3.9 \(\int \sinh ^{\frac{3}{2}}(a+b x) \, dx\)

Optimal. Leaf size=80 \[ \frac{2 \sqrt{\sinh (a+b x)} \cosh (a+b x)}{3 b}+\frac{2 i \sqrt{i \sinh (a+b x)} \text{EllipticF}\left (\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right ),2\right )}{3 b \sqrt{\sinh (a+b x)}} \]

[Out]

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

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Rubi [A]  time = 0.0318454, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {2635, 2642, 2641} \[ \frac{2 \sqrt{\sinh (a+b x)} \cosh (a+b x)}{3 b}+\frac{2 i \sqrt{i \sinh (a+b x)} F\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{3 b \sqrt{\sinh (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b*x]^(3/2),x]

[Out]

(((2*I)/3)*EllipticF[(I*a - Pi/2 + I*b*x)/2, 2]*Sqrt[I*Sinh[a + b*x]])/(b*Sqrt[Sinh[a + b*x]]) + (2*Cosh[a + b
*x]*Sqrt[Sinh[a + b*x]])/(3*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 2642

Int[1/Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[Sin[c + d*x]]/Sqrt[b*Sin[c + d*x]], Int[1/Sqr
t[Sin[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

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 \sinh ^{\frac{3}{2}}(a+b x) \, dx &=\frac{2 \cosh (a+b x) \sqrt{\sinh (a+b x)}}{3 b}-\frac{1}{3} \int \frac{1}{\sqrt{\sinh (a+b x)}} \, dx\\ &=\frac{2 \cosh (a+b x) \sqrt{\sinh (a+b x)}}{3 b}-\frac{\sqrt{i \sinh (a+b x)} \int \frac{1}{\sqrt{i \sinh (a+b x)}} \, dx}{3 \sqrt{\sinh (a+b x)}}\\ &=\frac{2 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)}}{3 b \sqrt{\sinh (a+b x)}}+\frac{2 \cosh (a+b x) \sqrt{\sinh (a+b x)}}{3 b}\\ \end{align*}

Mathematica [C]  time = 0.0872205, size = 83, normalized size = 1.04 \[ \frac{\sinh (2 (a+b x))-2 \sqrt{-\sinh (2 a+2 b x)-\cosh (2 a+2 b x)+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\cosh (2 (a+b x))+\sinh (2 (a+b x))\right )}{3 b \sqrt{\sinh (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[a + b*x]^(3/2),x]

[Out]

(Sinh[2*(a + b*x)] - 2*Hypergeometric2F1[1/4, 1/2, 5/4, Cosh[2*(a + b*x)] + Sinh[2*(a + b*x)]]*Sqrt[1 - Cosh[2
*a + 2*b*x] - Sinh[2*a + 2*b*x]])/(3*b*Sqrt[Sinh[a + b*x]])

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Maple [A]  time = 0.042, size = 100, normalized size = 1.3 \begin{align*}{\frac{1}{b\cosh \left ( bx+a \right ) } \left ( -{\frac{i}{3}}\sqrt{1-i\sinh \left ( bx+a \right ) }\sqrt{2}\sqrt{1+i\sinh \left ( bx+a \right ) }\sqrt{i\sinh \left ( bx+a \right ) }{\it EllipticF} \left ( \sqrt{1-i\sinh \left ( bx+a \right ) },{\frac{\sqrt{2}}{2}} \right ) +{\frac{2\,\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{3}} \right ){\frac{1}{\sqrt{\sinh \left ( bx+a \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(b*x+a)^(3/2),x)

[Out]

(-1/3*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))+2/3*sinh(b*x+a)*cosh(b*x+a)^2)/cosh(b*x+a)/sinh(b*x+a)^(1/2)/b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sinh \left (b x + a\right )^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(b*x+a)^(3/2),x, algorithm="maxima")

[Out]

integrate(sinh(b*x + a)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sinh \left (b x + a\right )^{\frac{3}{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(b*x+a)^(3/2),x, algorithm="fricas")

[Out]

integral(sinh(b*x + a)^(3/2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sinh ^{\frac{3}{2}}{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(b*x+a)**(3/2),x)

[Out]

Integral(sinh(a + b*x)**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sinh \left (b x + a\right )^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(b*x+a)^(3/2),x, algorithm="giac")

[Out]

integrate(sinh(b*x + a)^(3/2), x)