Optimal. Leaf size=325 \[ -\frac {32 \sqrt {2} a b \cosh (2 c+2 d x)}{3 d \left (4 a^2+b^2\right )^2 \sqrt {2 a+b \sinh (2 c+2 d x)}}-\frac {4 \sqrt {2} b \cosh (2 c+2 d x)}{3 d \left (4 a^2+b^2\right ) (2 a+b \sinh (2 c+2 d x))^{3/2}}+\frac {4 i \sqrt {2} \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}} F\left (\frac {1}{2} \left (2 i c+2 i d x-\frac {\pi }{2}\right )|\frac {2 b}{2 i a+b}\right )}{3 d \left (4 a^2+b^2\right ) \sqrt {2 a+b \sinh (2 c+2 d x)}}-\frac {32 i \sqrt {2} a \sqrt {2 a+b \sinh (2 c+2 d x)} E\left (\frac {1}{2} \left (2 i c+2 i d x-\frac {\pi }{2}\right )|\frac {2 b}{2 i a+b}\right )}{3 d \left (4 a^2+b^2\right )^2 \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}} \]
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Rubi [A] time = 0.38, antiderivative size = 325, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2666, 2664, 2754, 2752, 2663, 2661, 2655, 2653} \[ -\frac {32 \sqrt {2} a b \cosh (2 c+2 d x)}{3 d \left (4 a^2+b^2\right )^2 \sqrt {2 a+b \sinh (2 c+2 d x)}}-\frac {4 \sqrt {2} b \cosh (2 c+2 d x)}{3 d \left (4 a^2+b^2\right ) (2 a+b \sinh (2 c+2 d x))^{3/2}}+\frac {4 i \sqrt {2} \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}} F\left (\frac {1}{2} \left (2 i c+2 i d x-\frac {\pi }{2}\right )|\frac {2 b}{2 i a+b}\right )}{3 d \left (4 a^2+b^2\right ) \sqrt {2 a+b \sinh (2 c+2 d x)}}-\frac {32 i \sqrt {2} a \sqrt {2 a+b \sinh (2 c+2 d x)} E\left (\frac {1}{2} \left (2 i c+2 i d x-\frac {\pi }{2}\right )|\frac {2 b}{2 i a+b}\right )}{3 d \left (4 a^2+b^2\right )^2 \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2664
Rule 2666
Rule 2752
Rule 2754
Rubi steps
\begin {align*} \int \frac {1}{(a+b \cosh (c+d x) \sinh (c+d x))^{5/2}} \, dx &=\int \frac {1}{\left (a+\frac {1}{2} b \sinh (2 c+2 d x)\right )^{5/2}} \, dx\\ &=-\frac {4 \sqrt {2} b \cosh (2 c+2 d x)}{3 \left (4 a^2+b^2\right ) d (2 a+b \sinh (2 c+2 d x))^{3/2}}-\frac {8 \int \frac {-\frac {3 a}{2}+\frac {1}{4} b \sinh (2 c+2 d x)}{\left (a+\frac {1}{2} b \sinh (2 c+2 d x)\right )^{3/2}} \, dx}{3 \left (4 a^2+b^2\right )}\\ &=-\frac {4 \sqrt {2} b \cosh (2 c+2 d x)}{3 \left (4 a^2+b^2\right ) d (2 a+b \sinh (2 c+2 d x))^{3/2}}-\frac {32 \sqrt {2} a b \cosh (2 c+2 d x)}{3 \left (4 a^2+b^2\right )^2 d \sqrt {2 a+b \sinh (2 c+2 d x)}}+\frac {64 \int \frac {\frac {1}{16} \left (12 a^2-b^2\right )+\frac {1}{2} a b \sinh (2 c+2 d x)}{\sqrt {a+\frac {1}{2} b \sinh (2 c+2 d x)}} \, dx}{3 \left (4 a^2+b^2\right )^2}\\ &=-\frac {4 \sqrt {2} b \cosh (2 c+2 d x)}{3 \left (4 a^2+b^2\right ) d (2 a+b \sinh (2 c+2 d x))^{3/2}}-\frac {32 \sqrt {2} a b \cosh (2 c+2 d x)}{3 \left (4 a^2+b^2\right )^2 d \sqrt {2 a+b \sinh (2 c+2 d x)}}+\frac {(64 a) \int \sqrt {a+\frac {1}{2} b \sinh (2 c+2 d x)} \, dx}{3 \left (4 a^2+b^2\right )^2}-\frac {4 \int \frac {1}{\sqrt {a+\frac {1}{2} b \sinh (2 c+2 d x)}} \, dx}{3 \left (4 a^2+b^2\right )}\\ &=-\frac {4 \sqrt {2} b \cosh (2 c+2 d x)}{3 \left (4 a^2+b^2\right ) d (2 a+b \sinh (2 c+2 d x))^{3/2}}-\frac {32 \sqrt {2} a b \cosh (2 c+2 d x)}{3 \left (4 a^2+b^2\right )^2 d \sqrt {2 a+b \sinh (2 c+2 d x)}}+\frac {\left (64 a \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}{3 \left (4 a^2+b^2\right )^2 \sqrt {\frac {a+\frac {1}{2} b \sinh (2 c+2 d x)}{a-\frac {i b}{2}}}}-\frac {\left (4 \sqrt {\frac {a+\frac {1}{2} b \sinh (2 c+2 d x)}{a-\frac {i b}{2}}}\right ) \int \frac {1}{\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}{3 \left (4 a^2+b^2\right ) \sqrt {a+\frac {1}{2} b \sinh (2 c+2 d x)}}\\ &=-\frac {4 \sqrt {2} b \cosh (2 c+2 d x)}{3 \left (4 a^2+b^2\right ) d (2 a+b \sinh (2 c+2 d x))^{3/2}}-\frac {32 \sqrt {2} a b \cosh (2 c+2 d x)}{3 \left (4 a^2+b^2\right )^2 d \sqrt {2 a+b \sinh (2 c+2 d x)}}-\frac {32 i \sqrt {2} a 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)}}{3 \left (4 a^2+b^2\right )^2 d \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}+\frac {4 i \sqrt {2} F\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 {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}{3 \left (4 a^2+b^2\right ) d \sqrt {2 a+b \sinh (2 c+2 d x)}}\\ \end {align*}
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Mathematica [A] time = 1.65, size = 237, normalized size = 0.73 \[ \frac {4 \sqrt {2} \left (-b \cosh (2 (c+d x)) \left (20 a^2+8 a b \sinh (2 (c+d x))+b^2\right )+(b-2 i a) (2 a-i b)^2 \left (\frac {2 a+b \sinh (2 (c+d x))}{2 a-i b}\right )^{3/2} F\left (\frac {1}{4} (-4 i c-4 i d x+\pi )|-\frac {2 i b}{2 a-i b}\right )+8 i a (2 a-i b)^2 \left (\frac {2 a+b \sinh (2 (c+d x))}{2 a-i b}\right )^{3/2} E\left (\frac {1}{4} (-4 i c-4 i d x+\pi )|-\frac {2 i b}{2 a-i b}\right )\right )}{3 d \left (4 a^2+b^2\right )^2 (2 a+b \sinh (2 (c+d x)))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a}}{b^{3} \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right )^{3} + 3 \, a b^{2} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{2} + 3 \, a^{2} b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.86, size = 641, normalized size = 1.97 \[ \frac {4 \sqrt {\left (2 a +b \sinh \left (2 d x +2 c \right )\right ) \left (\cosh ^{2}\left (2 d x +2 c \right )\right )}\, \left (-\frac {2 \sqrt {\left (2 a +b \sinh \left (2 d x +2 c \right )\right ) \left (\cosh ^{2}\left (2 d x +2 c \right )\right )}}{3 b \left (4 a^{2}+b^{2}\right ) \left (\sinh \left (2 d x +2 c \right )+\frac {2 a}{b}\right )^{2}}-\frac {16 b \left (\cosh ^{2}\left (2 d x +2 c \right )\right ) a}{3 \left (4 a^{2}+b^{2}\right )^{2} \sqrt {\left (2 a +b \sinh \left (2 d x +2 c \right )\right ) \left (\cosh ^{2}\left (2 d x +2 c \right )\right )}}+\frac {2 \left (12 a^{2}-b^{2}\right ) \left (\frac {2 a}{b}-i\right ) \sqrt {\frac {-b \sinh \left (2 d x +2 c \right )-2 a}{i b -2 a}}\, \sqrt {\frac {\left (-\sinh \left (2 d x +2 c \right )+i\right ) b}{i b +2 a}}\, \sqrt {\frac {\left (\sinh \left (2 d x +2 c \right )+i\right ) b}{i b -2 a}}\, \EllipticF \left (\sqrt {\frac {-b \sinh \left (2 d x +2 c \right )-2 a}{i b -2 a}}, \sqrt {\frac {-i b +2 a}{i b +2 a}}\right )}{\left (48 a^{4}+24 a^{2} b^{2}+3 b^{4}\right ) \sqrt {\left (2 a +b \sinh \left (2 d x +2 c \right )\right ) \left (\cosh ^{2}\left (2 d x +2 c \right )\right )}}+\frac {16 a b \left (\frac {2 a}{b}-i\right ) \sqrt {\frac {-b \sinh \left (2 d x +2 c \right )-2 a}{i b -2 a}}\, \sqrt {\frac {\left (-\sinh \left (2 d x +2 c \right )+i\right ) b}{i b +2 a}}\, \sqrt {\frac {\left (\sinh \left (2 d x +2 c \right )+i\right ) b}{i b -2 a}}\, \left (\left (-\frac {2 a}{b}-i\right ) \EllipticE \left (\sqrt {\frac {-b \sinh \left (2 d x +2 c \right )-2 a}{i b -2 a}}, \sqrt {\frac {-i b +2 a}{i b +2 a}}\right )+i \EllipticF \left (\sqrt {\frac {-b \sinh \left (2 d x +2 c \right )-2 a}{i b -2 a}}, \sqrt {\frac {-i b +2 a}{i b +2 a}}\right )\right )}{3 \left (4 a^{2}+b^{2}\right )^{2} \sqrt {\left (2 a +b \sinh \left (2 d x +2 c \right )\right ) \left (\cosh ^{2}\left (2 d x +2 c \right )\right )}}\right )}{\cosh \left (2 d x +2 c \right ) \sqrt {4 a +2 b \sinh \left (2 d x +2 c \right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (a+b\,\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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