Optimal. Leaf size=246 \[ -\frac {\coth (c+d x) \sqrt {a+b \text {sech}(c+d x)}}{d}+\frac {\sqrt {a+b} \coth (c+d x) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{d}+\frac {2 \coth (c+d x) \sqrt {-\frac {b (1-\text {sech}(c+d x))}{a+b \text {sech}(c+d x)}} \sqrt {\frac {b (\text {sech}(c+d x)+1)}{a+b \text {sech}(c+d x)}} (a+b \text {sech}(c+d x)) \Pi \left (\frac {a}{a+b};\sin ^{-1}\left (\frac {\sqrt {a+b}}{\sqrt {a+b \text {sech}(c+d x)}}\right )|\frac {a-b}{a+b}\right )}{d \sqrt {a+b}} \]
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Rubi [A] time = 0.22, antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3896, 3780, 3875, 3832} \[ -\frac {\coth (c+d x) \sqrt {a+b \text {sech}(c+d x)}}{d}+\frac {\sqrt {a+b} \coth (c+d x) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{d}+\frac {2 \coth (c+d x) \sqrt {-\frac {b (1-\text {sech}(c+d x))}{a+b \text {sech}(c+d x)}} \sqrt {\frac {b (\text {sech}(c+d x)+1)}{a+b \text {sech}(c+d x)}} (a+b \text {sech}(c+d x)) \Pi \left (\frac {a}{a+b};\sin ^{-1}\left (\frac {\sqrt {a+b}}{\sqrt {a+b \text {sech}(c+d x)}}\right )|\frac {a-b}{a+b}\right )}{d \sqrt {a+b}} \]
Antiderivative was successfully verified.
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Rule 3780
Rule 3832
Rule 3875
Rule 3896
Rubi steps
\begin {align*} \int \coth ^2(c+d x) \sqrt {a+b \text {sech}(c+d x)} \, dx &=-\int \left (-\sqrt {a+b \text {sech}(c+d x)}-\text {csch}^2(c+d x) \sqrt {a+b \text {sech}(c+d x)}\right ) \, dx\\ &=\int \sqrt {a+b \text {sech}(c+d x)} \, dx+\int \text {csch}^2(c+d x) \sqrt {a+b \text {sech}(c+d x)} \, dx\\ &=-\frac {\coth (c+d x) \sqrt {a+b \text {sech}(c+d x)}}{d}+\frac {2 \coth (c+d x) \Pi \left (\frac {a}{a+b};\sin ^{-1}\left (\frac {\sqrt {a+b}}{\sqrt {a+b \text {sech}(c+d x)}}\right )|\frac {a-b}{a+b}\right ) \sqrt {-\frac {b (1-\text {sech}(c+d x))}{a+b \text {sech}(c+d x)}} \sqrt {\frac {b (1+\text {sech}(c+d x))}{a+b \text {sech}(c+d x)}} (a+b \text {sech}(c+d x))}{\sqrt {a+b} d}-\frac {1}{2} b \int \frac {\text {sech}(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx\\ &=\frac {\sqrt {a+b} \coth (c+d x) F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{d}-\frac {\coth (c+d x) \sqrt {a+b \text {sech}(c+d x)}}{d}+\frac {2 \coth (c+d x) \Pi \left (\frac {a}{a+b};\sin ^{-1}\left (\frac {\sqrt {a+b}}{\sqrt {a+b \text {sech}(c+d x)}}\right )|\frac {a-b}{a+b}\right ) \sqrt {-\frac {b (1-\text {sech}(c+d x))}{a+b \text {sech}(c+d x)}} \sqrt {\frac {b (1+\text {sech}(c+d x))}{a+b \text {sech}(c+d x)}} (a+b \text {sech}(c+d x))}{\sqrt {a+b} d}\\ \end {align*}
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Mathematica [B] time = 18.23, size = 539, normalized size = 2.19 \[ \frac {\sqrt {a+b \text {sech}(c+d x)} \left (\frac {2 \sqrt {b} \sinh (c+d x) (a-a \cosh (c+d x))^{3/2} \sqrt {\frac {(a+b) (a \cosh (c+d x)+a)}{(a-b) (a-a \cosh (c+d x))}} F\left (\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {b+a \cosh (c+d x)}}{\sqrt {b} \sqrt {a-a \cosh (c+d x)}}\right )|-\frac {2 b}{a-b}\right )}{a^{3/2} \sqrt {\cosh (c+d x)-1} \sqrt {\cosh (c+d x)+1} \sqrt {\text {sech}(c+d x)} \left (-\frac {a-a \cosh (c+d x)}{a}\right )^{3/2} \sqrt {\frac {a \cosh (c+d x)+a}{a}} \sqrt {-\frac {a (a+b) \cosh (c+d x)}{b (a-a \cosh (c+d x))}}}-\frac {4 b \sinh (c+d x) (a-a \cosh (c+d x)) \sqrt {-\frac {b \text {sech}(c+d x) (a \cosh (c+d x)+a)}{a (a-b)}} \Pi \left (\frac {a+b}{a};\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {b+a \cosh (c+d x)}}{\sqrt {a+b} \sqrt {a \cosh (c+d x)}}\right )|\frac {a+b}{a-b}\right )}{\sqrt {a} \sqrt {a+b} \sqrt {\cosh (c+d x)-1} \sqrt {\cosh (c+d x)+1} \sqrt {\text {sech}(c+d x)} \sqrt {a \cosh (c+d x)} \sqrt {-\frac {a-a \cosh (c+d x)}{a}} \sqrt {\frac {a \cosh (c+d x)+a}{a}} \sqrt {-\frac {b \text {sech}(c+d x) (a-a \cosh (c+d x))}{a (a+b)}}}\right )}{2 d \sqrt {\text {sech}(c+d x)} \sqrt {a \cosh (c+d x)+b}}-\frac {\coth (c+d x) \sqrt {a+b \text {sech}(c+d x)}}{d} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.27, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b \operatorname {sech}\left (d x + c\right ) + a} \coth \left (d x + c\right )^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \operatorname {sech}\left (d x + c\right ) + a} \coth \left (d x + c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.55, size = 0, normalized size = 0.00 \[ \int \left (\coth ^{2}\left (d x +c \right )\right ) \sqrt {a +b \,\mathrm {sech}\left (d x +c \right )}\, dx \]
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 \[ \int \sqrt {b \operatorname {sech}\left (d x + c\right ) + a} \coth \left (d x + c\right )^{2}\,{d x} \]
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 \[ \int {\mathrm {coth}\left (c+d\,x\right )}^2\,\sqrt {a+\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a + b \operatorname {sech}{\left (c + d x \right )}} \coth ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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