Integrand size = 17, antiderivative size = 36 \[ \int \cosh ^3(x) \left (a+b \cosh ^2(x)\right )^3 \sinh (x) \, dx=-\frac {a \left (a+b \cosh ^2(x)\right )^4}{8 b^2}+\frac {\left (a+b \cosh ^2(x)\right )^5}{10 b^2} \] Output:
-1/8*a*(a+b*cosh(x)^2)^4/b^2+1/10*(a+b*cosh(x)^2)^5/b^2
Leaf count is larger than twice the leaf count of optimal. \(136\) vs. \(2(36)=72\).
Time = 0.27 (sec) , antiderivative size = 136, normalized size of antiderivative = 3.78 \[ \int \cosh ^3(x) \left (a+b \cosh ^2(x)\right )^3 \sinh (x) \, dx=\frac {1}{32} \left (12 a^2 b \cosh ^4(x)+8 a b^2 \cosh ^6(x)+2 b^3 \cosh ^8(x)+4 a^3 \cosh (2 x)+4 a^2 b \cosh ^3(x) \cosh (3 x)+a^3 \cosh (4 x)+\frac {1}{32} a b^2 (48 \cosh (2 x)+36 \cosh (4 x)+16 \cosh (6 x)+3 \cosh (8 x))+\frac {1}{320} b^3 (140 \cosh (2 x)+100 \cosh (4 x)+50 \cosh (6 x)+15 \cosh (8 x)+2 \cosh (10 x))\right ) \] Input:
Integrate[Cosh[x]^3*(a + b*Cosh[x]^2)^3*Sinh[x],x]
Output:
(12*a^2*b*Cosh[x]^4 + 8*a*b^2*Cosh[x]^6 + 2*b^3*Cosh[x]^8 + 4*a^3*Cosh[2*x ] + 4*a^2*b*Cosh[x]^3*Cosh[3*x] + a^3*Cosh[4*x] + (a*b^2*(48*Cosh[2*x] + 3 6*Cosh[4*x] + 16*Cosh[6*x] + 3*Cosh[8*x]))/32 + (b^3*(140*Cosh[2*x] + 100* Cosh[4*x] + 50*Cosh[6*x] + 15*Cosh[8*x] + 2*Cosh[10*x]))/320)/32
Time = 0.29 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {3042, 26, 4835, 243, 49, 2009}
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.
\(\displaystyle \int \sinh (x) \cosh ^3(x) \left (a+b \cosh ^2(x)\right )^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -i \sin (i x) \cos (i x)^3 \left (a+b \cos (i x)^2\right )^3dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \cos (i x)^3 \left (b \cos (i x)^2+a\right )^3 \sin (i x)dx\) |
\(\Big \downarrow \) 4835 |
\(\displaystyle \int \cosh ^3(x) \left (a+b \cosh ^2(x)\right )^3d\cosh (x)\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \cosh ^2(x) \left (b \cosh ^2(x)+a\right )^3d\cosh ^2(x)\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {1}{2} \int \left (\frac {\left (b \cosh ^2(x)+a\right )^4}{b}-\frac {a \left (b \cosh ^2(x)+a\right )^3}{b}\right )d\cosh ^2(x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {\left (a+b \cosh ^2(x)\right )^5}{5 b^2}-\frac {a \left (a+b \cosh ^2(x)\right )^4}{4 b^2}\right )\) |
Input:
Int[Cosh[x]^3*(a + b*Cosh[x]^2)^3*Sinh[x],x]
Output:
(-1/4*(a*(a + b*Cosh[x]^2)^4)/b^2 + (a + b*Cosh[x]^2)^5/(5*b^2))/2
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFacto rs[Cos[c*(a + b*x)], x]}, Simp[-d/(b*c) Subst[Int[SubstFor[1, Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d], x] /; FunctionOfQ[Cos[c*(a + b* x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Sin] || EqQ[F, sin])
Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.11
\[\frac {b^{3} \cosh \left (x \right )^{10}}{10}+\frac {3 a \,b^{2} \cosh \left (x \right )^{8}}{8}+\frac {a^{2} b \cosh \left (x \right )^{6}}{2}+\frac {a^{3} \cosh \left (x \right )^{4}}{4}\]
Input:
int(cosh(x)^3*(a+b*cosh(x)^2)^3*sinh(x),x)
Output:
1/10*b^3*cosh(x)^10+3/8*a*b^2*cosh(x)^8+1/2*a^2*b*cosh(x)^6+1/4*a^3*cosh(x )^4
Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (32) = 64\).
Time = 0.09 (sec) , antiderivative size = 386, normalized size of antiderivative = 10.72 \[ \int \cosh ^3(x) \left (a+b \cosh ^2(x)\right )^3 \sinh (x) \, dx=\frac {1}{5120} \, b^{3} \cosh \left (x\right )^{10} + \frac {1}{5120} \, b^{3} \sinh \left (x\right )^{10} + \frac {1}{1024} \, {\left (3 \, a b^{2} + 2 \, b^{3}\right )} \cosh \left (x\right )^{8} + \frac {1}{1024} \, {\left (9 \, b^{3} \cosh \left (x\right )^{2} + 3 \, a b^{2} + 2 \, b^{3}\right )} \sinh \left (x\right )^{8} + \frac {1}{1024} \, {\left (16 \, a^{2} b + 24 \, a b^{2} + 9 \, b^{3}\right )} \cosh \left (x\right )^{6} + \frac {1}{1024} \, {\left (42 \, b^{3} \cosh \left (x\right )^{4} + 16 \, a^{2} b + 24 \, a b^{2} + 9 \, b^{3} + 28 \, {\left (3 \, a b^{2} + 2 \, b^{3}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{6} + \frac {1}{256} \, {\left (8 \, a^{3} + 24 \, a^{2} b + 21 \, a b^{2} + 6 \, b^{3}\right )} \cosh \left (x\right )^{4} + \frac {1}{1024} \, {\left (42 \, b^{3} \cosh \left (x\right )^{6} + 70 \, {\left (3 \, a b^{2} + 2 \, b^{3}\right )} \cosh \left (x\right )^{4} + 32 \, a^{3} + 96 \, a^{2} b + 84 \, a b^{2} + 24 \, b^{3} + 15 \, {\left (16 \, a^{2} b + 24 \, a b^{2} + 9 \, b^{3}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{4} + \frac {1}{512} \, {\left (64 \, a^{3} + 120 \, a^{2} b + 84 \, a b^{2} + 21 \, b^{3}\right )} \cosh \left (x\right )^{2} + \frac {1}{1024} \, {\left (9 \, b^{3} \cosh \left (x\right )^{8} + 28 \, {\left (3 \, a b^{2} + 2 \, b^{3}\right )} \cosh \left (x\right )^{6} + 15 \, {\left (16 \, a^{2} b + 24 \, a b^{2} + 9 \, b^{3}\right )} \cosh \left (x\right )^{4} + 128 \, a^{3} + 240 \, a^{2} b + 168 \, a b^{2} + 42 \, b^{3} + 24 \, {\left (8 \, a^{3} + 24 \, a^{2} b + 21 \, a b^{2} + 6 \, b^{3}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} \] Input:
integrate(cosh(x)^3*(a+b*cosh(x)^2)^3*sinh(x),x, algorithm="fricas")
Output:
1/5120*b^3*cosh(x)^10 + 1/5120*b^3*sinh(x)^10 + 1/1024*(3*a*b^2 + 2*b^3)*c osh(x)^8 + 1/1024*(9*b^3*cosh(x)^2 + 3*a*b^2 + 2*b^3)*sinh(x)^8 + 1/1024*( 16*a^2*b + 24*a*b^2 + 9*b^3)*cosh(x)^6 + 1/1024*(42*b^3*cosh(x)^4 + 16*a^2 *b + 24*a*b^2 + 9*b^3 + 28*(3*a*b^2 + 2*b^3)*cosh(x)^2)*sinh(x)^6 + 1/256* (8*a^3 + 24*a^2*b + 21*a*b^2 + 6*b^3)*cosh(x)^4 + 1/1024*(42*b^3*cosh(x)^6 + 70*(3*a*b^2 + 2*b^3)*cosh(x)^4 + 32*a^3 + 96*a^2*b + 84*a*b^2 + 24*b^3 + 15*(16*a^2*b + 24*a*b^2 + 9*b^3)*cosh(x)^2)*sinh(x)^4 + 1/512*(64*a^3 + 120*a^2*b + 84*a*b^2 + 21*b^3)*cosh(x)^2 + 1/1024*(9*b^3*cosh(x)^8 + 28*(3 *a*b^2 + 2*b^3)*cosh(x)^6 + 15*(16*a^2*b + 24*a*b^2 + 9*b^3)*cosh(x)^4 + 1 28*a^3 + 240*a^2*b + 168*a*b^2 + 42*b^3 + 24*(8*a^3 + 24*a^2*b + 21*a*b^2 + 6*b^3)*cosh(x)^2)*sinh(x)^2
Time = 0.91 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.22 \[ \int \cosh ^3(x) \left (a+b \cosh ^2(x)\right )^3 \sinh (x) \, dx=\frac {a^{3} \cosh ^{4}{\left (x \right )}}{4} + \frac {a^{2} b \cosh ^{6}{\left (x \right )}}{2} + \frac {3 a b^{2} \cosh ^{8}{\left (x \right )}}{8} + \frac {b^{3} \cosh ^{10}{\left (x \right )}}{10} \] Input:
integrate(cosh(x)**3*(a+b*cosh(x)**2)**3*sinh(x),x)
Output:
a**3*cosh(x)**4/4 + a**2*b*cosh(x)**6/2 + 3*a*b**2*cosh(x)**8/8 + b**3*cos h(x)**10/10
Time = 0.04 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.08 \[ \int \cosh ^3(x) \left (a+b \cosh ^2(x)\right )^3 \sinh (x) \, dx=\frac {1}{10} \, b^{3} \cosh \left (x\right )^{10} + \frac {3}{8} \, a b^{2} \cosh \left (x\right )^{8} + \frac {1}{2} \, a^{2} b \cosh \left (x\right )^{6} + \frac {1}{4} \, a^{3} \cosh \left (x\right )^{4} \] Input:
integrate(cosh(x)^3*(a+b*cosh(x)^2)^3*sinh(x),x, algorithm="maxima")
Output:
1/10*b^3*cosh(x)^10 + 3/8*a*b^2*cosh(x)^8 + 1/2*a^2*b*cosh(x)^6 + 1/4*a^3* cosh(x)^4
Leaf count of result is larger than twice the leaf count of optimal. 224 vs. \(2 (32) = 64\).
Time = 0.11 (sec) , antiderivative size = 224, normalized size of antiderivative = 6.22 \[ \int \cosh ^3(x) \left (a+b \cosh ^2(x)\right )^3 \sinh (x) \, dx=\frac {1}{10240} \, b^{3} {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{5} + \frac {3}{2048} \, a b^{2} {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{4} + \frac {1}{1024} \, b^{3} {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{4} + \frac {1}{128} \, a^{2} b {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{3} + \frac {3}{256} \, a b^{2} {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{3} + \frac {1}{256} \, b^{3} {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{3} + \frac {1}{64} \, a^{3} {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{2} + \frac {3}{64} \, a^{2} b {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{2} + \frac {9}{256} \, a b^{2} {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{2} + \frac {1}{128} \, b^{3} {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{2} + \frac {1}{16} \, a^{3} {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )} + \frac {3}{32} \, a^{2} b {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )} + \frac {3}{64} \, a b^{2} {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )} + \frac {1}{128} \, b^{3} {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )} \] Input:
integrate(cosh(x)^3*(a+b*cosh(x)^2)^3*sinh(x),x, algorithm="giac")
Output:
1/10240*b^3*(e^(2*x) + e^(-2*x))^5 + 3/2048*a*b^2*(e^(2*x) + e^(-2*x))^4 + 1/1024*b^3*(e^(2*x) + e^(-2*x))^4 + 1/128*a^2*b*(e^(2*x) + e^(-2*x))^3 + 3/256*a*b^2*(e^(2*x) + e^(-2*x))^3 + 1/256*b^3*(e^(2*x) + e^(-2*x))^3 + 1/ 64*a^3*(e^(2*x) + e^(-2*x))^2 + 3/64*a^2*b*(e^(2*x) + e^(-2*x))^2 + 9/256* a*b^2*(e^(2*x) + e^(-2*x))^2 + 1/128*b^3*(e^(2*x) + e^(-2*x))^2 + 1/16*a^3 *(e^(2*x) + e^(-2*x)) + 3/32*a^2*b*(e^(2*x) + e^(-2*x)) + 3/64*a*b^2*(e^(2 *x) + e^(-2*x)) + 1/128*b^3*(e^(2*x) + e^(-2*x))
Time = 0.19 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.08 \[ \int \cosh ^3(x) \left (a+b \cosh ^2(x)\right )^3 \sinh (x) \, dx=\frac {a^3\,{\mathrm {cosh}\left (x\right )}^4}{4}+\frac {a^2\,b\,{\mathrm {cosh}\left (x\right )}^6}{2}+\frac {3\,a\,b^2\,{\mathrm {cosh}\left (x\right )}^8}{8}+\frac {b^3\,{\mathrm {cosh}\left (x\right )}^{10}}{10} \] Input:
int(cosh(x)^3*sinh(x)*(a + b*cosh(x)^2)^3,x)
Output:
(a^3*cosh(x)^4)/4 + (b^3*cosh(x)^10)/10 + (a^2*b*cosh(x)^6)/2 + (3*a*b^2*c osh(x)^8)/8
Time = 0.24 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.14 \[ \int \cosh ^3(x) \left (a+b \cosh ^2(x)\right )^3 \sinh (x) \, dx=\frac {\cosh \left (x \right )^{4} \left (4 \cosh \left (x \right )^{6} b^{3}+15 \cosh \left (x \right )^{4} a \,b^{2}+20 \cosh \left (x \right )^{2} a^{2} b +10 a^{3}\right )}{40} \] Input:
int(cosh(x)^3*(a+b*cosh(x)^2)^3*sinh(x),x)
Output:
(cosh(x)**4*(4*cosh(x)**6*b**3 + 15*cosh(x)**4*a*b**2 + 20*cosh(x)**2*a**2 *b + 10*a**3))/40