3.14 \(\int \frac {1}{\cosh ^{\frac {7}{2}}(a+b x)} \, dx\)

Optimal. Leaf size=69 \[ \frac {6 i E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{5 b}+\frac {2 \sinh (a+b x)}{5 b \cosh ^{\frac {5}{2}}(a+b x)}+\frac {6 \sinh (a+b x)}{5 b \sqrt {\cosh (a+b x)}} \]

[Out]

6/5*I*(cosh(1/2*a+1/2*b*x)^2)^(1/2)/cosh(1/2*a+1/2*b*x)*EllipticE(I*sinh(1/2*a+1/2*b*x),2^(1/2))/b+2/5*sinh(b*
x+a)/b/cosh(b*x+a)^(5/2)+6/5*sinh(b*x+a)/b/cosh(b*x+a)^(1/2)

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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 = {2636, 2639} \[ \frac {6 i E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{5 b}+\frac {2 \sinh (a+b x)}{5 b \cosh ^{\frac {5}{2}}(a+b x)}+\frac {6 \sinh (a+b x)}{5 b \sqrt {\cosh (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x]^(-7/2),x]

[Out]

(((6*I)/5)*EllipticE[(I/2)*(a + b*x), 2])/b + (2*Sinh[a + b*x])/(5*b*Cosh[a + b*x]^(5/2)) + (6*Sinh[a + b*x])/
(5*b*Sqrt[Cosh[a + b*x]])

Rule 2636

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

Rule 2639

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

Rubi steps

\begin {align*} \int \frac {1}{\cosh ^{\frac {7}{2}}(a+b x)} \, dx &=\frac {2 \sinh (a+b x)}{5 b \cosh ^{\frac {5}{2}}(a+b x)}+\frac {3}{5} \int \frac {1}{\cosh ^{\frac {3}{2}}(a+b x)} \, dx\\ &=\frac {2 \sinh (a+b x)}{5 b \cosh ^{\frac {5}{2}}(a+b x)}+\frac {6 \sinh (a+b x)}{5 b \sqrt {\cosh (a+b x)}}-\frac {3}{5} \int \sqrt {\cosh (a+b x)} \, dx\\ &=\frac {6 i E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{5 b}+\frac {2 \sinh (a+b x)}{5 b \cosh ^{\frac {5}{2}}(a+b x)}+\frac {6 \sinh (a+b x)}{5 b \sqrt {\cosh (a+b x)}}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 63, normalized size = 0.91 \[ \frac {3 \sinh (2 (a+b x))+2 \tanh (a+b x)+6 i \cosh ^{\frac {3}{2}}(a+b x) E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{5 b \cosh ^{\frac {3}{2}}(a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + b*x]^(-7/2),x]

[Out]

((6*I)*Cosh[a + b*x]^(3/2)*EllipticE[(I/2)*(a + b*x), 2] + 3*Sinh[2*(a + b*x)] + 2*Tanh[a + b*x])/(5*b*Cosh[a
+ b*x]^(3/2))

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fricas [F]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\cosh \left (b x + a\right )^{\frac {7}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/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 \[ \int \frac {1}{\cosh \left (b x + a\right )^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/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 = 0.63, size = 363, normalized size = 5.26 \[ \frac {2 \sqrt {\left (2 \left (\cosh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \left (12 \sqrt {-\left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \sqrt {-2 \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticE \left (\cosh \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \left (\sinh ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+24 \left (\sinh ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )+12 \sqrt {-\left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \sqrt {-2 \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticE \left (\cosh \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+24 \left (\sinh ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )+3 \sqrt {-2 \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticE \left (\cosh \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \sqrt {-\left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}+8 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )\right ) \sqrt {2 \left (\sinh ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}}{5 \left (8 \left (\sinh ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+12 \left (\sinh ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+6 \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+1\right ) \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{3} \sqrt {2 \left (\cosh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/cosh(b*x+a)^(7/2),x)

[Out]

2/5*((2*cosh(1/2*b*x+1/2*a)^2-1)*sinh(1/2*b*x+1/2*a)^2)^(1/2)/(8*sinh(1/2*b*x+1/2*a)^6+12*sinh(1/2*b*x+1/2*a)^
4+6*sinh(1/2*b*x+1/2*a)^2+1)/sinh(1/2*b*x+1/2*a)^3*(12*(-sinh(1/2*b*x+1/2*a)^2)^(1/2)*(-2*sinh(1/2*b*x+1/2*a)^
2-1)^(1/2)*EllipticE(cosh(1/2*b*x+1/2*a),2^(1/2))*sinh(1/2*b*x+1/2*a)^4+24*sinh(1/2*b*x+1/2*a)^6*cosh(1/2*b*x+
1/2*a)+12*(-sinh(1/2*b*x+1/2*a)^2)^(1/2)*(-2*sinh(1/2*b*x+1/2*a)^2-1)^(1/2)*EllipticE(cosh(1/2*b*x+1/2*a),2^(1
/2))*sinh(1/2*b*x+1/2*a)^2+24*sinh(1/2*b*x+1/2*a)^4*cosh(1/2*b*x+1/2*a)+3*(-2*sinh(1/2*b*x+1/2*a)^2-1)^(1/2)*E
llipticE(cosh(1/2*b*x+1/2*a),2^(1/2))*(-sinh(1/2*b*x+1/2*a)^2)^(1/2)+8*cosh(1/2*b*x+1/2*a)*sinh(1/2*b*x+1/2*a)
^2)*(2*sinh(1/2*b*x+1/2*a)^4+sinh(1/2*b*x+1/2*a)^2)^(1/2)/(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 \[ \int \frac {1}{\cosh \left (b x + a\right )^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/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 \[ \int \frac {1}{{\mathrm {cosh}\left (a+b\,x\right )}^{7/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/cosh(a + b*x)^(7/2),x)

[Out]

int(1/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(1/cosh(b*x+a)**(7/2),x)

[Out]

Timed out

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