3.2.8 \(\int \frac {\sinh (e+f x)}{(a+b \sinh ^2(e+f x))^{3/2}} \, dx\) [108]

Optimal. Leaf size=36 \[ \frac {\cosh (e+f x)}{(a-b) f \sqrt {a-b+b \cosh ^2(e+f x)}} \]

[Out]

cosh(f*x+e)/(a-b)/f/(a-b+b*cosh(f*x+e)^2)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3265, 197} \begin {gather*} \frac {\cosh (e+f x)}{f (a-b) \sqrt {a+b \cosh ^2(e+f x)-b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 3265

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]

Rubi steps

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

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Mathematica [A]
time = 0.10, size = 43, normalized size = 1.19 \begin {gather*} \frac {\sqrt {2} \cosh (e+f x)}{(a-b) f \sqrt {2 a-b+b \cosh (2 (e+f x))}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2]*Cosh[e + f*x])/((a - b)*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])

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Maple [A]
time = 0.70, size = 32, normalized size = 0.89

method	result	size
default	\(\frac {\cosh \left (f x +e \right )}{\left (a -b \right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, f}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 246 vs. \(2 (36) = 72\).
time = 0.50, size = 246, normalized size = 6.83 \begin {gather*} \frac {b^{2} e^{\left (-6 \, f x - 6 \, e\right )} + 2 \, a b - b^{2} + {\left (8 \, a^{2} - 8 \, a b + 3 \, b^{2}\right )} e^{\left (-2 \, f x - 2 \, e\right )} + 3 \, {\left (2 \, a b - b^{2}\right )} e^{\left (-4 \, f x - 4 \, e\right )}}{2 \, {\left (a^{2} - a b\right )} {\left (2 \, {\left (2 \, a - b\right )} e^{\left (-2 \, f x - 2 \, e\right )} + b e^{\left (-4 \, f x - 4 \, e\right )} + b\right )}^{\frac {3}{2}} f} + \frac {b^{2} + 3 \, {\left (2 \, a b - b^{2}\right )} e^{\left (-2 \, f x - 2 \, e\right )} + {\left (8 \, a^{2} - 8 \, a b + 3 \, b^{2}\right )} e^{\left (-4 \, f x - 4 \, e\right )} + {\left (2 \, a b - b^{2}\right )} e^{\left (-6 \, f x - 6 \, e\right )}}{2 \, {\left (a^{2} - a b\right )} {\left (2 \, {\left (2 \, a - b\right )} e^{\left (-2 \, f x - 2 \, e\right )} + b e^{\left (-4 \, f x - 4 \, e\right )} + b\right )}^{\frac {3}{2}} f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^2*e^(-6*f*x - 6*e) + 2*a*b - b^2 + (8*a^2 - 8*a*b + 3*b^2)*e^(-2*f*x - 2*e) + 3*(2*a*b - b^2)*e^(-4*f*x
- 4*e))/((a^2 - a*b)*(2*(2*a - b)*e^(-2*f*x - 2*e) + b*e^(-4*f*x - 4*e) + b)^(3/2)*f) + 1/2*(b^2 + 3*(2*a*b -
b^2)*e^(-2*f*x - 2*e) + (8*a^2 - 8*a*b + 3*b^2)*e^(-4*f*x - 4*e) + (2*a*b - b^2)*e^(-6*f*x - 6*e))/((a^2 - a*
b)*(2*(2*a - b)*e^(-2*f*x - 2*e) + b*e^(-4*f*x - 4*e) + b)^(3/2)*f)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 296 vs. \(2 (34) = 68\).
time = 0.45, size = 296, normalized size = 8.22 \begin {gather*} \frac {\sqrt {2} {\left (\cosh \left (f x + e\right )^{2} + 2 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + \sinh \left (f x + e\right )^{2} + 1\right )} \sqrt {\frac {b \cosh \left (f x + e\right )^{2} + b \sinh \left (f x + e\right )^{2} + 2 \, a - b}{\cosh \left (f x + e\right )^{2} - 2 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + \sinh \left (f x + e\right )^{2}}}}{{\left (a b - b^{2}\right )} f \cosh \left (f x + e\right )^{4} + 4 \, {\left (a b - b^{2}\right )} f \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + {\left (a b - b^{2}\right )} f \sinh \left (f x + e\right )^{4} + 2 \, {\left (2 \, a^{2} - 3 \, a b + b^{2}\right )} f \cosh \left (f x + e\right )^{2} + 2 \, {\left (3 \, {\left (a b - b^{2}\right )} f \cosh \left (f x + e\right )^{2} + {\left (2 \, a^{2} - 3 \, a b + b^{2}\right )} f\right )} \sinh \left (f x + e\right )^{2} + {\left (a b - b^{2}\right )} f + 4 \, {\left ({\left (a b - b^{2}\right )} f \cosh \left (f x + e\right )^{3} + {\left (2 \, a^{2} - 3 \, a b + b^{2}\right )} f \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2)*(cosh(f*x + e)^2 + 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2 + 1)*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)*f*c
osh(f*x + e)^4 + 4*(a*b - b^2)*f*cosh(f*x + e)*sinh(f*x + e)^3 + (a*b - b^2)*f*sinh(f*x + e)^4 + 2*(2*a^2 - 3*
a*b + b^2)*f*cosh(f*x + e)^2 + 2*(3*(a*b - b^2)*f*cosh(f*x + e)^2 + (2*a^2 - 3*a*b + b^2)*f)*sinh(f*x + e)^2 +
(a*b - b^2)*f + 4*((a*b - b^2)*f*cosh(f*x + e)^3 + (2*a^2 - 3*a*b + b^2)*f*cosh(f*x + e))*sinh(f*x + e))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sinh {\left (e + f x \right )}}{\left (a + b \sinh ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (34) = 68\).
time = 0.62, size = 99, normalized size = 2.75 \begin {gather*} \frac {\frac {a e^{\left (2 \, f x + 4 \, e\right )}}{a^{2} e^{\left (2 \, e\right )} - a b e^{\left (2 \, e\right )}} + \frac {a e^{\left (2 \, e\right )}}{a^{2} e^{\left (2 \, e\right )} - a b e^{\left (2 \, e\right )}}}{\sqrt {b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b} f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.92, size = 191, normalized size = 5.31 \begin {gather*} -\frac {{\mathrm {e}}^{e+f\,x}\,\sqrt {b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a}\,\left (\frac {2\,{\mathrm {e}}^{e+f\,x}\,\mathrm {sinh}\left (e+f\,x\right )\,\left (b\,\left (2\,a-b\right )-b\,\left (4\,a-2\,b\right )\right )}{f\,\left (a\,b^2-a^2\,b\right )}+\frac {2\,b^2\,\mathrm {cosh}\left (e+f\,x\right )\,{\mathrm {e}}^{e+f\,x}}{f\,\left (a\,b^2-a^2\,b\right )}+\frac {b\,{\mathrm {e}}^{2\,e+2\,f\,x}\,\left (4\,a-2\,b\right )}{f\,\left (a\,b^2-a^2\,b\right )}\right )}{4\,a\,{\mathrm {e}}^{2\,e+2\,f\,x}-2\,b\,{\mathrm {e}}^{2\,e+2\,f\,x}+2\,b\,{\mathrm {e}}^{2\,e+2\,f\,x}\,\mathrm {cosh}\left (2\,e+2\,f\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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