\(\int x \cosh (a+b x) \sinh ^{\frac {5}{2}}(a+b x) \, dx\) [544]

Optimal result

Integrand size = 18, antiderivative size = 121 \[ \int x \cosh (a+b x) \sinh ^{\frac {5}{2}}(a+b x) \, dx=\frac {20 i \operatorname {EllipticF}\left (\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right ),2\right ) \sqrt {i \sinh (a+b x)}}{147 b^2 \sqrt {\sinh (a+b x)}}+\frac {20 \cosh (a+b x) \sqrt {\sinh (a+b x)}}{147 b^2}-\frac {4 \cosh (a+b x) \sinh ^{\frac {5}{2}}(a+b x)}{49 b^2}+\frac {2 x \sinh ^{\frac {7}{2}}(a+b x)}{7 b} \]

[Out]

-4/49*cosh(b*x+a)*sinh(b*x+a)^(5/2)/b^2+2/7*x*sinh(b*x+a)^(7/2)/b-20/147*I*(sin(1/2*I*a+1/4*Pi+1/2*I*b*x)^2)^(
1/2)/sin(1/2*I*a+1/4*Pi+1/2*I*b*x)*EllipticF(cos(1/2*I*a+1/4*Pi+1/2*I*b*x),2^(1/2))*(I*sinh(b*x+a))^(1/2)/b^2/
sinh(b*x+a)^(1/2)+20/147*cosh(b*x+a)*sinh(b*x+a)^(1/2)/b^2

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5480, 2715, 2721, 2720} \[ \int x \cosh (a+b x) \sinh ^{\frac {5}{2}}(a+b x) \, dx=-\frac {4 \sinh ^{\frac {5}{2}}(a+b x) \cosh (a+b x)}{49 b^2}+\frac {20 \sqrt {\sinh (a+b x)} \cosh (a+b x)}{147 b^2}+\frac {20 i \sqrt {i \sinh (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right ),2\right )}{147 b^2 \sqrt {\sinh (a+b x)}}+\frac {2 x \sinh ^{\frac {7}{2}}(a+b x)}{7 b} \]

[In]

Int[x*Cosh[a + b*x]*Sinh[a + b*x]^(5/2),x]

[Out]

(((20*I)/147)*EllipticF[(I*a - Pi/2 + I*b*x)/2, 2]*Sqrt[I*Sinh[a + b*x]])/(b^2*Sqrt[Sinh[a + b*x]]) + (20*Cosh
[a + b*x]*Sqrt[Sinh[a + b*x]])/(147*b^2) - (4*Cosh[a + b*x]*Sinh[a + b*x]^(5/2))/(49*b^2) + (2*x*Sinh[a + b*x]
^(7/2))/(7*b)

Rule 2715

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] && Integ
erQ[2*n]

Rule 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ
[{c, d}, x]

Rule 2721

Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[(b*Sin[c + d*x])^n/Sin[c + d*x]^n, Int[Sin[c + d*x]
^n, x], x] /; FreeQ[{b, c, d}, x] && LtQ[-1, n, 1] && IntegerQ[2*n]

Rule 5480

Int[Cosh[(a_.) + (b_.)*(x_)^(n_.)]*(x_)^(m_.)*Sinh[(a_.) + (b_.)*(x_)^(n_.)]^(p_.), x_Symbol] :> Simp[x^(m - n
+ 1)*(Sinh[a + b*x^n]^(p + 1)/(b*n*(p + 1))), x] - Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*Sinh[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*} \text {integral}& = \frac {2 x \sinh ^{\frac {7}{2}}(a+b x)}{7 b}-\frac {2 \int \sinh ^{\frac {7}{2}}(a+b x) \, dx}{7 b} \\ & = -\frac {4 \cosh (a+b x) \sinh ^{\frac {5}{2}}(a+b x)}{49 b^2}+\frac {2 x \sinh ^{\frac {7}{2}}(a+b x)}{7 b}+\frac {10 \int \sinh ^{\frac {3}{2}}(a+b x) \, dx}{49 b} \\ & = \frac {20 \cosh (a+b x) \sqrt {\sinh (a+b x)}}{147 b^2}-\frac {4 \cosh (a+b x) \sinh ^{\frac {5}{2}}(a+b x)}{49 b^2}+\frac {2 x \sinh ^{\frac {7}{2}}(a+b x)}{7 b}-\frac {10 \int \frac {1}{\sqrt {\sinh (a+b x)}} \, dx}{147 b} \\ & = \frac {20 \cosh (a+b x) \sqrt {\sinh (a+b x)}}{147 b^2}-\frac {4 \cosh (a+b x) \sinh ^{\frac {5}{2}}(a+b x)}{49 b^2}+\frac {2 x \sinh ^{\frac {7}{2}}(a+b x)}{7 b}-\frac {\left (10 \sqrt {i \sinh (a+b x)}\right ) \int \frac {1}{\sqrt {i \sinh (a+b x)}} \, dx}{147 b \sqrt {\sinh (a+b x)}} \\ & = \frac {20 i \operatorname {EllipticF}\left (\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right ),2\right ) \sqrt {i \sinh (a+b x)}}{147 b^2 \sqrt {\sinh (a+b x)}}+\frac {20 \cosh (a+b x) \sqrt {\sinh (a+b x)}}{147 b^2}-\frac {4 \cosh (a+b x) \sinh ^{\frac {5}{2}}(a+b x)}{49 b^2}+\frac {2 x \sinh ^{\frac {7}{2}}(a+b x)}{7 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.85 \[ \int x \cosh (a+b x) \sinh ^{\frac {5}{2}}(a+b x) \, dx=\frac {63 b x-84 b x \cosh (2 (a+b x))+21 b x \cosh (4 (a+b x))-80 i \operatorname {EllipticF}\left (\frac {1}{4} (-2 i a+\pi -2 i b x),2\right ) \sqrt {i \sinh (a+b x)}+52 \sinh (2 (a+b x))-6 \sinh (4 (a+b x))}{588 b^2 \sqrt {\sinh (a+b x)}} \]

[In]

Integrate[x*Cosh[a + b*x]*Sinh[a + b*x]^(5/2),x]

[Out]

(63*b*x - 84*b*x*Cosh[2*(a + b*x)] + 21*b*x*Cosh[4*(a + b*x)] - (80*I)*EllipticF[((-2*I)*a + Pi - (2*I)*b*x)/4
, 2]*Sqrt[I*Sinh[a + b*x]] + 52*Sinh[2*(a + b*x)] - 6*Sinh[4*(a + b*x)])/(588*b^2*Sqrt[Sinh[a + b*x]])

Maple [F]

\[\int x \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{\frac {5}{2}}d x\]

[In]

int(x*cosh(b*x+a)*sinh(b*x+a)^(5/2),x)

[Out]

int(x*cosh(b*x+a)*sinh(b*x+a)^(5/2),x)

Fricas [F(-2)]

Exception generated. \[ \int x \cosh (a+b x) \sinh ^{\frac {5}{2}}(a+b x) \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x*cosh(b*x+a)*sinh(b*x+a)^(5/2),x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int x \cosh (a+b x) \sinh ^{\frac {5}{2}}(a+b x) \, dx=\text {Timed out} \]

[In]

integrate(x*cosh(b*x+a)*sinh(b*x+a)**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int x \cosh (a+b x) \sinh ^{\frac {5}{2}}(a+b x) \, dx=\int { x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{\frac {5}{2}} \,d x } \]

[In]

integrate(x*cosh(b*x+a)*sinh(b*x+a)^(5/2),x, algorithm="maxima")

[Out]

integrate(x*cosh(b*x + a)*sinh(b*x + a)^(5/2), x)

Giac [F(-2)]

Exception generated. \[ \int x \cosh (a+b x) \sinh ^{\frac {5}{2}}(a+b x) \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(x*cosh(b*x+a)*sinh(b*x+a)^(5/2),x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:Unable to divide,
perhaps due to rounding error%%%{1,[0,1,1,0]%%%} / %%%{1,[0,0,0,2]%%%} Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int x \cosh (a+b x) \sinh ^{\frac {5}{2}}(a+b x) \, dx=\int x\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^{5/2} \,d x \]

[In]

int(x*cosh(a + b*x)*sinh(a + b*x)^(5/2),x)

[Out]

int(x*cosh(a + b*x)*sinh(a + b*x)^(5/2), x)