Optimal. Leaf size=158 \[ -\frac{2 \sqrt{2} b \cosh (2 c+2 d x)}{d \left (4 a^2+b^2\right ) \sqrt{2 a+b \sinh (2 c+2 d x)}}-\frac{2 i \sqrt{2} \sqrt{2 a+b \sinh (2 c+2 d x)} E\left (\frac{1}{2} \left (2 i c+2 i d x-\frac{\pi }{2}\right )|\frac{2 b}{2 i a+b}\right )}{d \left (4 a^2+b^2\right ) \sqrt{\frac{2 a+b \sinh (2 c+2 d x)}{2 a-i b}}} \]
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Rubi [A] time = 0.113346, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2666, 2664, 21, 2655, 2653} \[ -\frac{2 \sqrt{2} b \cosh (2 c+2 d x)}{d \left (4 a^2+b^2\right ) \sqrt{2 a+b \sinh (2 c+2 d x)}}-\frac{2 i \sqrt{2} \sqrt{2 a+b \sinh (2 c+2 d x)} E\left (\frac{1}{2} \left (2 i c+2 i d x-\frac{\pi }{2}\right )|\frac{2 b}{2 i a+b}\right )}{d \left (4 a^2+b^2\right ) \sqrt{\frac{2 a+b \sinh (2 c+2 d x)}{2 a-i b}}} \]
Antiderivative was successfully verified.
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Rule 2666
Rule 2664
Rule 21
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \frac{1}{(a+b \cosh (c+d x) \sinh (c+d x))^{3/2}} \, dx &=\int \frac{1}{\left (a+\frac{1}{2} b \sinh (2 c+2 d x)\right )^{3/2}} \, dx\\ &=-\frac{2 \sqrt{2} b \cosh (2 c+2 d x)}{\left (4 a^2+b^2\right ) d \sqrt{2 a+b \sinh (2 c+2 d x)}}-\frac{8 \int \frac{-\frac{a}{2}-\frac{1}{4} b \sinh (2 c+2 d x)}{\sqrt{a+\frac{1}{2} b \sinh (2 c+2 d x)}} \, dx}{4 a^2+b^2}\\ &=-\frac{2 \sqrt{2} b \cosh (2 c+2 d x)}{\left (4 a^2+b^2\right ) d \sqrt{2 a+b \sinh (2 c+2 d x)}}+\frac{4 \int \sqrt{a+\frac{1}{2} b \sinh (2 c+2 d x)} \, dx}{4 a^2+b^2}\\ &=-\frac{2 \sqrt{2} b \cosh (2 c+2 d x)}{\left (4 a^2+b^2\right ) d \sqrt{2 a+b \sinh (2 c+2 d x)}}+\frac{\left (4 \sqrt{a+\frac{1}{2} b \sinh (2 c+2 d x)}\right ) \int \sqrt{\frac{a}{a-\frac{i b}{2}}+\frac{b \sinh (2 c+2 d x)}{2 \left (a-\frac{i b}{2}\right )}} \, dx}{\left (4 a^2+b^2\right ) \sqrt{\frac{a+\frac{1}{2} b \sinh (2 c+2 d x)}{a-\frac{i b}{2}}}}\\ &=-\frac{2 \sqrt{2} b \cosh (2 c+2 d x)}{\left (4 a^2+b^2\right ) d \sqrt{2 a+b \sinh (2 c+2 d x)}}-\frac{2 i \sqrt{2} E\left (\frac{1}{2} \left (2 i c-\frac{\pi }{2}+2 i d x\right )|\frac{2 b}{2 i a+b}\right ) \sqrt{2 a+b \sinh (2 c+2 d x)}}{\left (4 a^2+b^2\right ) d \sqrt{\frac{2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}\\ \end{align*}
Mathematica [A] time = 0.473617, size = 119, normalized size = 0.75 \[ \frac{2 \left (-b \cosh (2 (c+d x))+(b+2 i a) \sqrt{\frac{2 a+b \sinh (2 (c+d x))}{2 a-i b}} E\left (\frac{1}{4} (-4 i c-4 i d x+\pi )|-\frac{2 i b}{2 a-i b}\right )\right )}{d \left (4 a^2+b^2\right ) \sqrt{a+\frac{1}{2} b \sinh (2 (c+d x))}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.452, size = 630, normalized size = 4. \begin{align*} 4\,{\frac{1}{ \left ( 4\,{a}^{2}+{b}^{2} \right ) b\cosh \left ( 2\,dx+2\,c \right ) \sqrt{4\,a+2\,b\sinh \left ( 2\,dx+2\,c \right ) }d} \left ( 4\,\sqrt{-{\frac{2\,a+b\sinh \left ( 2\,dx+2\,c \right ) }{ib-2\,a}}}\sqrt{{\frac{ \left ( -\sinh \left ( 2\,dx+2\,c \right ) +i \right ) b}{ib+2\,a}}}\sqrt{{\frac{ \left ( i+\sinh \left ( 2\,dx+2\,c \right ) \right ) b}{ib-2\,a}}}{\it EllipticF} \left ( \sqrt{-{\frac{2\,a+b\sinh \left ( 2\,dx+2\,c \right ) }{ib-2\,a}}},\sqrt{-{\frac{ib-2\,a}{ib+2\,a}}} \right ){a}^{2}+\sqrt{-{\frac{2\,a+b\sinh \left ( 2\,dx+2\,c \right ) }{ib-2\,a}}}\sqrt{{\frac{ \left ( -\sinh \left ( 2\,dx+2\,c \right ) +i \right ) b}{ib+2\,a}}}\sqrt{{\frac{ \left ( i+\sinh \left ( 2\,dx+2\,c \right ) \right ) b}{ib-2\,a}}}{\it EllipticF} \left ( \sqrt{-{\frac{2\,a+b\sinh \left ( 2\,dx+2\,c \right ) }{ib-2\,a}}},\sqrt{-{\frac{ib-2\,a}{ib+2\,a}}} \right ){b}^{2}-4\,\sqrt{-{\frac{2\,a+b\sinh \left ( 2\,dx+2\,c \right ) }{ib-2\,a}}}\sqrt{{\frac{ \left ( -\sinh \left ( 2\,dx+2\,c \right ) +i \right ) b}{ib+2\,a}}}\sqrt{{\frac{ \left ( i+\sinh \left ( 2\,dx+2\,c \right ) \right ) b}{ib-2\,a}}}{\it EllipticE} \left ( \sqrt{-{\frac{2\,a+b\sinh \left ( 2\,dx+2\,c \right ) }{ib-2\,a}}},\sqrt{-{\frac{ib-2\,a}{ib+2\,a}}} \right ){a}^{2}-\sqrt{-{\frac{2\,a+b\sinh \left ( 2\,dx+2\,c \right ) }{ib-2\,a}}}\sqrt{{\frac{ \left ( -\sinh \left ( 2\,dx+2\,c \right ) +i \right ) b}{ib+2\,a}}}\sqrt{{\frac{ \left ( i+\sinh \left ( 2\,dx+2\,c \right ) \right ) b}{ib-2\,a}}}{\it EllipticE} \left ( \sqrt{-{\frac{2\,a+b\sinh \left ( 2\,dx+2\,c \right ) }{ib-2\,a}}},\sqrt{-{\frac{ib-2\,a}{ib+2\,a}}} \right ){b}^{2}-{b}^{2} \left ( \sinh \left ( 2\,dx+2\,c \right ) \right ) ^{2}-{b}^{2} \right ) } \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 (d x + c\right ) \sinh \left (d x + c\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{\sqrt{b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a}}{b^{2} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) \sinh \left (d x + c\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{1}{{\left (b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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