3.2.14 \(\int \frac {1}{(a+b \sinh ^2(e+f x))^{3/2}} \, dx\) [114]

Optimal. Leaf size=115 \[ -\frac {b \cosh (e+f x) \sinh (e+f x)}{a (a-b) f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {i E\left (i e+i f x\left |\frac {b}{a}\right .\right ) \sqrt {a+b \sinh ^2(e+f x)}}{a (a-b) f \sqrt {1+\frac {b \sinh ^2(e+f x)}{a}}} \]

[Out]

-b*cosh(f*x+e)*sinh(f*x+e)/a/(a-b)/f/(a+b*sinh(f*x+e)^2)^(1/2)-I*(cos(I*e+I*f*x)^2)^(1/2)/cos(I*e+I*f*x)*Ellip
ticE(sin(I*e+I*f*x),(b/a)^(1/2))*(a+b*sinh(f*x+e)^2)^(1/2)/a/(a-b)/f/(1+b*sinh(f*x+e)^2/a)^(1/2)

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Rubi [A]
time = 0.05, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3263, 21, 3257, 3256} \begin {gather*} -\frac {b \sinh (e+f x) \cosh (e+f x)}{a f (a-b) \sqrt {a+b \sinh ^2(e+f x)}}-\frac {i \sqrt {a+b \sinh ^2(e+f x)} E\left (i e+i f x\left |\frac {b}{a}\right .\right )}{a f (a-b) \sqrt {\frac {b \sinh ^2(e+f x)}{a}+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Sinh[e + f*x]^2)^(-3/2),x]

[Out]

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

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 3256

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[(Sqrt[a]/f)*EllipticE[e + f*x, -b/a], x] /
; FreeQ[{a, b, e, f}, x] && GtQ[a, 0]

Rule 3257

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Dist[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[1 + b*(Sin
[e + f*x]^2/a)], Int[Sqrt[1 + (b*Sin[e + f*x]^2)/a], x], x] /; FreeQ[{a, b, e, f}, x] &&  !GtQ[a, 0]

Rule 3263

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[(-b)*Cos[e + f*x]*Sin[e + f*x]*((a + b*Si
n[e + f*x]^2)^(p + 1)/(2*a*f*(p + 1)*(a + b))), x] + Dist[1/(2*a*(p + 1)*(a + b)), Int[(a + b*Sin[e + f*x]^2)^
(p + 1)*Simp[2*a*(p + 1) + b*(2*p + 3) - 2*b*(p + 2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f}, x] && N
eQ[a + b, 0] && LtQ[p, -1]

Rubi steps

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

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Mathematica [A]
time = 0.12, size = 100, normalized size = 0.87 \begin {gather*} \frac {-2 i 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 )-\sqrt {2} b \sinh (2 (e+f x))}{2 a (a-b) f \sqrt {2 a-b+b \cosh (2 (e+f x))}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 1.18, size = 253, normalized size = 2.20

method	result	size
default	\(-\frac {\sqrt {-\frac {b}{a}}\, b \left (\cosh ^{2}\left (f x +e \right )\right ) \sinh \left (f x +e \right )-a \sqrt {\frac {b \left (\cosh ^{2}\left (f x +e \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )+\sqrt {\frac {b \left (\cosh ^{2}\left (f x +e \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b -\sqrt {\frac {b \left (\cosh ^{2}\left (f x +e \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b}{a \left (a -b \right ) \sqrt {-\frac {b}{a}}\, \cosh \left (f x +e \right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, f}\)	\(253\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*sinh(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-((-1/a*b)^(1/2)*b*cosh(f*x+e)^2*sinh(f*x+e)-a*(b/a*cosh(f*x+e)^2+(a-b)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*Ellipti
cF(sinh(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)*EllipticF(s
inh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*b-(b/a*cosh(f*x+e)^2+(a-b)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticE(sin
h(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*b)/a/(a-b)/(-1/a*b)^(1/2)/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(1/(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="maxima")

[Out]

integrate((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. 1464 vs. \(2 (123) = 246\).
time = 0.11, size = 1464, normalized size = 12.73 \begin {gather*} \frac {{\left ({\left (2 \, a b^{2} - b^{3}\right )} \cosh \left (f x + e\right )^{4} + 4 \, {\left (2 \, a b^{2} - b^{3}\right )} \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + {\left (2 \, a b^{2} - b^{3}\right )} \sinh \left (f x + e\right )^{4} + 2 \, a b^{2} - b^{3} + 2 \, {\left (4 \, a^{2} b - 4 \, a b^{2} + b^{3}\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (4 \, a^{2} b - 4 \, a b^{2} + b^{3} + 3 \, {\left (2 \, a b^{2} - b^{3}\right )} \cosh \left (f x + e\right )^{2}\right )} \sinh \left (f x + e\right )^{2} + 4 \, {\left ({\left (2 \, a b^{2} - b^{3}\right )} \cosh \left (f x + e\right )^{3} + {\left (4 \, a^{2} b - 4 \, a b^{2} + b^{3}\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right ) - 2 \, {\left (b^{3} \cosh \left (f x + e\right )^{4} + 4 \, b^{3} \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + b^{3} \sinh \left (f x + e\right )^{4} + b^{3} + 2 \, {\left (2 \, a b^{2} - b^{3}\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (3 \, b^{3} \cosh \left (f x + e\right )^{2} + 2 \, a b^{2} - b^{3}\right )} \sinh \left (f x + e\right )^{2} + 4 \, {\left (b^{3} \cosh \left (f x + e\right )^{3} + {\left (2 \, a b^{2} - b^{3}\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}}) - 2 \, {\left ({\left (2 \, a^{2} b - a b^{2}\right )} \cosh \left (f x + e\right )^{4} + 4 \, {\left (2 \, a^{2} b - a b^{2}\right )} \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + {\left (2 \, a^{2} b - a b^{2}\right )} \sinh \left (f x + e\right )^{4} + 2 \, a^{2} b - a b^{2} + 2 \, {\left (4 \, a^{3} - 4 \, a^{2} b + a b^{2}\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (4 \, a^{3} - 4 \, a^{2} b + a b^{2} + 3 \, {\left (2 \, a^{2} b - a b^{2}\right )} \cosh \left (f x + e\right )^{2}\right )} \sinh \left (f x + e\right )^{2} + 4 \, {\left ({\left (2 \, a^{2} b - a b^{2}\right )} \cosh \left (f x + e\right )^{3} + {\left (4 \, a^{3} - 4 \, a^{2} b + a b^{2}\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right ) + 2 \, {\left ({\left (a b^{2} - b^{3}\right )} \cosh \left (f x + e\right )^{4} + 4 \, {\left (a b^{2} - b^{3}\right )} \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + {\left (a b^{2} - b^{3}\right )} \sinh \left (f x + e\right )^{4} + a b^{2} - b^{3} + 2 \, {\left (2 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (2 \, a^{2} b - 3 \, a b^{2} + b^{3} + 3 \, {\left (a b^{2} - b^{3}\right )} \cosh \left (f x + e\right )^{2}\right )} \sinh \left (f x + e\right )^{2} + 4 \, {\left ({\left (a b^{2} - b^{3}\right )} \cosh \left (f x + e\right )^{3} + {\left (2 \, a^{2} b - 3 \, a b^{2} + b^{3}\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}} 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}}) - \sqrt {2} {\left (b^{3} \cosh \left (f x + e\right )^{3} + 3 \, b^{3} \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{2} + b^{3} \sinh \left (f x + e\right )^{3} + {\left (2 \, a b^{2} - b^{3}\right )} \cosh \left (f x + e\right ) + {\left (3 \, b^{3} \cosh \left (f x + e\right )^{2} + 2 \, a b^{2} - b^{3}\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}}}}{{\left (a^{2} b^{3} - a b^{4}\right )} f \cosh \left (f x + e\right )^{4} + 4 \, {\left (a^{2} b^{3} - a b^{4}\right )} f \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + {\left (a^{2} b^{3} - a b^{4}\right )} f \sinh \left (f x + e\right )^{4} + 2 \, {\left (2 \, a^{3} b^{2} - 3 \, a^{2} b^{3} + a b^{4}\right )} f \cosh \left (f x + e\right )^{2} + 2 \, {\left (3 \, {\left (a^{2} b^{3} - a b^{4}\right )} f \cosh \left (f x + e\right )^{2} + {\left (2 \, a^{3} b^{2} - 3 \, a^{2} b^{3} + a b^{4}\right )} f\right )} \sinh \left (f x + e\right )^{2} + {\left (a^{2} b^{3} - a b^{4}\right )} f + 4 \, {\left ({\left (a^{2} b^{3} - a b^{4}\right )} f \cosh \left (f x + e\right )^{3} + {\left (2 \, a^{3} b^{2} - 3 \, a^{2} b^{3} + a b^{4}\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(1/(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="fricas")

[Out]

(((2*a*b^2 - b^3)*cosh(f*x + e)^4 + 4*(2*a*b^2 - b^3)*cosh(f*x + e)*sinh(f*x + e)^3 + (2*a*b^2 - b^3)*sinh(f*x
+ e)^4 + 2*a*b^2 - b^3 + 2*(4*a^2*b - 4*a*b^2 + b^3)*cosh(f*x + e)^2 + 2*(4*a^2*b - 4*a*b^2 + b^3 + 3*(2*a*b^
2 - b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 4*((2*a*b^2 - b^3)*cosh(f*x + e)^3 + (4*a^2*b - 4*a*b^2 + b^3)*cos
h(f*x + e))*sinh(f*x + e) - 2*(b^3*cosh(f*x + e)^4 + 4*b^3*cosh(f*x + e)*sinh(f*x + e)^3 + b^3*sinh(f*x + e)^4
+ b^3 + 2*(2*a*b^2 - b^3)*cosh(f*x + e)^2 + 2*(3*b^3*cosh(f*x + e)^2 + 2*a*b^2 - b^3)*sinh(f*x + e)^2 + 4*(b^
3*cosh(f*x + e)^3 + (2*a*b^2 - b^3)*cosh(f*x + e))*sinh(f*x + e))*sqrt((a^2 - a*b)/b^2))*sqrt(b)*sqrt((2*b*sqr
t((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) - 2*((2*a^2*b - a*b^2)
*cosh(f*x + e)^4 + 4*(2*a^2*b - a*b^2)*cosh(f*x + e)*sinh(f*x + e)^3 + (2*a^2*b - a*b^2)*sinh(f*x + e)^4 + 2*a
^2*b - a*b^2 + 2*(4*a^3 - 4*a^2*b + a*b^2)*cosh(f*x + e)^2 + 2*(4*a^3 - 4*a^2*b + a*b^2 + 3*(2*a^2*b - a*b^2)*
cosh(f*x + e)^2)*sinh(f*x + e)^2 + 4*((2*a^2*b - a*b^2)*cosh(f*x + e)^3 + (4*a^3 - 4*a^2*b + a*b^2)*cosh(f*x +
e))*sinh(f*x + e) + 2*((a*b^2 - b^3)*cosh(f*x + e)^4 + 4*(a*b^2 - b^3)*cosh(f*x + e)*sinh(f*x + e)^3 + (a*b^2
- b^3)*sinh(f*x + e)^4 + a*b^2 - b^3 + 2*(2*a^2*b - 3*a*b^2 + b^3)*cosh(f*x + e)^2 + 2*(2*a^2*b - 3*a*b^2 + b
^3 + 3*(a*b^2 - b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 4*((a*b^2 - b^3)*cosh(f*x + e)^3 + (2*a^2*b - 3*a*b^2
+ b^3)*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_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) - sqrt(2)*(b^3*cosh(f*x + e)^3 + 3*b^3*cosh(f*x + e
)*sinh(f*x + e)^2 + b^3*sinh(f*x + e)^3 + (2*a*b^2 - b^3)*cosh(f*x + e) + (3*b^3*cosh(f*x + e)^2 + 2*a*b^2 - b
^3)*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)*s
inh(f*x + e) + sinh(f*x + e)^2)))/((a^2*b^3 - a*b^4)*f*cosh(f*x + e)^4 + 4*(a^2*b^3 - a*b^4)*f*cosh(f*x + e)*s
inh(f*x + e)^3 + (a^2*b^3 - a*b^4)*f*sinh(f*x + e)^4 + 2*(2*a^3*b^2 - 3*a^2*b^3 + a*b^4)*f*cosh(f*x + e)^2 + 2
*(3*(a^2*b^3 - a*b^4)*f*cosh(f*x + e)^2 + (2*a^3*b^2 - 3*a^2*b^3 + a*b^4)*f)*sinh(f*x + e)^2 + (a^2*b^3 - a*b^
4)*f + 4*((a^2*b^3 - a*b^4)*f*cosh(f*x + e)^3 + (2*a^3*b^2 - 3*a^2*b^3 + a*b^4)*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 {1}{\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(1/(a+b*sinh(f*x+e)**2)**(3/2),x)

[Out]

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

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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(1/(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 {1}{{\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(1/(a + b*sinh(e + f*x)^2)^(3/2),x)

[Out]

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

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