Integrand size = 17, antiderivative size = 69 \[ \int \cosh ^2(a+b x) \sinh ^4(a+b x) \, dx=\frac {x}{16}+\frac {\cosh (a+b x) \sinh (a+b x)}{16 b}-\frac {\cosh ^3(a+b x) \sinh (a+b x)}{8 b}+\frac {\cosh ^3(a+b x) \sinh ^3(a+b x)}{6 b} \] Output:
1/16*x+1/16*cosh(b*x+a)*sinh(b*x+a)/b-1/8*cosh(b*x+a)^3*sinh(b*x+a)/b+1/6* cosh(b*x+a)^3*sinh(b*x+a)^3/b
Time = 0.06 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.58 \[ \int \cosh ^2(a+b x) \sinh ^4(a+b x) \, dx=\frac {12 b x-3 \sinh (2 (a+b x))-3 \sinh (4 (a+b x))+\sinh (6 (a+b x))}{192 b} \] Input:
Integrate[Cosh[a + b*x]^2*Sinh[a + b*x]^4,x]
Output:
(12*b*x - 3*Sinh[2*(a + b*x)] - 3*Sinh[4*(a + b*x)] + Sinh[6*(a + b*x)])/( 192*b)
Time = 0.36 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.14, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {3042, 3048, 25, 3042, 25, 3048, 3042, 3115, 24}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sinh ^4(a+b x) \cosh ^2(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (i a+i b x)^4 \cos (i a+i b x)^2dx\) |
\(\Big \downarrow \) 3048 |
\(\displaystyle \frac {1}{2} \int -\cosh ^2(a+b x) \sinh ^2(a+b x)dx+\frac {\sinh ^3(a+b x) \cosh ^3(a+b x)}{6 b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\sinh ^3(a+b x) \cosh ^3(a+b x)}{6 b}-\frac {1}{2} \int \cosh ^2(a+b x) \sinh ^2(a+b x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sinh ^3(a+b x) \cosh ^3(a+b x)}{6 b}-\frac {1}{2} \int -\cos (i a+i b x)^2 \sin (i a+i b x)^2dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\sinh ^3(a+b x) \cosh ^3(a+b x)}{6 b}+\frac {1}{2} \int \cos (i a+i b x)^2 \sin (i a+i b x)^2dx\) |
\(\Big \downarrow \) 3048 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{4} \int \cosh ^2(a+b x)dx-\frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}\right )+\frac {\sinh ^3(a+b x) \cosh ^3(a+b x)}{6 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sinh ^3(a+b x) \cosh ^3(a+b x)}{6 b}+\frac {1}{2} \left (-\frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}+\frac {1}{4} \int \sin \left (i a+i b x+\frac {\pi }{2}\right )^2dx\right )\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{4} \left (\frac {\int 1dx}{2}+\frac {\sinh (a+b x) \cosh (a+b x)}{2 b}\right )-\frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}\right )+\frac {\sinh ^3(a+b x) \cosh ^3(a+b x)}{6 b}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {\sinh ^3(a+b x) \cosh ^3(a+b x)}{6 b}+\frac {1}{2} \left (\frac {1}{4} \left (\frac {\sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x}{2}\right )-\frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}\right )\) |
Input:
Int[Cosh[a + b*x]^2*Sinh[a + b*x]^4,x]
Output:
(Cosh[a + b*x]^3*Sinh[a + b*x]^3)/(6*b) + (-1/4*(Cosh[a + b*x]^3*Sinh[a + b*x])/b + (x/2 + (Cosh[a + b*x]*Sinh[a + b*x])/(2*b))/4)/2
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(-a)*(b*Cos[e + f*x])^(n + 1)*((a*Sin[e + f*x])^(m - 1)/(b*f*(m + n))), x] + Simp[a^2*((m - 1)/(m + n)) Int[(b*Cos[e + f*x])^n *(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
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] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Time = 127.85 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(\frac {\frac {\sinh \left (b x +a \right )^{3} \cosh \left (b x +a \right )^{3}}{6}-\frac {\sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{8}+\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{16}+\frac {b x}{16}+\frac {a}{16}}{b}\) | \(61\) |
default | \(\frac {\frac {\sinh \left (b x +a \right )^{3} \cosh \left (b x +a \right )^{3}}{6}-\frac {\sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{8}+\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{16}+\frac {b x}{16}+\frac {a}{16}}{b}\) | \(61\) |
risch | \(\frac {x}{16}+\frac {{\mathrm e}^{6 b x +6 a}}{384 b}-\frac {{\mathrm e}^{4 b x +4 a}}{128 b}-\frac {{\mathrm e}^{2 b x +2 a}}{128 b}+\frac {{\mathrm e}^{-2 b x -2 a}}{128 b}+\frac {{\mathrm e}^{-4 b x -4 a}}{128 b}-\frac {{\mathrm e}^{-6 b x -6 a}}{384 b}\) | \(89\) |
orering | \(x \cosh \left (b x +a \right )^{2} \sinh \left (b x +a \right )^{4}+\frac {\frac {49 \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{5} b}{72}+\frac {49 \cosh \left (b x +a \right )^{3} \sinh \left (b x +a \right )^{3} b}{36}}{b^{2}}-\frac {49 x \left (2 b^{2} \sinh \left (b x +a \right )^{6}+22 \cosh \left (b x +a \right )^{2} \sinh \left (b x +a \right )^{4} b^{2}+12 \cosh \left (b x +a \right )^{4} \sinh \left (b x +a \right )^{2} b^{2}\right )}{144 b^{2}}-\frac {7 \left (56 b^{3} \sinh \left (b x +a \right )^{5} \cosh \left (b x +a \right )+136 \cosh \left (b x +a \right )^{3} \sinh \left (b x +a \right )^{3} b^{3}+24 \cosh \left (b x +a \right )^{5} \sinh \left (b x +a \right ) b^{3}\right )}{288 b^{4}}+\frac {7 x \left (688 \cosh \left (b x +a \right )^{2} \sinh \left (b x +a \right )^{4} b^{4}+56 \sinh \left (b x +a \right )^{6} b^{4}+528 \cosh \left (b x +a \right )^{4} \sinh \left (b x +a \right )^{2} b^{4}+24 \cosh \left (b x +a \right )^{6} b^{4}\right )}{288 b^{4}}+\frac {1712 \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{5} b^{5}+4864 \cosh \left (b x +a \right )^{3} \sinh \left (b x +a \right )^{3} b^{5}+1200 \cosh \left (b x +a \right )^{5} \sinh \left (b x +a \right ) b^{5}}{2304 b^{6}}-\frac {x \left (1712 b^{6} \sinh \left (b x +a \right )^{6}+23152 \cosh \left (b x +a \right )^{2} \sinh \left (b x +a \right )^{4} b^{6}+20592 \cosh \left (b x +a \right )^{4} \sinh \left (b x +a \right )^{2} b^{6}+1200 \cosh \left (b x +a \right )^{6} b^{6}\right )}{2304 b^{6}}\) | \(404\) |
Input:
int(cosh(b*x+a)^2*sinh(b*x+a)^4,x,method=_RETURNVERBOSE)
Output:
1/b*(1/6*sinh(b*x+a)^3*cosh(b*x+a)^3-1/8*sinh(b*x+a)*cosh(b*x+a)^3+1/16*co sh(b*x+a)*sinh(b*x+a)+1/16*b*x+1/16*a)
Time = 0.10 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.30 \[ \int \cosh ^2(a+b x) \sinh ^4(a+b x) \, dx=\frac {3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + 2 \, {\left (5 \, \cosh \left (b x + a\right )^{3} - 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 6 \, b x + 3 \, {\left (\cosh \left (b x + a\right )^{5} - 2 \, \cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{96 \, b} \] Input:
integrate(cosh(b*x+a)^2*sinh(b*x+a)^4,x, algorithm="fricas")
Output:
1/96*(3*cosh(b*x + a)*sinh(b*x + a)^5 + 2*(5*cosh(b*x + a)^3 - 3*cosh(b*x + a))*sinh(b*x + a)^3 + 6*b*x + 3*(cosh(b*x + a)^5 - 2*cosh(b*x + a)^3 - c osh(b*x + a))*sinh(b*x + a))/b
Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (58) = 116\).
Time = 0.33 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.97 \[ \int \cosh ^2(a+b x) \sinh ^4(a+b x) \, dx=\begin {cases} - \frac {x \sinh ^{6}{\left (a + b x \right )}}{16} + \frac {3 x \sinh ^{4}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{16} - \frac {3 x \sinh ^{2}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{16} + \frac {x \cosh ^{6}{\left (a + b x \right )}}{16} + \frac {\sinh ^{5}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{16 b} + \frac {\sinh ^{3}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{6 b} - \frac {\sinh {\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{16 b} & \text {for}\: b \neq 0 \\x \sinh ^{4}{\left (a \right )} \cosh ^{2}{\left (a \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cosh(b*x+a)**2*sinh(b*x+a)**4,x)
Output:
Piecewise((-x*sinh(a + b*x)**6/16 + 3*x*sinh(a + b*x)**4*cosh(a + b*x)**2/ 16 - 3*x*sinh(a + b*x)**2*cosh(a + b*x)**4/16 + x*cosh(a + b*x)**6/16 + si nh(a + b*x)**5*cosh(a + b*x)/(16*b) + sinh(a + b*x)**3*cosh(a + b*x)**3/(6 *b) - sinh(a + b*x)*cosh(a + b*x)**5/(16*b), Ne(b, 0)), (x*sinh(a)**4*cosh (a)**2, True))
Time = 0.04 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.28 \[ \int \cosh ^2(a+b x) \sinh ^4(a+b x) \, dx=-\frac {{\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} + 3 \, e^{\left (-4 \, b x - 4 \, a\right )} - 1\right )} e^{\left (6 \, b x + 6 \, a\right )}}{384 \, b} + \frac {b x + a}{16 \, b} + \frac {3 \, e^{\left (-2 \, b x - 2 \, a\right )} + 3 \, e^{\left (-4 \, b x - 4 \, a\right )} - e^{\left (-6 \, b x - 6 \, a\right )}}{384 \, b} \] Input:
integrate(cosh(b*x+a)^2*sinh(b*x+a)^4,x, algorithm="maxima")
Output:
-1/384*(3*e^(-2*b*x - 2*a) + 3*e^(-4*b*x - 4*a) - 1)*e^(6*b*x + 6*a)/b + 1 /16*(b*x + a)/b + 1/384*(3*e^(-2*b*x - 2*a) + 3*e^(-4*b*x - 4*a) - e^(-6*b *x - 6*a))/b
Time = 0.12 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.28 \[ \int \cosh ^2(a+b x) \sinh ^4(a+b x) \, dx=\frac {1}{16} \, x + \frac {e^{\left (6 \, b x + 6 \, a\right )}}{384 \, b} - \frac {e^{\left (4 \, b x + 4 \, a\right )}}{128 \, b} - \frac {e^{\left (2 \, b x + 2 \, a\right )}}{128 \, b} + \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{128 \, b} + \frac {e^{\left (-4 \, b x - 4 \, a\right )}}{128 \, b} - \frac {e^{\left (-6 \, b x - 6 \, a\right )}}{384 \, b} \] Input:
integrate(cosh(b*x+a)^2*sinh(b*x+a)^4,x, algorithm="giac")
Output:
1/16*x + 1/384*e^(6*b*x + 6*a)/b - 1/128*e^(4*b*x + 4*a)/b - 1/128*e^(2*b* x + 2*a)/b + 1/128*e^(-2*b*x - 2*a)/b + 1/128*e^(-4*b*x - 4*a)/b - 1/384*e ^(-6*b*x - 6*a)/b
Time = 1.09 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.62 \[ \int \cosh ^2(a+b x) \sinh ^4(a+b x) \, dx=\frac {x}{16}-\frac {\frac {\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{64}+\frac {\mathrm {sinh}\left (4\,a+4\,b\,x\right )}{64}-\frac {\mathrm {sinh}\left (6\,a+6\,b\,x\right )}{192}}{b} \] Input:
int(cosh(a + b*x)^2*sinh(a + b*x)^4,x)
Output:
x/16 - (sinh(2*a + 2*b*x)/64 + sinh(4*a + 4*b*x)/64 - sinh(6*a + 6*b*x)/19 2)/b
Time = 0.22 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.32 \[ \int \cosh ^2(a+b x) \sinh ^4(a+b x) \, dx=\frac {e^{12 b x +12 a}-3 e^{10 b x +10 a}-3 e^{8 b x +8 a}+24 e^{6 b x +6 a} b x +3 e^{4 b x +4 a}+3 e^{2 b x +2 a}-1}{384 e^{6 b x +6 a} b} \] Input:
int(cosh(b*x+a)^2*sinh(b*x+a)^4,x)
Output:
(e**(12*a + 12*b*x) - 3*e**(10*a + 10*b*x) - 3*e**(8*a + 8*b*x) + 24*e**(6 *a + 6*b*x)*b*x + 3*e**(4*a + 4*b*x) + 3*e**(2*a + 2*b*x) - 1)/(384*e**(6* a + 6*b*x)*b)