Integrand size = 21, antiderivative size = 89 \[ \int \cosh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx=\frac {1}{16} (6 a-b) x+\frac {(6 a-b) \cosh (c+d x) \sinh (c+d x)}{16 d}+\frac {(6 a-b) \cosh ^3(c+d x) \sinh (c+d x)}{24 d}+\frac {b \cosh ^5(c+d x) \sinh (c+d x)}{6 d} \]
1/16*(6*a-b)*x+1/16*(6*a-b)*cosh(d*x+c)*sinh(d*x+c)/d+1/24*(6*a-b)*cosh(d* x+c)^3*sinh(d*x+c)/d+1/6*b*cosh(d*x+c)^5*sinh(d*x+c)/d
Time = 0.19 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.71 \[ \int \cosh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx=\frac {72 a c+72 a d x-12 b d x+(48 a-3 b) \sinh (2 (c+d x))+3 (2 a+b) \sinh (4 (c+d x))+b \sinh (6 (c+d x))}{192 d} \]
(72*a*c + 72*a*d*x - 12*b*d*x + (48*a - 3*b)*Sinh[2*(c + d*x)] + 3*(2*a + b)*Sinh[4*(c + d*x)] + b*Sinh[6*(c + d*x)])/(192*d)
Time = 0.24 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.19, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 3670, 298, 215, 215, 219}
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.
\(\displaystyle \int \cosh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \cos (i c+i d x)^4 \left (a-b \sin (i c+i d x)^2\right )dx\) |
\(\Big \downarrow \) 3670 |
\(\displaystyle \frac {\int \frac {a-(a-b) \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right )^4}d\tanh (c+d x)}{d}\) |
\(\Big \downarrow \) 298 |
\(\displaystyle \frac {\frac {1}{6} (6 a-b) \int \frac {1}{\left (1-\tanh ^2(c+d x)\right )^3}d\tanh (c+d x)+\frac {b \tanh (c+d x)}{6 \left (1-\tanh ^2(c+d x)\right )^3}}{d}\) |
\(\Big \downarrow \) 215 |
\(\displaystyle \frac {\frac {1}{6} (6 a-b) \left (\frac {3}{4} \int \frac {1}{\left (1-\tanh ^2(c+d x)\right )^2}d\tanh (c+d x)+\frac {\tanh (c+d x)}{4 \left (1-\tanh ^2(c+d x)\right )^2}\right )+\frac {b \tanh (c+d x)}{6 \left (1-\tanh ^2(c+d x)\right )^3}}{d}\) |
\(\Big \downarrow \) 215 |
\(\displaystyle \frac {\frac {1}{6} (6 a-b) \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)+\frac {\tanh (c+d x)}{2 \left (1-\tanh ^2(c+d x)\right )}\right )+\frac {\tanh (c+d x)}{4 \left (1-\tanh ^2(c+d x)\right )^2}\right )+\frac {b \tanh (c+d x)}{6 \left (1-\tanh ^2(c+d x)\right )^3}}{d}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\frac {1}{6} (6 a-b) \left (\frac {3}{4} \left (\frac {1}{2} \text {arctanh}(\tanh (c+d x))+\frac {\tanh (c+d x)}{2 \left (1-\tanh ^2(c+d x)\right )}\right )+\frac {\tanh (c+d x)}{4 \left (1-\tanh ^2(c+d x)\right )^2}\right )+\frac {b \tanh (c+d x)}{6 \left (1-\tanh ^2(c+d x)\right )^3}}{d}\) |
((b*Tanh[c + d*x])/(6*(1 - Tanh[c + d*x]^2)^3) + ((6*a - b)*(Tanh[c + d*x] /(4*(1 - Tanh[c + d*x]^2)^2) + (3*(ArcTanh[Tanh[c + d*x]]/2 + Tanh[c + d*x ]/(2*(1 - Tanh[c + d*x]^2))))/4))/6)/d
3.3.83.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 2*p + 3))/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
Int[cos[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^( p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff/f Su bst[Int[(a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^(m/2 + p + 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]
Time = 31.02 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.07
method | result | size |
derivativedivides | \(\frac {a \left (\left (\frac {\cosh \left (d x +c \right )^{3}}{4}+\frac {3 \cosh \left (d x +c \right )}{8}\right ) \sinh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )+b \left (\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )^{5}}{6}-\frac {\left (\frac {\cosh \left (d x +c \right )^{3}}{4}+\frac {3 \cosh \left (d x +c \right )}{8}\right ) \sinh \left (d x +c \right )}{6}-\frac {d x}{16}-\frac {c}{16}\right )}{d}\) | \(95\) |
default | \(\frac {a \left (\left (\frac {\cosh \left (d x +c \right )^{3}}{4}+\frac {3 \cosh \left (d x +c \right )}{8}\right ) \sinh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )+b \left (\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )^{5}}{6}-\frac {\left (\frac {\cosh \left (d x +c \right )^{3}}{4}+\frac {3 \cosh \left (d x +c \right )}{8}\right ) \sinh \left (d x +c \right )}{6}-\frac {d x}{16}-\frac {c}{16}\right )}{d}\) | \(95\) |
risch | \(\frac {3 a x}{8}-\frac {b x}{16}+\frac {b \,{\mathrm e}^{6 d x +6 c}}{384 d}+\frac {{\mathrm e}^{4 d x +4 c} a}{64 d}+\frac {{\mathrm e}^{4 d x +4 c} b}{128 d}+\frac {{\mathrm e}^{2 d x +2 c} a}{8 d}-\frac {{\mathrm e}^{2 d x +2 c} b}{128 d}-\frac {{\mathrm e}^{-2 d x -2 c} a}{8 d}+\frac {{\mathrm e}^{-2 d x -2 c} b}{128 d}-\frac {{\mathrm e}^{-4 d x -4 c} a}{64 d}-\frac {{\mathrm e}^{-4 d x -4 c} b}{128 d}-\frac {b \,{\mathrm e}^{-6 d x -6 c}}{384 d}\) | \(160\) |
1/d*(a*((1/4*cosh(d*x+c)^3+3/8*cosh(d*x+c))*sinh(d*x+c)+3/8*d*x+3/8*c)+b*( 1/6*sinh(d*x+c)*cosh(d*x+c)^5-1/6*(1/4*cosh(d*x+c)^3+3/8*cosh(d*x+c))*sinh (d*x+c)-1/16*d*x-1/16*c))
Time = 0.25 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.31 \[ \int \cosh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx=\frac {3 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + 2 \, {\left (5 \, b \cosh \left (d x + c\right )^{3} + 3 \, {\left (2 \, a + b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 6 \, {\left (6 \, a - b\right )} d x + 3 \, {\left (b \cosh \left (d x + c\right )^{5} + 2 \, {\left (2 \, a + b\right )} \cosh \left (d x + c\right )^{3} + {\left (16 \, a - b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{96 \, d} \]
1/96*(3*b*cosh(d*x + c)*sinh(d*x + c)^5 + 2*(5*b*cosh(d*x + c)^3 + 3*(2*a + b)*cosh(d*x + c))*sinh(d*x + c)^3 + 6*(6*a - b)*d*x + 3*(b*cosh(d*x + c) ^5 + 2*(2*a + b)*cosh(d*x + c)^3 + (16*a - b)*cosh(d*x + c))*sinh(d*x + c) )/d
Leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (76) = 152\).
Time = 0.35 (sec) , antiderivative size = 250, normalized size of antiderivative = 2.81 \[ \int \cosh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx=\begin {cases} \frac {3 a x \sinh ^{4}{\left (c + d x \right )}}{8} - \frac {3 a x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac {3 a x \cosh ^{4}{\left (c + d x \right )}}{8} - \frac {3 a \sinh ^{3}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{8 d} + \frac {5 a \sinh {\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} + \frac {b x \sinh ^{6}{\left (c + d x \right )}}{16} - \frac {3 b x \sinh ^{4}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{16} + \frac {3 b x \sinh ^{2}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{16} - \frac {b x \cosh ^{6}{\left (c + d x \right )}}{16} - \frac {b \sinh ^{5}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{16 d} + \frac {b \sinh ^{3}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{6 d} + \frac {b \sinh {\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{16 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\left (c \right )}\right ) \cosh ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((3*a*x*sinh(c + d*x)**4/8 - 3*a*x*sinh(c + d*x)**2*cosh(c + d*x) **2/4 + 3*a*x*cosh(c + d*x)**4/8 - 3*a*sinh(c + d*x)**3*cosh(c + d*x)/(8*d ) + 5*a*sinh(c + d*x)*cosh(c + d*x)**3/(8*d) + b*x*sinh(c + d*x)**6/16 - 3 *b*x*sinh(c + d*x)**4*cosh(c + d*x)**2/16 + 3*b*x*sinh(c + d*x)**2*cosh(c + d*x)**4/16 - b*x*cosh(c + d*x)**6/16 - b*sinh(c + d*x)**5*cosh(c + d*x)/ (16*d) + b*sinh(c + d*x)**3*cosh(c + d*x)**3/(6*d) + b*sinh(c + d*x)*cosh( c + d*x)**5/(16*d), Ne(d, 0)), (x*(a + b*sinh(c)**2)*cosh(c)**4, True))
Time = 0.21 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.71 \[ \int \cosh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx=\frac {1}{64} \, a {\left (24 \, x + \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} + \frac {8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} + \frac {1}{384} \, b {\left (\frac {{\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )} e^{\left (6 \, d x + 6 \, c\right )}}{d} - \frac {24 \, {\left (d x + c\right )}}{d} + \frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} - e^{\left (-6 \, d x - 6 \, c\right )}}{d}\right )} \]
1/64*a*(24*x + e^(4*d*x + 4*c)/d + 8*e^(2*d*x + 2*c)/d - 8*e^(-2*d*x - 2*c )/d - e^(-4*d*x - 4*c)/d) + 1/384*b*((3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4 *c) + 1)*e^(6*d*x + 6*c)/d - 24*(d*x + c)/d + (3*e^(-2*d*x - 2*c) - 3*e^(- 4*d*x - 4*c) - e^(-6*d*x - 6*c))/d)
Time = 0.28 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.36 \[ \int \cosh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx=\frac {1}{16} \, {\left (6 \, a - b\right )} x + \frac {b e^{\left (6 \, d x + 6 \, c\right )}}{384 \, d} + \frac {{\left (2 \, a + b\right )} e^{\left (4 \, d x + 4 \, c\right )}}{128 \, d} + \frac {{\left (16 \, a - b\right )} e^{\left (2 \, d x + 2 \, c\right )}}{128 \, d} - \frac {{\left (16 \, a - b\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{128 \, d} - \frac {{\left (2 \, a + b\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{128 \, d} - \frac {b e^{\left (-6 \, d x - 6 \, c\right )}}{384 \, d} \]
1/16*(6*a - b)*x + 1/384*b*e^(6*d*x + 6*c)/d + 1/128*(2*a + b)*e^(4*d*x + 4*c)/d + 1/128*(16*a - b)*e^(2*d*x + 2*c)/d - 1/128*(16*a - b)*e^(-2*d*x - 2*c)/d - 1/128*(2*a + b)*e^(-4*d*x - 4*c)/d - 1/384*b*e^(-6*d*x - 6*c)/d
Time = 2.66 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.85 \[ \int \cosh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx=\frac {12\,a\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )+\frac {3\,a\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )}{2}-\frac {3\,b\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )}{4}+\frac {3\,b\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )}{4}+\frac {b\,\mathrm {sinh}\left (6\,c+6\,d\,x\right )}{4}+18\,a\,d\,x-3\,b\,d\,x}{48\,d} \]