Integrand size = 23, antiderivative size = 98 \[ \int \cosh ^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {a^3 \sinh (c+d x)}{d}+\frac {a^2 (a+3 b) \sinh ^3(c+d x)}{3 d}+\frac {3 a b (a+b) \sinh ^5(c+d x)}{5 d}+\frac {b^2 (3 a+b) \sinh ^7(c+d x)}{7 d}+\frac {b^3 \sinh ^9(c+d x)}{9 d} \] Output:
a^3*sinh(d*x+c)/d+1/3*a^2*(a+3*b)*sinh(d*x+c)^3/d+3/5*a*b*(a+b)*sinh(d*x+c )^5/d+1/7*b^2*(3*a+b)*sinh(d*x+c)^7/d+1/9*b^3*sinh(d*x+c)^9/d
Time = 0.18 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.85 \[ \int \cosh ^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {315 a^3 \sinh (c+d x)+105 a^2 (a+3 b) \sinh ^3(c+d x)+189 a b (a+b) \sinh ^5(c+d x)+45 b^2 (3 a+b) \sinh ^7(c+d x)+35 b^3 \sinh ^9(c+d x)}{315 d} \] Input:
Integrate[Cosh[c + d*x]^3*(a + b*Sinh[c + d*x]^2)^3,x]
Output:
(315*a^3*Sinh[c + d*x] + 105*a^2*(a + 3*b)*Sinh[c + d*x]^3 + 189*a*b*(a + b)*Sinh[c + d*x]^5 + 45*b^2*(3*a + b)*Sinh[c + d*x]^7 + 35*b^3*Sinh[c + d* x]^9)/(315*d)
Time = 0.31 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.89, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 3669, 290, 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.
\(\displaystyle \int \cosh ^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \cos (i c+i d x)^3 \left (a-b \sin (i c+i d x)^2\right )^3dx\) |
\(\Big \downarrow \) 3669 |
\(\displaystyle \frac {\int \left (\sinh ^2(c+d x)+1\right ) \left (b \sinh ^2(c+d x)+a\right )^3d\sinh (c+d x)}{d}\) |
\(\Big \downarrow \) 290 |
\(\displaystyle \frac {\int \left (b^3 \sinh ^8(c+d x)+b^2 (3 a+b) \sinh ^6(c+d x)+3 a b (a+b) \sinh ^4(c+d x)+a^2 (a+3 b) \sinh ^2(c+d x)+a^3\right )d\sinh (c+d x)}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^3 \sinh (c+d x)+\frac {1}{3} a^2 (a+3 b) \sinh ^3(c+d x)+\frac {1}{7} b^2 (3 a+b) \sinh ^7(c+d x)+\frac {3}{5} a b (a+b) \sinh ^5(c+d x)+\frac {1}{9} b^3 \sinh ^9(c+d x)}{d}\) |
Input:
Int[Cosh[c + d*x]^3*(a + b*Sinh[c + d*x]^2)^3,x]
Output:
(a^3*Sinh[c + d*x] + (a^2*(a + 3*b)*Sinh[c + d*x]^3)/3 + (3*a*b*(a + b)*Si nh[c + d*x]^5)/5 + (b^2*(3*a + b)*Sinh[c + d*x]^7)/7 + (b^3*Sinh[c + d*x]^ 9)/9)/d
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> I nt[ExpandIntegrand[(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d }, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && IGtQ[q, 0]
Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f S ubst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x] /ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Time = 0.12 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.94
\[\frac {\frac {b^{3} \sinh \left (d x +c \right )^{9}}{9}+\frac {\left (3 b^{2} a +b^{3}\right ) \sinh \left (d x +c \right )^{7}}{7}+\frac {\left (3 a^{2} b +3 b^{2} a \right ) \sinh \left (d x +c \right )^{5}}{5}+\frac {\left (a^{3}+3 a^{2} b \right ) \sinh \left (d x +c \right )^{3}}{3}+a^{3} \sinh \left (d x +c \right )}{d}\]
Input:
int(cosh(d*x+c)^3*(a+b*sinh(d*x+c)^2)^3,x)
Output:
1/d*(1/9*b^3*sinh(d*x+c)^9+1/7*(3*a*b^2+b^3)*sinh(d*x+c)^7+1/5*(3*a^2*b+3* a*b^2)*sinh(d*x+c)^5+1/3*(a^3+3*a^2*b)*sinh(d*x+c)^3+a^3*sinh(d*x+c))
Leaf count of result is larger than twice the leaf count of optimal. 324 vs. \(2 (90) = 180\).
Time = 0.10 (sec) , antiderivative size = 324, normalized size of antiderivative = 3.31 \[ \int \cosh ^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {35 \, b^{3} \sinh \left (d x + c\right )^{9} + 45 \, {\left (28 \, b^{3} \cosh \left (d x + c\right )^{2} + 12 \, a b^{2} - 3 \, b^{3}\right )} \sinh \left (d x + c\right )^{7} + 63 \, {\left (70 \, b^{3} \cosh \left (d x + c\right )^{4} + 48 \, a^{2} b - 12 \, a b^{2} + 45 \, {\left (4 \, a b^{2} - b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{5} + 105 \, {\left (28 \, b^{3} \cosh \left (d x + c\right )^{6} + 45 \, {\left (4 \, a b^{2} - b^{3}\right )} \cosh \left (d x + c\right )^{4} + 64 \, a^{3} + 48 \, a^{2} b - 36 \, a b^{2} + 8 \, b^{3} + 72 \, {\left (4 \, a^{2} b - a b^{2}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{3} + 315 \, {\left (b^{3} \cosh \left (d x + c\right )^{8} + 3 \, {\left (4 \, a b^{2} - b^{3}\right )} \cosh \left (d x + c\right )^{6} + 12 \, {\left (4 \, a^{2} b - a b^{2}\right )} \cosh \left (d x + c\right )^{4} + 192 \, a^{3} - 96 \, a^{2} b + 36 \, a b^{2} - 6 \, b^{3} + 4 \, {\left (16 \, a^{3} + 12 \, a^{2} b - 9 \, a b^{2} + 2 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )}{80640 \, d} \] Input:
integrate(cosh(d*x+c)^3*(a+b*sinh(d*x+c)^2)^3,x, algorithm="fricas")
Output:
1/80640*(35*b^3*sinh(d*x + c)^9 + 45*(28*b^3*cosh(d*x + c)^2 + 12*a*b^2 - 3*b^3)*sinh(d*x + c)^7 + 63*(70*b^3*cosh(d*x + c)^4 + 48*a^2*b - 12*a*b^2 + 45*(4*a*b^2 - b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 105*(28*b^3*cosh(d *x + c)^6 + 45*(4*a*b^2 - b^3)*cosh(d*x + c)^4 + 64*a^3 + 48*a^2*b - 36*a* b^2 + 8*b^3 + 72*(4*a^2*b - a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 315* (b^3*cosh(d*x + c)^8 + 3*(4*a*b^2 - b^3)*cosh(d*x + c)^6 + 12*(4*a^2*b - a *b^2)*cosh(d*x + c)^4 + 192*a^3 - 96*a^2*b + 36*a*b^2 - 6*b^3 + 4*(16*a^3 + 12*a^2*b - 9*a*b^2 + 2*b^3)*cosh(d*x + c)^2)*sinh(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (87) = 174\).
Time = 0.92 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.86 \[ \int \cosh ^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\begin {cases} - \frac {2 a^{3} \sinh ^{3}{\left (c + d x \right )}}{3 d} + \frac {a^{3} \sinh {\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{d} - \frac {2 a^{2} b \sinh ^{5}{\left (c + d x \right )}}{5 d} + \frac {a^{2} b \sinh ^{3}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{d} - \frac {6 a b^{2} \sinh ^{7}{\left (c + d x \right )}}{35 d} + \frac {3 a b^{2} \sinh ^{5}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{5 d} - \frac {2 b^{3} \sinh ^{9}{\left (c + d x \right )}}{63 d} + \frac {b^{3} \sinh ^{7}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{7 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\left (c \right )}\right )^{3} \cosh ^{3}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cosh(d*x+c)**3*(a+b*sinh(d*x+c)**2)**3,x)
Output:
Piecewise((-2*a**3*sinh(c + d*x)**3/(3*d) + a**3*sinh(c + d*x)*cosh(c + d* x)**2/d - 2*a**2*b*sinh(c + d*x)**5/(5*d) + a**2*b*sinh(c + d*x)**3*cosh(c + d*x)**2/d - 6*a*b**2*sinh(c + d*x)**7/(35*d) + 3*a*b**2*sinh(c + d*x)** 5*cosh(c + d*x)**2/(5*d) - 2*b**3*sinh(c + d*x)**9/(63*d) + b**3*sinh(c + d*x)**7*cosh(c + d*x)**2/(7*d), Ne(d, 0)), (x*(a + b*sinh(c)**2)**3*cosh(c )**3, True))
Leaf count of result is larger than twice the leaf count of optimal. 349 vs. \(2 (90) = 180\).
Time = 0.05 (sec) , antiderivative size = 349, normalized size of antiderivative = 3.56 \[ \int \cosh ^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=-\frac {1}{32256} \, b^{3} {\left (\frac {{\left (27 \, e^{\left (-2 \, d x - 2 \, c\right )} - 168 \, e^{\left (-6 \, d x - 6 \, c\right )} + 378 \, e^{\left (-8 \, d x - 8 \, c\right )} - 7\right )} e^{\left (9 \, d x + 9 \, c\right )}}{d} - \frac {378 \, e^{\left (-d x - c\right )} - 168 \, e^{\left (-3 \, d x - 3 \, c\right )} + 27 \, e^{\left (-7 \, d x - 7 \, c\right )} - 7 \, e^{\left (-9 \, d x - 9 \, c\right )}}{d}\right )} - \frac {3}{4480} \, a b^{2} {\left (\frac {{\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} + 35 \, e^{\left (-4 \, d x - 4 \, c\right )} - 105 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5\right )} e^{\left (7 \, d x + 7 \, c\right )}}{d} + \frac {105 \, e^{\left (-d x - c\right )} - 35 \, e^{\left (-3 \, d x - 3 \, c\right )} - 7 \, e^{\left (-5 \, d x - 5 \, c\right )} + 5 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d}\right )} + \frac {1}{160} \, a^{2} b {\left (\frac {{\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 30 \, e^{\left (-4 \, d x - 4 \, c\right )} + 3\right )} e^{\left (5 \, d x + 5 \, c\right )}}{d} + \frac {30 \, e^{\left (-d x - c\right )} - 5 \, e^{\left (-3 \, d x - 3 \, c\right )} - 3 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d}\right )} + \frac {1}{24} \, a^{3} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} - \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} \] Input:
integrate(cosh(d*x+c)^3*(a+b*sinh(d*x+c)^2)^3,x, algorithm="maxima")
Output:
-1/32256*b^3*((27*e^(-2*d*x - 2*c) - 168*e^(-6*d*x - 6*c) + 378*e^(-8*d*x - 8*c) - 7)*e^(9*d*x + 9*c)/d - (378*e^(-d*x - c) - 168*e^(-3*d*x - 3*c) + 27*e^(-7*d*x - 7*c) - 7*e^(-9*d*x - 9*c))/d) - 3/4480*a*b^2*((7*e^(-2*d*x - 2*c) + 35*e^(-4*d*x - 4*c) - 105*e^(-6*d*x - 6*c) - 5)*e^(7*d*x + 7*c)/ d + (105*e^(-d*x - c) - 35*e^(-3*d*x - 3*c) - 7*e^(-5*d*x - 5*c) + 5*e^(-7 *d*x - 7*c))/d) + 1/160*a^2*b*((5*e^(-2*d*x - 2*c) - 30*e^(-4*d*x - 4*c) + 3)*e^(5*d*x + 5*c)/d + (30*e^(-d*x - c) - 5*e^(-3*d*x - 3*c) - 3*e^(-5*d* x - 5*c))/d) + 1/24*a^3*(e^(3*d*x + 3*c)/d + 9*e^(d*x + c)/d - 9*e^(-d*x - c)/d - e^(-3*d*x - 3*c)/d)
Leaf count of result is larger than twice the leaf count of optimal. 286 vs. \(2 (90) = 180\).
Time = 0.16 (sec) , antiderivative size = 286, normalized size of antiderivative = 2.92 \[ \int \cosh ^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {b^{3} e^{\left (9 \, d x + 9 \, c\right )}}{4608 \, d} - \frac {b^{3} e^{\left (-9 \, d x - 9 \, c\right )}}{4608 \, d} + \frac {3 \, {\left (4 \, a b^{2} - b^{3}\right )} e^{\left (7 \, d x + 7 \, c\right )}}{3584 \, d} + \frac {3 \, {\left (4 \, a^{2} b - a b^{2}\right )} e^{\left (5 \, d x + 5 \, c\right )}}{640 \, d} + \frac {{\left (16 \, a^{3} + 12 \, a^{2} b - 9 \, a b^{2} + 2 \, b^{3}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{384 \, d} + \frac {3 \, {\left (32 \, a^{3} - 16 \, a^{2} b + 6 \, a b^{2} - b^{3}\right )} e^{\left (d x + c\right )}}{256 \, d} - \frac {3 \, {\left (32 \, a^{3} - 16 \, a^{2} b + 6 \, a b^{2} - b^{3}\right )} e^{\left (-d x - c\right )}}{256 \, d} - \frac {{\left (16 \, a^{3} + 12 \, a^{2} b - 9 \, a b^{2} + 2 \, b^{3}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{384 \, d} - \frac {3 \, {\left (4 \, a^{2} b - a b^{2}\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{640 \, d} - \frac {3 \, {\left (4 \, a b^{2} - b^{3}\right )} e^{\left (-7 \, d x - 7 \, c\right )}}{3584 \, d} \] Input:
integrate(cosh(d*x+c)^3*(a+b*sinh(d*x+c)^2)^3,x, algorithm="giac")
Output:
1/4608*b^3*e^(9*d*x + 9*c)/d - 1/4608*b^3*e^(-9*d*x - 9*c)/d + 3/3584*(4*a *b^2 - b^3)*e^(7*d*x + 7*c)/d + 3/640*(4*a^2*b - a*b^2)*e^(5*d*x + 5*c)/d + 1/384*(16*a^3 + 12*a^2*b - 9*a*b^2 + 2*b^3)*e^(3*d*x + 3*c)/d + 3/256*(3 2*a^3 - 16*a^2*b + 6*a*b^2 - b^3)*e^(d*x + c)/d - 3/256*(32*a^3 - 16*a^2*b + 6*a*b^2 - b^3)*e^(-d*x - c)/d - 1/384*(16*a^3 + 12*a^2*b - 9*a*b^2 + 2* b^3)*e^(-3*d*x - 3*c)/d - 3/640*(4*a^2*b - a*b^2)*e^(-5*d*x - 5*c)/d - 3/3 584*(4*a*b^2 - b^3)*e^(-7*d*x - 7*c)/d
Time = 0.30 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.14 \[ \int \cosh ^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {105\,a^3\,{\mathrm {sinh}\left (c+d\,x\right )}^3+315\,a^3\,\mathrm {sinh}\left (c+d\,x\right )+189\,a^2\,b\,{\mathrm {sinh}\left (c+d\,x\right )}^5+315\,a^2\,b\,{\mathrm {sinh}\left (c+d\,x\right )}^3+135\,a\,b^2\,{\mathrm {sinh}\left (c+d\,x\right )}^7+189\,a\,b^2\,{\mathrm {sinh}\left (c+d\,x\right )}^5+35\,b^3\,{\mathrm {sinh}\left (c+d\,x\right )}^9+45\,b^3\,{\mathrm {sinh}\left (c+d\,x\right )}^7}{315\,d} \] Input:
int(cosh(c + d*x)^3*(a + b*sinh(c + d*x)^2)^3,x)
Output:
(315*a^3*sinh(c + d*x) + 105*a^3*sinh(c + d*x)^3 + 45*b^3*sinh(c + d*x)^7 + 35*b^3*sinh(c + d*x)^9 + 315*a^2*b*sinh(c + d*x)^3 + 189*a*b^2*sinh(c + d*x)^5 + 189*a^2*b*sinh(c + d*x)^5 + 135*a*b^2*sinh(c + d*x)^7)/(315*d)
Time = 0.17 (sec) , antiderivative size = 412, normalized size of antiderivative = 4.20 \[ \int \cosh ^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {-35 b^{3}+135 e^{2 d x +2 c} b^{3}+60480 e^{10 d x +10 c} a^{3}-1890 e^{10 d x +10 c} b^{3}+540 e^{16 d x +16 c} a \,b^{2}+3024 e^{14 d x +14 c} a^{2} b -756 e^{14 d x +14 c} a \,b^{2}+5040 e^{12 d x +12 c} a^{2} b -3780 e^{12 d x +12 c} a \,b^{2}+30240 e^{8 d x +8 c} a^{2} b -11340 e^{8 d x +8 c} a \,b^{2}-5040 e^{6 d x +6 c} a^{2} b +3780 e^{6 d x +6 c} a \,b^{2}-3024 e^{4 d x +4 c} a^{2} b +756 e^{4 d x +4 c} a \,b^{2}-840 e^{6 d x +6 c} b^{3}+35 e^{18 d x +18 c} b^{3}-135 e^{16 d x +16 c} b^{3}+6720 e^{12 d x +12 c} a^{3}+840 e^{12 d x +12 c} b^{3}-60480 e^{8 d x +8 c} a^{3}+1890 e^{8 d x +8 c} b^{3}-6720 e^{6 d x +6 c} a^{3}-30240 e^{10 d x +10 c} a^{2} b +11340 e^{10 d x +10 c} a \,b^{2}-540 e^{2 d x +2 c} a \,b^{2}}{161280 e^{9 d x +9 c} d} \] Input:
int(cosh(d*x+c)^3*(a+b*sinh(d*x+c)^2)^3,x)
Output:
(35*e**(18*c + 18*d*x)*b**3 + 540*e**(16*c + 16*d*x)*a*b**2 - 135*e**(16*c + 16*d*x)*b**3 + 3024*e**(14*c + 14*d*x)*a**2*b - 756*e**(14*c + 14*d*x)* a*b**2 + 6720*e**(12*c + 12*d*x)*a**3 + 5040*e**(12*c + 12*d*x)*a**2*b - 3 780*e**(12*c + 12*d*x)*a*b**2 + 840*e**(12*c + 12*d*x)*b**3 + 60480*e**(10 *c + 10*d*x)*a**3 - 30240*e**(10*c + 10*d*x)*a**2*b + 11340*e**(10*c + 10* d*x)*a*b**2 - 1890*e**(10*c + 10*d*x)*b**3 - 60480*e**(8*c + 8*d*x)*a**3 + 30240*e**(8*c + 8*d*x)*a**2*b - 11340*e**(8*c + 8*d*x)*a*b**2 + 1890*e**( 8*c + 8*d*x)*b**3 - 6720*e**(6*c + 6*d*x)*a**3 - 5040*e**(6*c + 6*d*x)*a** 2*b + 3780*e**(6*c + 6*d*x)*a*b**2 - 840*e**(6*c + 6*d*x)*b**3 - 3024*e**( 4*c + 4*d*x)*a**2*b + 756*e**(4*c + 4*d*x)*a*b**2 - 540*e**(2*c + 2*d*x)*a *b**2 + 135*e**(2*c + 2*d*x)*b**3 - 35*b**3)/(161280*e**(9*c + 9*d*x)*d)