3.530 \(\int x \sqrt {\cosh (a+b x)} \sinh (a+b x) \, dx\)

Optimal. Leaf size=64 \[ \frac {4 i F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{9 b^2}-\frac {4 \sinh (a+b x) \sqrt {\cosh (a+b x)}}{9 b^2}+\frac {2 x \cosh ^{\frac {3}{2}}(a+b x)}{3 b} \]

[Out]

2/3*x*cosh(b*x+a)^(3/2)/b+4/9*I*(cosh(1/2*a+1/2*b*x)^2)^(1/2)/cosh(1/2*a+1/2*b*x)*EllipticF(I*sinh(1/2*a+1/2*b
*x),2^(1/2))/b^2-4/9*sinh(b*x+a)*cosh(b*x+a)^(1/2)/b^2

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Rubi [A]  time = 0.04, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5373, 2635, 2641} \[ \frac {4 i F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{9 b^2}-\frac {4 \sinh (a+b x) \sqrt {\cosh (a+b x)}}{9 b^2}+\frac {2 x \cosh ^{\frac {3}{2}}(a+b x)}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[Cosh[a + b*x]]*Sinh[a + b*x],x]

[Out]

(2*x*Cosh[a + b*x]^(3/2))/(3*b) + (((4*I)/9)*EllipticF[(I/2)*(a + b*x), 2])/b^2 - (4*Sqrt[Cosh[a + b*x]]*Sinh[
a + b*x])/(9*b^2)

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 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]

Rule 5373

Int[Cosh[(a_.) + (b_.)*(x_)^(n_.)]^(p_.)*(x_)^(m_.)*Sinh[(a_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Simp[(x^(m -
n + 1)*Cosh[a + b*x^n]^(p + 1))/(b*n*(p + 1)), x] - Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*Cosh[a + b*x
^n]^(p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]

Rubi steps

\begin {align*} \int x \sqrt {\cosh (a+b x)} \sinh (a+b x) \, dx &=\frac {2 x \cosh ^{\frac {3}{2}}(a+b x)}{3 b}-\frac {2 \int \cosh ^{\frac {3}{2}}(a+b x) \, dx}{3 b}\\ &=\frac {2 x \cosh ^{\frac {3}{2}}(a+b x)}{3 b}-\frac {4 \sqrt {\cosh (a+b x)} \sinh (a+b x)}{9 b^2}-\frac {2 \int \frac {1}{\sqrt {\cosh (a+b x)}} \, dx}{9 b}\\ &=\frac {2 x \cosh ^{\frac {3}{2}}(a+b x)}{3 b}+\frac {4 i F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{9 b^2}-\frac {4 \sqrt {\cosh (a+b x)} \sinh (a+b x)}{9 b^2}\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 56, normalized size = 0.88 \[ \frac {2 \sqrt {\cosh (a+b x)} (3 b x \cosh (a+b x)-2 \sinh (a+b x))+4 i F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{9 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Sqrt[Cosh[a + b*x]]*Sinh[a + b*x],x]

[Out]

((4*I)*EllipticF[(I/2)*(a + b*x), 2] + 2*Sqrt[Cosh[a + b*x]]*(3*b*x*Cosh[a + b*x] - 2*Sinh[a + b*x]))/(9*b^2)

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sinh(b*x+a)*cosh(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sqrt {\cosh \left (b x + a\right )} \sinh \left (b x + a\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sinh(b*x+a)*cosh(b*x+a)^(1/2),x, algorithm="giac")

[Out]

integrate(x*sqrt(cosh(b*x + a))*sinh(b*x + a), x)

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maple [F]  time = 0.11, size = 0, normalized size = 0.00 \[ \int x \sinh \left (b x +a \right ) \left (\sqrt {\cosh }\left (b x +a \right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*sinh(b*x+a)*cosh(b*x+a)^(1/2),x)

[Out]

int(x*sinh(b*x+a)*cosh(b*x+a)^(1/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sqrt {\cosh \left (b x + a\right )} \sinh \left (b x + a\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sinh(b*x+a)*cosh(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

integrate(x*sqrt(cosh(b*x + a))*sinh(b*x + a), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,\sqrt {\mathrm {cosh}\left (a+b\,x\right )}\,\mathrm {sinh}\left (a+b\,x\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*cosh(a + b*x)^(1/2)*sinh(a + b*x),x)

[Out]

int(x*cosh(a + b*x)^(1/2)*sinh(a + b*x), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sinh {\left (a + b x \right )} \sqrt {\cosh {\left (a + b x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sinh(b*x+a)*cosh(b*x+a)**(1/2),x)

[Out]

Integral(x*sinh(a + b*x)*sqrt(cosh(a + b*x)), x)

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