3.22 \(\int \frac{1}{(b \sinh (c+d x))^{7/2}} \, dx\)

Optimal. Leaf size=118 \[ \frac{6 \cosh (c+d x)}{5 b^3 d \sqrt{b \sinh (c+d x)}}+\frac{6 i E\left (\left .\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{b \sinh (c+d x)}}{5 b^4 d \sqrt{i \sinh (c+d x)}}-\frac{2 \cosh (c+d x)}{5 b d (b \sinh (c+d x))^{5/2}} \]

[Out]

(-2*Cosh[c + d*x])/(5*b*d*(b*Sinh[c + d*x])^(5/2)) + (6*Cosh[c + d*x])/(5*b^3*d*Sqrt[b*Sinh[c + d*x]]) + (((6*
I)/5)*EllipticE[(I*c - Pi/2 + I*d*x)/2, 2]*Sqrt[b*Sinh[c + d*x]])/(b^4*d*Sqrt[I*Sinh[c + d*x]])

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Rubi [A]  time = 0.0560453, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2636, 2640, 2639} \[ \frac{6 \cosh (c+d x)}{5 b^3 d \sqrt{b \sinh (c+d x)}}+\frac{6 i E\left (\left .\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{b \sinh (c+d x)}}{5 b^4 d \sqrt{i \sinh (c+d x)}}-\frac{2 \cosh (c+d x)}{5 b d (b \sinh (c+d x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(b*Sinh[c + d*x])^(-7/2),x]

[Out]

(-2*Cosh[c + d*x])/(5*b*d*(b*Sinh[c + d*x])^(5/2)) + (6*Cosh[c + d*x])/(5*b^3*d*Sqrt[b*Sinh[c + d*x]]) + (((6*
I)/5)*EllipticE[(I*c - Pi/2 + I*d*x)/2, 2]*Sqrt[b*Sinh[c + d*x]])/(b^4*d*Sqrt[I*Sinh[c + d*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 2640

Int[Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[b*Sin[c + d*x]]/Sqrt[Sin[c + d*x]], Int[Sqrt[Si
n[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

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}{(b \sinh (c+d x))^{7/2}} \, dx &=-\frac{2 \cosh (c+d x)}{5 b d (b \sinh (c+d x))^{5/2}}-\frac{3 \int \frac{1}{(b \sinh (c+d x))^{3/2}} \, dx}{5 b^2}\\ &=-\frac{2 \cosh (c+d x)}{5 b d (b \sinh (c+d x))^{5/2}}+\frac{6 \cosh (c+d x)}{5 b^3 d \sqrt{b \sinh (c+d x)}}-\frac{3 \int \sqrt{b \sinh (c+d x)} \, dx}{5 b^4}\\ &=-\frac{2 \cosh (c+d x)}{5 b d (b \sinh (c+d x))^{5/2}}+\frac{6 \cosh (c+d x)}{5 b^3 d \sqrt{b \sinh (c+d x)}}-\frac{\left (3 \sqrt{b \sinh (c+d x)}\right ) \int \sqrt{i \sinh (c+d x)} \, dx}{5 b^4 \sqrt{i \sinh (c+d x)}}\\ &=-\frac{2 \cosh (c+d x)}{5 b d (b \sinh (c+d x))^{5/2}}+\frac{6 \cosh (c+d x)}{5 b^3 d \sqrt{b \sinh (c+d x)}}+\frac{6 i E\left (\left .\frac{1}{2} \left (i c-\frac{\pi }{2}+i d x\right )\right |2\right ) \sqrt{b \sinh (c+d x)}}{5 b^4 d \sqrt{i \sinh (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.163265, size = 79, normalized size = 0.67 \[ -\frac{2 \left (-3 \cosh (c+d x)+\coth (c+d x) \text{csch}(c+d x)+3 \sqrt{i \sinh (c+d x)} E\left (\left .\frac{1}{4} (-2 i c-2 i d x+\pi )\right |2\right )\right )}{5 b^3 d \sqrt{b \sinh (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Sinh[c + d*x])^(-7/2),x]

[Out]

(-2*(-3*Cosh[c + d*x] + Coth[c + d*x]*Csch[c + d*x] + 3*EllipticE[((-2*I)*c + Pi - (2*I)*d*x)/4, 2]*Sqrt[I*Sin
h[c + d*x]]))/(5*b^3*d*Sqrt[b*Sinh[c + d*x]])

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Maple [A]  time = 0.049, size = 205, normalized size = 1.7 \begin{align*} -{\frac{1}{5\,{b}^{3} \left ( \sinh \left ( dx+c \right ) \right ) ^{2}\cosh \left ( dx+c \right ) d} \left ( 6\,\sqrt{-i \left ( \sinh \left ( dx+c \right ) +i \right ) }\sqrt{2}\sqrt{-i \left ( i-\sinh \left ( dx+c \right ) \right ) }\sqrt{i\sinh \left ( dx+c \right ) } \left ( \sinh \left ( dx+c \right ) \right ) ^{2}{\it EllipticE} \left ( \sqrt{-i \left ( \sinh \left ( dx+c \right ) +i \right ) },1/2\,\sqrt{2} \right ) -3\,\sqrt{-i \left ( \sinh \left ( dx+c \right ) +i \right ) }\sqrt{2}\sqrt{-i \left ( i-\sinh \left ( dx+c \right ) \right ) }\sqrt{i\sinh \left ( dx+c \right ) } \left ( \sinh \left ( dx+c \right ) \right ) ^{2}{\it EllipticF} \left ( \sqrt{-i \left ( \sinh \left ( dx+c \right ) +i \right ) },1/2\,\sqrt{2} \right ) -6\, \left ( \sinh \left ( dx+c \right ) \right ) ^{4}-4\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}+2 \right ){\frac{1}{\sqrt{b\sinh \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*sinh(d*x+c))^(7/2),x)

[Out]

-1/5/b^3/sinh(d*x+c)^2*(6*(-I*(sinh(d*x+c)+I))^(1/2)*2^(1/2)*(-I*(I-sinh(d*x+c)))^(1/2)*(I*sinh(d*x+c))^(1/2)*
sinh(d*x+c)^2*EllipticE((-I*(sinh(d*x+c)+I))^(1/2),1/2*2^(1/2))-3*(-I*(sinh(d*x+c)+I))^(1/2)*2^(1/2)*(-I*(I-si
nh(d*x+c)))^(1/2)*(I*sinh(d*x+c))^(1/2)*sinh(d*x+c)^2*EllipticF((-I*(sinh(d*x+c)+I))^(1/2),1/2*2^(1/2))-6*sinh
(d*x+c)^4-4*sinh(d*x+c)^2+2)/cosh(d*x+c)/(b*sinh(d*x+c))^(1/2)/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*sinh(d*x+c))^(7/2),x, algorithm="maxima")

[Out]

integrate((b*sinh(d*x + c))^(-7/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b \sinh \left (d x + c\right )}}{b^{4} \sinh \left (d x + c\right )^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*sinh(d*x+c))^(7/2),x, algorithm="fricas")

[Out]

integral(sqrt(b*sinh(d*x + c))/(b^4*sinh(d*x + c)^4), 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/(b*sinh(d*x+c))**(7/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*sinh(d*x+c))^(7/2),x, algorithm="giac")

[Out]

integrate((b*sinh(d*x + c))^(-7/2), x)