3.7 \(\int \cosh ^{\frac {7}{2}}(a+b x) \, dx\)

Optimal. Leaf size=69 \[ -\frac {10 i F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{21 b}+\frac {2 \sinh (a+b x) \cosh ^{\frac {5}{2}}(a+b x)}{7 b}+\frac {10 \sinh (a+b x) \sqrt {\cosh (a+b x)}}{21 b} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2635, 2641} \[ -\frac {10 i F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{21 b}+\frac {2 \sinh (a+b x) \cosh ^{\frac {5}{2}}(a+b x)}{7 b}+\frac {10 \sinh (a+b x) \sqrt {\cosh (a+b x)}}{21 b} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x]^(7/2),x]

[Out]

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

Rule 2635

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] && Integer
Q[2*n]

Rule 2641

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

Rubi steps

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

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Mathematica [A]  time = 0.12, size = 55, normalized size = 0.80 \[ \frac {(23 \sinh (a+b x)+3 \sinh (3 (a+b x))) \sqrt {\cosh (a+b x)}-20 i F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{42 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + b*x]^(7/2),x]

[Out]

((-20*I)*EllipticF[(I/2)*(a + b*x), 2] + Sqrt[Cosh[a + b*x]]*(23*Sinh[a + b*x] + 3*Sinh[3*(a + b*x)]))/(42*b)

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fricas [F]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\cosh \left (b x + a\right )^{\frac {7}{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(cosh(b*x + a)^(7/2), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cosh \left (b x + a\right )^{\frac {7}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(cosh(b*x + a)^(7/2), x)

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maple [B]  time = 0.35, size = 201, normalized size = 2.91 \[ \frac {2 \sqrt {\left (2 \left (\cosh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \left (48 \left (\cosh ^{9}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-120 \left (\cosh ^{7}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+128 \left (\cosh ^{5}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-72 \left (\cosh ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+5 \sqrt {-\left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \sqrt {-2 \left (\cosh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+1}\, \EllipticF \left (\cosh \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )+16 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{21 \sqrt {2 \left (\sinh ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}\, \sinh \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {2 \left (\cosh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(b*x+a)^(7/2),x)

[Out]

2/21*((2*cosh(1/2*b*x+1/2*a)^2-1)*sinh(1/2*b*x+1/2*a)^2)^(1/2)*(48*cosh(1/2*b*x+1/2*a)^9-120*cosh(1/2*b*x+1/2*
a)^7+128*cosh(1/2*b*x+1/2*a)^5-72*cosh(1/2*b*x+1/2*a)^3+5*(-sinh(1/2*b*x+1/2*a)^2)^(1/2)*(-2*cosh(1/2*b*x+1/2*
a)^2+1)^(1/2)*EllipticF(cosh(1/2*b*x+1/2*a),2^(1/2))+16*cosh(1/2*b*x+1/2*a))/(2*sinh(1/2*b*x+1/2*a)^4+sinh(1/2
*b*x+1/2*a)^2)^(1/2)/sinh(1/2*b*x+1/2*a)/(2*cosh(1/2*b*x+1/2*a)^2-1)^(1/2)/b

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cosh \left (b x + a\right )^{\frac {7}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(cosh(b*x + a)^(7/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {cosh}\left (a+b\,x\right )}^{7/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*x)^(7/2),x)

[Out]

int(cosh(a + b*x)^(7/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)**(7/2),x)

[Out]

Timed out

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