3.509 \(\int \frac {1}{(a+b \sinh ^2(e+f x))^{5/2}} \, dx\)

Optimal. Leaf size=251 \[ -\frac {2 b (2 a-b) \sinh (e+f x) \cosh (e+f x)}{3 a^2 f (a-b)^2 \sqrt {a+b \sinh ^2(e+f x)}}-\frac {2 i (2 a-b) \sqrt {a+b \sinh ^2(e+f x)} E\left (i e+i f x\left |\frac {b}{a}\right .\right )}{3 a^2 f (a-b)^2 \sqrt {\frac {b \sinh ^2(e+f x)}{a}+1}}-\frac {b \sinh (e+f x) \cosh (e+f x)}{3 a f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {i \sqrt {\frac {b \sinh ^2(e+f x)}{a}+1} F\left (i e+i f x\left |\frac {b}{a}\right .\right )}{3 a f (a-b) \sqrt {a+b \sinh ^2(e+f x)}} \]

[Out]

-1/3*b*cosh(f*x+e)*sinh(f*x+e)/a/(a-b)/f/(a+b*sinh(f*x+e)^2)^(3/2)-2/3*(2*a-b)*b*cosh(f*x+e)*sinh(f*x+e)/a^2/(
a-b)^2/f/(a+b*sinh(f*x+e)^2)^(1/2)-2/3*I*(2*a-b)*(cos(I*e+I*f*x)^2)^(1/2)/cos(I*e+I*f*x)*EllipticE(sin(I*e+I*f
*x),(b/a)^(1/2))*(a+b*sinh(f*x+e)^2)^(1/2)/a^2/(a-b)^2/f/(1+b*sinh(f*x+e)^2/a)^(1/2)+1/3*I*(cos(I*e+I*f*x)^2)^
(1/2)/cos(I*e+I*f*x)*EllipticF(sin(I*e+I*f*x),(b/a)^(1/2))*(1+b*sinh(f*x+e)^2/a)^(1/2)/a/(a-b)/f/(a+b*sinh(f*x
+e)^2)^(1/2)

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Rubi [A]  time = 0.29, antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {3184, 3173, 3172, 3178, 3177, 3183, 3182} \[ -\frac {2 b (2 a-b) \sinh (e+f x) \cosh (e+f x)}{3 a^2 f (a-b)^2 \sqrt {a+b \sinh ^2(e+f x)}}-\frac {2 i (2 a-b) \sqrt {a+b \sinh ^2(e+f x)} E\left (i e+i f x\left |\frac {b}{a}\right .\right )}{3 a^2 f (a-b)^2 \sqrt {\frac {b \sinh ^2(e+f x)}{a}+1}}-\frac {b \sinh (e+f x) \cosh (e+f x)}{3 a f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {i \sqrt {\frac {b \sinh ^2(e+f x)}{a}+1} F\left (i e+i f x\left |\frac {b}{a}\right .\right )}{3 a f (a-b) \sqrt {a+b \sinh ^2(e+f x)}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sinh[e + f*x]^2)^(-5/2),x]

[Out]

-(b*Cosh[e + f*x]*Sinh[e + f*x])/(3*a*(a - b)*f*(a + b*Sinh[e + f*x]^2)^(3/2)) - (2*(2*a - b)*b*Cosh[e + f*x]*
Sinh[e + f*x])/(3*a^2*(a - b)^2*f*Sqrt[a + b*Sinh[e + f*x]^2]) - (((2*I)/3)*(2*a - b)*EllipticE[I*e + I*f*x, b
/a]*Sqrt[a + b*Sinh[e + f*x]^2])/(a^2*(a - b)^2*f*Sqrt[1 + (b*Sinh[e + f*x]^2)/a]) + ((I/3)*EllipticF[I*e + I*
f*x, b/a]*Sqrt[1 + (b*Sinh[e + f*x]^2)/a])/(a*(a - b)*f*Sqrt[a + b*Sinh[e + f*x]^2])

Rule 3172

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]^2)/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Dist[
B/b, Int[Sqrt[a + b*Sin[e + f*x]^2], x], x] + Dist[(A*b - a*B)/b, Int[1/Sqrt[a + b*Sin[e + f*x]^2], x], x] /;
FreeQ[{a, b, e, f, A, B}, x]

Rule 3173

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

Rule 3177

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

Rule 3178

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 3182

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

Rule 3183

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

Rule 3184

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> -Simp[(b*Cos[e + f*x]*Sin[e + f*x]*(a + b*Sin[
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] && NeQ
[a + b, 0] && LtQ[p, -1]

Rubi steps

\begin {align*} \int \frac {1}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx &=-\frac {b \cosh (e+f x) \sinh (e+f x)}{3 a (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\int \frac {-3 a+2 b+b \sinh ^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx}{3 a (a-b)}\\ &=-\frac {b \cosh (e+f x) \sinh (e+f x)}{3 a (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {2 (2 a-b) b \cosh (e+f x) \sinh (e+f x)}{3 a^2 (a-b)^2 f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {\int \frac {-a (3 a-b)-2 (2 a-b) b \sinh ^2(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx}{3 a^2 (a-b)^2}\\ &=-\frac {b \cosh (e+f x) \sinh (e+f x)}{3 a (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {2 (2 a-b) b \cosh (e+f x) \sinh (e+f x)}{3 a^2 (a-b)^2 f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {\int \frac {1}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx}{3 a (a-b)}+\frac {(2 (2 a-b)) \int \sqrt {a+b \sinh ^2(e+f x)} \, dx}{3 a^2 (a-b)^2}\\ &=-\frac {b \cosh (e+f x) \sinh (e+f x)}{3 a (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {2 (2 a-b) b \cosh (e+f x) \sinh (e+f x)}{3 a^2 (a-b)^2 f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {\left (2 (2 a-b) \sqrt {a+b \sinh ^2(e+f x)}\right ) \int \sqrt {1+\frac {b \sinh ^2(e+f x)}{a}} \, dx}{3 a^2 (a-b)^2 \sqrt {1+\frac {b \sinh ^2(e+f x)}{a}}}-\frac {\sqrt {1+\frac {b \sinh ^2(e+f x)}{a}} \int \frac {1}{\sqrt {1+\frac {b \sinh ^2(e+f x)}{a}}} \, dx}{3 a (a-b) \sqrt {a+b \sinh ^2(e+f x)}}\\ &=-\frac {b \cosh (e+f x) \sinh (e+f x)}{3 a (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {2 (2 a-b) b \cosh (e+f x) \sinh (e+f x)}{3 a^2 (a-b)^2 f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {2 i (2 a-b) E\left (i e+i f x\left |\frac {b}{a}\right .\right ) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^2 (a-b)^2 f \sqrt {1+\frac {b \sinh ^2(e+f x)}{a}}}+\frac {i F\left (i e+i f x\left |\frac {b}{a}\right .\right ) \sqrt {1+\frac {b \sinh ^2(e+f x)}{a}}}{3 a (a-b) f \sqrt {a+b \sinh ^2(e+f x)}}\\ \end {align*}

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Mathematica [A]  time = 0.96, size = 190, normalized size = 0.76 \[ \frac {\sqrt {2} b \sinh (2 (e+f x)) \left (-5 a^2+b (b-2 a) \cosh (2 (e+f x))+5 a b-b^2\right )+i a^2 (a-b) \left (\frac {2 a+b \cosh (2 (e+f x))-b}{a}\right )^{3/2} F\left (i (e+f x)\left |\frac {b}{a}\right .\right )-2 i a^2 (2 a-b) \left (\frac {2 a+b \cosh (2 (e+f x))-b}{a}\right )^{3/2} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )}{3 a^2 f (a-b)^2 (2 a+b \cosh (2 (e+f x))-b)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sinh[e + f*x]^2)^(-5/2),x]

[Out]

((-2*I)*a^2*(2*a - b)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticE[I*(e + f*x), b/a] + I*a^2*(a - b)*((
2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticF[I*(e + f*x), b/a] + Sqrt[2]*b*(-5*a^2 + 5*a*b - b^2 + b*(-2*
a + b)*Cosh[2*(e + f*x)])*Sinh[2*(e + f*x)])/(3*a^2*(a - b)^2*f*(2*a - b + b*Cosh[2*(e + f*x)])^(3/2))

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fricas [F]  time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \sinh \left (f x + e\right )^{2} + a}}{b^{3} \sinh \left (f x + e\right )^{6} + 3 \, a b^{2} \sinh \left (f x + e\right )^{4} + 3 \, a^{2} b \sinh \left (f x + e\right )^{2} + a^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sinh(f*x+e)^2)^(5/2),x, algorithm="fricas")

[Out]

integral(sqrt(b*sinh(f*x + e)^2 + a)/(b^3*sinh(f*x + e)^6 + 3*a*b^2*sinh(f*x + e)^4 + 3*a^2*b*sinh(f*x + e)^2
+ a^3), x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sinh(f*x+e)^2)^(5/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Eval
uation time: 0.45Error: Bad Argument Type

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maple [A]  time = 0.58, size = 406, normalized size = 1.62 \[ \frac {\sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}\, \left (-\frac {\sinh \left (f x +e \right ) \sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}}{3 a b \left (a -b \right ) \left (\sinh ^{2}\left (f x +e \right )+\frac {a}{b}\right )^{2}}-\frac {2 b \left (\cosh ^{2}\left (f x +e \right )\right ) \sinh \left (f x +e \right ) \left (2 a -b \right )}{3 a^{2} \left (a -b \right )^{2} \sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}}+\frac {\left (3 a -b \right ) \sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )}{\left (3 a^{3}-6 a^{2} b +3 a \,b^{2}\right ) \sqrt {-\frac {b}{a}}\, \sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}}-\frac {2 b \left (2 a -b \right ) \sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \left (\EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-\EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )\right )}{3 a^{2} \left (a -b \right )^{2} \sqrt {-\frac {b}{a}}\, \sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}}\right )}{\cosh \left (f x +e \right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*sinh(f*x+e)^2)^(5/2),x)

[Out]

((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)*(-1/3/a/b/(a-b)*sinh(f*x+e)*((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2
)/(sinh(f*x+e)^2+a/b)^2-2/3*b*cosh(f*x+e)^2/a^2/(a-b)^2*sinh(f*x+e)*(2*a-b)/((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2
)^(1/2)+(3*a-b)/(3*a^3-6*a^2*b+3*a*b^2)/(-1/a*b)^(1/2)*((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)/((a
+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))-2/3*b*(2*a-b)/a^2/(a-
b)^2/(-1/a*b)^(1/2)*((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)/((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1
/2)*(EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))-EllipticE(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))))/cos
h(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 \[ \int \frac {1}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sinh(f*x+e)^2)^(5/2),x, algorithm="maxima")

[Out]

integrate((b*sinh(f*x + e)^2 + a)^(-5/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*sinh(e + f*x)^2)^(5/2),x)

[Out]

int(1/(a + b*sinh(e + f*x)^2)^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b \sinh ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sinh(f*x+e)**2)**(5/2),x)

[Out]

Integral((a + b*sinh(e + f*x)**2)**(-5/2), x)

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