3.1.97 \(\int (a+b \sinh ^2(x))^{3/2} \, dx\) [97]

Optimal. Leaf size=123 \[ \frac {1}{3} b \cosh (x) \sinh (x) \sqrt {a+b \sinh ^2(x)}-\frac {2 i (2 a-b) E\left (i x\left |\frac {b}{a}\right .\right ) \sqrt {a+b \sinh ^2(x)}}{3 \sqrt {1+\frac {b \sinh ^2(x)}{a}}}+\frac {i a (a-b) F\left (i x\left |\frac {b}{a}\right .\right ) \sqrt {1+\frac {b \sinh ^2(x)}{a}}}{3 \sqrt {a+b \sinh ^2(x)}} \]

[Out]

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

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Rubi [A]
time = 0.12, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3259, 3251, 3257, 3256, 3262, 3261} \begin {gather*} \frac {1}{3} b \sinh (x) \cosh (x) \sqrt {a+b \sinh ^2(x)}+\frac {i a (a-b) \sqrt {\frac {b \sinh ^2(x)}{a}+1} F\left (i x\left |\frac {b}{a}\right .\right )}{3 \sqrt {a+b \sinh ^2(x)}}-\frac {2 i (2 a-b) \sqrt {a+b \sinh ^2(x)} E\left (i x\left |\frac {b}{a}\right .\right )}{3 \sqrt {\frac {b \sinh ^2(x)}{a}+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Sinh[x]^2)^(3/2),x]

[Out]

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

Rule 3251

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 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 3259

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*f*p)), x] + Dist[1/(2*p), Int[(a + b*Sin[e + f*x]^2)^(p - 2)*Simp[a*(b + 2*a*p) + b*(
2*a + b)*(2*p - 1)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a + b, 0] && GtQ[p, 1]

Rule 3261

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

Rule 3262

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]

Rubi steps

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

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Mathematica [A]
time = 0.28, size = 132, normalized size = 1.07 \begin {gather*} \frac {-8 i a (2 a-b) \sqrt {\frac {2 a-b+b \cosh (2 x)}{a}} E\left (i x\left |\frac {b}{a}\right .\right )+4 i a (a-b) \sqrt {\frac {2 a-b+b \cosh (2 x)}{a}} F\left (i x\left |\frac {b}{a}\right .\right )+\sqrt {2} b (2 a-b+b \cosh (2 x)) \sinh (2 x)}{12 \sqrt {2 a-b+b \cosh (2 x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sinh[x]^2)^(3/2),x]

[Out]

((-8*I)*a*(2*a - b)*Sqrt[(2*a - b + b*Cosh[2*x])/a]*EllipticE[I*x, b/a] + (4*I)*a*(a - b)*Sqrt[(2*a - b + b*Co
sh[2*x])/a]*EllipticF[I*x, b/a] + Sqrt[2]*b*(2*a - b + b*Cosh[2*x])*Sinh[2*x])/(12*Sqrt[2*a - b + b*Cosh[2*x]]
)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(332\) vs. \(2(129)=258\).
time = 1.14, size = 333, normalized size = 2.71

method	result	size
default	\(\frac {\sqrt {-\frac {b}{a}}\, b^{2} \left (\cosh ^{4}\left (x \right )\right ) \sinh \left (x \right )+\sqrt {-\frac {b}{a}}\, a b \left (\cosh ^{2}\left (x \right )\right ) \sinh \left (x \right )-\sqrt {-\frac {b}{a}}\, b^{2} \left (\cosh ^{2}\left (x \right )\right ) \sinh \left (x \right )+3 a^{2} \sqrt {\frac {b \left (\cosh ^{2}\left (x \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {1}{2}+\frac {\cosh \left (2 x \right )}{2}}\, \EllipticF \left (\sinh \left (x \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-5 a b \sqrt {\frac {b \left (\cosh ^{2}\left (x \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {1}{2}+\frac {\cosh \left (2 x \right )}{2}}\, \EllipticF \left (\sinh \left (x \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )+2 \sqrt {\frac {b \left (\cosh ^{2}\left (x \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {1}{2}+\frac {\cosh \left (2 x \right )}{2}}\, \EllipticF \left (\sinh \left (x \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2}+4 a b \sqrt {\frac {b \left (\cosh ^{2}\left (x \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {1}{2}+\frac {\cosh \left (2 x \right )}{2}}\, \EllipticE \left (\sinh \left (x \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-2 \sqrt {\frac {b \left (\cosh ^{2}\left (x \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {1}{2}+\frac {\cosh \left (2 x \right )}{2}}\, \EllipticE \left (\sinh \left (x \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2}}{3 \sqrt {-\frac {b}{a}}\, \cosh \left (x \right ) \sqrt {a +b \left (\sinh ^{2}\left (x \right )\right )}}\)	\(333\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*((-1/a*b)^(1/2)*b^2*cosh(x)^4*sinh(x)+(-1/a*b)^(1/2)*a*b*cosh(x)^2*sinh(x)-(-1/a*b)^(1/2)*b^2*cosh(x)^2*si
nh(x)+3*a^2*(b/a*cosh(x)^2+(a-b)/a)^(1/2)*(cosh(x)^2)^(1/2)*EllipticF(sinh(x)*(-1/a*b)^(1/2),(a/b)^(1/2))-5*a*
b*(b/a*cosh(x)^2+(a-b)/a)^(1/2)*(cosh(x)^2)^(1/2)*EllipticF(sinh(x)*(-1/a*b)^(1/2),(a/b)^(1/2))+2*(b/a*cosh(x)
^2+(a-b)/a)^(1/2)*(cosh(x)^2)^(1/2)*EllipticF(sinh(x)*(-1/a*b)^(1/2),(a/b)^(1/2))*b^2+4*a*b*(b/a*cosh(x)^2+(a-
b)/a)^(1/2)*(cosh(x)^2)^(1/2)*EllipticE(sinh(x)*(-1/a*b)^(1/2),(a/b)^(1/2))-2*(b/a*cosh(x)^2+(a-b)/a)^(1/2)*(c
osh(x)^2)^(1/2)*EllipticE(sinh(x)*(-1/a*b)^(1/2),(a/b)^(1/2))*b^2)/(-1/a*b)^(1/2)/cosh(x)/(a+b*sinh(x)^2)^(1/2
)

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

[Out]

integrate((b*sinh(x)^2 + a)^(3/2), x)

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Fricas [F]
time = 0.10, size = 12, normalized size = 0.10 \begin {gather*} {\rm integral}\left ({\left (b \sinh \left (x\right )^{2} + a\right )}^{\frac {3}{2}}, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*sinh(x)^2 + a)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*sinh(x)^2 + a)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*sinh(x)^2)^(3/2),x)

[Out]

int((a + b*sinh(x)^2)^(3/2), x)

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