3.93 \(\int (1+\sinh ^2(x))^{3/2} \, dx\)

Optimal. Leaf size=29 \[ \frac{1}{3} \cosh ^2(x)^{3/2} \tanh (x)+\frac{2}{3} \sqrt{\cosh ^2(x)} \tanh (x) \]

[Out]

(2*Sqrt[Cosh[x]^2]*Tanh[x])/3 + ((Cosh[x]^2)^(3/2)*Tanh[x])/3

________________________________________________________________________________________

Rubi [A]  time = 0.026434, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {3176, 3203, 3207, 2637} \[ \frac{1}{3} \cosh ^2(x)^{3/2} \tanh (x)+\frac{2}{3} \sqrt{\cosh ^2(x)} \tanh (x) \]

Antiderivative was successfully verified.

[In]

Int[(1 + Sinh[x]^2)^(3/2),x]

[Out]

(2*Sqrt[Cosh[x]^2]*Tanh[x])/3 + ((Cosh[x]^2)^(3/2)*Tanh[x])/3

Rule 3176

Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(a*cos[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0]

Rule 3203

Int[((b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> -Simp[(Cot[e + f*x]*(b*Sin[e + f*x]^2)^p)/(2*f*p), x]
+ Dist[(b*(2*p - 1))/(2*p), Int[(b*Sin[e + f*x]^2)^(p - 1), x], x] /; FreeQ[{b, e, f}, x] &&  !IntegerQ[p] &&
GtQ[p, 1]

Rule 3207

Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Di
st[((b*ff^n)^IntPart[p]*(b*Sin[e + f*x]^n)^FracPart[p])/(Sin[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]
*(Sin[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \left (1+\sinh ^2(x)\right )^{3/2} \, dx &=\int \cosh ^2(x)^{3/2} \, dx\\ &=\frac{1}{3} \cosh ^2(x)^{3/2} \tanh (x)+\frac{2}{3} \int \sqrt{\cosh ^2(x)} \, dx\\ &=\frac{1}{3} \cosh ^2(x)^{3/2} \tanh (x)+\frac{1}{3} \left (2 \sqrt{\cosh ^2(x)} \text{sech}(x)\right ) \int \cosh (x) \, dx\\ &=\frac{2}{3} \sqrt{\cosh ^2(x)} \tanh (x)+\frac{1}{3} \cosh ^2(x)^{3/2} \tanh (x)\\ \end{align*}

Mathematica [A]  time = 0.0203646, size = 23, normalized size = 0.79 \[ \frac{1}{12} (9 \sinh (x)+\sinh (3 x)) \sqrt{\cosh ^2(x)} \text{sech}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Sinh[x]^2)^(3/2),x]

[Out]

(Sqrt[Cosh[x]^2]*Sech[x]*(9*Sinh[x] + Sinh[3*x]))/12

________________________________________________________________________________________

Maple [A]  time = 0.041, size = 21, normalized size = 0.7 \begin{align*}{\frac{\sinh \left ( x \right ) \left ( \left ( \sinh \left ( x \right ) \right ) ^{2}+3 \right ) }{3\,\cosh \left ( x \right ) }\sqrt{ \left ( \cosh \left ( x \right ) \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+sinh(x)^2)^(3/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.5582, size = 31, normalized size = 1.07 \begin{align*} \frac{1}{24} \, e^{\left (3 \, x\right )} - \frac{3}{8} \, e^{\left (-x\right )} - \frac{1}{24} \, e^{\left (-3 \, x\right )} + \frac{3}{8} \, e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+sinh(x)^2)^(3/2),x, algorithm="maxima")

[Out]

1/24*e^(3*x) - 3/8*e^(-x) - 1/24*e^(-3*x) + 3/8*e^x

________________________________________________________________________________________

Fricas [A]  time = 1.78809, size = 62, normalized size = 2.14 \begin{align*} \frac{1}{12} \, \sinh \left (x\right )^{3} + \frac{1}{4} \,{\left (\cosh \left (x\right )^{2} + 3\right )} \sinh \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+sinh(x)^2)^(3/2),x, algorithm="fricas")

[Out]

1/12*sinh(x)^3 + 1/4*(cosh(x)^2 + 3)*sinh(x)

________________________________________________________________________________________

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+sinh(x)**2)**(3/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.22696, size = 34, normalized size = 1.17 \begin{align*} -\frac{1}{24} \,{\left (9 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-3 \, x\right )} + \frac{1}{24} \, e^{\left (3 \, x\right )} + \frac{3}{8} \, e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+sinh(x)^2)^(3/2),x, algorithm="giac")

[Out]

-1/24*(9*e^(2*x) + 1)*e^(-3*x) + 1/24*e^(3*x) + 3/8*e^x