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*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]])

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Rubi [A]  time = 0.03007, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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*}

Mathematica [A]  time = 0.127899, 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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Maple [B]  time = 0.068, size = 363, normalized size = 5.3 \begin{align*}{\frac{2}{5\,b}\sqrt{ \left ( 2\, \left ( \cosh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) \left ( \sinh \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}} \left ( 12\,\sqrt{-2\, \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}\sqrt{- \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}{\it EllipticE} \left ( \cosh \left ( 1/2\,bx+a/2 \right ) ,\sqrt{2} \right ) \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}+24\, \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{6}\cosh \left ( 1/2\,bx+a/2 \right ) +12\,\sqrt{-2\, \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}\sqrt{- \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}{\it EllipticE} \left ( \cosh \left ( 1/2\,bx+a/2 \right ) ,\sqrt{2} \right ) \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}+24\, \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}\cosh \left ( 1/2\,bx+a/2 \right ) +3\,\sqrt{-2\, \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}\sqrt{- \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}{\it EllipticE} \left ( \cosh \left ( 1/2\,bx+a/2 \right ) ,\sqrt{2} \right ) +8\,\cosh \left ( 1/2\,bx+a/2 \right ) \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2} \right ) \sqrt{2\, \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}+ \left ( \sinh \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}} \left ( 8\, \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{6}+12\, \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}+6\, \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}+1 \right ) ^{-1} \left ( \sinh \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{-3}{\frac{1}{\sqrt{2\, \left ( \cosh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}}}} \end{align*}

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*(-2*sinh(1/2*b*x+1/2*a)^2-1)^(1/2)*(-sinh(1/2*b*x+1/2*a
)^2)^(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*(-2*sinh(1/2*b*x+1/2*a)^2-1)^(1/2)*(-sinh(1/2*b*x+1/2*a)^2)^(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)*(
-sinh(1/2*b*x+1/2*a)^2)^(1/2)*EllipticE(cosh(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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\cosh \left (b x + a\right )^{\frac{7}{2}}}\,{d x} \end{align*}

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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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{\cosh \left (b x + a\right )^{\frac{7}{2}}}, x\right ) \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\cosh \left (b x + a\right )^{\frac{7}{2}}}\,{d x} \end{align*}

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)