3.514 \(\int (a+b \sinh ^2(c+d x))^p \tanh (c+d x) \, dx\)

Optimal. Leaf size=63 \[ -\frac{\left (a+b \sinh ^2(c+d x)\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b \sinh ^2(c+d x)+a}{a-b}\right )}{2 d (p+1) (a-b)} \]

[Out]

-(Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Sinh[c + d*x]^2)/(a - b)]*(a + b*Sinh[c + d*x]^2)^(1 + p))/(2*(a -
b)*d*(1 + p))

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Rubi [A]  time = 0.0547016, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3194, 68} \[ -\frac{\left (a+b \sinh ^2(c+d x)\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b \sinh ^2(c+d x)+a}{a-b}\right )}{2 d (p+1) (a-b)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sinh[c + d*x]^2)^p*Tanh[c + d*x],x]

[Out]

-(Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Sinh[c + d*x]^2)/(a - b)]*(a + b*Sinh[c + d*x]^2)^(1 + p))/(2*(a -
b)*d*(1 + p))

Rule 3194

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = Free
Factors[Sin[e + f*x]^2, x]}, Dist[ff^((m + 1)/2)/(2*f), Subst[Int[(x^((m - 1)/2)*(a + b*ff*x)^p)/(1 - ff*x)^((
m + 1)/2), x], x, Sin[e + f*x]^2/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rule 68

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((b*c - a*d)^n*(a + b*x)^(m + 1)*Hype
rgeometric2F1[-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b^(n + 1)*(m + 1)), x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rubi steps

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

Mathematica [A]  time = 0.070719, size = 65, normalized size = 1.03 \[ -\frac{\left (a+b \cosh ^2(c+d x)-b\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b \cosh ^2(c+d x)}{a-b}+1\right )}{2 d (p+1) (a-b)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sinh[c + d*x]^2)^p*Tanh[c + d*x],x]

[Out]

-((a - b + b*Cosh[c + d*x]^2)^(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Cosh[c + d*x]^2)/(a - b)])/(2*
(a - b)*d*(1 + p))

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Maple [F]  time = 0.255, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b \left ( \sinh \left ( dx+c \right ) \right ) ^{2} \right ) ^{p}\tanh \left ( dx+c \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sinh(d*x+c)^2)^p*tanh(d*x+c),x)

[Out]

int((a+b*sinh(d*x+c)^2)^p*tanh(d*x+c),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sinh \left (d x + c\right )^{2} + a\right )}^{p} \tanh \left (d x + c\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sinh(d*x+c)^2)^p*tanh(d*x+c),x, algorithm="maxima")

[Out]

integrate((b*sinh(d*x + c)^2 + a)^p*tanh(d*x + c), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \sinh \left (d x + c\right )^{2} + a\right )}^{p} \tanh \left (d x + c\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sinh(d*x+c)^2)^p*tanh(d*x+c),x, algorithm="fricas")

[Out]

integral((b*sinh(d*x + c)^2 + a)^p*tanh(d*x + c), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sinh(d*x+c)**2)**p*tanh(d*x+c),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sinh \left (d x + c\right )^{2} + a\right )}^{p} \tanh \left (d x + c\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sinh(d*x+c)^2)^p*tanh(d*x+c),x, algorithm="giac")

[Out]

integrate((b*sinh(d*x + c)^2 + a)^p*tanh(d*x + c), x)