Optimal. Leaf size=88 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {2 \left (a^2-b^2\right )}{a b^2 d \sqrt {a+b \text {sech}(c+d x)}}+\frac {2 \sqrt {a+b \text {sech}(c+d x)}}{b^2 d} \]
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Rubi [A] time = 0.14, antiderivative size = 88, 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 = {3885, 898, 1261, 206} \[ \frac {2 \left (a^2-b^2\right )}{a b^2 d \sqrt {a+b \text {sech}(c+d x)}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {2 \sqrt {a+b \text {sech}(c+d x)}}{b^2 d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 898
Rule 1261
Rule 3885
Rubi steps
\begin {align*} \int \frac {\tanh ^3(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {b^2-x^2}{x (a+x)^{3/2}} \, dx,x,b \text {sech}(c+d x)\right )}{b^2 d}\\ &=-\frac {2 \operatorname {Subst}\left (\int \frac {-a^2+b^2+2 a x^2-x^4}{x^2 \left (-a+x^2\right )} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{b^2 d}\\ &=-\frac {2 \operatorname {Subst}\left (\int \left (-1+\frac {a^2-b^2}{a x^2}-\frac {b^2}{a \left (a-x^2\right )}\right ) \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{b^2 d}\\ &=\frac {2 \left (a^2-b^2\right )}{a b^2 d \sqrt {a+b \text {sech}(c+d x)}}+\frac {2 \sqrt {a+b \text {sech}(c+d x)}}{b^2 d}+\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 \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {2 \left (a^2-b^2\right )}{a b^2 d \sqrt {a+b \text {sech}(c+d x)}}+\frac {2 \sqrt {a+b \text {sech}(c+d x)}}{b^2 d}\\ \end {align*}
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Mathematica [A] time = 0.67, size = 103, normalized size = 1.17 \[ \frac {2 \left (2 a^2+\frac {b^2 \sqrt {a \cosh (c+d x)+b} \tanh ^{-1}\left (\frac {\sqrt {a \cosh (c+d x)+b}}{\sqrt {a \cosh (c+d x)}}\right )}{\sqrt {a \cosh (c+d x)}}+a b \text {sech}(c+d x)-b^2\right )}{a b^2 d \sqrt {a+b \text {sech}(c+d x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.24, size = 1107, normalized size = 12.58 \[ \text {result too large to display} \]
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 \[ \int \frac {\tanh \left (d x + c\right )^{3}}{{\left (b \operatorname {sech}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.56, size = 0, normalized size = 0.00 \[ \int \frac {\tanh ^{3}\left (d x +c \right )}{\left (a +b \,\mathrm {sech}\left (d x +c \right )\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {\tanh \left (d x + c\right )^{3}}{{\left (b \operatorname {sech}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {tanh}\left (c+d\,x\right )}^3}{{\left (a+\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\tanh ^{3}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}{\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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