3.1.87 \(\int (a+b \sinh ^2(c+d x))^{5/2} \, dx\) [87]

Optimal. Leaf size=232 \[ \frac {4 (2 a-b) b \cosh (c+d x) \sinh (c+d x) \sqrt {a+b \sinh ^2(c+d x)}}{15 d}+\frac {b \cosh (c+d x) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^{3/2}}{5 d}-\frac {i \left (23 a^2-23 a b+8 b^2\right ) E\left (i c+i d x\left |\frac {b}{a}\right .\right ) \sqrt {a+b \sinh ^2(c+d x)}}{15 d \sqrt {1+\frac {b \sinh ^2(c+d x)}{a}}}+\frac {4 i a (a-b) (2 a-b) F\left (i c+i d x\left |\frac {b}{a}\right .\right ) \sqrt {1+\frac {b \sinh ^2(c+d x)}{a}}}{15 d \sqrt {a+b \sinh ^2(c+d x)}} \]

[Out]

1/5*b*cosh(d*x+c)*sinh(d*x+c)*(a+b*sinh(d*x+c)^2)^(3/2)/d+4/15*(2*a-b)*b*cosh(d*x+c)*sinh(d*x+c)*(a+b*sinh(d*x
+c)^2)^(1/2)/d-1/15*I*(23*a^2-23*a*b+8*b^2)*(cos(I*c+I*d*x)^2)^(1/2)/cos(I*c+I*d*x)*EllipticE(sin(I*c+I*d*x),(
b/a)^(1/2))*(a+b*sinh(d*x+c)^2)^(1/2)/d/(1+b*sinh(d*x+c)^2/a)^(1/2)+4/15*I*a*(a-b)*(2*a-b)*(cos(I*c+I*d*x)^2)^
(1/2)/cos(I*c+I*d*x)*EllipticF(sin(I*c+I*d*x),(b/a)^(1/2))*(1+b*sinh(d*x+c)^2/a)^(1/2)/d/(a+b*sinh(d*x+c)^2)^(
1/2)

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Rubi [A]
time = 0.21, antiderivative size = 232, 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 = {3259, 3249, 3251, 3257, 3256, 3262, 3261} \begin {gather*} -\frac {i \left (23 a^2-23 a b+8 b^2\right ) \sqrt {a+b \sinh ^2(c+d x)} E\left (i c+i d x\left |\frac {b}{a}\right .\right )}{15 d \sqrt {\frac {b \sinh ^2(c+d x)}{a}+1}}+\frac {b \sinh (c+d x) \cosh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^{3/2}}{5 d}+\frac {4 b (2 a-b) \sinh (c+d x) \cosh (c+d x) \sqrt {a+b \sinh ^2(c+d x)}}{15 d}+\frac {4 i a (a-b) (2 a-b) \sqrt {\frac {b \sinh ^2(c+d x)}{a}+1} F\left (i c+i d x\left |\frac {b}{a}\right .\right )}{15 d \sqrt {a+b \sinh ^2(c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Sinh[c + d*x]^2)^(5/2),x]

[Out]

(4*(2*a - b)*b*Cosh[c + d*x]*Sinh[c + d*x]*Sqrt[a + b*Sinh[c + d*x]^2])/(15*d) + (b*Cosh[c + d*x]*Sinh[c + d*x
]*(a + b*Sinh[c + d*x]^2)^(3/2))/(5*d) - ((I/15)*(23*a^2 - 23*a*b + 8*b^2)*EllipticE[I*c + I*d*x, b/a]*Sqrt[a
+ b*Sinh[c + d*x]^2])/(d*Sqrt[1 + (b*Sinh[c + d*x]^2)/a]) + (((4*I)/15)*a*(a - b)*(2*a - b)*EllipticF[I*c + I*
d*x, b/a]*Sqrt[1 + (b*Sinh[c + d*x]^2)/a])/(d*Sqrt[a + b*Sinh[c + d*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 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(c+d x)\right )^{5/2} \, dx &=\frac {b \cosh (c+d x) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^{3/2}}{5 d}+\frac {1}{5} \int \sqrt {a+b \sinh ^2(c+d x)} \left (a (5 a-b)+4 (2 a-b) b \sinh ^2(c+d x)\right ) \, dx\\ &=\frac {4 (2 a-b) b \cosh (c+d x) \sinh (c+d x) \sqrt {a+b \sinh ^2(c+d x)}}{15 d}+\frac {b \cosh (c+d x) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^{3/2}}{5 d}+\frac {1}{15} \int \frac {a \left (15 a^2-11 a b+4 b^2\right )+b \left (23 a^2-23 a b+8 b^2\right ) \sinh ^2(c+d x)}{\sqrt {a+b \sinh ^2(c+d x)}} \, dx\\ &=\frac {4 (2 a-b) b \cosh (c+d x) \sinh (c+d x) \sqrt {a+b \sinh ^2(c+d x)}}{15 d}+\frac {b \cosh (c+d x) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^{3/2}}{5 d}-\frac {1}{15} (4 a (a-b) (2 a-b)) \int \frac {1}{\sqrt {a+b \sinh ^2(c+d x)}} \, dx+\frac {1}{15} \left (23 a^2-23 a b+8 b^2\right ) \int \sqrt {a+b \sinh ^2(c+d x)} \, dx\\ &=\frac {4 (2 a-b) b \cosh (c+d x) \sinh (c+d x) \sqrt {a+b \sinh ^2(c+d x)}}{15 d}+\frac {b \cosh (c+d x) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^{3/2}}{5 d}+\frac {\left (\left (23 a^2-23 a b+8 b^2\right ) \sqrt {a+b \sinh ^2(c+d x)}\right ) \int \sqrt {1+\frac {b \sinh ^2(c+d x)}{a}} \, dx}{15 \sqrt {1+\frac {b \sinh ^2(c+d x)}{a}}}-\frac {\left (4 a (a-b) (2 a-b) \sqrt {1+\frac {b \sinh ^2(c+d x)}{a}}\right ) \int \frac {1}{\sqrt {1+\frac {b \sinh ^2(c+d x)}{a}}} \, dx}{15 \sqrt {a+b \sinh ^2(c+d x)}}\\ &=\frac {4 (2 a-b) b \cosh (c+d x) \sinh (c+d x) \sqrt {a+b \sinh ^2(c+d x)}}{15 d}+\frac {b \cosh (c+d x) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^{3/2}}{5 d}-\frac {i \left (23 a^2-23 a b+8 b^2\right ) E\left (i c+i d x\left |\frac {b}{a}\right .\right ) \sqrt {a+b \sinh ^2(c+d x)}}{15 d \sqrt {1+\frac {b \sinh ^2(c+d x)}{a}}}+\frac {4 i a (a-b) (2 a-b) F\left (i c+i d x\left |\frac {b}{a}\right .\right ) \sqrt {1+\frac {b \sinh ^2(c+d x)}{a}}}{15 d \sqrt {a+b \sinh ^2(c+d x)}}\\ \end {align*}

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Mathematica [A]
time = 0.95, size = 208, normalized size = 0.90 \begin {gather*} \frac {-16 i a \left (23 a^2-23 a b+8 b^2\right ) \sqrt {\frac {2 a-b+b \cosh (2 (c+d x))}{a}} E\left (i (c+d x)\left |\frac {b}{a}\right .\right )+64 i a \left (2 a^2-3 a b+b^2\right ) \sqrt {\frac {2 a-b+b \cosh (2 (c+d x))}{a}} F\left (i (c+d x)\left |\frac {b}{a}\right .\right )+\sqrt {2} b \left (88 a^2-88 a b+25 b^2+28 (2 a-b) b \cosh (2 (c+d x))+3 b^2 \cosh (4 (c+d x))\right ) \sinh (2 (c+d x))}{240 d \sqrt {2 a-b+b \cosh (2 (c+d x))}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sinh[c + d*x]^2)^(5/2),x]

[Out]

((-16*I)*a*(23*a^2 - 23*a*b + 8*b^2)*Sqrt[(2*a - b + b*Cosh[2*(c + d*x)])/a]*EllipticE[I*(c + d*x), b/a] + (64
*I)*a*(2*a^2 - 3*a*b + b^2)*Sqrt[(2*a - b + b*Cosh[2*(c + d*x)])/a]*EllipticF[I*(c + d*x), b/a] + Sqrt[2]*b*(8
8*a^2 - 88*a*b + 25*b^2 + 28*(2*a - b)*b*Cosh[2*(c + d*x)] + 3*b^2*Cosh[4*(c + d*x)])*Sinh[2*(c + d*x)])/(240*
d*Sqrt[2*a - b + b*Cosh[2*(c + d*x)]])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(608\) vs. \(2(268)=536\).
time = 1.40, size = 609, normalized size = 2.62

method	result	size
default	\(\frac {3 \sqrt {-\frac {b}{a}}\, b^{3} \left (\cosh ^{6}\left (d x +c \right )\right ) \sinh \left (d x +c \right )+\left (14 \sqrt {-\frac {b}{a}}\, a \,b^{2}-10 \sqrt {-\frac {b}{a}}\, b^{3}\right ) \left (\cosh ^{4}\left (d x +c \right )\right ) \sinh \left (d x +c \right )+\left (11 \sqrt {-\frac {b}{a}}\, a^{2} b -18 \sqrt {-\frac {b}{a}}\, a \,b^{2}+7 \sqrt {-\frac {b}{a}}\, b^{3}\right ) \left (\cosh ^{2}\left (d x +c \right )\right ) \sinh \left (d x +c \right )+15 a^{3} \sqrt {\frac {b \left (\cosh ^{2}\left (d x +c \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 d x +2 c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (d x +c \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-34 a^{2} b \sqrt {\frac {b \left (\cosh ^{2}\left (d x +c \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 d x +2 c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (d x +c \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )+27 \sqrt {\frac {b \left (\cosh ^{2}\left (d x +c \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 d x +2 c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (d x +c \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a \,b^{2}-8 \sqrt {\frac {b \left (\cosh ^{2}\left (d x +c \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 d x +2 c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (d x +c \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{3}+23 a^{2} b \sqrt {\frac {b \left (\cosh ^{2}\left (d x +c \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 d x +2 c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sinh \left (d x +c \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-23 \sqrt {\frac {b \left (\cosh ^{2}\left (d x +c \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 d x +2 c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sinh \left (d x +c \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a \,b^{2}+8 \sqrt {\frac {b \left (\cosh ^{2}\left (d x +c \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 d x +2 c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sinh \left (d x +c \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{3}}{15 \sqrt {-\frac {b}{a}}\, \cosh \left (d x +c \right ) \sqrt {a +b \left (\sinh ^{2}\left (d x +c \right )\right )}\, d}\)	\(609\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sinh(d*x+c)^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/15*(3*(-1/a*b)^(1/2)*b^3*cosh(d*x+c)^6*sinh(d*x+c)+(14*(-1/a*b)^(1/2)*a*b^2-10*(-1/a*b)^(1/2)*b^3)*cosh(d*x+
c)^4*sinh(d*x+c)+(11*(-1/a*b)^(1/2)*a^2*b-18*(-1/a*b)^(1/2)*a*b^2+7*(-1/a*b)^(1/2)*b^3)*cosh(d*x+c)^2*sinh(d*x
+c)+15*a^3*(b/a*cosh(d*x+c)^2+(a-b)/a)^(1/2)*(cosh(d*x+c)^2)^(1/2)*EllipticF(sinh(d*x+c)*(-1/a*b)^(1/2),(a/b)^
(1/2))-34*a^2*b*(b/a*cosh(d*x+c)^2+(a-b)/a)^(1/2)*(cosh(d*x+c)^2)^(1/2)*EllipticF(sinh(d*x+c)*(-1/a*b)^(1/2),(
a/b)^(1/2))+27*(b/a*cosh(d*x+c)^2+(a-b)/a)^(1/2)*(cosh(d*x+c)^2)^(1/2)*EllipticF(sinh(d*x+c)*(-1/a*b)^(1/2),(a
/b)^(1/2))*a*b^2-8*(b/a*cosh(d*x+c)^2+(a-b)/a)^(1/2)*(cosh(d*x+c)^2)^(1/2)*EllipticF(sinh(d*x+c)*(-1/a*b)^(1/2
),(a/b)^(1/2))*b^3+23*a^2*b*(b/a*cosh(d*x+c)^2+(a-b)/a)^(1/2)*(cosh(d*x+c)^2)^(1/2)*EllipticE(sinh(d*x+c)*(-1/
a*b)^(1/2),(a/b)^(1/2))-23*(b/a*cosh(d*x+c)^2+(a-b)/a)^(1/2)*(cosh(d*x+c)^2)^(1/2)*EllipticE(sinh(d*x+c)*(-1/a
*b)^(1/2),(a/b)^(1/2))*a*b^2+8*(b/a*cosh(d*x+c)^2+(a-b)/a)^(1/2)*(cosh(d*x+c)^2)^(1/2)*EllipticE(sinh(d*x+c)*(
-1/a*b)^(1/2),(a/b)^(1/2))*b^3)/(-1/a*b)^(1/2)/cosh(d*x+c)/(a+b*sinh(d*x+c)^2)^(1/2)/d

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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(d*x+c)^2)^(5/2),x, algorithm="maxima")

[Out]

integrate((b*sinh(d*x + c)^2 + a)^(5/2), x)

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Fricas [F]
time = 0.15, size = 45, normalized size = 0.19 \begin {gather*} {\rm integral}\left ({\left (b^{2} \sinh \left (d x + c\right )^{4} + 2 \, a b \sinh \left (d x + c\right )^{2} + a^{2}\right )} \sqrt {b \sinh \left (d x + c\right )^{2} + a}, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sinh(d*x+c)^2)^(5/2),x, algorithm="fricas")

[Out]

integral((b^2*sinh(d*x + c)^4 + 2*a*b*sinh(d*x + c)^2 + a^2)*sqrt(b*sinh(d*x + c)^2 + a), x)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sinh(d*x+c)**2)**(5/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3062 deep

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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((a+b*sinh(d*x+c)^2)^(5/2),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 {\left (b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+a\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*sinh(c + d*x)^2)^(5/2),x)

[Out]

int((a + b*sinh(c + d*x)^2)^(5/2), x)

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