Integrand size = 14, antiderivative size = 70 \[ \int (c+d x)^3 \cosh (a+b x) \, dx=-\frac {6 d^3 \cosh (a+b x)}{b^4}-\frac {3 d (c+d x)^2 \cosh (a+b x)}{b^2}+\frac {6 d^2 (c+d x) \sinh (a+b x)}{b^3}+\frac {(c+d x)^3 \sinh (a+b x)}{b} \]
-6*d^3*cosh(b*x+a)/b^4-3*d*(d*x+c)^2*cosh(b*x+a)/b^2+6*d^2*(d*x+c)*sinh(b* x+a)/b^3+(d*x+c)^3*sinh(b*x+a)/b
Time = 0.12 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.87 \[ \int (c+d x)^3 \cosh (a+b x) \, dx=\frac {-3 d \left (2 d^2+b^2 (c+d x)^2\right ) \cosh (a+b x)+b (c+d x) \left (6 d^2+b^2 (c+d x)^2\right ) \sinh (a+b x)}{b^4} \]
(-3*d*(2*d^2 + b^2*(c + d*x)^2)*Cosh[a + b*x] + b*(c + d*x)*(6*d^2 + b^2*( c + d*x)^2)*Sinh[a + b*x])/b^4
Result contains complex when optimal does not.
Time = 0.46 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.19, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {3042, 3777, 26, 3042, 26, 3777, 3042, 3777, 26, 3042, 26, 3118}
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.
| \(\displaystyle \int (c+d x)^3 \cosh (a+b x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (c+d x)^3 \sin \left (i a+i b x+\frac {\pi }{2}\right )dx\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {(c+d x)^3 \sinh (a+b x)}{b}-\frac {3 i d \int -i (c+d x)^2 \sinh (a+b x)dx}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {(c+d x)^3 \sinh (a+b x)}{b}-\frac {3 d \int (c+d x)^2 \sinh (a+b x)dx}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(c+d x)^3 \sinh (a+b x)}{b}-\frac {3 d \int -i (c+d x)^2 \sin (i a+i b x)dx}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {(c+d x)^3 \sinh (a+b x)}{b}+\frac {3 i d \int (c+d x)^2 \sin (i a+i b x)dx}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {(c+d x)^3 \sinh (a+b x)}{b}+\frac {3 i d \left (\frac {i (c+d x)^2 \cosh (a+b x)}{b}-\frac {2 i d \int (c+d x) \cosh (a+b x)dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(c+d x)^3 \sinh (a+b x)}{b}+\frac {3 i d \left (\frac {i (c+d x)^2 \cosh (a+b x)}{b}-\frac {2 i d \int (c+d x) \sin \left (i a+i b x+\frac {\pi }{2}\right )dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {(c+d x)^3 \sinh (a+b x)}{b}+\frac {3 i d \left (\frac {i (c+d x)^2 \cosh (a+b x)}{b}-\frac {2 i d \left (\frac {(c+d x) \sinh (a+b x)}{b}-\frac {i d \int -i \sinh (a+b x)dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {(c+d x)^3 \sinh (a+b x)}{b}+\frac {3 i d \left (\frac {i (c+d x)^2 \cosh (a+b x)}{b}-\frac {2 i d \left (\frac {(c+d x) \sinh (a+b x)}{b}-\frac {d \int \sinh (a+b x)dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(c+d x)^3 \sinh (a+b x)}{b}+\frac {3 i d \left (\frac {i (c+d x)^2 \cosh (a+b x)}{b}-\frac {2 i d \left (\frac {(c+d x) \sinh (a+b x)}{b}-\frac {d \int -i \sin (i a+i b x)dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {(c+d x)^3 \sinh (a+b x)}{b}+\frac {3 i d \left (\frac {i (c+d x)^2 \cosh (a+b x)}{b}-\frac {2 i d \left (\frac {(c+d x) \sinh (a+b x)}{b}+\frac {i d \int \sin (i a+i b x)dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle \frac {(c+d x)^3 \sinh (a+b x)}{b}+\frac {3 i d \left (\frac {i (c+d x)^2 \cosh (a+b x)}{b}-\frac {2 i d \left (\frac {(c+d x) \sinh (a+b x)}{b}-\frac {d \cosh (a+b x)}{b^2}\right )}{b}\right )}{b}\) |
((c + d*x)^3*Sinh[a + b*x])/b + ((3*I)*d*((I*(c + d*x)^2*Cosh[a + b*x])/b - ((2*I)*d*(-((d*Cosh[a + b*x])/b^2) + ((c + d*x)*Sinh[a + b*x])/b))/b))/b
3.1.2.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Time = 0.20 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.59
| method | result | size |
| parallelrisch | \(\frac {6 d^{2} \left (\frac {d x}{2}+c \right ) x \,b^{2} \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-2 \left (\left (d x +c \right )^{2} b^{2}+6 d^{2}\right ) \left (d x +c \right ) b \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )+6 d \left (\left (\frac {1}{2} x^{2} d^{2}+c d x +c^{2}\right ) b^{2}+2 d^{2}\right )}{b^{4} \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1\right )}\) | \(111\) |
| risch | \(\frac {\left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x -3 b^{2} d^{3} x^{2}+b^{3} c^{3}-6 b^{2} c \,d^{2} x -3 b^{2} c^{2} d +6 b \,d^{3} x +6 b c \,d^{2}-6 d^{3}\right ) {\mathrm e}^{b x +a}}{2 b^{4}}-\frac {\left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +3 b^{2} d^{3} x^{2}+b^{3} c^{3}+6 b^{2} c \,d^{2} x +3 b^{2} c^{2} d +6 b \,d^{3} x +6 b c \,d^{2}+6 d^{3}\right ) {\mathrm e}^{-b x -a}}{2 b^{4}}\) | \(205\) |
| parts | \(\frac {\sinh \left (b x +a \right ) d^{3} x^{3}}{b}+\frac {3 \sinh \left (b x +a \right ) c \,d^{2} x^{2}}{b}+\frac {3 \sinh \left (b x +a \right ) c^{2} d x}{b}+\frac {\sinh \left (b x +a \right ) c^{3}}{b}-\frac {3 d \left (\frac {d^{2} \left (\left (b x +a \right )^{2} \cosh \left (b x +a \right )-2 \left (b x +a \right ) \sinh \left (b x +a \right )+2 \cosh \left (b x +a \right )\right )}{b^{2}}-\frac {2 d^{2} a \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b^{2}}+\frac {2 d c \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b}+\frac {d^{2} a^{2} \cosh \left (b x +a \right )}{b^{2}}-\frac {2 d a c \cosh \left (b x +a \right )}{b}+c^{2} \cosh \left (b x +a \right )\right )}{b^{2}}\) | \(213\) |
| derivativedivides | \(\frac {\frac {d^{3} \left (\left (b x +a \right )^{3} \sinh \left (b x +a \right )-3 \left (b x +a \right )^{2} \cosh \left (b x +a \right )+6 \left (b x +a \right ) \sinh \left (b x +a \right )-6 \cosh \left (b x +a \right )\right )}{b^{3}}-\frac {3 d^{3} a \left (\left (b x +a \right )^{2} \sinh \left (b x +a \right )-2 \left (b x +a \right ) \cosh \left (b x +a \right )+2 \sinh \left (b x +a \right )\right )}{b^{3}}+\frac {3 d^{2} c \left (\left (b x +a \right )^{2} \sinh \left (b x +a \right )-2 \left (b x +a \right ) \cosh \left (b x +a \right )+2 \sinh \left (b x +a \right )\right )}{b^{2}}+\frac {3 d^{3} a^{2} \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b^{3}}-\frac {6 d^{2} a c \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b^{2}}+\frac {3 d \,c^{2} \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b}-\frac {d^{3} a^{3} \sinh \left (b x +a \right )}{b^{3}}+\frac {3 d^{2} a^{2} c \sinh \left (b x +a \right )}{b^{2}}-\frac {3 d a \,c^{2} \sinh \left (b x +a \right )}{b}+c^{3} \sinh \left (b x +a \right )}{b}\) | \(308\) |
| default | \(\frac {\frac {d^{3} \left (\left (b x +a \right )^{3} \sinh \left (b x +a \right )-3 \left (b x +a \right )^{2} \cosh \left (b x +a \right )+6 \left (b x +a \right ) \sinh \left (b x +a \right )-6 \cosh \left (b x +a \right )\right )}{b^{3}}-\frac {3 d^{3} a \left (\left (b x +a \right )^{2} \sinh \left (b x +a \right )-2 \left (b x +a \right ) \cosh \left (b x +a \right )+2 \sinh \left (b x +a \right )\right )}{b^{3}}+\frac {3 d^{2} c \left (\left (b x +a \right )^{2} \sinh \left (b x +a \right )-2 \left (b x +a \right ) \cosh \left (b x +a \right )+2 \sinh \left (b x +a \right )\right )}{b^{2}}+\frac {3 d^{3} a^{2} \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b^{3}}-\frac {6 d^{2} a c \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b^{2}}+\frac {3 d \,c^{2} \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b}-\frac {d^{3} a^{3} \sinh \left (b x +a \right )}{b^{3}}+\frac {3 d^{2} a^{2} c \sinh \left (b x +a \right )}{b^{2}}-\frac {3 d a \,c^{2} \sinh \left (b x +a \right )}{b}+c^{3} \sinh \left (b x +a \right )}{b}\) | \(308\) |
| meijerg | \(\frac {8 d^{3} \cosh \left (a \right ) \sqrt {\pi }\, \left (\frac {3}{4 \sqrt {\pi }}-\frac {\left (\frac {3 x^{2} b^{2}}{2}+3\right ) \cosh \left (b x \right )}{4 \sqrt {\pi }}+\frac {x b \left (\frac {x^{2} b^{2}}{2}+3\right ) \sinh \left (b x \right )}{4 \sqrt {\pi }}\right )}{b^{4}}-\frac {8 i d^{3} \sinh \left (a \right ) \sqrt {\pi }\, \left (\frac {i x b \left (\frac {5 x^{2} b^{2}}{2}+15\right ) \cosh \left (b x \right )}{20 \sqrt {\pi }}-\frac {i \left (\frac {15 x^{2} b^{2}}{2}+15\right ) \sinh \left (b x \right )}{20 \sqrt {\pi }}\right )}{b^{4}}+\frac {12 i d^{2} c \cosh \left (a \right ) \sqrt {\pi }\, \left (\frac {i x b \cosh \left (b x \right )}{2 \sqrt {\pi }}-\frac {i \left (\frac {3 x^{2} b^{2}}{2}+3\right ) \sinh \left (b x \right )}{6 \sqrt {\pi }}\right )}{b^{3}}+\frac {12 d^{2} c \sinh \left (a \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\left (\frac {x^{2} b^{2}}{2}+1\right ) \cosh \left (b x \right )}{2 \sqrt {\pi }}-\frac {x b \sinh \left (b x \right )}{2 \sqrt {\pi }}\right )}{b^{3}}-\frac {6 d \,c^{2} \cosh \left (a \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\cosh \left (b x \right )}{2 \sqrt {\pi }}-\frac {x b \sinh \left (b x \right )}{2 \sqrt {\pi }}\right )}{b^{2}}+\frac {3 d \,c^{2} \sinh \left (a \right ) \left (\cosh \left (b x \right ) x b -\sinh \left (b x \right )\right )}{b^{2}}+\frac {c^{3} \cosh \left (a \right ) \sinh \left (b x \right )}{b}-\frac {c^{3} \sinh \left (a \right ) \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cosh \left (b x \right )}{\sqrt {\pi }}\right )}{b}\) | \(320\) |
2*(3*d^2*(1/2*d*x+c)*x*b^2*tanh(1/2*b*x+1/2*a)^2-((d*x+c)^2*b^2+6*d^2)*(d* x+c)*b*tanh(1/2*b*x+1/2*a)+3*d*((1/2*x^2*d^2+c*d*x+c^2)*b^2+2*d^2))/b^4/(t anh(1/2*b*x+1/2*a)^2-1)
Time = 0.25 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.59 \[ \int (c+d x)^3 \cosh (a+b x) \, dx=-\frac {3 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d + 2 \, d^{3}\right )} \cosh \left (b x + a\right ) - {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + b^{3} c^{3} + 6 \, b c d^{2} + 3 \, {\left (b^{3} c^{2} d + 2 \, b d^{3}\right )} x\right )} \sinh \left (b x + a\right )}{b^{4}} \]
-(3*(b^2*d^3*x^2 + 2*b^2*c*d^2*x + b^2*c^2*d + 2*d^3)*cosh(b*x + a) - (b^3 *d^3*x^3 + 3*b^3*c*d^2*x^2 + b^3*c^3 + 6*b*c*d^2 + 3*(b^3*c^2*d + 2*b*d^3) *x)*sinh(b*x + a))/b^4
Leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (70) = 140\).
Time = 0.29 (sec) , antiderivative size = 202, normalized size of antiderivative = 2.89 \[ \int (c+d x)^3 \cosh (a+b x) \, dx=\begin {cases} \frac {c^{3} \sinh {\left (a + b x \right )}}{b} + \frac {3 c^{2} d x \sinh {\left (a + b x \right )}}{b} + \frac {3 c d^{2} x^{2} \sinh {\left (a + b x \right )}}{b} + \frac {d^{3} x^{3} \sinh {\left (a + b x \right )}}{b} - \frac {3 c^{2} d \cosh {\left (a + b x \right )}}{b^{2}} - \frac {6 c d^{2} x \cosh {\left (a + b x \right )}}{b^{2}} - \frac {3 d^{3} x^{2} \cosh {\left (a + b x \right )}}{b^{2}} + \frac {6 c d^{2} \sinh {\left (a + b x \right )}}{b^{3}} + \frac {6 d^{3} x \sinh {\left (a + b x \right )}}{b^{3}} - \frac {6 d^{3} \cosh {\left (a + b x \right )}}{b^{4}} & \text {for}\: b \neq 0 \\\left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) \cosh {\left (a \right )} & \text {otherwise} \end {cases} \]
Piecewise((c**3*sinh(a + b*x)/b + 3*c**2*d*x*sinh(a + b*x)/b + 3*c*d**2*x* *2*sinh(a + b*x)/b + d**3*x**3*sinh(a + b*x)/b - 3*c**2*d*cosh(a + b*x)/b* *2 - 6*c*d**2*x*cosh(a + b*x)/b**2 - 3*d**3*x**2*cosh(a + b*x)/b**2 + 6*c* d**2*sinh(a + b*x)/b**3 + 6*d**3*x*sinh(a + b*x)/b**3 - 6*d**3*cosh(a + b* x)/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), True))
Leaf count of result is larger than twice the leaf count of optimal. 222 vs. \(2 (70) = 140\).
Time = 0.19 (sec) , antiderivative size = 222, normalized size of antiderivative = 3.17 \[ \int (c+d x)^3 \cosh (a+b x) \, dx=\frac {c^{3} e^{\left (b x + a\right )}}{2 \, b} + \frac {3 \, {\left (b x e^{a} - e^{a}\right )} c^{2} d e^{\left (b x\right )}}{2 \, b^{2}} - \frac {c^{3} e^{\left (-b x - a\right )}}{2 \, b} - \frac {3 \, {\left (b x + 1\right )} c^{2} d e^{\left (-b x - a\right )}}{2 \, b^{2}} + \frac {3 \, {\left (b^{2} x^{2} e^{a} - 2 \, b x e^{a} + 2 \, e^{a}\right )} c d^{2} e^{\left (b x\right )}}{2 \, b^{3}} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, b x + 2\right )} c d^{2} e^{\left (-b x - a\right )}}{2 \, b^{3}} + \frac {{\left (b^{3} x^{3} e^{a} - 3 \, b^{2} x^{2} e^{a} + 6 \, b x e^{a} - 6 \, e^{a}\right )} d^{3} e^{\left (b x\right )}}{2 \, b^{4}} - \frac {{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} d^{3} e^{\left (-b x - a\right )}}{2 \, b^{4}} \]
1/2*c^3*e^(b*x + a)/b + 3/2*(b*x*e^a - e^a)*c^2*d*e^(b*x)/b^2 - 1/2*c^3*e^ (-b*x - a)/b - 3/2*(b*x + 1)*c^2*d*e^(-b*x - a)/b^2 + 3/2*(b^2*x^2*e^a - 2 *b*x*e^a + 2*e^a)*c*d^2*e^(b*x)/b^3 - 3/2*(b^2*x^2 + 2*b*x + 2)*c*d^2*e^(- b*x - a)/b^3 + 1/2*(b^3*x^3*e^a - 3*b^2*x^2*e^a + 6*b*x*e^a - 6*e^a)*d^3*e ^(b*x)/b^4 - 1/2*(b^3*x^3 + 3*b^2*x^2 + 6*b*x + 6)*d^3*e^(-b*x - a)/b^4
Leaf count of result is larger than twice the leaf count of optimal. 204 vs. \(2 (70) = 140\).
Time = 0.26 (sec) , antiderivative size = 204, normalized size of antiderivative = 2.91 \[ \int (c+d x)^3 \cosh (a+b x) \, dx=\frac {{\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x - 3 \, b^{2} d^{3} x^{2} + b^{3} c^{3} - 6 \, b^{2} c d^{2} x - 3 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} - 6 \, d^{3}\right )} e^{\left (b x + a\right )}}{2 \, b^{4}} - \frac {{\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + 3 \, b^{2} d^{3} x^{2} + b^{3} c^{3} + 6 \, b^{2} c d^{2} x + 3 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} + 6 \, d^{3}\right )} e^{\left (-b x - a\right )}}{2 \, b^{4}} \]
1/2*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x - 3*b^2*d^3*x^2 + b^3*c ^3 - 6*b^2*c*d^2*x - 3*b^2*c^2*d + 6*b*d^3*x + 6*b*c*d^2 - 6*d^3)*e^(b*x + a)/b^4 - 1/2*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + 3*b^2*d^3*x ^2 + b^3*c^3 + 6*b^2*c*d^2*x + 3*b^2*c^2*d + 6*b*d^3*x + 6*b*c*d^2 + 6*d^3 )*e^(-b*x - a)/b^4
Time = 1.78 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.04 \[ \int (c+d x)^3 \cosh (a+b x) \, dx=\frac {\mathrm {sinh}\left (a+b\,x\right )\,\left (b^2\,c^3+6\,c\,d^2\right )}{b^3}-\frac {3\,\mathrm {cosh}\left (a+b\,x\right )\,\left (b^2\,c^2\,d+2\,d^3\right )}{b^4}-\frac {3\,d^3\,x^2\,\mathrm {cosh}\left (a+b\,x\right )}{b^2}+\frac {d^3\,x^3\,\mathrm {sinh}\left (a+b\,x\right )}{b}+\frac {3\,x\,\mathrm {sinh}\left (a+b\,x\right )\,\left (b^2\,c^2\,d+2\,d^3\right )}{b^3}-\frac {6\,c\,d^2\,x\,\mathrm {cosh}\left (a+b\,x\right )}{b^2}+\frac {3\,c\,d^2\,x^2\,\mathrm {sinh}\left (a+b\,x\right )}{b} \]