3.4.88 \(\int \frac {\cosh ^2(e+f x)}{(a+b \sinh ^2(e+f x))^{3/2}} \, dx\) [388]

Optimal. Leaf size=91 \[ \frac {\cosh (e+f x) E\left (\text {ArcTan}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{\sqrt {a} \sqrt {b} f \sqrt {\frac {a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}} \sqrt {a+b \sinh ^2(e+f x)}} \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {3271, 422} \begin {gather*} \frac {\cosh (e+f x) E\left (\text {ArcTan}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{\sqrt {a} \sqrt {b} f \sqrt {a+b \sinh ^2(e+f x)} \sqrt {\frac {a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Cosh[e + f*x]*EllipticE[ArcTan[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a]], 1 - a/b])/(Sqrt[a]*Sqrt[b]*f*Sqrt[(a*Cosh[e
+ f*x]^2)/(a + b*Sinh[e + f*x]^2)]*Sqrt[a + b*Sinh[e + f*x]^2])

Rule 422

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sq
rt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /;
FreeQ[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]

Rule 3271

Int[cos[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Sin[e + f*x], x]}, Dist[ff*(Sqrt[Cos[e + f*x]^2]/(f*Cos[e + f*x])), 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/2] &&  !IntegerQ
[p]

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.23, size = 143, normalized size = 1.57 \begin {gather*} \frac {i \sqrt {2} a \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )-i \sqrt {2} a \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} F\left (i (e+f x)\left |\frac {b}{a}\right .\right )+b \sinh (2 (e+f x))}{a b f \sqrt {4 a-2 b+2 b \cosh (2 (e+f x))}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(I*Sqrt[2]*a*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] - I*Sqrt[2]*a*Sqrt[(2*a - b +
b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] + b*Sinh[2*(e + f*x)])/(a*b*f*Sqrt[4*a - 2*b + 2*b*Cosh[2
*(e + f*x)]])

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Maple [A]
time = 1.47, size = 181, normalized size = 1.99

method	result	size
default	\(\frac {\sqrt {-\frac {b}{a}}\, \left (\cosh ^{2}\left (f x +e \right )\right ) \sinh \left (f x +e \right )+\sqrt {\frac {b \left (\cosh ^{2}\left (f x +e \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-\sqrt {\frac {b \left (\cosh ^{2}\left (f x +e \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )}{\sqrt {-\frac {b}{a}}\, a \cosh \left (f x +e \right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, f}\)	\(181\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((-1/a*b)^(1/2)*cosh(f*x+e)^2*sinh(f*x+e)+(b/a*cosh(f*x+e)^2+(a-b)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticF(si
nh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))-(b/a*cosh(f*x+e)^2+(a-b)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticE(sinh(f
*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2)))/(-1/a*b)^(1/2)/a/cosh(f*x+e)/(a+b*sinh(f*x+e)^2)^(1/2)/f

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(cosh(f*x + e)^2/(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. 1068 vs. \(2 (97) = 194\).
time = 0.16, size = 1068, normalized size = 11.74 \begin {gather*} -\frac {4 \, {\left (b^{2} \cosh \left (f x + e\right )^{4} + 4 \, b^{2} \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + b^{2} \sinh \left (f x + e\right )^{4} + 2 \, {\left (2 \, a b - b^{2}\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (3 \, b^{2} \cosh \left (f x + e\right )^{2} + 2 \, a b - b^{2}\right )} \sinh \left (f x + e\right )^{2} + b^{2} + 4 \, {\left (b^{2} \cosh \left (f x + e\right )^{3} + {\left (2 \, a b - b^{2}\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )\right )} \sqrt {b} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} - 2 \, a + b}{b}} \sqrt {\frac {a^{2} - a b}{b^{2}}} F(\arcsin \left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} - 2 \, a + b}{b}} {\left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right )}\right )\,|\,\frac {8 \, a^{2} - 8 \, a b + b^{2} + 4 \, {\left (2 \, a b - b^{2}\right )} \sqrt {\frac {a^{2} - a b}{b^{2}}}}{b^{2}}) + {\left ({\left (2 \, a b - b^{2}\right )} \cosh \left (f x + e\right )^{4} + 4 \, {\left (2 \, a b - b^{2}\right )} \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + {\left (2 \, a b - b^{2}\right )} \sinh \left (f x + e\right )^{4} + 2 \, {\left (4 \, a^{2} - 4 \, a b + b^{2}\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (3 \, {\left (2 \, a b - b^{2}\right )} \cosh \left (f x + e\right )^{2} + 4 \, a^{2} - 4 \, a b + b^{2}\right )} \sinh \left (f x + e\right )^{2} + 2 \, a b - b^{2} + 4 \, {\left ({\left (2 \, a b - b^{2}\right )} \cosh \left (f x + e\right )^{3} + {\left (4 \, a^{2} - 4 \, a b + b^{2}\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right ) - 2 \, {\left (b^{2} \cosh \left (f x + e\right )^{4} + 4 \, b^{2} \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + b^{2} \sinh \left (f x + e\right )^{4} + 2 \, {\left (2 \, a b - b^{2}\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (3 \, b^{2} \cosh \left (f x + e\right )^{2} + 2 \, a b - b^{2}\right )} \sinh \left (f x + e\right )^{2} + b^{2} + 4 \, {\left (b^{2} \cosh \left (f x + e\right )^{3} + {\left (2 \, a b - b^{2}\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )\right )} \sqrt {\frac {a^{2} - a b}{b^{2}}}\right )} \sqrt {b} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} - 2 \, a + b}{b}} E(\arcsin \left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} - 2 \, a + b}{b}} {\left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right )}\right )\,|\,\frac {8 \, a^{2} - 8 \, a b + b^{2} + 4 \, {\left (2 \, a b - b^{2}\right )} \sqrt {\frac {a^{2} - a b}{b^{2}}}}{b^{2}}) - \sqrt {2} {\left (b^{2} \cosh \left (f x + e\right )^{3} + 3 \, b^{2} \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{2} + b^{2} \sinh \left (f x + e\right )^{3} + {\left (2 \, a b - b^{2}\right )} \cosh \left (f x + e\right ) + {\left (3 \, b^{2} \cosh \left (f x + e\right )^{2} + 2 \, a b - b^{2}\right )} \sinh \left (f x + e\right )\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}}}}{a b^{3} f \cosh \left (f x + e\right )^{4} + 4 \, a b^{3} f \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + a b^{3} f \sinh \left (f x + e\right )^{4} + a b^{3} f + 2 \, {\left (2 \, a^{2} b^{2} - a b^{3}\right )} f \cosh \left (f x + e\right )^{2} + 2 \, {\left (3 \, a b^{3} f \cosh \left (f x + e\right )^{2} + {\left (2 \, a^{2} b^{2} - a b^{3}\right )} f\right )} \sinh \left (f x + e\right )^{2} + 4 \, {\left (a b^{3} f \cosh \left (f x + e\right )^{3} + {\left (2 \, a^{2} b^{2} - a b^{3}\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(cosh(f*x+e)^2/(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="fricas")

[Out]

-(4*(b^2*cosh(f*x + e)^4 + 4*b^2*cosh(f*x + e)*sinh(f*x + e)^3 + b^2*sinh(f*x + e)^4 + 2*(2*a*b - b^2)*cosh(f*
x + e)^2 + 2*(3*b^2*cosh(f*x + e)^2 + 2*a*b - b^2)*sinh(f*x + e)^2 + b^2 + 4*(b^2*cosh(f*x + e)^3 + (2*a*b - b
^2)*cosh(f*x + e))*sinh(f*x + e))*sqrt(b)*sqrt((2*b*sqrt((a^2 - a*b)/b^2) - 2*a + b)/b)*sqrt((a^2 - a*b)/b^2)*
elliptic_f(arcsin(sqrt((2*b*sqrt((a^2 - a*b)/b^2) - 2*a + b)/b)*(cosh(f*x + e) + sinh(f*x + e))), (8*a^2 - 8*a
*b + b^2 + 4*(2*a*b - b^2)*sqrt((a^2 - a*b)/b^2))/b^2) + ((2*a*b - b^2)*cosh(f*x + e)^4 + 4*(2*a*b - b^2)*cosh
(f*x + e)*sinh(f*x + e)^3 + (2*a*b - b^2)*sinh(f*x + e)^4 + 2*(4*a^2 - 4*a*b + b^2)*cosh(f*x + e)^2 + 2*(3*(2*
a*b - b^2)*cosh(f*x + e)^2 + 4*a^2 - 4*a*b + b^2)*sinh(f*x + e)^2 + 2*a*b - b^2 + 4*((2*a*b - b^2)*cosh(f*x +
e)^3 + (4*a^2 - 4*a*b + b^2)*cosh(f*x + e))*sinh(f*x + e) - 2*(b^2*cosh(f*x + e)^4 + 4*b^2*cosh(f*x + e)*sinh(
f*x + e)^3 + b^2*sinh(f*x + e)^4 + 2*(2*a*b - b^2)*cosh(f*x + e)^2 + 2*(3*b^2*cosh(f*x + e)^2 + 2*a*b - b^2)*s
inh(f*x + e)^2 + b^2 + 4*(b^2*cosh(f*x + e)^3 + (2*a*b - b^2)*cosh(f*x + e))*sinh(f*x + e))*sqrt((a^2 - a*b)/b
^2))*sqrt(b)*sqrt((2*b*sqrt((a^2 - a*b)/b^2) - 2*a + b)/b)*elliptic_e(arcsin(sqrt((2*b*sqrt((a^2 - a*b)/b^2) -
2*a + b)/b)*(cosh(f*x + e) + sinh(f*x + e))), (8*a^2 - 8*a*b + b^2 + 4*(2*a*b - b^2)*sqrt((a^2 - a*b)/b^2))/b
^2) - sqrt(2)*(b^2*cosh(f*x + e)^3 + 3*b^2*cosh(f*x + e)*sinh(f*x + e)^2 + b^2*sinh(f*x + e)^3 + (2*a*b - b^2)
*cosh(f*x + e) + (3*b^2*cosh(f*x + e)^2 + 2*a*b - b^2)*sinh(f*x + e))*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))

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Error: Bad Argume
nt Type

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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 )}^2}{{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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