3.1.7 \(\int \sinh ^{\frac {7}{2}}(a+b x) \, dx\) [7]

Optimal. Leaf size=103 \[ -\frac {10 i F\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right ) \sqrt {i \sinh (a+b x)}}{21 b \sqrt {\sinh (a+b x)}}-\frac {10 \cosh (a+b x) \sqrt {\sinh (a+b x)}}{21 b}+\frac {2 \cosh (a+b x) \sinh ^{\frac {5}{2}}(a+b x)}{7 b} \]

[Out]

2/7*cosh(b*x+a)*sinh(b*x+a)^(5/2)/b+10/21*I*(sin(1/2*I*a+1/4*Pi+1/2*I*b*x)^2)^(1/2)/sin(1/2*I*a+1/4*Pi+1/2*I*b
*x)*EllipticF(cos(1/2*I*a+1/4*Pi+1/2*I*b*x),2^(1/2))*(I*sinh(b*x+a))^(1/2)/b/sinh(b*x+a)^(1/2)-10/21*cosh(b*x+
a)*sinh(b*x+a)^(1/2)/b

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Rubi [A]
time = 0.03, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2715, 2721, 2720} \begin {gather*} \frac {2 \sinh ^{\frac {5}{2}}(a+b x) \cosh (a+b x)}{7 b}-\frac {10 \sqrt {\sinh (a+b x)} \cosh (a+b x)}{21 b}-\frac {10 i \sqrt {i \sinh (a+b x)} F\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{21 b \sqrt {\sinh (a+b x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b*x]^(7/2),x]

[Out]

(((-10*I)/21)*EllipticF[(I*a - Pi/2 + I*b*x)/2, 2]*Sqrt[I*Sinh[a + b*x]])/(b*Sqrt[Sinh[a + b*x]]) - (10*Cosh[a
+ b*x]*Sqrt[Sinh[a + b*x]])/(21*b) + (2*Cosh[a + b*x]*Sinh[a + b*x]^(5/2))/(7*b)

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ
[{c, d}, x]

Rule 2721

Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[(b*Sin[c + d*x])^n/Sin[c + d*x]^n, Int[Sin[c + d*x]
^n, x], x] /; FreeQ[{b, c, d}, x] && LtQ[-1, n, 1] && IntegerQ[2*n]

Rubi steps

\begin {align*} \int \sinh ^{\frac {7}{2}}(a+b x) \, dx &=\frac {2 \cosh (a+b x) \sinh ^{\frac {5}{2}}(a+b x)}{7 b}-\frac {5}{7} \int \sinh ^{\frac {3}{2}}(a+b x) \, dx\\ &=-\frac {10 \cosh (a+b x) \sqrt {\sinh (a+b x)}}{21 b}+\frac {2 \cosh (a+b x) \sinh ^{\frac {5}{2}}(a+b x)}{7 b}+\frac {5}{21} \int \frac {1}{\sqrt {\sinh (a+b x)}} \, dx\\ &=-\frac {10 \cosh (a+b x) \sqrt {\sinh (a+b x)}}{21 b}+\frac {2 \cosh (a+b x) \sinh ^{\frac {5}{2}}(a+b x)}{7 b}+\frac {\left (5 \sqrt {i \sinh (a+b x)}\right ) \int \frac {1}{\sqrt {i \sinh (a+b x)}} \, dx}{21 \sqrt {\sinh (a+b x)}}\\ &=-\frac {10 i F\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right ) \sqrt {i \sinh (a+b x)}}{21 b \sqrt {\sinh (a+b x)}}-\frac {10 \cosh (a+b x) \sqrt {\sinh (a+b x)}}{21 b}+\frac {2 \cosh (a+b x) \sinh ^{\frac {5}{2}}(a+b x)}{7 b}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 75, normalized size = 0.73 \begin {gather*} \frac {40 i F\left (\left .\frac {1}{4} (-2 i a+\pi -2 i b x)\right |2\right ) \sqrt {i \sinh (a+b x)}-26 \sinh (2 (a+b x))+3 \sinh (4 (a+b x))}{84 b \sqrt {\sinh (a+b x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sinh[a + b*x]^(7/2),x]

[Out]

((40*I)*EllipticF[((-2*I)*a + Pi - (2*I)*b*x)/4, 2]*Sqrt[I*Sinh[a + b*x]] - 26*Sinh[2*(a + b*x)] + 3*Sinh[4*(a
+ b*x)])/(84*b*Sqrt[Sinh[a + b*x]])

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Maple [A]
time = 0.66, size = 116, normalized size = 1.13

method	result	size
default	\(\frac {\frac {5 i \sqrt {1-i \sinh \left (b x +a \right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (b x +a \right )}\, \sqrt {i \sinh \left (b x +a \right )}\, \EllipticF \left (\sqrt {1-i \sinh \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )}{21}+\frac {2 \left (\cosh ^{4}\left (b x +a \right )\right ) \sinh \left (b x +a \right )}{7}-\frac {16 \left (\cosh ^{2}\left (b x +a \right )\right ) \sinh \left (b x +a \right )}{21}}{\cosh \left (b x +a \right ) \sqrt {\sinh \left (b x +a \right )}\, b}\)	\(116\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(b*x+a)^(7/2),x,method=_RETURNVERBOSE)

[Out]

(5/21*I*(1-I*sinh(b*x+a))^(1/2)*2^(1/2)*(1+I*sinh(b*x+a))^(1/2)*(I*sinh(b*x+a))^(1/2)*EllipticF((1-I*sinh(b*x+
a))^(1/2),1/2*2^(1/2))+2/7*cosh(b*x+a)^4*sinh(b*x+a)-16/21*cosh(b*x+a)^2*sinh(b*x+a))/cosh(b*x+a)/sinh(b*x+a)^
(1/2)/b

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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(b*x+a)^(7/2),x, algorithm="maxima")

[Out]

integrate(sinh(b*x + a)^(7/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.08, size = 326, normalized size = 3.17 \begin {gather*} \frac {40 \, {\left (\sqrt {2} \cosh \left (b x + a\right )^{3} + 3 \, \sqrt {2} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right ) + 3 \, \sqrt {2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sqrt {2} \sinh \left (b x + a\right )^{3}\right )} {\rm weierstrassPInverse}\left (4, 0, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + {\left (3 \, \cosh \left (b x + a\right )^{6} + 18 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + 3 \, \sinh \left (b x + a\right )^{6} + {\left (45 \, \cosh \left (b x + a\right )^{2} - 23\right )} \sinh \left (b x + a\right )^{4} - 23 \, \cosh \left (b x + a\right )^{4} + 4 \, {\left (15 \, \cosh \left (b x + a\right )^{3} - 23 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + {\left (45 \, \cosh \left (b x + a\right )^{4} - 138 \, \cosh \left (b x + a\right )^{2} - 23\right )} \sinh \left (b x + a\right )^{2} - 23 \, \cosh \left (b x + a\right )^{2} + 2 \, {\left (9 \, \cosh \left (b x + a\right )^{5} - 46 \, \cosh \left (b x + a\right )^{3} - 23 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 3\right )} \sqrt {\sinh \left (b x + a\right )}}{84 \, {\left (b \cosh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right ) + 3 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b \sinh \left (b x + a\right )^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(b*x+a)^(7/2),x, algorithm="fricas")

[Out]

1/84*(40*(sqrt(2)*cosh(b*x + a)^3 + 3*sqrt(2)*cosh(b*x + a)^2*sinh(b*x + a) + 3*sqrt(2)*cosh(b*x + a)*sinh(b*x
+ a)^2 + sqrt(2)*sinh(b*x + a)^3)*weierstrassPInverse(4, 0, cosh(b*x + a) + sinh(b*x + a)) + (3*cosh(b*x + a)
^6 + 18*cosh(b*x + a)*sinh(b*x + a)^5 + 3*sinh(b*x + a)^6 + (45*cosh(b*x + a)^2 - 23)*sinh(b*x + a)^4 - 23*cos
h(b*x + a)^4 + 4*(15*cosh(b*x + a)^3 - 23*cosh(b*x + a))*sinh(b*x + a)^3 + (45*cosh(b*x + a)^4 - 138*cosh(b*x
+ a)^2 - 23)*sinh(b*x + a)^2 - 23*cosh(b*x + a)^2 + 2*(9*cosh(b*x + a)^5 - 46*cosh(b*x + a)^3 - 23*cosh(b*x +
a))*sinh(b*x + a) + 3)*sqrt(sinh(b*x + a)))/(b*cosh(b*x + a)^3 + 3*b*cosh(b*x + a)^2*sinh(b*x + a) + 3*b*cosh(
b*x + a)*sinh(b*x + a)^2 + b*sinh(b*x + a)^3)

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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(sinh(b*x+a)**(7/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3060 deep

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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(sinh(b*x+a)^(7/2),x, algorithm="giac")

[Out]

integrate(sinh(b*x + a)^(7/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {sinh}\left (a+b\,x\right )}^{7/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(a + b*x)^(7/2),x)

[Out]

int(sinh(a + b*x)^(7/2), x)

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