Integrand size = 10, antiderivative size = 48 \[ \int (b \tanh (c+d x))^n \, dx=\frac {\operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},\tanh ^2(c+d x)\right ) (b \tanh (c+d x))^{1+n}}{b d (1+n)} \]
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Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3557, 371} \[ \int (b \tanh (c+d x))^n \, dx=\frac {(b \tanh (c+d x))^{n+1} \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},\tanh ^2(c+d x)\right )}{b d (n+1)} \]
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Rule 371
Rule 3557
Rubi steps \begin{align*} \text {integral}& = -\frac {b \text {Subst}\left (\int \frac {x^n}{-b^2+x^2} \, dx,x,b \tanh (c+d x)\right )}{d} \\ & = \frac {\operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},\tanh ^2(c+d x)\right ) (b \tanh (c+d x))^{1+n}}{b d (1+n)} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.02 \[ \int (b \tanh (c+d x))^n \, dx=\frac {\operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},\tanh ^2(c+d x)\right ) \tanh (c+d x) (b \tanh (c+d x))^n}{d (1+n)} \]
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\[\int \left (b \tanh \left (d x +c \right )\right )^{n}d x\]
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\[ \int (b \tanh (c+d x))^n \, dx=\int { \left (b \tanh \left (d x + c\right )\right )^{n} \,d x } \]
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\[ \int (b \tanh (c+d x))^n \, dx=\int \left (b \tanh {\left (c + d x \right )}\right )^{n}\, dx \]
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\[ \int (b \tanh (c+d x))^n \, dx=\int { \left (b \tanh \left (d x + c\right )\right )^{n} \,d x } \]
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\[ \int (b \tanh (c+d x))^n \, dx=\int { \left (b \tanh \left (d x + c\right )\right )^{n} \,d x } \]
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Timed out. \[ \int (b \tanh (c+d x))^n \, dx=\int {\left (b\,\mathrm {tanh}\left (c+d\,x\right )\right )}^n \,d x \]
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