3.1.83 \(\int \sinh ^2(e+f x) (a+b \sinh ^2(e+f x))^{3/2} \, dx\) [83]

Optimal. Leaf size=236 \[ \frac {(3 a-4 b) \cosh (e+f x) \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{15 f}+\frac {\cosh (e+f x) \sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{5 f}-\frac {i \left (3 a^2-13 a b+8 b^2\right ) E\left (i e+i f x\left |\frac {b}{a}\right .\right ) \sqrt {a+b \sinh ^2(e+f x)}}{15 b f \sqrt {1+\frac {b \sinh ^2(e+f x)}{a}}}+\frac {i a (3 a-4 b) (a-b) F\left (i e+i f x\left |\frac {b}{a}\right .\right ) \sqrt {1+\frac {b \sinh ^2(e+f x)}{a}}}{15 b f \sqrt {a+b \sinh ^2(e+f x)}} \]

[Out]

1/5*cosh(f*x+e)*sinh(f*x+e)*(a+b*sinh(f*x+e)^2)^(3/2)/f+1/15*(3*a-4*b)*cosh(f*x+e)*sinh(f*x+e)*(a+b*sinh(f*x+e
)^2)^(1/2)/f-1/15*I*(3*a^2-13*a*b+8*b^2)*(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)/b/f/(1+b*sinh(f*x+e)^2/a)^(1/2)+1/15*I*a*(3*a-4*b)*(a-b)*(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)/b/f/(a+b*sinh(f*x+e)^2
)^(1/2)

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Rubi [A]
time = 0.23, antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3249, 3251, 3257, 3256, 3262, 3261} \begin {gather*} -\frac {i \left (3 a^2-13 a b+8 b^2\right ) \sqrt {a+b \sinh ^2(e+f x)} E\left (i e+i f x\left |\frac {b}{a}\right .\right )}{15 b f \sqrt {\frac {b \sinh ^2(e+f x)}{a}+1}}+\frac {\sinh (e+f x) \cosh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{5 f}+\frac {(3 a-4 b) \sinh (e+f x) \cosh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{15 f}+\frac {i a (3 a-4 b) (a-b) \sqrt {\frac {b \sinh ^2(e+f x)}{a}+1} F\left (i e+i f x\left |\frac {b}{a}\right .\right )}{15 b f \sqrt {a+b \sinh ^2(e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((3*a - 4*b)*Cosh[e + f*x]*Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(15*f) + (Cosh[e + f*x]*Sinh[e + f*x]*(a
+ b*Sinh[e + f*x]^2)^(3/2))/(5*f) - ((I/15)*(3*a^2 - 13*a*b + 8*b^2)*EllipticE[I*e + I*f*x, b/a]*Sqrt[a + b*S
inh[e + f*x]^2])/(b*f*Sqrt[1 + (b*Sinh[e + f*x]^2)/a]) + ((I/15)*a*(3*a - 4*b)*(a - b)*EllipticF[I*e + I*f*x,
b/a]*Sqrt[1 + (b*Sinh[e + f*x]^2)/a])/(b*f*Sqrt[a + b*Sinh[e + f*x]^2])

Rule 3249

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

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 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 \sinh ^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx &=\frac {\cosh (e+f x) \sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{5 f}-\frac {1}{5} \int \left (a-(3 a-4 b) \sinh ^2(e+f x)\right ) \sqrt {a+b \sinh ^2(e+f x)} \, dx\\ &=\frac {(3 a-4 b) \cosh (e+f x) \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{15 f}+\frac {\cosh (e+f x) \sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{5 f}-\frac {1}{15} \int \frac {2 a (3 a-2 b)-\left (3 a^2-13 a b+8 b^2\right ) \sinh ^2(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx\\ &=\frac {(3 a-4 b) \cosh (e+f x) \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{15 f}+\frac {\cosh (e+f x) \sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{5 f}-\frac {(a (3 a-4 b) (a-b)) \int \frac {1}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx}{15 b}+\frac {\left (3 a^2-13 a b+8 b^2\right ) \int \sqrt {a+b \sinh ^2(e+f x)} \, dx}{15 b}\\ &=\frac {(3 a-4 b) \cosh (e+f x) \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{15 f}+\frac {\cosh (e+f x) \sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{5 f}+\frac {\left (\left (3 a^2-13 a b+8 b^2\right ) \sqrt {a+b \sinh ^2(e+f x)}\right ) \int \sqrt {1+\frac {b \sinh ^2(e+f x)}{a}} \, dx}{15 b \sqrt {1+\frac {b \sinh ^2(e+f x)}{a}}}-\frac {\left (a (3 a-4 b) (a-b) \sqrt {1+\frac {b \sinh ^2(e+f x)}{a}}\right ) \int \frac {1}{\sqrt {1+\frac {b \sinh ^2(e+f x)}{a}}} \, dx}{15 b \sqrt {a+b \sinh ^2(e+f x)}}\\ &=\frac {(3 a-4 b) \cosh (e+f x) \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{15 f}+\frac {\cosh (e+f x) \sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{5 f}-\frac {i \left (3 a^2-13 a b+8 b^2\right ) E\left (i e+i f x\left |\frac {b}{a}\right .\right ) \sqrt {a+b \sinh ^2(e+f x)}}{15 b f \sqrt {1+\frac {b \sinh ^2(e+f x)}{a}}}+\frac {i a (3 a-4 b) (a-b) F\left (i e+i f x\left |\frac {b}{a}\right .\right ) \sqrt {1+\frac {b \sinh ^2(e+f x)}{a}}}{15 b f \sqrt {a+b \sinh ^2(e+f x)}}\\ \end {align*}

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Mathematica [A]
time = 0.94, size = 213, normalized size = 0.90 \begin {gather*} \frac {-16 i a \left (3 a^2-13 a b+8 b^2\right ) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )+16 i a \left (3 a^2-7 a b+4 b^2\right ) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} F\left (i (e+f x)\left |\frac {b}{a}\right .\right )+\sqrt {2} b \left (48 a^2-68 a b+25 b^2+4 (9 a-7 b) b \cosh (2 (e+f x))+3 b^2 \cosh (4 (e+f x))\right ) \sinh (2 (e+f x))}{240 b f \sqrt {2 a-b+b \cosh (2 (e+f x))}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-16*I)*a*(3*a^2 - 13*a*b + 8*b^2)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] + (16*
I)*a*(3*a^2 - 7*a*b + 4*b^2)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] + Sqrt[2]*b*(
48*a^2 - 68*a*b + 25*b^2 + 4*(9*a - 7*b)*b*Cosh[2*(e + f*x)] + 3*b^2*Cosh[4*(e + f*x)])*Sinh[2*(e + f*x)])/(24
0*b*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])

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Maple [A]
time = 1.18, size = 535, normalized size = 2.27

method	result	size
default	\(\frac {3 \sqrt {-\frac {b}{a}}\, b^{2} \left (\cosh ^{6}\left (f x +e \right )\right ) \sinh \left (f x +e \right )+\left (9 \sqrt {-\frac {b}{a}}\, a b -10 \sqrt {-\frac {b}{a}}\, b^{2}\right ) \left (\cosh ^{4}\left (f x +e \right )\right ) \sinh \left (f x +e \right )+\left (6 \sqrt {-\frac {b}{a}}\, a^{2}-13 \sqrt {-\frac {b}{a}}\, a b +7 \sqrt {-\frac {b}{a}}\, b^{2}\right ) \left (\cosh ^{2}\left (f x +e \right )\right ) \sinh \left (f x +e \right )-9 a^{2} \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 )+17 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 ) b -8 \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^{2}+3 \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 ) a^{2}-13 \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 ) a b +8 \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^{2}}{15 \sqrt {-\frac {b}{a}}\, \cosh \left (f x +e \right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, f}\)	\(535\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(3*(-1/a*b)^(1/2)*b^2*cosh(f*x+e)^6*sinh(f*x+e)+(9*(-1/a*b)^(1/2)*a*b-10*(-1/a*b)^(1/2)*b^2)*cosh(f*x+e)^
4*sinh(f*x+e)+(6*(-1/a*b)^(1/2)*a^2-13*(-1/a*b)^(1/2)*a*b+7*(-1/a*b)^(1/2)*b^2)*cosh(f*x+e)^2*sinh(f*x+e)-9*a^
2*(b/a*cosh(f*x+e)^2+(a-b)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))+17
*a*(b/a*cosh(f*x+e)^2+(a-b)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*b
-8*(b/a*cosh(f*x+e)^2+(a-b)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*b
^2+3*(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))
*a^2-13*(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))*a*b+8*(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))*b^2)/(-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(sinh(f*x+e)^2*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [F]
time = 0.12, size = 38, normalized size = 0.16 \begin {gather*} {\rm integral}\left ({\left (b \sinh \left (f x + e\right )^{4} + a \sinh \left (f x + e\right )^{2}\right )} \sqrt {b \sinh \left (f x + e\right )^{2} + a}, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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(sinh(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: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Error: Bad Argument Type

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

[Out]

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

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