Optimal. Leaf size=142 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {2 b^2}{a d \left (a^2-b^2\right ) \sqrt {a+b \text {sech}(c+d x)}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{d (a-b)^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{d (a+b)^{3/2}} \]
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Rubi [A] time = 0.22, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3885, 898, 1287, 206} \[ \frac {2 b^2}{a d \left (a^2-b^2\right ) \sqrt {a+b \text {sech}(c+d x)}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{d (a-b)^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{d (a+b)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 898
Rule 1287
Rule 3885
Rubi steps
\begin {align*} \int \frac {\coth (c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx &=-\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{x (a+x)^{3/2} \left (b^2-x^2\right )} \, dx,x,b \text {sech}(c+d x)\right )}{d}\\ &=-\frac {\left (2 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \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 {\left (2 b^2\right ) \operatorname {Subst}\left (\int \left (\frac {1}{a \left (a^2-b^2\right ) x^2}-\frac {1}{a b^2 \left (a-x^2\right )}+\frac {1}{2 (a-b) b^2 \left (a-b-x^2\right )}+\frac {1}{2 b^2 (a+b) \left (a+b-x^2\right )}\right ) \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{d}\\ &=\frac {2 b^2}{a \left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{a d}-\frac {\operatorname {Subst}\left (\int \frac {1}{a-b-x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{(a-b) d}-\frac {\operatorname {Subst}\left (\int \frac {1}{a+b-x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{(a+b) d}\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{(a-b)^{3/2} d}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{(a+b)^{3/2} d}+\frac {2 b^2}{a \left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}}\\ \end {align*}
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Mathematica [B] time = 7.37, size = 904, normalized size = 6.37 \[ \frac {(b+a \cosh (c+d x))^2 \left (-\frac {2 b^3}{a^2 \left (a^2-b^2\right ) (b+a \cosh (c+d x))}-\frac {2 b^2}{a^2 \left (b^2-a^2\right )}\right ) \text {sech}^2(c+d x)}{d (a+b \text {sech}(c+d x))^{3/2}}-\frac {(b+a \cosh (c+d x))^{3/2} \left (\frac {\left (a^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 (a^2+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 {2 \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 ) \text {sech}^{\frac {3}{2}}(c+d x)}{2 a (b-a) (a+b) d (a+b \text {sech}(c+d x))^{3/2}} \]
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 \[ \int \frac {\coth \left (d x + c\right )}{{\left (b \operatorname {sech}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.57, size = 0, normalized size = 0.00 \[ \int \frac {\coth \left (d x +c \right )}{\left (a +b \,\mathrm {sech}\left (d x +c \right )\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {\coth \left (d x + c\right )}{{\left (b \operatorname {sech}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {coth}\left (c+d\,x\right )}{{\left (a+\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\coth {\left (c + d x \right )}}{\left (a + b \operatorname {sech}{\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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