3.1.43 \(\int (a+a \cosh (c+d x))^{3/2} \, dx\) [43]

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

[Out]

8/3*a^2*sinh(d*x+c)/d/(a+a*cosh(d*x+c))^(1/2)+2/3*a*sinh(d*x+c)*(a+a*cosh(d*x+c))^(1/2)/d

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Rubi [A]
time = 0.02, antiderivative size = 59, normalized size of antiderivative = 1.00, 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 = {2726, 2725} \begin {gather*} \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} \end {gather*}

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 2725

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]

Rule 2726

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] &&
EqQ[a^2 - b^2, 0] && IGtQ[n - 1/2, 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*}

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Mathematica [A]
time = 0.05, size = 55, normalized size = 0.93 \begin {gather*} \frac {a \sqrt {a (1+\cosh (c+d x))} \text {sech}\left (\frac {1}{2} (c+d x)\right ) \left (9 \sinh \left (\frac {1}{2} (c+d x)\right )+\sinh \left (\frac {3}{2} (c+d x)\right )\right )}{3 d} \end {gather*}

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 = 0.94, size = 58, normalized size = 0.98

method	result	size
default	\(\frac {4 a^{2} \cosh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sinh \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\cosh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+2\right ) \sqrt {2}}{3 \sqrt {a \left (\cosh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.47, size = 81, normalized size = 1.37 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (51) = 102\).
time = 0.36, size = 140, normalized size = 2.37 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a \cosh {\left (c + d x \right )} + a\right )^{\frac {3}{2}}\, dx \end {gather*}

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 = 0.43, size = 75, normalized size = 1.27 \begin {gather*} -\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 {gather*}

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (a+a\,\mathrm {cosh}\left (c+d\,x\right )\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*cosh(c + d*x))^(3/2),x)

[Out]

int((a + a*cosh(c + d*x))^(3/2), x)

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