Optimal. Leaf size=316 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}-\frac {2 b^4}{a d \left (a^2-b^2\right )^2 \sqrt {a+b \text {sech}(c+d x)}}-\frac {\sqrt {a+b \text {sech}(c+d x)}}{4 d (a+b)^2 (1-\text {sech}(c+d x))}-\frac {\sqrt {a+b \text {sech}(c+d x)}}{4 d (a-b)^2 (\text {sech}(c+d x)+1)}+\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{4 d (a-b)^{5/2}}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{4 d (a+b)^{5/2}}-\frac {(2 a-3 b) \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{2 d (a-b)^{5/2}}-\frac {(2 a+3 b) \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{2 d (a+b)^{5/2}} \]
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Rubi [A] time = 0.43, antiderivative size = 316, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3885, 898, 1335, 206, 199} \[ -\frac {2 b^4}{a d \left (a^2-b^2\right )^2 \sqrt {a+b \text {sech}(c+d x)}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}-\frac {\sqrt {a+b \text {sech}(c+d x)}}{4 d (a+b)^2 (1-\text {sech}(c+d x))}-\frac {\sqrt {a+b \text {sech}(c+d x)}}{4 d (a-b)^2 (\text {sech}(c+d x)+1)}+\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{4 d (a-b)^{5/2}}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{4 d (a+b)^{5/2}}-\frac {(2 a-3 b) \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{2 d (a-b)^{5/2}}-\frac {(2 a+3 b) \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{2 d (a+b)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 199
Rule 206
Rule 898
Rule 1335
Rule 3885
Rubi steps
\begin {align*} \int \frac {\coth ^3(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx &=-\frac {b^4 \operatorname {Subst}\left (\int \frac {1}{x (a+x)^{3/2} \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}{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^4\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{a (a-b)^2 (a+b)^2 x^2}-\frac {1}{a b^4 \left (a-x^2\right )}-\frac {1}{4 (a-b) b^3 \left (a-b-x^2\right )^2}+\frac {2 a-3 b}{4 (a-b)^2 b^4 \left (a-b-x^2\right )}+\frac {1}{4 b^3 (a+b) \left (a+b-x^2\right )^2}+\frac {2 a+3 b}{4 b^4 (a+b)^2 \left (a+b-x^2\right )}\right ) \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{d}\\ &=-\frac {2 b^4}{a \left (a^2-b^2\right )^2 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 {(2 a-3 b) \operatorname {Subst}\left (\int \frac {1}{a-b-x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{2 (a-b)^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 (a-b) 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 (a+b) d}-\frac {(2 a+3 b) \operatorname {Subst}\left (\int \frac {1}{a+b-x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{2 (a+b)^2 d}\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}-\frac {(2 a-3 b) \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{2 (a-b)^{5/2} d}-\frac {(2 a+3 b) \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{2 (a+b)^{5/2} d}-\frac {2 b^4}{a \left (a^2-b^2\right )^2 d \sqrt {a+b \text {sech}(c+d x)}}-\frac {\sqrt {a+b \text {sech}(c+d x)}}{4 (a+b)^2 d (1-\text {sech}(c+d x))}-\frac {\sqrt {a+b \text {sech}(c+d x)}}{4 (a-b)^2 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)^2 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)^2 d}\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}-\frac {(2 a-3 b) \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{2 (a-b)^{5/2} d}+\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{4 (a-b)^{5/2} d}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{4 (a+b)^{5/2} d}-\frac {(2 a+3 b) \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{2 (a+b)^{5/2} d}-\frac {2 b^4}{a \left (a^2-b^2\right )^2 d \sqrt {a+b \text {sech}(c+d x)}}-\frac {\sqrt {a+b \text {sech}(c+d x)}}{4 (a+b)^2 d (1-\text {sech}(c+d x))}-\frac {\sqrt {a+b \text {sech}(c+d x)}}{4 (a-b)^2 d (1+\text {sech}(c+d x))}\\ \end {align*}
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Mathematica [B] time = 7.61, size = 996, normalized size = 3.15 \[ \frac {(b+a \cosh (c+d x))^2 \left (\frac {2 b^5}{a^2 \left (a^2-b^2\right )^2 (b+a \cosh (c+d x))}+\frac {\left (-a^2+2 b \cosh (c+d x) a-b^2\right ) \text {csch}^2(c+d x)}{2 \left (b^2-a^2\right )^2}-\frac {a^4+b^2 a^2+4 b^4}{2 a^2 \left (b^2-a^2\right )^2}\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 (2 a^4-4 b^2 a^2+2 b^4\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^4-6 b^2 a^2-2 b^4\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 {\left (7 a b^3-a^3 b\right ) \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} \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)}{4 a (a-b)^2 (a+b)^2 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 \[ \int \frac {\coth \left (d x + c\right )^{3}}{{\left (b \operatorname {sech}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.70, size = 0, normalized size = 0.00 \[ \int \frac {\coth ^{3}\left (d x +c \right )}{\left (a +b \,\mathrm {sech}\left (d x +c \right )\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {\coth \left (d x + c\right )^{3}}{{\left (b \operatorname {sech}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {coth}\left (c+d\,x\right )}^3}{{\left (a+\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\coth ^{3}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}{\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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