Integrand size = 21, antiderivative size = 83 \[ \int \sinh ^2(c+d x) \left (a+b \sinh ^4(c+d x)\right ) \, dx=-\frac {1}{16} (8 a+5 b) x+\frac {(8 a+11 b) \cosh (c+d x) \sinh (c+d x)}{16 d}-\frac {13 b \cosh ^3(c+d x) \sinh (c+d x)}{24 d}+\frac {b \cosh ^5(c+d x) \sinh (c+d x)}{6 d} \]
-1/16*(8*a+5*b)*x+1/16*(8*a+11*b)*cosh(d*x+c)*sinh(d*x+c)/d-13/24*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 = 0.05 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.76 \[ \int \sinh ^2(c+d x) \left (a+b \sinh ^4(c+d x)\right ) \, dx=\frac {-96 a c-60 b c-96 a d x-60 b d x+(48 a+45 b) \sinh (2 (c+d x))-9 b \sinh (4 (c+d x))+b \sinh (6 (c+d x))}{192 d} \]
(-96*a*c - 60*b*c - 96*a*d*x - 60*b*d*x + (48*a + 45*b)*Sinh[2*(c + d*x)] - 9*b*Sinh[4*(c + d*x)] + b*Sinh[6*(c + d*x)])/(192*d)
Time = 0.33 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.37, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 25, 3696, 1580, 25, 1471, 27, 298, 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.
| \(\displaystyle \int \sinh ^2(c+d x) \left (a+b \sinh ^4(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\sin (i c+i d x)^2 \left (a+b \sin (i c+i d x)^4\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \sin (i c+i d x)^2 \left (b \sin (i c+i d x)^4+a\right )dx\) |
| \(\Big \downarrow \) 3696 |
| \(\displaystyle \frac {\int \frac {\tanh ^2(c+d x) \left ((a+b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}{\left (1-\tanh ^2(c+d x)\right )^4}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 1580 |
| \(\displaystyle \frac {\frac {1}{6} \int -\frac {6 (a+b) \tanh ^4(c+d x)-6 (a-b) \tanh ^2(c+d x)+b}{\left (1-\tanh ^2(c+d x)\right )^3}d\tanh (c+d x)+\frac {b \tanh (c+d x)}{6 \left (1-\tanh ^2(c+d x)\right )^3}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {b \tanh (c+d x)}{6 \left (1-\tanh ^2(c+d x)\right )^3}-\frac {1}{6} \int \frac {6 (a+b) \tanh ^4(c+d x)-6 (a-b) \tanh ^2(c+d x)+b}{\left (1-\tanh ^2(c+d x)\right )^3}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 1471 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{4} \int \frac {3 \left (8 (a+b) \tanh ^2(c+d x)+3 b\right )}{\left (1-\tanh ^2(c+d x)\right )^2}d\tanh (c+d x)-\frac {13 b \tanh (c+d x)}{4 \left (1-\tanh ^2(c+d x)\right )^2}\right )+\frac {b \tanh (c+d x)}{6 \left (1-\tanh ^2(c+d x)\right )^3}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{4} \int \frac {8 (a+b) \tanh ^2(c+d x)+3 b}{\left (1-\tanh ^2(c+d x)\right )^2}d\tanh (c+d x)-\frac {13 b \tanh (c+d x)}{4 \left (1-\tanh ^2(c+d x)\right )^2}\right )+\frac {b \tanh (c+d x)}{6 \left (1-\tanh ^2(c+d x)\right )^3}}{d}\) |
| \(\Big \downarrow \) 298 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{4} \left (\frac {(8 a+11 b) \tanh (c+d x)}{2 \left (1-\tanh ^2(c+d x)\right )}-\frac {1}{2} (8 a+5 b) \int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)\right )-\frac {13 b \tanh (c+d x)}{4 \left (1-\tanh ^2(c+d x)\right )^2}\right )+\frac {b \tanh (c+d x)}{6 \left (1-\tanh ^2(c+d x)\right )^3}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{4} \left (\frac {(8 a+11 b) \tanh (c+d x)}{2 \left (1-\tanh ^2(c+d x)\right )}-\frac {1}{2} (8 a+5 b) \text {arctanh}(\tanh (c+d x))\right )-\frac {13 b \tanh (c+d x)}{4 \left (1-\tanh ^2(c+d x)\right )^2}\right )+\frac {b \tanh (c+d x)}{6 \left (1-\tanh ^2(c+d x)\right )^3}}{d}\) |
((b*Tanh[c + d*x])/(6*(1 - Tanh[c + d*x]^2)^3) + ((-13*b*Tanh[c + d*x])/(4 *(1 - Tanh[c + d*x]^2)^2) + (3*(-1/2*((8*a + 5*b)*ArcTanh[Tanh[c + d*x]]) + ((8*a + 11*b)*Tanh[c + d*x])/(2*(1 - Tanh[c + d*x]^2))))/4)/6)/d
3.2.86.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, d + e*x^2 , x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], x , 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Simp[1/(2*d*(q + 1)) Int[(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^ 2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_) ^4)^(p_.), x_Symbol] :> Simp[(-d)^(m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*((d + e*x^2)^(q + 1)/(2*e^(2*p + m/2)*(q + 1))), x] + Simp[1/(2*e^(2*p + m/2)* (q + 1)) Int[(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1/(d + e*x^2))*(2* e^(2*p + m/2)*(q + 1)*x^m*(a + b*x^2 + c*x^4)^p - (-d)^(m/2 - 1)*(c*d^2 - b *d*e + a*e^2)^p*(d + e*(2*q + 3)*x^2))], x], x], x] /; FreeQ[{a, b, c, d, e }, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && IGtQ[m/2, 0]
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^( p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff^(m + 1 )/f Subst[Int[x^m*((a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p/(1 + ff^2*x^2) ^(m/2 + 2*p + 1)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] & & IntegerQ[m/2] && IntegerQ[p]
Time = 0.44 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.67
| method | result | size |
| parallelrisch | \(\frac {\left (48 a +45 b \right ) \sinh \left (2 d x +2 c \right )-9 b \sinh \left (4 d x +4 c \right )+b \sinh \left (6 d x +6 c \right )-96 d \left (a +\frac {5 b}{8}\right ) x}{192 d}\) | \(56\) |
| derivativedivides | \(\frac {a \left (\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )+b \left (\left (\frac {\sinh \left (d x +c \right )^{5}}{6}-\frac {5 \sinh \left (d x +c \right )^{3}}{24}+\frac {5 \sinh \left (d x +c \right )}{16}\right ) \cosh \left (d x +c \right )-\frac {5 d x}{16}-\frac {5 c}{16}\right )}{d}\) | \(76\) |
| default | \(\frac {a \left (\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )+b \left (\left (\frac {\sinh \left (d x +c \right )^{5}}{6}-\frac {5 \sinh \left (d x +c \right )^{3}}{24}+\frac {5 \sinh \left (d x +c \right )}{16}\right ) \cosh \left (d x +c \right )-\frac {5 d x}{16}-\frac {5 c}{16}\right )}{d}\) | \(76\) |
| parts | \(\frac {a \left (\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )}{d}+\frac {b \left (\left (\frac {\sinh \left (d x +c \right )^{5}}{6}-\frac {5 \sinh \left (d x +c \right )^{3}}{24}+\frac {5 \sinh \left (d x +c \right )}{16}\right ) \cosh \left (d x +c \right )-\frac {5 d x}{16}-\frac {5 c}{16}\right )}{d}\) | \(78\) |
| risch | \(-\frac {a x}{2}-\frac {5 b x}{16}+\frac {b \,{\mathrm e}^{6 d x +6 c}}{384 d}-\frac {3 \,{\mathrm e}^{4 d x +4 c} b}{128 d}+\frac {{\mathrm e}^{2 d x +2 c} a}{8 d}+\frac {15 \,{\mathrm e}^{2 d x +2 c} b}{128 d}-\frac {{\mathrm e}^{-2 d x -2 c} a}{8 d}-\frac {15 \,{\mathrm e}^{-2 d x -2 c} b}{128 d}+\frac {3 \,{\mathrm e}^{-4 d x -4 c} b}{128 d}-\frac {b \,{\mathrm e}^{-6 d x -6 c}}{384 d}\) | \(130\) |
1/192*((48*a+45*b)*sinh(2*d*x+2*c)-9*b*sinh(4*d*x+4*c)+b*sinh(6*d*x+6*c)-9 6*d*(a+5/8*b)*x)/d
Time = 0.26 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.31 \[ \int \sinh ^2(c+d x) \left (a+b \sinh ^4(c+d x)\right ) \, dx=\frac {3 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + 2 \, {\left (5 \, b \cosh \left (d x + c\right )^{3} - 9 \, b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - 6 \, {\left (8 \, a + 5 \, b\right )} d x + 3 \, {\left (b \cosh \left (d x + c\right )^{5} - 6 \, b \cosh \left (d x + c\right )^{3} + {\left (16 \, a + 15 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{96 \, d} \]
1/96*(3*b*cosh(d*x + c)*sinh(d*x + c)^5 + 2*(5*b*cosh(d*x + c)^3 - 9*b*cos h(d*x + c))*sinh(d*x + c)^3 - 6*(8*a + 5*b)*d*x + 3*(b*cosh(d*x + c)^5 - 6 *b*cosh(d*x + c)^3 + (16*a + 15*b)*cosh(d*x + c))*sinh(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (76) = 152\).
Time = 0.35 (sec) , antiderivative size = 206, normalized size of antiderivative = 2.48 \[ \int \sinh ^2(c+d x) \left (a+b \sinh ^4(c+d x)\right ) \, dx=\begin {cases} \frac {a x \sinh ^{2}{\left (c + d x \right )}}{2} - \frac {a x \cosh ^{2}{\left (c + d x \right )}}{2} + \frac {a \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{2 d} + \frac {5 b x \sinh ^{6}{\left (c + d x \right )}}{16} - \frac {15 b x \sinh ^{4}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{16} + \frac {15 b x \sinh ^{2}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{16} - \frac {5 b x \cosh ^{6}{\left (c + d x \right )}}{16} + \frac {11 b \sinh ^{5}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{16 d} - \frac {5 b \sinh ^{3}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{6 d} + \frac {5 b \sinh {\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{16 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{4}{\left (c \right )}\right ) \sinh ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((a*x*sinh(c + d*x)**2/2 - a*x*cosh(c + d*x)**2/2 + a*sinh(c + d* x)*cosh(c + d*x)/(2*d) + 5*b*x*sinh(c + d*x)**6/16 - 15*b*x*sinh(c + d*x)* *4*cosh(c + d*x)**2/16 + 15*b*x*sinh(c + d*x)**2*cosh(c + d*x)**4/16 - 5*b *x*cosh(c + d*x)**6/16 + 11*b*sinh(c + d*x)**5*cosh(c + d*x)/(16*d) - 5*b* sinh(c + d*x)**3*cosh(c + d*x)**3/(6*d) + 5*b*sinh(c + d*x)*cosh(c + d*x)* *5/(16*d), Ne(d, 0)), (x*(a + b*sinh(c)**4)*sinh(c)**2, True))
Time = 0.19 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.47 \[ \int \sinh ^2(c+d x) \left (a+b \sinh ^4(c+d x)\right ) \, dx=-\frac {1}{8} \, a {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac {1}{384} \, b {\left (\frac {{\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 45 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )} e^{\left (6 \, d x + 6 \, c\right )}}{d} + \frac {120 \, {\left (d x + c\right )}}{d} + \frac {45 \, e^{\left (-2 \, d x - 2 \, c\right )} - 9 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}}{d}\right )} \]
-1/8*a*(4*x - e^(2*d*x + 2*c)/d + e^(-2*d*x - 2*c)/d) - 1/384*b*((9*e^(-2* d*x - 2*c) - 45*e^(-4*d*x - 4*c) - 1)*e^(6*d*x + 6*c)/d + 120*(d*x + c)/d + (45*e^(-2*d*x - 2*c) - 9*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c))/d)
Time = 0.28 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.36 \[ \int \sinh ^2(c+d x) \left (a+b \sinh ^4(c+d x)\right ) \, dx=-\frac {1}{16} \, {\left (8 \, a + 5 \, b\right )} x + \frac {b e^{\left (6 \, d x + 6 \, c\right )}}{384 \, d} - \frac {3 \, b e^{\left (4 \, d x + 4 \, c\right )}}{128 \, d} + \frac {{\left (16 \, a + 15 \, b\right )} e^{\left (2 \, d x + 2 \, c\right )}}{128 \, d} - \frac {{\left (16 \, a + 15 \, b\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{128 \, d} + \frac {3 \, b e^{\left (-4 \, d x - 4 \, c\right )}}{128 \, d} - \frac {b e^{\left (-6 \, d x - 6 \, c\right )}}{384 \, d} \]
-1/16*(8*a + 5*b)*x + 1/384*b*e^(6*d*x + 6*c)/d - 3/128*b*e^(4*d*x + 4*c)/ d + 1/128*(16*a + 15*b)*e^(2*d*x + 2*c)/d - 1/128*(16*a + 15*b)*e^(-2*d*x - 2*c)/d + 3/128*b*e^(-4*d*x - 4*c)/d - 1/384*b*e^(-6*d*x - 6*c)/d
Time = 0.16 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.77 \[ \int \sinh ^2(c+d x) \left (a+b \sinh ^4(c+d x)\right ) \, dx=\frac {12\,a\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )+\frac {45\,b\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )}{4}-\frac {9\,b\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )}{4}+\frac {b\,\mathrm {sinh}\left (6\,c+6\,d\,x\right )}{4}-24\,a\,d\,x-15\,b\,d\,x}{48\,d} \]