Optimal. Leaf size=217 \[ -\frac {\coth ^2(c+d x) \sqrt {a+b \text {sech}(c+d x)}}{2 d}+\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{d}+\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{4 d \sqrt {a-b}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{d \sqrt {a-b}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{d \sqrt {a+b}}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{4 d \sqrt {a+b}} \]
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Rubi [A] time = 0.33, antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {3885, 898, 1315, 1178, 12, 1093, 206, 1170, 207} \[ -\frac {\coth ^2(c+d x) \sqrt {a+b \text {sech}(c+d x)}}{2 d}+\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{d}+\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{4 d \sqrt {a-b}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{d \sqrt {a-b}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{d \sqrt {a+b}}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{4 d \sqrt {a+b}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 207
Rule 898
Rule 1093
Rule 1170
Rule 1178
Rule 1315
Rule 3885
Rubi steps
\begin {align*} \int \coth ^3(c+d x) \sqrt {a+b \text {sech}(c+d x)} \, dx &=-\frac {b^4 \operatorname {Subst}\left (\int \frac {\sqrt {a+x}}{x \left (b^2-x^2\right )^2} \, dx,x,b \text {sech}(c+d x)\right )}{d}\\ &=-\frac {\left (2 b^4\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-a+x^2\right ) \left (-a^2+b^2+2 a x^2-x^4\right )^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{d}\\ &=-\frac {\left (2 b^2\right ) \operatorname {Subst}\left (\int \frac {-a^2+b^2+a x^2}{\left (-a^2+b^2+2 a x^2-x^4\right )^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{d}-\frac {\left (2 a b^2\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-a+x^2\right ) \left (-a^2+b^2+2 a x^2-x^4\right )} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{d}\\ &=\frac {b^2 \sqrt {a+b \text {sech}(c+d x)}}{2 d \left (a^2-b^2-2 a (a+b \text {sech}(c+d x))+(a+b \text {sech}(c+d x))^2\right )}-\frac {\left (2 a b^2\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{b^2 \left (a-x^2\right )}+\frac {1}{2 b^2 \left (a+b-x^2\right )}-\frac {1}{2 b^2 \left (-a+b+x^2\right )}\right ) \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{d}-\frac {\operatorname {Subst}\left (\int \frac {6 b^2 \left (a^2-b^2\right )}{-a^2+b^2+2 a x^2-x^4} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{4 \left (a^2-b^2\right ) d}\\ &=\frac {b^2 \sqrt {a+b \text {sech}(c+d x)}}{2 d \left (a^2-b^2-2 a (a+b \text {sech}(c+d x))+(a+b \text {sech}(c+d x))^2\right )}-\frac {a \operatorname {Subst}\left (\int \frac {1}{a+b-x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{d}+\frac {a \operatorname {Subst}\left (\int \frac {1}{-a+b+x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{d}+\frac {(2 a) \operatorname {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{d}-\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{-a^2+b^2+2 a x^2-x^4} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{2 d}\\ &=\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{d}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b} d}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{\sqrt {a+b} d}+\frac {b^2 \sqrt {a+b \text {sech}(c+d x)}}{2 d \left (a^2-b^2-2 a (a+b \text {sech}(c+d x))+(a+b \text {sech}(c+d x))^2\right )}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{a-b-x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{4 d}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{a+b-x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{4 d}\\ &=\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{d}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b} d}+\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{4 \sqrt {a-b} d}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{\sqrt {a+b} d}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{4 \sqrt {a+b} d}+\frac {b^2 \sqrt {a+b \text {sech}(c+d x)}}{2 d \left (a^2-b^2-2 a (a+b \text {sech}(c+d x))+(a+b \text {sech}(c+d x))^2\right )}\\ \end {align*}
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Mathematica [B] time = 20.69, size = 518, normalized size = 2.39 \[ \frac {\sqrt {a+b \text {sech}(c+d x)} \left (\frac {8 \sqrt {-a \cosh (c+d x)} \tan ^{-1}\left (\frac {\sqrt {a \cosh (c+d x)+b}}{\sqrt {-a \cosh (c+d x)}}\right )}{\sqrt {a \cosh (c+d x)+b}}-\frac {2 \sqrt {a} \sqrt {-a \cosh (c+d x)} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {a \cosh (c+d x)+b}}{\sqrt {a-b} \sqrt {-a \cosh (c+d x)}}\right )}{\sqrt {a-b} \sqrt {a \cosh (c+d x)+b}}-\frac {2 \sqrt {a} \sqrt {-a \cosh (c+d x)} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {a \cosh (c+d x)+b}}{\sqrt {a+b} \sqrt {-a \cosh (c+d x)}}\right )}{\sqrt {a+b} \sqrt {a \cosh (c+d x)+b}}+\frac {3 b \sqrt {a \cosh (c+d x)} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {a \cosh (c+d x)+b}}{\sqrt {-a-b} \sqrt {a \cosh (c+d x)}}\right )}{\sqrt {a} \sqrt {-a-b} \sqrt {a \cosh (c+d x)+b}}-\frac {(2 a-3 b) \sqrt {a \cosh (c+d x)} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {a \cosh (c+d x)+b}}{\sqrt {a-b} \sqrt {a \cosh (c+d x)}}\right )}{\sqrt {a} \sqrt {a-b} \sqrt {a \cosh (c+d x)+b}}-\frac {2 \sqrt {a} \sqrt {a \cosh (c+d x)} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {a \cosh (c+d x)+b}}{\sqrt {a+b} \sqrt {a \cosh (c+d x)}}\right )}{\sqrt {a+b} \sqrt {a \cosh (c+d x)+b}}-2 \coth ^2(c+d x)\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \operatorname {sech}\left (d x + c\right ) + a} \coth \left (d x + c\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.61, size = 0, normalized size = 0.00 \[ \int \left (\coth ^{3}\left (d x +c \right )\right ) \sqrt {a +b \,\mathrm {sech}\left (d x +c \right )}\, dx \]
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 \[ \int \sqrt {b \operatorname {sech}\left (d x + c\right ) + a} \coth \left (d x + c\right )^{3}\,{d x} \]
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 \[ \int {\mathrm {coth}\left (c+d\,x\right )}^3\,\sqrt {a+\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}} \,d x \]
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 \[ \int \sqrt {a + b \operatorname {sech}{\left (c + d x \right )}} \coth ^{3}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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