Integrand size = 16, antiderivative size = 134 \[ \int (c+d x)^3 \cosh ^2(a+b x) \, dx=\frac {3 c d^2 x}{4 b^2}+\frac {3 d^3 x^2}{8 b^2}+\frac {(c+d x)^4}{8 d}-\frac {3 d^3 \cosh ^2(a+b x)}{8 b^4}-\frac {3 d (c+d x)^2 \cosh ^2(a+b x)}{4 b^2}+\frac {3 d^2 (c+d x) \cosh (a+b x) \sinh (a+b x)}{4 b^3}+\frac {(c+d x)^3 \cosh (a+b x) \sinh (a+b x)}{2 b} \]
3/4*c*d^2*x/b^2+3/8*d^3*x^2/b^2+1/8*(d*x+c)^4/d-3/8*d^3*cosh(b*x+a)^2/b^4- 3/4*d*(d*x+c)^2*cosh(b*x+a)^2/b^2+3/4*d^2*(d*x+c)*cosh(b*x+a)*sinh(b*x+a)/ b^3+1/2*(d*x+c)^3*cosh(b*x+a)*sinh(b*x+a)/b
Time = 0.27 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.78 \[ \int (c+d x)^3 \cosh ^2(a+b x) \, dx=\frac {2 b^4 x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )-3 d \left (d^2+2 b^2 (c+d x)^2\right ) \cosh (2 (a+b x))+2 b (c+d x) \left (3 d^2+2 b^2 (c+d x)^2\right ) \sinh (2 (a+b x))}{16 b^4} \]
(2*b^4*x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3) - 3*d*(d^2 + 2*b^2*(c + d*x)^2)*Cosh[2*(a + b*x)] + 2*b*(c + d*x)*(3*d^2 + 2*b^2*(c + d*x)^2)*S inh[2*(a + b*x)])/(16*b^4)
Time = 0.36 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3042, 3792, 17, 3042, 3791, 17}
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.
\(\displaystyle \int (c+d x)^3 \cosh ^2(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (c+d x)^3 \sin \left (i a+i b x+\frac {\pi }{2}\right )^2dx\) |
\(\Big \downarrow \) 3792 |
\(\displaystyle \frac {3 d^2 \int (c+d x) \cosh ^2(a+b x)dx}{2 b^2}+\frac {1}{2} \int (c+d x)^3dx-\frac {3 d (c+d x)^2 \cosh ^2(a+b x)}{4 b^2}+\frac {(c+d x)^3 \sinh (a+b x) \cosh (a+b x)}{2 b}\) |
\(\Big \downarrow \) 17 |
\(\displaystyle \frac {3 d^2 \int (c+d x) \cosh ^2(a+b x)dx}{2 b^2}-\frac {3 d (c+d x)^2 \cosh ^2(a+b x)}{4 b^2}+\frac {(c+d x)^3 \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {(c+d x)^4}{8 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3 d^2 \int (c+d x) \sin \left (i a+i b x+\frac {\pi }{2}\right )^2dx}{2 b^2}-\frac {3 d (c+d x)^2 \cosh ^2(a+b x)}{4 b^2}+\frac {(c+d x)^3 \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {(c+d x)^4}{8 d}\) |
\(\Big \downarrow \) 3791 |
\(\displaystyle \frac {3 d^2 \left (\frac {1}{2} \int (c+d x)dx-\frac {d \cosh ^2(a+b x)}{4 b^2}+\frac {(c+d x) \sinh (a+b x) \cosh (a+b x)}{2 b}\right )}{2 b^2}-\frac {3 d (c+d x)^2 \cosh ^2(a+b x)}{4 b^2}+\frac {(c+d x)^3 \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {(c+d x)^4}{8 d}\) |
\(\Big \downarrow \) 17 |
\(\displaystyle \frac {3 d^2 \left (-\frac {d \cosh ^2(a+b x)}{4 b^2}+\frac {(c+d x) \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {(c+d x)^2}{4 d}\right )}{2 b^2}-\frac {3 d (c+d x)^2 \cosh ^2(a+b x)}{4 b^2}+\frac {(c+d x)^3 \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {(c+d x)^4}{8 d}\) |
(c + d*x)^4/(8*d) - (3*d*(c + d*x)^2*Cosh[a + b*x]^2)/(4*b^2) + ((c + d*x) ^3*Cosh[a + b*x]*Sinh[a + b*x])/(2*b) + (3*d^2*((c + d*x)^2/(4*d) - (d*Cos h[a + b*x]^2)/(4*b^2) + ((c + d*x)*Cosh[a + b*x]*Sinh[a + b*x])/(2*b)))/(2 *b^2)
3.1.9.3.1 Defintions of rubi rules used
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Simp[b*(c + d*x)*Cos[e + f*x ]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^2*((n - 1)/n) Int[(c + d* x)*(b*Sin[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]
Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 2*((n - 1)/n) Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 *m*((m - 1)/(f^2*n^2)) Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
Time = 0.28 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.90
method | result | size |
parallelrisch | \(\frac {4 \left (d x +c \right ) \sinh \left (2 b x +2 a \right ) b \left (\left (d x +c \right )^{2} b^{2}+\frac {3 d^{2}}{2}\right )-6 \left (\left (d x +c \right )^{2} b^{2}+\frac {d^{2}}{2}\right ) d \cosh \left (2 b x +2 a \right )+2 \left (d^{3} x^{4}+4 d^{2} c \,x^{3}+6 d \,c^{2} x^{2}+4 c^{3} x \right ) b^{4}+6 b^{2} c^{2} d +3 d^{3}}{16 b^{4}}\) | \(121\) |
risch | \(\frac {d^{3} x^{4}}{8}+\frac {d^{2} c \,x^{3}}{2}+\frac {3 d \,c^{2} x^{2}}{4}+\frac {c^{3} x}{2}+\frac {c^{4}}{8 d}+\frac {\left (4 d^{3} x^{3} b^{3}+12 b^{3} c \,d^{2} x^{2}+12 b^{3} c^{2} d x -6 b^{2} d^{3} x^{2}+4 b^{3} c^{3}-12 b^{2} c \,d^{2} x -6 b^{2} c^{2} d +6 b \,d^{3} x +6 b c \,d^{2}-3 d^{3}\right ) {\mathrm e}^{2 b x +2 a}}{32 b^{4}}-\frac {\left (4 d^{3} x^{3} b^{3}+12 b^{3} c \,d^{2} x^{2}+12 b^{3} c^{2} d x +6 b^{2} d^{3} x^{2}+4 b^{3} c^{3}+12 b^{2} c \,d^{2} x +6 b^{2} c^{2} d +6 b \,d^{3} x +6 b c \,d^{2}+3 d^{3}\right ) {\mathrm e}^{-2 b x -2 a}}{32 b^{4}}\) | \(252\) |
derivativedivides | \(\frac {\frac {d^{3} \left (\frac {\left (b x +a \right )^{3} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {\left (b x +a \right )^{4}}{8}-\frac {3 \left (b x +a \right )^{2} \cosh \left (b x +a \right )^{2}}{4}+\frac {3 \left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{4}+\frac {3 \left (b x +a \right )^{2}}{8}-\frac {3 \cosh \left (b x +a \right )^{2}}{8}\right )}{b^{3}}-\frac {3 d^{3} a \left (\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {\left (b x +a \right )^{3}}{6}-\frac {\left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{2}+\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{4}+\frac {b x}{4}+\frac {a}{4}\right )}{b^{3}}+\frac {3 d^{2} c \left (\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {\left (b x +a \right )^{3}}{6}-\frac {\left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{2}+\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{4}+\frac {b x}{4}+\frac {a}{4}\right )}{b^{2}}+\frac {3 d^{3} a^{2} \left (\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {\left (b x +a \right )^{2}}{4}-\frac {\cosh \left (b x +a \right )^{2}}{4}\right )}{b^{3}}-\frac {6 d^{2} a c \left (\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {\left (b x +a \right )^{2}}{4}-\frac {\cosh \left (b x +a \right )^{2}}{4}\right )}{b^{2}}+\frac {3 d \,c^{2} \left (\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {\left (b x +a \right )^{2}}{4}-\frac {\cosh \left (b x +a \right )^{2}}{4}\right )}{b}-\frac {d^{3} a^{3} \left (\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{b^{3}}+\frac {3 d^{2} a^{2} c \left (\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{b^{2}}-\frac {3 d a \,c^{2} \left (\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{b}+c^{3} \left (\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{b}\) | \(523\) |
default | \(\frac {\frac {d^{3} \left (\frac {\left (b x +a \right )^{3} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {\left (b x +a \right )^{4}}{8}-\frac {3 \left (b x +a \right )^{2} \cosh \left (b x +a \right )^{2}}{4}+\frac {3 \left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{4}+\frac {3 \left (b x +a \right )^{2}}{8}-\frac {3 \cosh \left (b x +a \right )^{2}}{8}\right )}{b^{3}}-\frac {3 d^{3} a \left (\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {\left (b x +a \right )^{3}}{6}-\frac {\left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{2}+\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{4}+\frac {b x}{4}+\frac {a}{4}\right )}{b^{3}}+\frac {3 d^{2} c \left (\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {\left (b x +a \right )^{3}}{6}-\frac {\left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{2}+\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{4}+\frac {b x}{4}+\frac {a}{4}\right )}{b^{2}}+\frac {3 d^{3} a^{2} \left (\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {\left (b x +a \right )^{2}}{4}-\frac {\cosh \left (b x +a \right )^{2}}{4}\right )}{b^{3}}-\frac {6 d^{2} a c \left (\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {\left (b x +a \right )^{2}}{4}-\frac {\cosh \left (b x +a \right )^{2}}{4}\right )}{b^{2}}+\frac {3 d \,c^{2} \left (\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {\left (b x +a \right )^{2}}{4}-\frac {\cosh \left (b x +a \right )^{2}}{4}\right )}{b}-\frac {d^{3} a^{3} \left (\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{b^{3}}+\frac {3 d^{2} a^{2} c \left (\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{b^{2}}-\frac {3 d a \,c^{2} \left (\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{b}+c^{3} \left (\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{b}\) | \(523\) |
1/16*(4*(d*x+c)*sinh(2*b*x+2*a)*b*((d*x+c)^2*b^2+3/2*d^2)-6*((d*x+c)^2*b^2 +1/2*d^2)*d*cosh(2*b*x+2*a)+2*(d^3*x^4+4*c*d^2*x^3+6*c^2*d*x^2+4*c^3*x)*b^ 4+6*b^2*c^2*d+3*d^3)/b^4
Time = 0.26 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.56 \[ \int (c+d x)^3 \cosh ^2(a+b x) \, dx=\frac {2 \, b^{4} d^{3} x^{4} + 8 \, b^{4} c d^{2} x^{3} + 12 \, b^{4} c^{2} d x^{2} + 8 \, b^{4} c^{3} x - 3 \, {\left (2 \, b^{2} d^{3} x^{2} + 4 \, b^{2} c d^{2} x + 2 \, b^{2} c^{2} d + d^{3}\right )} \cosh \left (b x + a\right )^{2} + 4 \, {\left (2 \, b^{3} d^{3} x^{3} + 6 \, b^{3} c d^{2} x^{2} + 2 \, b^{3} c^{3} + 3 \, b c d^{2} + 3 \, {\left (2 \, b^{3} c^{2} d + b d^{3}\right )} x\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - 3 \, {\left (2 \, b^{2} d^{3} x^{2} + 4 \, b^{2} c d^{2} x + 2 \, b^{2} c^{2} d + d^{3}\right )} \sinh \left (b x + a\right )^{2}}{16 \, b^{4}} \]
1/16*(2*b^4*d^3*x^4 + 8*b^4*c*d^2*x^3 + 12*b^4*c^2*d*x^2 + 8*b^4*c^3*x - 3 *(2*b^2*d^3*x^2 + 4*b^2*c*d^2*x + 2*b^2*c^2*d + d^3)*cosh(b*x + a)^2 + 4*( 2*b^3*d^3*x^3 + 6*b^3*c*d^2*x^2 + 2*b^3*c^3 + 3*b*c*d^2 + 3*(2*b^3*c^2*d + b*d^3)*x)*cosh(b*x + a)*sinh(b*x + a) - 3*(2*b^2*d^3*x^2 + 4*b^2*c*d^2*x + 2*b^2*c^2*d + d^3)*sinh(b*x + a)^2)/b^4
Leaf count of result is larger than twice the leaf count of optimal. 456 vs. \(2 (131) = 262\).
Time = 0.39 (sec) , antiderivative size = 456, normalized size of antiderivative = 3.40 \[ \int (c+d x)^3 \cosh ^2(a+b x) \, dx=\begin {cases} - \frac {c^{3} x \sinh ^{2}{\left (a + b x \right )}}{2} + \frac {c^{3} x \cosh ^{2}{\left (a + b x \right )}}{2} - \frac {3 c^{2} d x^{2} \sinh ^{2}{\left (a + b x \right )}}{4} + \frac {3 c^{2} d x^{2} \cosh ^{2}{\left (a + b x \right )}}{4} - \frac {c d^{2} x^{3} \sinh ^{2}{\left (a + b x \right )}}{2} + \frac {c d^{2} x^{3} \cosh ^{2}{\left (a + b x \right )}}{2} - \frac {d^{3} x^{4} \sinh ^{2}{\left (a + b x \right )}}{8} + \frac {d^{3} x^{4} \cosh ^{2}{\left (a + b x \right )}}{8} + \frac {c^{3} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{2 b} + \frac {3 c^{2} d x \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{2 b} + \frac {3 c d^{2} x^{2} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{2 b} + \frac {d^{3} x^{3} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{2 b} - \frac {3 c^{2} d \cosh ^{2}{\left (a + b x \right )}}{4 b^{2}} - \frac {3 c d^{2} x \sinh ^{2}{\left (a + b x \right )}}{4 b^{2}} - \frac {3 c d^{2} x \cosh ^{2}{\left (a + b x \right )}}{4 b^{2}} - \frac {3 d^{3} x^{2} \sinh ^{2}{\left (a + b x \right )}}{8 b^{2}} - \frac {3 d^{3} x^{2} \cosh ^{2}{\left (a + b x \right )}}{8 b^{2}} + \frac {3 c d^{2} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{4 b^{3}} + \frac {3 d^{3} x \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{4 b^{3}} - \frac {3 d^{3} \cosh ^{2}{\left (a + b x \right )}}{8 b^{4}} & \text {for}\: b \neq 0 \\\left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) \cosh ^{2}{\left (a \right )} & \text {otherwise} \end {cases} \]
Piecewise((-c**3*x*sinh(a + b*x)**2/2 + c**3*x*cosh(a + b*x)**2/2 - 3*c**2 *d*x**2*sinh(a + b*x)**2/4 + 3*c**2*d*x**2*cosh(a + b*x)**2/4 - c*d**2*x** 3*sinh(a + b*x)**2/2 + c*d**2*x**3*cosh(a + b*x)**2/2 - d**3*x**4*sinh(a + b*x)**2/8 + d**3*x**4*cosh(a + b*x)**2/8 + c**3*sinh(a + b*x)*cosh(a + b* x)/(2*b) + 3*c**2*d*x*sinh(a + b*x)*cosh(a + b*x)/(2*b) + 3*c*d**2*x**2*si nh(a + b*x)*cosh(a + b*x)/(2*b) + d**3*x**3*sinh(a + b*x)*cosh(a + b*x)/(2 *b) - 3*c**2*d*cosh(a + b*x)**2/(4*b**2) - 3*c*d**2*x*sinh(a + b*x)**2/(4* b**2) - 3*c*d**2*x*cosh(a + b*x)**2/(4*b**2) - 3*d**3*x**2*sinh(a + b*x)** 2/(8*b**2) - 3*d**3*x**2*cosh(a + b*x)**2/(8*b**2) + 3*c*d**2*sinh(a + b*x )*cosh(a + b*x)/(4*b**3) + 3*d**3*x*sinh(a + b*x)*cosh(a + b*x)/(4*b**3) - 3*d**3*cosh(a + b*x)**2/(8*b**4), Ne(b, 0)), ((c**3*x + 3*c**2*d*x**2/2 + c*d**2*x**3 + d**3*x**4/4)*cosh(a)**2, True))
Leaf count of result is larger than twice the leaf count of optimal. 263 vs. \(2 (120) = 240\).
Time = 0.19 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.96 \[ \int (c+d x)^3 \cosh ^2(a+b x) \, dx=\frac {3}{16} \, {\left (4 \, x^{2} + \frac {{\left (2 \, b x e^{\left (2 \, a\right )} - e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{b^{2}} - \frac {{\left (2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{b^{2}}\right )} c^{2} d + \frac {1}{16} \, {\left (8 \, x^{3} + \frac {3 \, {\left (2 \, b^{2} x^{2} e^{\left (2 \, a\right )} - 2 \, b x e^{\left (2 \, a\right )} + e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{b^{3}} - \frac {3 \, {\left (2 \, b^{2} x^{2} + 2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{b^{3}}\right )} c d^{2} + \frac {1}{32} \, {\left (4 \, x^{4} + \frac {{\left (4 \, b^{3} x^{3} e^{\left (2 \, a\right )} - 6 \, b^{2} x^{2} e^{\left (2 \, a\right )} + 6 \, b x e^{\left (2 \, a\right )} - 3 \, e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{b^{4}} - \frac {{\left (4 \, b^{3} x^{3} + 6 \, b^{2} x^{2} + 6 \, b x + 3\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{b^{4}}\right )} d^{3} + \frac {1}{8} \, c^{3} {\left (4 \, x + \frac {e^{\left (2 \, b x + 2 \, a\right )}}{b} - \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{b}\right )} \]
3/16*(4*x^2 + (2*b*x*e^(2*a) - e^(2*a))*e^(2*b*x)/b^2 - (2*b*x + 1)*e^(-2* b*x - 2*a)/b^2)*c^2*d + 1/16*(8*x^3 + 3*(2*b^2*x^2*e^(2*a) - 2*b*x*e^(2*a) + e^(2*a))*e^(2*b*x)/b^3 - 3*(2*b^2*x^2 + 2*b*x + 1)*e^(-2*b*x - 2*a)/b^3 )*c*d^2 + 1/32*(4*x^4 + (4*b^3*x^3*e^(2*a) - 6*b^2*x^2*e^(2*a) + 6*b*x*e^( 2*a) - 3*e^(2*a))*e^(2*b*x)/b^4 - (4*b^3*x^3 + 6*b^2*x^2 + 6*b*x + 3)*e^(- 2*b*x - 2*a)/b^4)*d^3 + 1/8*c^3*(4*x + e^(2*b*x + 2*a)/b - e^(-2*b*x - 2*a )/b)
Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (120) = 240\).
Time = 0.27 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.81 \[ \int (c+d x)^3 \cosh ^2(a+b x) \, dx=\frac {1}{8} \, d^{3} x^{4} + \frac {1}{2} \, c d^{2} x^{3} + \frac {3}{4} \, c^{2} d x^{2} + \frac {1}{2} \, c^{3} x + \frac {{\left (4 \, b^{3} d^{3} x^{3} + 12 \, b^{3} c d^{2} x^{2} + 12 \, b^{3} c^{2} d x - 6 \, b^{2} d^{3} x^{2} + 4 \, b^{3} c^{3} - 12 \, b^{2} c d^{2} x - 6 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} - 3 \, d^{3}\right )} e^{\left (2 \, b x + 2 \, a\right )}}{32 \, b^{4}} - \frac {{\left (4 \, b^{3} d^{3} x^{3} + 12 \, b^{3} c d^{2} x^{2} + 12 \, b^{3} c^{2} d x + 6 \, b^{2} d^{3} x^{2} + 4 \, b^{3} c^{3} + 12 \, b^{2} c d^{2} x + 6 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} + 3 \, d^{3}\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{32 \, b^{4}} \]
1/8*d^3*x^4 + 1/2*c*d^2*x^3 + 3/4*c^2*d*x^2 + 1/2*c^3*x + 1/32*(4*b^3*d^3* x^3 + 12*b^3*c*d^2*x^2 + 12*b^3*c^2*d*x - 6*b^2*d^3*x^2 + 4*b^3*c^3 - 12*b ^2*c*d^2*x - 6*b^2*c^2*d + 6*b*d^3*x + 6*b*c*d^2 - 3*d^3)*e^(2*b*x + 2*a)/ b^4 - 1/32*(4*b^3*d^3*x^3 + 12*b^3*c*d^2*x^2 + 12*b^3*c^2*d*x + 6*b^2*d^3* x^2 + 4*b^3*c^3 + 12*b^2*c*d^2*x + 6*b^2*c^2*d + 6*b*d^3*x + 6*b*c*d^2 + 3 *d^3)*e^(-2*b*x - 2*a)/b^4
Time = 0.44 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.71 \[ \int (c+d x)^3 \cosh ^2(a+b x) \, dx=\frac {4\,b^4\,c^3\,x-\frac {3\,d^3\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{2}+2\,b^3\,c^3\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )+b^4\,d^3\,x^4-3\,b^2\,c^2\,d\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )+6\,b^4\,c^2\,d\,x^2+4\,b^4\,c\,d^2\,x^3-3\,b^2\,d^3\,x^2\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )+2\,b^3\,d^3\,x^3\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )+3\,b\,c\,d^2\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )+3\,b\,d^3\,x\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )-6\,b^2\,c\,d^2\,x\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )+6\,b^3\,c^2\,d\,x\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )+6\,b^3\,c\,d^2\,x^2\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{8\,b^4} \]
(4*b^4*c^3*x - (3*d^3*cosh(2*a + 2*b*x))/2 + 2*b^3*c^3*sinh(2*a + 2*b*x) + b^4*d^3*x^4 - 3*b^2*c^2*d*cosh(2*a + 2*b*x) + 6*b^4*c^2*d*x^2 + 4*b^4*c*d ^2*x^3 - 3*b^2*d^3*x^2*cosh(2*a + 2*b*x) + 2*b^3*d^3*x^3*sinh(2*a + 2*b*x) + 3*b*c*d^2*sinh(2*a + 2*b*x) + 3*b*d^3*x*sinh(2*a + 2*b*x) - 6*b^2*c*d^2 *x*cosh(2*a + 2*b*x) + 6*b^3*c^2*d*x*sinh(2*a + 2*b*x) + 6*b^3*c*d^2*x^2*s inh(2*a + 2*b*x))/(8*b^4)