Integrand size = 23, antiderivative size = 74 \[ \int \cosh ^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {a^2 \sinh (c+d x)}{d}+\frac {a (a+2 b) \sinh ^3(c+d x)}{3 d}+\frac {b (2 a+b) \sinh ^5(c+d x)}{5 d}+\frac {b^2 \sinh ^7(c+d x)}{7 d} \] Output:
a^2*sinh(d*x+c)/d+1/3*a*(a+2*b)*sinh(d*x+c)^3/d+1/5*b*(2*a+b)*sinh(d*x+c)^ 5/d+1/7*b^2*sinh(d*x+c)^7/d
Time = 0.09 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.86 \[ \int \cosh ^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {105 a^2 \sinh (c+d x)+35 a (a+2 b) \sinh ^3(c+d x)+21 b (2 a+b) \sinh ^5(c+d x)+15 b^2 \sinh ^7(c+d x)}{105 d} \] Input:
Integrate[Cosh[c + d*x]^3*(a + b*Sinh[c + d*x]^2)^2,x]
Output:
(105*a^2*Sinh[c + d*x] + 35*a*(a + 2*b)*Sinh[c + d*x]^3 + 21*b*(2*a + b)*S inh[c + d*x]^5 + 15*b^2*Sinh[c + d*x]^7)/(105*d)
Time = 0.29 (sec) , antiderivative size = 66, 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.
\(\displaystyle \int \cosh ^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \cos (i c+i d x)^3 \left (a-b \sin (i c+i d x)^2\right )^2dx\) |
\(\Big \downarrow \) 3669 |
\(\displaystyle \frac {\int \left (\sinh ^2(c+d x)+1\right ) \left (b \sinh ^2(c+d x)+a\right )^2d\sinh (c+d x)}{d}\) |
\(\Big \downarrow \) 290 |
\(\displaystyle \frac {\int \left (b^2 \sinh ^6(c+d x)+b (2 a+b) \sinh ^4(c+d x)+a (a+2 b) \sinh ^2(c+d x)+a^2\right )d\sinh (c+d x)}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^2 \sinh (c+d x)+\frac {1}{5} b (2 a+b) \sinh ^5(c+d x)+\frac {1}{3} a (a+2 b) \sinh ^3(c+d x)+\frac {1}{7} b^2 \sinh ^7(c+d x)}{d}\) |
Input:
Int[Cosh[c + d*x]^3*(a + b*Sinh[c + d*x]^2)^2,x]
Output:
(a^2*Sinh[c + d*x] + (a*(a + 2*b)*Sinh[c + d*x]^3)/3 + (b*(2*a + b)*Sinh[c + d*x]^5)/5 + (b^2*Sinh[c + d*x]^7)/7)/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 = 65, normalized size of antiderivative = 0.88
\[\frac {\frac {b^{2} \sinh \left (d x +c \right )^{7}}{7}+\frac {\left (2 a b +b^{2}\right ) \sinh \left (d x +c \right )^{5}}{5}+\frac {\left (a^{2}+2 a b \right ) \sinh \left (d x +c \right )^{3}}{3}+\sinh \left (d x +c \right ) a^{2}}{d}\]
Input:
int(cosh(d*x+c)^3*(a+b*sinh(d*x+c)^2)^2,x)
Output:
1/d*(1/7*b^2*sinh(d*x+c)^7+1/5*(2*a*b+b^2)*sinh(d*x+c)^5+1/3*(a^2+2*a*b)*s inh(d*x+c)^3+sinh(d*x+c)*a^2)
Leaf count of result is larger than twice the leaf count of optimal. 188 vs. \(2 (68) = 136\).
Time = 0.10 (sec) , antiderivative size = 188, normalized size of antiderivative = 2.54 \[ \int \cosh ^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {15 \, b^{2} \sinh \left (d x + c\right )^{7} + 21 \, {\left (15 \, b^{2} \cosh \left (d x + c\right )^{2} + 8 \, a b - b^{2}\right )} \sinh \left (d x + c\right )^{5} + 35 \, {\left (15 \, b^{2} \cosh \left (d x + c\right )^{4} + 6 \, {\left (8 \, a b - b^{2}\right )} \cosh \left (d x + c\right )^{2} + 16 \, a^{2} + 8 \, a b - 3 \, b^{2}\right )} \sinh \left (d x + c\right )^{3} + 105 \, {\left (b^{2} \cosh \left (d x + c\right )^{6} + {\left (8 \, a b - b^{2}\right )} \cosh \left (d x + c\right )^{4} + {\left (16 \, a^{2} + 8 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 48 \, a^{2} - 16 \, a b + 3 \, b^{2}\right )} \sinh \left (d x + c\right )}{6720 \, d} \] Input:
integrate(cosh(d*x+c)^3*(a+b*sinh(d*x+c)^2)^2,x, algorithm="fricas")
Output:
1/6720*(15*b^2*sinh(d*x + c)^7 + 21*(15*b^2*cosh(d*x + c)^2 + 8*a*b - b^2) *sinh(d*x + c)^5 + 35*(15*b^2*cosh(d*x + c)^4 + 6*(8*a*b - b^2)*cosh(d*x + c)^2 + 16*a^2 + 8*a*b - 3*b^2)*sinh(d*x + c)^3 + 105*(b^2*cosh(d*x + c)^6 + (8*a*b - b^2)*cosh(d*x + c)^4 + (16*a^2 + 8*a*b - 3*b^2)*cosh(d*x + c)^ 2 + 48*a^2 - 16*a*b + 3*b^2)*sinh(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (63) = 126\).
Time = 0.45 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.84 \[ \int \cosh ^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\begin {cases} - \frac {2 a^{2} \sinh ^{3}{\left (c + d x \right )}}{3 d} + \frac {a^{2} \sinh {\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{d} - \frac {4 a b \sinh ^{5}{\left (c + d x \right )}}{15 d} + \frac {2 a b \sinh ^{3}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{3 d} - \frac {2 b^{2} \sinh ^{7}{\left (c + d x \right )}}{35 d} + \frac {b^{2} \sinh ^{5}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{5 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\left (c \right )}\right )^{2} \cosh ^{3}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cosh(d*x+c)**3*(a+b*sinh(d*x+c)**2)**2,x)
Output:
Piecewise((-2*a**2*sinh(c + d*x)**3/(3*d) + a**2*sinh(c + d*x)*cosh(c + d* x)**2/d - 4*a*b*sinh(c + d*x)**5/(15*d) + 2*a*b*sinh(c + d*x)**3*cosh(c + d*x)**2/(3*d) - 2*b**2*sinh(c + d*x)**7/(35*d) + b**2*sinh(c + d*x)**5*cos h(c + d*x)**2/(5*d), Ne(d, 0)), (x*(a + b*sinh(c)**2)**2*cosh(c)**3, True) )
Leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (68) = 136\).
Time = 0.04 (sec) , antiderivative size = 242, normalized size of antiderivative = 3.27 \[ \int \cosh ^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=-\frac {1}{4480} \, b^{2} {\left (\frac {{\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} + 35 \, e^{\left (-4 \, d x - 4 \, c\right )} - 105 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5\right )} e^{\left (7 \, d x + 7 \, c\right )}}{d} + \frac {105 \, e^{\left (-d x - c\right )} - 35 \, e^{\left (-3 \, d x - 3 \, c\right )} - 7 \, e^{\left (-5 \, d x - 5 \, c\right )} + 5 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d}\right )} + \frac {1}{240} \, a b {\left (\frac {{\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 30 \, e^{\left (-4 \, d x - 4 \, c\right )} + 3\right )} e^{\left (5 \, d x + 5 \, c\right )}}{d} + \frac {30 \, e^{\left (-d x - c\right )} - 5 \, e^{\left (-3 \, d x - 3 \, c\right )} - 3 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d}\right )} + \frac {1}{24} \, a^{2} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} - \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} \] Input:
integrate(cosh(d*x+c)^3*(a+b*sinh(d*x+c)^2)^2,x, algorithm="maxima")
Output:
-1/4480*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) + 1/240*a*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^2*(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. 196 vs. \(2 (68) = 136\).
Time = 0.14 (sec) , antiderivative size = 196, normalized size of antiderivative = 2.65 \[ \int \cosh ^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {b^{2} e^{\left (7 \, d x + 7 \, c\right )}}{896 \, d} - \frac {b^{2} e^{\left (-7 \, d x - 7 \, c\right )}}{896 \, d} + \frac {{\left (8 \, a b - b^{2}\right )} e^{\left (5 \, d x + 5 \, c\right )}}{640 \, d} + \frac {{\left (16 \, a^{2} + 8 \, a b - 3 \, b^{2}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{384 \, d} + \frac {{\left (48 \, a^{2} - 16 \, a b + 3 \, b^{2}\right )} e^{\left (d x + c\right )}}{128 \, d} - \frac {{\left (48 \, a^{2} - 16 \, a b + 3 \, b^{2}\right )} e^{\left (-d x - c\right )}}{128 \, d} - \frac {{\left (16 \, a^{2} + 8 \, a b - 3 \, b^{2}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{384 \, d} - \frac {{\left (8 \, a b - b^{2}\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{640 \, d} \] Input:
integrate(cosh(d*x+c)^3*(a+b*sinh(d*x+c)^2)^2,x, algorithm="giac")
Output:
1/896*b^2*e^(7*d*x + 7*c)/d - 1/896*b^2*e^(-7*d*x - 7*c)/d + 1/640*(8*a*b - b^2)*e^(5*d*x + 5*c)/d + 1/384*(16*a^2 + 8*a*b - 3*b^2)*e^(3*d*x + 3*c)/ d + 1/128*(48*a^2 - 16*a*b + 3*b^2)*e^(d*x + c)/d - 1/128*(48*a^2 - 16*a*b + 3*b^2)*e^(-d*x - c)/d - 1/384*(16*a^2 + 8*a*b - 3*b^2)*e^(-3*d*x - 3*c) /d - 1/640*(8*a*b - b^2)*e^(-5*d*x - 5*c)/d
Time = 0.22 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.08 \[ \int \cosh ^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {35\,a^2\,{\mathrm {sinh}\left (c+d\,x\right )}^3+105\,a^2\,\mathrm {sinh}\left (c+d\,x\right )+42\,a\,b\,{\mathrm {sinh}\left (c+d\,x\right )}^5+70\,a\,b\,{\mathrm {sinh}\left (c+d\,x\right )}^3+15\,b^2\,{\mathrm {sinh}\left (c+d\,x\right )}^7+21\,b^2\,{\mathrm {sinh}\left (c+d\,x\right )}^5}{105\,d} \] Input:
int(cosh(c + d*x)^3*(a + b*sinh(c + d*x)^2)^2,x)
Output:
(105*a^2*sinh(c + d*x) + 35*a^2*sinh(c + d*x)^3 + 21*b^2*sinh(c + d*x)^5 + 15*b^2*sinh(c + d*x)^7 + 70*a*b*sinh(c + d*x)^3 + 42*a*b*sinh(c + d*x)^5) /(105*d)
Time = 0.17 (sec) , antiderivative size = 272, normalized size of antiderivative = 3.68 \[ \int \cosh ^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {15 e^{14 d x +14 c} b^{2}+168 e^{12 d x +12 c} a b -21 e^{12 d x +12 c} b^{2}+560 e^{10 d x +10 c} a^{2}+280 e^{10 d x +10 c} a b -105 e^{10 d x +10 c} b^{2}+5040 e^{8 d x +8 c} a^{2}-1680 e^{8 d x +8 c} a b +315 e^{8 d x +8 c} b^{2}-5040 e^{6 d x +6 c} a^{2}+1680 e^{6 d x +6 c} a b -315 e^{6 d x +6 c} b^{2}-560 e^{4 d x +4 c} a^{2}-280 e^{4 d x +4 c} a b +105 e^{4 d x +4 c} b^{2}-168 e^{2 d x +2 c} a b +21 e^{2 d x +2 c} b^{2}-15 b^{2}}{13440 e^{7 d x +7 c} d} \] Input:
int(cosh(d*x+c)^3*(a+b*sinh(d*x+c)^2)^2,x)
Output:
(15*e**(14*c + 14*d*x)*b**2 + 168*e**(12*c + 12*d*x)*a*b - 21*e**(12*c + 1 2*d*x)*b**2 + 560*e**(10*c + 10*d*x)*a**2 + 280*e**(10*c + 10*d*x)*a*b - 1 05*e**(10*c + 10*d*x)*b**2 + 5040*e**(8*c + 8*d*x)*a**2 - 1680*e**(8*c + 8 *d*x)*a*b + 315*e**(8*c + 8*d*x)*b**2 - 5040*e**(6*c + 6*d*x)*a**2 + 1680* e**(6*c + 6*d*x)*a*b - 315*e**(6*c + 6*d*x)*b**2 - 560*e**(4*c + 4*d*x)*a* *2 - 280*e**(4*c + 4*d*x)*a*b + 105*e**(4*c + 4*d*x)*b**2 - 168*e**(2*c + 2*d*x)*a*b + 21*e**(2*c + 2*d*x)*b**2 - 15*b**2)/(13440*e**(7*c + 7*d*x)*d )