqrt{b^2-c^2} \left (-\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{5/2}}+\frac{3 (c \cosh (x)+b \sinh (x))}{16 \left (b^2-c^2\right ) \left (-\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}}\\ \end{align*}
Mathematica [F] time = 180.009, size = 0, normalized size = 0. \[ \text{\$Aborted} \]
Verification is Not applicable to the result.
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Maple [B] time = 1.115, size = 984, normalized size = 4.7 \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{1}{{\left (b \cosh \left (x\right ) + c \sinh \left (x\right ) - \sqrt{b^{2} - c^{2}}\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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