Optimal. Leaf size=248 \[ \frac {i \left (4 a^2+b^2\right ) \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}} F\left (\frac {1}{2} \left (2 i c+2 i d x-\frac {\pi }{2}\right )|\frac {2 b}{2 i a+b}\right )}{6 \sqrt {2} d \sqrt {2 a+b \sinh (2 c+2 d x)}}+\frac {b \cosh (2 c+2 d x) \sqrt {2 a+b \sinh (2 c+2 d x)}}{6 \sqrt {2} d}-\frac {2 i \sqrt {2} a \sqrt {2 a+b \sinh (2 c+2 d x)} E\left (\frac {1}{2} \left (2 i c+2 i d x-\frac {\pi }{2}\right )|\frac {2 b}{2 i a+b}\right )}{3 d \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}} \]
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Rubi [A] time = 0.25, antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {2666, 2656, 2752, 2663, 2661, 2655, 2653} \[ \frac {i \left (4 a^2+b^2\right ) \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}} F\left (\frac {1}{2} \left (2 i c+2 i d x-\frac {\pi }{2}\right )|\frac {2 b}{2 i a+b}\right )}{6 \sqrt {2} d \sqrt {2 a+b \sinh (2 c+2 d x)}}+\frac {b \cosh (2 c+2 d x) \sqrt {2 a+b \sinh (2 c+2 d x)}}{6 \sqrt {2} d}-\frac {2 i \sqrt {2} a \sqrt {2 a+b \sinh (2 c+2 d x)} E\left (\frac {1}{2} \left (2 i c+2 i d x-\frac {\pi }{2}\right )|\frac {2 b}{2 i a+b}\right )}{3 d \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2656
Rule 2661
Rule 2663
Rule 2666
Rule 2752
Rubi steps
\begin {align*} \int (a+b \cosh (c+d x) \sinh (c+d x))^{3/2} \, dx &=\int \left (a+\frac {1}{2} b \sinh (2 c+2 d x)\right )^{3/2} \, dx\\ &=\frac {b \cosh (2 c+2 d x) \sqrt {2 a+b \sinh (2 c+2 d x)}}{6 \sqrt {2} d}+\frac {2}{3} \int \frac {\frac {1}{8} \left (12 a^2-b^2\right )+a b \sinh (2 c+2 d x)}{\sqrt {a+\frac {1}{2} b \sinh (2 c+2 d x)}} \, dx\\ &=\frac {b \cosh (2 c+2 d x) \sqrt {2 a+b \sinh (2 c+2 d x)}}{6 \sqrt {2} d}+\frac {1}{3} (4 a) \int \sqrt {a+\frac {1}{2} b \sinh (2 c+2 d x)} \, dx+\frac {1}{12} \left (-4 a^2-b^2\right ) \int \frac {1}{\sqrt {a+\frac {1}{2} b \sinh (2 c+2 d x)}} \, dx\\ &=\frac {b \cosh (2 c+2 d x) \sqrt {2 a+b \sinh (2 c+2 d x)}}{6 \sqrt {2} d}+\frac {\left (4 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 \sqrt {\frac {a+\frac {1}{2} b \sinh (2 c+2 d x)}{a-\frac {i b}{2}}}}+\frac {\left (\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}}}\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}{12 \sqrt {a+\frac {1}{2} b \sinh (2 c+2 d x)}}\\ &=\frac {b \cosh (2 c+2 d x) \sqrt {2 a+b \sinh (2 c+2 d x)}}{6 \sqrt {2} d}-\frac {2 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 d \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}+\frac {i \left (4 a^2+b^2\right ) 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}}}{6 \sqrt {2} d \sqrt {2 a+b \sinh (2 c+2 d x)}}\\ \end {align*}
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Mathematica [A] time = 0.85, size = 202, normalized size = 0.81 \[ \frac {-2 i \left (4 a^2+b^2\right ) \sqrt {\frac {2 a+b \sinh (2 (c+d x))}{2 a-i b}} F\left (\frac {1}{4} (-4 i c-4 i d x+\pi )|-\frac {2 i b}{2 a-i b}\right )+b (4 a \cosh (2 (c+d x))+b \sinh (4 (c+d x)))+16 a (b+2 i a) \sqrt {\frac {2 a+b \sinh (2 (c+d x))}{2 a-i b}} E\left (\frac {1}{4} (-4 i c-4 i d x+\pi )|-\frac {2 i b}{2 a-i b}\right )}{12 d \sqrt {4 a+2 b \sinh (2 (c+d x))}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a\right )}^{\frac {3}{2}}, x\right ) \]
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 [B] time = 1.08, size = 935, normalized size = 3.77 \[ \frac {4 i \sqrt {-\frac {2 a +b \sinh \left (2 d x +2 c \right )}{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 {2 a +b \sinh \left (2 d x +2 c \right )}{i b -2 a}}, \sqrt {-\frac {i b -2 a}{i b +2 a}}\right ) a^{2} b +i \sqrt {-\frac {2 a +b \sinh \left (2 d x +2 c \right )}{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 {2 a +b \sinh \left (2 d x +2 c \right )}{i b -2 a}}, \sqrt {-\frac {i b -2 a}{i b +2 a}}\right ) b^{3}+24 \sqrt {-\frac {2 a +b \sinh \left (2 d x +2 c \right )}{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 {2 a +b \sinh \left (2 d x +2 c \right )}{i b -2 a}}, \sqrt {-\frac {i b -2 a}{i b +2 a}}\right ) a^{3}+6 \sqrt {-\frac {2 a +b \sinh \left (2 d x +2 c \right )}{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 {2 a +b \sinh \left (2 d x +2 c \right )}{i b -2 a}}, \sqrt {-\frac {i b -2 a}{i b +2 a}}\right ) a \,b^{2}-32 \sqrt {-\frac {2 a +b \sinh \left (2 d x +2 c \right )}{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}}\, \EllipticE \left (\sqrt {-\frac {2 a +b \sinh \left (2 d x +2 c \right )}{i b -2 a}}, \sqrt {-\frac {i b -2 a}{i b +2 a}}\right ) a^{3}-8 \sqrt {-\frac {2 a +b \sinh \left (2 d x +2 c \right )}{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}}\, \EllipticE \left (\sqrt {-\frac {2 a +b \sinh \left (2 d x +2 c \right )}{i b -2 a}}, \sqrt {-\frac {i b -2 a}{i b +2 a}}\right ) a \,b^{2}+b^{3} \left (\sinh ^{3}\left (2 d x +2 c \right )\right )+2 a \,b^{2} \left (\sinh ^{2}\left (2 d x +2 c \right )\right )+b^{3} \sinh \left (2 d x +2 c \right )+2 a \,b^{2}}{6 b \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 \[ \int {\left (b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a\right )}^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\left (a+b\,\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )\right )}^{3/2} \,d x \]
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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