3.134 \(\int \text{csch}(e+f x) (a+b \sinh ^2(e+f x))^p \, dx\)

Optimal. Leaf size=88 \[ -\frac{\cosh (e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^p \left (\frac{b \cosh ^2(e+f x)}{a-b}+1\right )^{-p} F_1\left (\frac{1}{2};1,-p;\frac{3}{2};\cosh ^2(e+f x),-\frac{b \cosh ^2(e+f x)}{a-b}\right )}{f} \]

[Out]

-((AppellF1[1/2, 1, -p, 3/2, Cosh[e + f*x]^2, -((b*Cosh[e + f*x]^2)/(a - b))]*Cosh[e + f*x]*(a - b + b*Cosh[e
+ f*x]^2)^p)/(f*(1 + (b*Cosh[e + f*x]^2)/(a - b))^p))

________________________________________________________________________________________

Rubi [A]  time = 0.0878811, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3186, 430, 429} \[ -\frac{\cosh (e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^p \left (\frac{b \cosh ^2(e+f x)}{a-b}+1\right )^{-p} F_1\left (\frac{1}{2};1,-p;\frac{3}{2};\cosh ^2(e+f x),-\frac{b \cosh ^2(e+f x)}{a-b}\right )}{f} \]

Antiderivative was successfully verified.

[In]

Int[Csch[e + f*x]*(a + b*Sinh[e + f*x]^2)^p,x]

[Out]

-((AppellF1[1/2, 1, -p, 3/2, Cosh[e + f*x]^2, -((b*Cosh[e + f*x]^2)/(a - b))]*Cosh[e + f*x]*(a - b + b*Cosh[e
+ f*x]^2)^p)/(f*(1 + (b*Cosh[e + f*x]^2)/(a - b))^p))

Rule 3186

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

Rule 430

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^F
racPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n,
p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 429

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rubi steps

\begin{align*} \int \text{csch}(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (a-b+b x^2\right )^p}{1-x^2} \, dx,x,\cosh (e+f x)\right )}{f}\\ &=-\frac{\left (\left (a-b+b \cosh ^2(e+f x)\right )^p \left (1+\frac{b \cosh ^2(e+f x)}{a-b}\right )^{-p}\right ) \operatorname{Subst}\left (\int \frac{\left (1+\frac{b x^2}{a-b}\right )^p}{1-x^2} \, dx,x,\cosh (e+f x)\right )}{f}\\ &=-\frac{F_1\left (\frac{1}{2};1,-p;\frac{3}{2};\cosh ^2(e+f x),-\frac{b \cosh ^2(e+f x)}{a-b}\right ) \cosh (e+f x) \left (a-b+b \cosh ^2(e+f x)\right )^p \left (1+\frac{b \cosh ^2(e+f x)}{a-b}\right )^{-p}}{f}\\ \end{align*}

Mathematica [F]  time = 4.1886, size = 0, normalized size = 0. \[ \int \text{csch}(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Csch[e + f*x]*(a + b*Sinh[e + f*x]^2)^p,x]

[Out]

Integrate[Csch[e + f*x]*(a + b*Sinh[e + f*x]^2)^p, x]

________________________________________________________________________________________

Maple [F]  time = 0.264, size = 0, normalized size = 0. \begin{align*} \int{\rm csch} \left (fx+e\right ) \left ( a+b \left ( \sinh \left ( fx+e \right ) \right ) ^{2} \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(f*x+e)*(a+b*sinh(f*x+e)^2)^p,x)

[Out]

int(csch(f*x+e)*(a+b*sinh(f*x+e)^2)^p,x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(f*x+e)*(a+b*sinh(f*x+e)^2)^p,x, algorithm="maxima")

[Out]

integrate((b*sinh(f*x + e)^2 + a)^p*csch(f*x + e), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(f*x+e)*(a+b*sinh(f*x+e)^2)^p,x, algorithm="fricas")

[Out]

integral((b*sinh(f*x + e)^2 + a)^p*csch(f*x + e), x)

________________________________________________________________________________________

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(csch(f*x+e)*(a+b*sinh(f*x+e)**2)**p,x)

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(f*x+e)*(a+b*sinh(f*x+e)^2)^p,x, algorithm="giac")

[Out]

integrate((b*sinh(f*x + e)^2 + a)^p*csch(f*x + e), x)