3.97 \(\int (a+b \sinh ^2(x))^{3/2} \, dx\)

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

[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]

________________________________________________________________________________________

Rubi [A]  time = 0.162822, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3180, 3172, 3178, 3177, 3183, 3182} \[ \frac{1}{3} b \sinh (x) \cosh (x) \sqrt{a+b \sinh ^2(x)}+\frac{i a (a-b) \sqrt{\frac{b \sinh ^2(x)}{a}+1} F\left (i x\left |\frac{b}{a}\right .\right )}{3 \sqrt{a+b \sinh ^2(x)}}-\frac{2 i (2 a-b) \sqrt{a+b \sinh ^2(x)} E\left (i x\left |\frac{b}{a}\right .\right )}{3 \sqrt{\frac{b \sinh ^2(x)}{a}+1}} \]

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 3180

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*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 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 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 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 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 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]

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*}

Mathematica [A]  time = 0.378756, size = 132, normalized size = 1.07 \[ \frac{4 i a (a-b) \sqrt{\frac{2 a+b \cosh (2 x)-b}{a}} \text{EllipticF}\left (i x,\frac{b}{a}\right )+\sqrt{2} b \sinh (2 x) (2 a+b \cosh (2 x)-b)-8 i a (2 a-b) \sqrt{\frac{2 a+b \cosh (2 x)-b}{a}} E\left (i x\left |\frac{b}{a}\right .\right )}{12 \sqrt{2 a+b \cosh (2 x)-b}} \]

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]]
)

________________________________________________________________________________________

Maple [B]  time = 0.07, size = 329, normalized size = 2.7 \begin{align*}{\frac{1}{3\,\cosh \left ( x \right ) } \left ( \sqrt{-{\frac{b}{a}}}{b}^{2}\sinh \left ( x \right ) \left ( \cosh \left ( x \right ) \right ) ^{4}+ \left ( \sqrt{-{\frac{b}{a}}}ab-\sqrt{-{\frac{b}{a}}}{b}^{2} \right ) \left ( \cosh \left ( x \right ) \right ) ^{2}\sinh \left ( x \right ) +3\,{a}^{2}\sqrt{{\frac{b \left ( \cosh \left ( x \right ) \right ) ^{2}}{a}}+{\frac{a-b}{a}}}\sqrt{ \left ( \cosh \left ( x \right ) \right ) ^{2}}{\it EllipticF} \left ( \sinh \left ( x \right ) \sqrt{-{\frac{b}{a}}},\sqrt{{\frac{a}{b}}} \right ) -5\,ab\sqrt{{\frac{b \left ( \cosh \left ( x \right ) \right ) ^{2}}{a}}+{\frac{a-b}{a}}}\sqrt{ \left ( \cosh \left ( x \right ) \right ) ^{2}}{\it EllipticF} \left ( \sinh \left ( x \right ) \sqrt{-{\frac{b}{a}}},\sqrt{{\frac{a}{b}}} \right ) +2\,\sqrt{{\frac{b \left ( \cosh \left ( x \right ) \right ) ^{2}}{a}}+{\frac{a-b}{a}}}\sqrt{ \left ( \cosh \left ( x \right ) \right ) ^{2}}{\it EllipticF} \left ( \sinh \left ( x \right ) \sqrt{-{\frac{b}{a}}},\sqrt{{\frac{a}{b}}} \right ){b}^{2}+4\,ab\sqrt{{\frac{b \left ( \cosh \left ( x \right ) \right ) ^{2}}{a}}+{\frac{a-b}{a}}}\sqrt{ \left ( \cosh \left ( x \right ) \right ) ^{2}}{\it EllipticE} \left ( \sinh \left ( x \right ) \sqrt{-{\frac{b}{a}}},\sqrt{{\frac{a}{b}}} \right ) -2\,\sqrt{{\frac{b \left ( \cosh \left ( x \right ) \right ) ^{2}}{a}}+{\frac{a-b}{a}}}\sqrt{ \left ( \cosh \left ( x \right ) \right ) ^{2}}{\it EllipticE} \left ( \sinh \left ( x \right ) \sqrt{-{\frac{b}{a}}},\sqrt{{\frac{a}{b}}} \right ){b}^{2} \right ){\frac{1}{\sqrt{-{\frac{b}{a}}}}}{\frac{1}{\sqrt{a+b \left ( \sinh \left ( x \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*((-1/a*b)^(1/2)*b^2*sinh(x)*cosh(x)^4+((-1/a*b)^(1/2)*a*b-(-1/a*b)^(1/2)*b^2)*cosh(x)^2*sinh(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)*(cos
h(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)*(cosh(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)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sinh \left (x\right )^{2} + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}

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)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \sinh \left (x\right )^{2} + a\right )}^{\frac{3}{2}}, x\right ) \end{align*}

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)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sinh \left (x\right )^{2} + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}

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)