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

Optimal. Leaf size=57 \[ \frac {\tanh (c+d x) \left (b \tanh ^m(c+d x)\right )^n \, _2F_1\left (1,\frac {1}{2} (m n+1);\frac {1}{2} (m n+3);\tanh ^2(c+d x)\right )}{d (m n+1)} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3659, 3476, 364} \[ \frac {\tanh (c+d x) \left (b \tanh ^m(c+d x)\right )^n \, _2F_1\left (1,\frac {1}{2} (m n+1);\frac {1}{2} (m n+3);\tanh ^2(c+d x)\right )}{d (m n+1)} \]

Antiderivative was successfully verified.

[In]

Int[(b*Tanh[c + d*x]^m)^n,x]

[Out]

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

Rule 3659

Int[(u_.)*((b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Dist[(b^IntPart[p]*(b*(c*Tan[e + f*x
])^n)^FracPart[p])/(c*Tan[e + f*x])^(n*FracPart[p]), Int[ActivateTrig[u]*(c*Tan[e + f*x])^(n*p), x], x] /; Fre
eQ[{b, c, e, f, n, p}, x] &&  !IntegerQ[p] &&  !IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x]
)^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])

Rubi steps

\begin {align*} \int \left (b \tanh ^m(c+d x)\right )^n \, dx &=\left (\tanh ^{-m n}(c+d x) \left (b \tanh ^m(c+d x)\right )^n\right ) \int \tanh ^{m n}(c+d x) \, dx\\ &=-\frac {\left (\tanh ^{-m n}(c+d x) \left (b \tanh ^m(c+d x)\right )^n\right ) \operatorname {Subst}\left (\int \frac {x^{m n}}{-1+x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\, _2F_1\left (1,\frac {1}{2} (1+m n);\frac {1}{2} (3+m n);\tanh ^2(c+d x)\right ) \tanh (c+d x) \left (b \tanh ^m(c+d x)\right )^n}{d (1+m n)}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 55, normalized size = 0.96 \[ \frac {\tanh (c+d x) \left (b \tanh ^m(c+d x)\right )^n \, _2F_1\left (1,\frac {1}{2} (m n+1);\frac {1}{2} (m n+3);\tanh ^2(c+d x)\right )}{d m n+d} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Tanh[c + d*x]^m)^n,x]

[Out]

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

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fricas [F]  time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (b \tanh \left (d x + c\right )^{m}\right )^{n}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*tanh(d*x + c)^m)^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*tanh(d*x + c)^m)^n, x)

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maple [F]  time = 6.17, size = 0, normalized size = 0.00 \[ \int \left (b \left (\tanh ^{m}\left (d x +c \right )\right )\right )^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*tanh(d*x + c)^m)^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*tanh(c + d*x)^m)^n,x)

[Out]

int((b*tanh(c + d*x)^m)^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((b*tanh(c + d*x)**m)**n, x)

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