3.43 \(\int (a+a \cosh (c+d x))^{3/2} \, dx\)

Optimal. Leaf size=59 \[ \frac{8 a^2 \sinh (c+d x)}{3 d \sqrt{a \cosh (c+d x)+a}}+\frac{2 a \sinh (c+d x) \sqrt{a \cosh (c+d x)+a}}{3 d} \]

[Out]

(8*a^2*Sinh[c + d*x])/(3*d*Sqrt[a + a*Cosh[c + d*x]]) + (2*a*Sqrt[a + a*Cosh[c + d*x]]*Sinh[c + d*x])/(3*d)

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Rubi [A]  time = 0.0287345, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2647, 2646} \[ \frac{8 a^2 \sinh (c+d x)}{3 d \sqrt{a \cosh (c+d x)+a}}+\frac{2 a \sinh (c+d x) \sqrt{a \cosh (c+d x)+a}}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Cosh[c + d*x])^(3/2),x]

[Out]

(8*a^2*Sinh[c + d*x])/(3*d*Sqrt[a + a*Cosh[c + d*x]]) + (2*a*Sqrt[a + a*Cosh[c + d*x]]*Sinh[c + d*x])/(3*d)

Rule 2647

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(a + b*Sin[c + d*x])^(n -
1))/(d*n), x] + Dist[(a*(2*n - 1))/n, Int[(a + b*Sin[c + d*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && Eq
Q[a^2 - b^2, 0] && IGtQ[n - 1/2, 0]

Rule 2646

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(-2*b*Cos[c + d*x])/(d*Sqrt[a + b*Sin[c + d*
x]]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

\begin{align*} \int (a+a \cosh (c+d x))^{3/2} \, dx &=\frac{2 a \sqrt{a+a \cosh (c+d x)} \sinh (c+d x)}{3 d}+\frac{1}{3} (4 a) \int \sqrt{a+a \cosh (c+d x)} \, dx\\ &=\frac{8 a^2 \sinh (c+d x)}{3 d \sqrt{a+a \cosh (c+d x)}}+\frac{2 a \sqrt{a+a \cosh (c+d x)} \sinh (c+d x)}{3 d}\\ \end{align*}

Mathematica [A]  time = 0.0633739, size = 55, normalized size = 0.93 \[ \frac{a \left (9 \sinh \left (\frac{1}{2} (c+d x)\right )+\sinh \left (\frac{3}{2} (c+d x)\right )\right ) \text{sech}\left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cosh (c+d x)+1)}}{3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Cosh[c + d*x])^(3/2),x]

[Out]

(a*Sqrt[a*(1 + Cosh[c + d*x])]*Sech[(c + d*x)/2]*(9*Sinh[(c + d*x)/2] + Sinh[(3*(c + d*x))/2]))/(3*d)

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Maple [A]  time = 0.03, size = 58, normalized size = 1. \begin{align*}{\frac{4\,{a}^{2}\sqrt{2}}{3\,d}\cosh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sinh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( \left ( \cosh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+2 \right ){\frac{1}{\sqrt{a \left ( \cosh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*cosh(d*x+c))^(3/2),x)

[Out]

4/3*a^2*cosh(1/2*d*x+1/2*c)*sinh(1/2*d*x+1/2*c)*(cosh(1/2*d*x+1/2*c)^2+2)*2^(1/2)/(a*cosh(1/2*d*x+1/2*c)^2)^(1
/2)/d

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Maxima [A]  time = 1.55873, size = 109, normalized size = 1.85 \begin{align*} \frac{\sqrt{2} a^{\frac{3}{2}} e^{\left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right )}}{6 \, d} + \frac{3 \, \sqrt{2} a^{\frac{3}{2}} e^{\left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}}{2 \, d} - \frac{3 \, \sqrt{2} a^{\frac{3}{2}} e^{\left (-\frac{1}{2} \, d x - \frac{1}{2} \, c\right )}}{2 \, d} - \frac{\sqrt{2} a^{\frac{3}{2}} e^{\left (-\frac{3}{2} \, d x - \frac{3}{2} \, c\right )}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cosh(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

1/6*sqrt(2)*a^(3/2)*e^(3/2*d*x + 3/2*c)/d + 3/2*sqrt(2)*a^(3/2)*e^(1/2*d*x + 1/2*c)/d - 3/2*sqrt(2)*a^(3/2)*e^
(-1/2*d*x - 1/2*c)/d - 1/6*sqrt(2)*a^(3/2)*e^(-3/2*d*x - 3/2*c)/d

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Fricas [B]  time = 1.85382, size = 385, normalized size = 6.53 \begin{align*} \frac{\sqrt{\frac{1}{2}}{\left (a \cosh \left (d x + c\right )^{3} + a \sinh \left (d x + c\right )^{3} + 9 \, a \cosh \left (d x + c\right )^{2} + 3 \,{\left (a \cosh \left (d x + c\right ) + 3 \, a\right )} \sinh \left (d x + c\right )^{2} - 9 \, a \cosh \left (d x + c\right ) + 3 \,{\left (a \cosh \left (d x + c\right )^{2} + 6 \, a \cosh \left (d x + c\right ) - 3 \, a\right )} \sinh \left (d x + c\right ) - a\right )} \sqrt{\frac{a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}}}{3 \,{\left (d \cosh \left (d x + c\right ) + d \sinh \left (d x + c\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cosh(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

1/3*sqrt(1/2)*(a*cosh(d*x + c)^3 + a*sinh(d*x + c)^3 + 9*a*cosh(d*x + c)^2 + 3*(a*cosh(d*x + c) + 3*a)*sinh(d*
x + c)^2 - 9*a*cosh(d*x + c) + 3*(a*cosh(d*x + c)^2 + 6*a*cosh(d*x + c) - 3*a)*sinh(d*x + c) - a)*sqrt(a/(cosh
(d*x + c) + sinh(d*x + c)))/(d*cosh(d*x + c) + d*sinh(d*x + c))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cosh(d*x+c))**(3/2),x)

[Out]

Integral((a*cosh(c + d*x) + a)**(3/2), x)

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Giac [A]  time = 1.11523, size = 101, normalized size = 1.71 \begin{align*} -\frac{\sqrt{2}{\left ({\left (9 \, a^{\frac{3}{2}} e^{\left (d x + \frac{3}{2} \, c\right )} + a^{\frac{3}{2}} e^{\left (\frac{1}{2} \, c\right )}\right )} e^{\left (-\frac{3}{2} \, d x - 2 \, c\right )} -{\left (a^{\frac{3}{2}} e^{\left (\frac{3}{2} \, d x + \frac{15}{2} \, c\right )} + 9 \, a^{\frac{3}{2}} e^{\left (\frac{1}{2} \, d x + \frac{13}{2} \, c\right )}\right )} e^{\left (-6 \, c\right )}\right )}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cosh(d*x+c))^(3/2),x, algorithm="giac")

[Out]

-1/6*sqrt(2)*((9*a^(3/2)*e^(d*x + 3/2*c) + a^(3/2)*e^(1/2*c))*e^(-3/2*d*x - 2*c) - (a^(3/2)*e^(3/2*d*x + 15/2*
c) + 9*a^(3/2)*e^(1/2*d*x + 13/2*c))*e^(-6*c))/d