3.23 \(\int (b \tanh (c+d x))^n \, dx\)

Optimal. Leaf size=48 \[ \frac {(b \tanh (c+d x))^{n+1} \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};\tanh ^2(c+d x)\right )}{b d (n+1)} \]

[Out]

hypergeom([1, 1/2+1/2*n],[3/2+1/2*n],tanh(d*x+c)^2)*(b*tanh(d*x+c))^(1+n)/b/d/(1+n)

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Rubi [A]  time = 0.03, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3476, 364} \[ \frac {(b \tanh (c+d x))^{n+1} \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};\tanh ^2(c+d x)\right )}{b d (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[(b*Tanh[c + d*x])^n,x]

[Out]

(Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, Tanh[c + d*x]^2]*(b*Tanh[c + d*x])^(1 + n))/(b*d*(1 + n))

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 3476

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[x^n/(b^2 + x^2), x], x, b*Tan[c + d
*x]], x] /; FreeQ[{b, c, d, n}, x] &&  !IntegerQ[n]

Rubi steps

\begin {align*} \int (b \tanh (c+d x))^n \, dx &=-\frac {b \operatorname {Subst}\left (\int \frac {x^n}{-b^2+x^2} \, dx,x,b \tanh (c+d x)\right )}{d}\\ &=\frac {\, _2F_1\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*}

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Mathematica [A]  time = 0.05, size = 51, normalized size = 1.06 \[ \frac {\tanh (c+d x) (b \tanh (c+d x))^n \, _2F_1\left (1,\frac {n+1}{2};\frac {n+1}{2}+1;\tanh ^2(c+d x)\right )}{d (n+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Tanh[c + d*x])^n,x]

[Out]

(Hypergeometric2F1[1, (1 + n)/2, 1 + (1 + n)/2, Tanh[c + d*x]^2]*Tanh[c + d*x]*(b*Tanh[c + d*x])^n)/(d*(1 + n)
)

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fricas [F]  time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (b \tanh \left (d x + c\right )\right )^{n}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*tanh(d*x+c))^n,x, algorithm="fricas")

[Out]

integral((b*tanh(d*x + c))^n, x)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*tanh(d*x+c))^n,x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 0.36, size = 0, normalized size = 0.00 \[ \int \left (b \tanh \left (d x +c \right )\right )^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*tanh(d*x+c))^n,x)

[Out]

int((b*tanh(d*x+c))^n,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \tanh \left (d x + c\right )\right )^{n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*tanh(d*x+c))^n,x, algorithm="maxima")

[Out]

integrate((b*tanh(d*x + c))^n, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (b\,\mathrm {tanh}\left (c+d\,x\right )\right )}^n \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*tanh(c + d*x))^n,x)

[Out]

int((b*tanh(c + d*x))^n, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \tanh {\left (c + d x \right )}\right )^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*tanh(d*x+c))**n,x)

[Out]

Integral((b*tanh(c + d*x))**n, x)

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