Optimal. Leaf size=79 \[ \frac {2 (a+b \text {sech}(c+d x))^{3/2}}{3 b^2 d}-\frac {2 a \sqrt {a+b \text {sech}(c+d x)}}{b^2 d}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d} \]
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Rubi [A] time = 0.11, antiderivative size = 79, 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, 1153, 207} \[ \frac {2 (a+b \text {sech}(c+d x))^{3/2}}{3 b^2 d}-\frac {2 a \sqrt {a+b \text {sech}(c+d x)}}{b^2 d}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d} \]
Antiderivative was successfully verified.
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Rule 207
Rule 898
Rule 1153
Rule 3885
Rubi steps
\begin {align*} \int \frac {\tanh ^3(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {b^2-x^2}{x \sqrt {a+x}} \, 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}{-a+x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{b^2 d}\\ &=-\frac {2 \operatorname {Subst}\left (\int \left (a-x^2+\frac {b^2}{-a+x^2}\right ) \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{b^2 d}\\ &=-\frac {2 a \sqrt {a+b \text {sech}(c+d x)}}{b^2 d}+\frac {2 (a+b \text {sech}(c+d x))^{3/2}}{3 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 )}{d}\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}-\frac {2 a \sqrt {a+b \text {sech}(c+d x)}}{b^2 d}+\frac {2 (a+b \text {sech}(c+d x))^{3/2}}{3 b^2 d}\\ \end {align*}
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Mathematica [A] time = 0.64, size = 111, normalized size = 1.41 \[ \frac {2 \left (-2 a^2+\frac {3 b^2 \sqrt {a \cosh (c+d x)+b} \tanh ^{-1}\left (\frac {\sqrt {a \cosh (c+d x)+b}}{\sqrt {a \cosh (c+d x)}}\right )}{\sqrt {a \cosh (c+d x)}}-a b \text {sech}(c+d x)+b^2 \text {sech}^2(c+d x)\right )}{3 b^2 d \sqrt {a+b \text {sech}(c+d x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.04, size = 925, normalized size = 11.71 \[ \text {result too large to display} \]
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 \[ \int \frac {\tanh \left (d x + c\right )^{3}}{\sqrt {b \operatorname {sech}\left (d x + c\right ) + a}}\,{d x} \]
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 \[ \int \frac {\tanh ^{3}\left (d x +c \right )}{\sqrt {a +b \,\mathrm {sech}\left (d x +c \right )}}\, dx \]
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 \[ \int \frac {\tanh \left (d x + c\right )^{3}}{\sqrt {b \operatorname {sech}\left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {tanh}\left (c+d\,x\right )}^3}{\sqrt {a+\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}}} \,d x \]
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 \[ \int \frac {\tanh ^{3}{\left (c + d x \right )}}{\sqrt {a + b \operatorname {sech}{\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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