∫ cosh3 (e+fx)___ (a+b sinh2(e+fx))3∕2 dx [382]

Optimal. Leaf size=77 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}}\right )}{b^{3/2} f}-\frac {(a-b) \sinh (e+f x)}{a b f \sqrt {a+b \sinh ^2(e+f x)}} \]

arctanh(sinh(f*x+e)*b^(1/2)/(a+b*sinh(f*x+e)^2)^(1/2))/b^(3/2)/f-(a-b)*sinh(f*x+e)/a/b/f/(a+b*sinh(f*x+e)^2)^(

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Rubi [A]
time = 0.07, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3269, 393, 223, 212} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}}\right )}{b^{3/2} f}-\frac {(a-b) \sinh (e+f x)}{a b f \sqrt {a+b \sinh ^2(e+f x)}} \end {gather*}

Int[Cosh[e + f*x]^3/(a + b*Sinh[e + f*x]^2)^(3/2),x]

ArcTanh[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]]/(b^(3/2)*f) - ((a - b)*Sinh[e + f*x])/(a*b*f*Sqrt

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]

/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d))*x*((a + b*x^n)^(p

+ 1)/(a*b*n*(p + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x]

/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free

Factors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e +

f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

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

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Mathematica [A]
time = 0.13, size = 89, normalized size = 1.16 \begin {gather*} \frac {\sqrt {b} (-a+b) \sinh (e+f x)+a^{3/2} \sinh ^{-1}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right ) \sqrt {1+\frac {b \sinh ^2(e+f x)}{a}}}{a b^{3/2} f \sqrt {a+b \sinh ^2(e+f x)}} \end {gather*}

Integrate[Cosh[e + f*x]^3/(a + b*Sinh[e + f*x]^2)^(3/2),x]

(Sqrt[b]*(-a + b)*Sinh[e + f*x] + a^(3/2)*ArcSinh[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a]]*Sqrt[1 + (b*Sinh[e + f*x]^2

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 1.14, size = 35, normalized size = 0.45


method	result	size

default	\(\frac {\mathit {`\,int/indef0`\,}\left (\frac {\cosh ^{2}\left (f x +e \right )}{\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}}, \sinh \left (f x +e \right )\right )}{f}\)	\(35\)

int(cosh(f*x+e)^3/(a+b*sinh(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)

`int/indef0`(cosh(f*x+e)^2/(a+b*sinh(f*x+e)^2)^(3/2),sinh(f*x+e))/f

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

integrate(cosh(f*x+e)^3/(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="maxima")

integrate(cosh(f*x + e)^3/(b*sinh(f*x + e)^2 + a)^(3/2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1108 vs. \(2 (69) = 138\).
time = 0.50, size = 3126, normalized size = 40.60 \begin {gather*} \text {Too large to display} \end {gather*}

integrate(cosh(f*x+e)^3/(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="fricas")

[1/4*((a*b*cosh(f*x + e)^4 + 4*a*b*cosh(f*x + e)*sinh(f*x + e)^3 + a*b*sinh(f*x + e)^4 + 2*(2*a^2 - a*b)*cosh(

f*x + e)^2 + 2*(3*a*b*cosh(f*x + e)^2 + 2*a^2 - a*b)*sinh(f*x + e)^2 + a*b + 4*(a*b*cosh(f*x + e)^3 + (2*a^2 -

a*b)*cosh(f*x + e))*sinh(f*x + e))*sqrt(b)*log(-((a^2*b - 2*a*b^2 + b^3)*cosh(f*x + e)^8 + 8*(a^2*b - 2*a*b^2

+ b^3)*cosh(f*x + e)*sinh(f*x + e)^7 + (a^2*b - 2*a*b^2 + b^3)*sinh(f*x + e)^8 + 2*(a^3 - 4*a^2*b + 5*a*b^2 -

2*b^3)*cosh(f*x + e)^6 + 2*(a^3 - 4*a^2*b + 5*a*b^2 - 2*b^3 + 14*(a^2*b - 2*a*b^2 + b^3)*cosh(f*x + e)^2)*sin

h(f*x + e)^6 + 4*(14*(a^2*b - 2*a*b^2 + b^3)*cosh(f*x + e)^3 + 3*(a^3 - 4*a^2*b + 5*a*b^2 - 2*b^3)*cosh(f*x +

e))*sinh(f*x + e)^5 + (9*a^2*b - 14*a*b^2 + 6*b^3)*cosh(f*x + e)^4 + (70*(a^2*b - 2*a*b^2 + b^3)*cosh(f*x + e)

^4 + 9*a^2*b - 14*a*b^2 + 6*b^3 + 30*(a^3 - 4*a^2*b + 5*a*b^2 - 2*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 4*(1

4*(a^2*b - 2*a*b^2 + b^3)*cosh(f*x + e)^5 + 10*(a^3 - 4*a^2*b + 5*a*b^2 - 2*b^3)*cosh(f*x + e)^3 + (9*a^2*b -

14*a*b^2 + 6*b^3)*cosh(f*x + e))*sinh(f*x + e)^3 + b^3 + 2*(3*a*b^2 - 2*b^3)*cosh(f*x + e)^2 + 2*(14*(a^2*b -

2*a*b^2 + b^3)*cosh(f*x + e)^6 + 15*(a^3 - 4*a^2*b + 5*a*b^2 - 2*b^3)*cosh(f*x + e)^4 + 3*a*b^2 - 2*b^3 + 3*(9

*a^2*b - 14*a*b^2 + 6*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + sqrt(2)*((a^2 - 2*a*b + b^2)*cosh(f*x + e)^6 + 6

*(a^2 - 2*a*b + b^2)*cosh(f*x + e)*sinh(f*x + e)^5 + (a^2 - 2*a*b + b^2)*sinh(f*x + e)^6 - 3*(a^2 - 2*a*b + b^

2)*cosh(f*x + e)^4 + 3*(5*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^2 - a^2 + 2*a*b - b^2)*sinh(f*x + e)^4 + 4*(5*(a^2

- 2*a*b + b^2)*cosh(f*x + e)^3 - 3*(a^2 - 2*a*b + b^2)*cosh(f*x + e))*sinh(f*x + e)^3 - (4*a*b - 3*b^2)*cosh(

f*x + e)^2 + (15*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^4 - 18*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^2 - 4*a*b + 3*b^2)

*sinh(f*x + e)^2 - b^2 + 2*(3*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^5 - 6*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^3 - (4

*a*b - 3*b^2)*cosh(f*x + e))*sinh(f*x + e))*sqrt(b)*sqrt((b*cosh(f*x + e)^2 + b*sinh(f*x + e)^2 + 2*a - b)/(co

sh(f*x + e)^2 - 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2)) + 4*(2*(a^2*b - 2*a*b^2 + b^3)*cosh(f*x + e)

^7 + 3*(a^3 - 4*a^2*b + 5*a*b^2 - 2*b^3)*cosh(f*x + e)^5 + (9*a^2*b - 14*a*b^2 + 6*b^3)*cosh(f*x + e)^3 + (3*a

*b^2 - 2*b^3)*cosh(f*x + e))*sinh(f*x + e))/(cosh(f*x + e)^6 + 6*cosh(f*x + e)^5*sinh(f*x + e) + 15*cosh(f*x +

e)^4*sinh(f*x + e)^2 + 20*cosh(f*x + e)^3*sinh(f*x + e)^3 + 15*cosh(f*x + e)^2*sinh(f*x + e)^4 + 6*cosh(f*x +

e)*sinh(f*x + e)^5 + sinh(f*x + e)^6)) + (a*b*cosh(f*x + e)^4 + 4*a*b*cosh(f*x + e)*sinh(f*x + e)^3 + a*b*sin

h(f*x + e)^4 + 2*(2*a^2 - a*b)*cosh(f*x + e)^2 + 2*(3*a*b*cosh(f*x + e)^2 + 2*a^2 - a*b)*sinh(f*x + e)^2 + a*b

+ 4*(a*b*cosh(f*x + e)^3 + (2*a^2 - a*b)*cosh(f*x + e))*sinh(f*x + e))*sqrt(b)*log((b*cosh(f*x + e)^4 + 4*b*c

osh(f*x + e)*sinh(f*x + e)^3 + b*sinh(f*x + e)^4 + 2*a*cosh(f*x + e)^2 + 2*(3*b*cosh(f*x + e)^2 + a)*sinh(f*x

+ e)^2 + sqrt(2)*(cosh(f*x + e)^2 + 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2 + 1)*sqrt(b)*sqrt((b*cosh(

f*x + e)^2 + b*sinh(f*x + e)^2 + 2*a - b)/(cosh(f*x + e)^2 - 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2))

+ 4*(b*cosh(f*x + e)^3 + a*cosh(f*x + e))*sinh(f*x + e) + b)/(cosh(f*x + e)^2 + 2*cosh(f*x + e)*sinh(f*x + e)

+ sinh(f*x + e)^2)) - 4*sqrt(2)*((a*b - b^2)*cosh(f*x + e)^2 + 2*(a*b - b^2)*cosh(f*x + e)*sinh(f*x + e) + (a

*b - b^2)*sinh(f*x + e)^2 - a*b + b^2)*sqrt((b*cosh(f*x + e)^2 + b*sinh(f*x + e)^2 + 2*a - b)/(cosh(f*x + e)^2

- 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2)))/(a*b^3*f*cosh(f*x + e)^4 + 4*a*b^3*f*cosh(f*x + e)*sinh(

f*x + e)^3 + a*b^3*f*sinh(f*x + e)^4 + a*b^3*f + 2*(2*a^2*b^2 - a*b^3)*f*cosh(f*x + e)^2 + 2*(3*a*b^3*f*cosh(f

*x + e)^2 + (2*a^2*b^2 - a*b^3)*f)*sinh(f*x + e)^2 + 4*(a*b^3*f*cosh(f*x + e)^3 + (2*a^2*b^2 - a*b^3)*f*cosh(f

*x + e))*sinh(f*x + e)), -1/2*((a*b*cosh(f*x + e)^4 + 4*a*b*cosh(f*x + e)*sinh(f*x + e)^3 + a*b*sinh(f*x + e)^

4 + 2*(2*a^2 - a*b)*cosh(f*x + e)^2 + 2*(3*a*b*cosh(f*x + e)^2 + 2*a^2 - a*b)*sinh(f*x + e)^2 + a*b + 4*(a*b*c

osh(f*x + e)^3 + (2*a^2 - a*b)*cosh(f*x + e))*sinh(f*x + e))*sqrt(-b)*arctan(sqrt(2)*((a - b)*cosh(f*x + e)^2

+ 2*(a - b)*cosh(f*x + e)*sinh(f*x + e) + (a - b)*sinh(f*x + e)^2 + b)*sqrt(-b)*sqrt((b*cosh(f*x + e)^2 + b*si

nh(f*x + e)^2 + 2*a - b)/(cosh(f*x + e)^2 - 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2))/((a*b - b^2)*cos

h(f*x + e)^4 + 4*(a*b - b^2)*cosh(f*x + e)*sinh(f*x + e)^3 + (a*b - b^2)*sinh(f*x + e)^4 - (3*a*b - 2*b^2)*cos

h(f*x + e)^2 + (6*(a*b - b^2)*cosh(f*x + e)^2 - 3*a*b + 2*b^2)*sinh(f*x + e)^2 - b^2 + 2*(2*(a*b - b^2)*cosh(f

*x + e)^3 - (3*a*b - 2*b^2)*cosh(f*x + e))*sinh(f*x + e))) + (a*b*cosh(f*x + e)^4 + 4*a*b*cosh(f*x + e)*sinh(f

*x + e)^3 + a*b*sinh(f*x + e)^4 + 2*(2*a^2 - a*b)*cosh(f*x + e)^2 + 2*(3*a*b*cosh(f*x + e)^2 + 2*a^2 - a*b)*si

nh(f*x + e)^2 + a*b + 4*(a*b*cosh(f*x + e)^3 + (2*a^2 - a*b)*cosh(f*x + e))*sinh(f*x + e))*sqrt(-b)*arctan(sqr

t(2)*(cosh(f*x + e)^2 + 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2 + 1)*sqrt(-b)*sqrt((b*cosh(f*x + e)^2

+ b*sinh(f*x + e)^2 + 2*a - b)/(cosh(f*x + e)^2...

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

integrate(cosh(f*x+e)**3/(a+b*sinh(f*x+e)**2)**(3/2),x)

Exception raised: SystemError >> excessive stack use: stack is 3063 deep

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

integrate(cosh(f*x+e)^3/(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="giac")

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {cosh}\left (e+f\,x\right )}^3}{{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2}} \,d x \end {gather*}

int(cosh(e + f*x)^3/(a + b*sinh(e + f*x)^2)^(3/2),x)

int(cosh(e + f*x)^3/(a + b*sinh(e + f*x)^2)^(3/2), x)

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3.4.82 \(\int \frac {\cosh ^3(e+f x)}{(a+b \sinh ^2(e+f x))^{3/2}} \, dx\) [382]