Optimal. Leaf size=262 \[ -\frac {\sqrt {a+b \text {sech}(c+d x)}}{4 d (a+b) (1-\text {sech}(c+d x))}-\frac {\sqrt {a+b \text {sech}(c+d x)}}{4 d (a-b) (\text {sech}(c+d x)+1)}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}+\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{4 d (a-b)^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{d \sqrt {a-b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{d \sqrt {a+b}}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{4 d (a+b)^{3/2}} \]
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Rubi [A] time = 0.30, antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3885, 898, 1238, 206, 199, 207} \[ -\frac {\sqrt {a+b \text {sech}(c+d x)}}{4 d (a+b) (1-\text {sech}(c+d x))}-\frac {\sqrt {a+b \text {sech}(c+d x)}}{4 d (a-b) (\text {sech}(c+d x)+1)}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}+\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{4 d (a-b)^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{d \sqrt {a-b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{d \sqrt {a+b}}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{4 d (a+b)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 199
Rule 206
Rule 207
Rule 898
Rule 1238
Rule 3885
Rubi steps
\begin {align*} \int \frac {\coth ^3(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx &=-\frac {b^4 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+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 {1}{\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^4\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{b^4 \left (a-x^2\right )}+\frac {1}{4 b^3 \left (a+b-x^2\right )^2}+\frac {1}{2 b^4 \left (a+b-x^2\right )}-\frac {1}{4 b^3 \left (-a+b+x^2\right )^2}-\frac {1}{2 b^4 \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 {1}{a+b-x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{d}+\frac {\operatorname {Subst}\left (\int \frac {1}{-a+b+x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{d}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{d}-\frac {b \operatorname {Subst}\left (\int \frac {1}{\left (a+b-x^2\right )^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{2 d}+\frac {b \operatorname {Subst}\left (\int \frac {1}{\left (-a+b+x^2\right )^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{2 d}\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b} d}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{\sqrt {a+b} d}-\frac {\sqrt {a+b \text {sech}(c+d x)}}{4 (a+b) d (1-\text {sech}(c+d x))}-\frac {\sqrt {a+b \text {sech}(c+d x)}}{4 (a-b) d (1+\text {sech}(c+d x))}-\frac {b \operatorname {Subst}\left (\int \frac {1}{-a+b+x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{4 (a-b) d}-\frac {b \operatorname {Subst}\left (\int \frac {1}{a+b-x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{4 (a+b) d}\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b} d}+\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{4 (a-b)^{3/2} d}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{4 (a+b)^{3/2} d}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{\sqrt {a+b} d}-\frac {\sqrt {a+b \text {sech}(c+d x)}}{4 (a+b) d (1-\text {sech}(c+d x))}-\frac {\sqrt {a+b \text {sech}(c+d x)}}{4 (a-b) d (1+\text {sech}(c+d x))}\\ \end {align*}
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Mathematica [B] time = 7.40, size = 902, normalized size = 3.44 \[ \frac {\sqrt {b+a \cosh (c+d x)} \sqrt {\text {sech}(c+d x)} \left (\frac {\left (2 a^2-2 b^2\right ) \left (\sqrt {a} \left (\sqrt {a+b} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {b+a \cosh (c+d x)}}{\sqrt {a-b} \sqrt {-a \cosh (c+d x)}}\right )+\sqrt {a-b} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {b+a \cosh (c+d x)}}{\sqrt {a+b} \sqrt {-a \cosh (c+d x)}}\right )\right )-4 \sqrt {a-b} \sqrt {a+b} \tan ^{-1}\left (\frac {\sqrt {b+a \cosh (c+d x)}}{\sqrt {-a \cosh (c+d x)}}\right )\right ) \sqrt {-a \cosh (c+d x)} \sqrt {\frac {a \cosh (c+d x)-a}{\cosh (c+d x) a+a}} \cosh (2 (c+d x)) \sqrt {\text {sech}(c+d x)} (\cosh (c+d x) a+a)}{\sqrt {a-b} \sqrt {a+b} \sqrt {\cosh (c+d x)-1} \sqrt {\cosh (c+d x)+1} \left (a^2-2 b^2-2 (b+a \cosh (c+d x))^2+4 b (b+a \cosh (c+d x))\right )}-\frac {\left (2 a^2-3 b^2\right ) \left (\sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {b+a \cosh (c+d x)}}{\sqrt {a-b} \sqrt {a \cosh (c+d x)}}\right )+\sqrt {a-b} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {b+a \cosh (c+d x)}}{\sqrt {a+b} \sqrt {a \cosh (c+d x)}}\right )\right ) \sqrt {a \cosh (c+d x)} \sqrt {\frac {a \cosh (c+d x)-a}{\cosh (c+d x) a+a}} \sqrt {\text {sech}(c+d x)} (\cosh (c+d x) a+a)}{a^{3/2} \sqrt {a-b} \sqrt {a+b} \sqrt {\cosh (c+d x)-1} \sqrt {\cosh (c+d x)+1}}+\frac {\sqrt {a} b \left (\sqrt {a-b} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {b+a \cosh (c+d x)}}{\sqrt {-a-b} \sqrt {a \cosh (c+d x)}}\right )+\sqrt {-a-b} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {b+a \cosh (c+d x)}}{\sqrt {a-b} \sqrt {a \cosh (c+d x)}}\right )\right ) \sqrt {\frac {a \cosh (c+d x)-a}{\cosh (c+d x) a+a}} (\cosh (c+d x) a+a)}{\sqrt {-a-b} \sqrt {a-b} \sqrt {\cosh (c+d x)-1} \sqrt {a \cosh (c+d x)} \sqrt {\cosh (c+d x)+1} \sqrt {\text {sech}(c+d x)}}\right )}{4 (a-b) (a+b) d \sqrt {a+b \text {sech}(c+d x)}}+\frac {(b+a \cosh (c+d x)) \left (\frac {(a-b \cosh (c+d x)) \text {csch}^2(c+d x)}{2 \left (b^2-a^2\right )}-\frac {a}{2 \left (a^2-b^2\right )}\right ) \text {sech}(c+d x)}{d \sqrt {a+b \text {sech}(c+d x)}} \]
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 \[ \int \frac {\coth \left (d x + c\right )^{3}}{\sqrt {b \operatorname {sech}\left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.65, size = 0, normalized size = 0.00 \[ \int \frac {\coth ^{3}\left (d x +c \right )}{\sqrt {a +b \,\mathrm {sech}\left (d x +c \right )}}\, dx \]
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 \[ \int \frac {\coth \left (d x + c\right )^{3}}{\sqrt {b \operatorname {sech}\left (d x + c\right ) + a}}\,{d x} \]
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 \[ \int \frac {{\mathrm {coth}\left (c+d\,x\right )}^3}{\sqrt {a+\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}}} \,d x \]
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 \[ \int \frac {\coth ^{3}{\left (c + d x \right )}}{\sqrt {a + b \operatorname {sech}{\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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