\(\int (c+d x)^4 \cosh (a+b x) \, dx\) [1]

Optimal result

Integrand size = 14, antiderivative size = 91 \[ \int (c+d x)^4 \cosh (a+b x) \, dx=-\frac {24 d^3 (c+d x) \cosh (a+b x)}{b^4}-\frac {4 d (c+d x)^3 \cosh (a+b x)}{b^2}+\frac {24 d^4 \sinh (a+b x)}{b^5}+\frac {12 d^2 (c+d x)^2 \sinh (a+b x)}{b^3}+\frac {(c+d x)^4 \sinh (a+b x)}{b} \]

[Out]

-24*d^3*(d*x+c)*cosh(b*x+a)/b^4-4*d*(d*x+c)^3*cosh(b*x+a)/b^2+24*d^4*sinh(b*x+a)/b^5+12*d^2*(d*x+c)^2*sinh(b*x
+a)/b^3+(d*x+c)^4*sinh(b*x+a)/b

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3377, 2717} \[ \int (c+d x)^4 \cosh (a+b x) \, dx=\frac {24 d^4 \sinh (a+b x)}{b^5}-\frac {24 d^3 (c+d x) \cosh (a+b x)}{b^4}+\frac {12 d^2 (c+d x)^2 \sinh (a+b x)}{b^3}-\frac {4 d (c+d x)^3 \cosh (a+b x)}{b^2}+\frac {(c+d x)^4 \sinh (a+b x)}{b} \]

[In]

Int[(c + d*x)^4*Cosh[a + b*x],x]

[Out]

(-24*d^3*(c + d*x)*Cosh[a + b*x])/b^4 - (4*d*(c + d*x)^3*Cosh[a + b*x])/b^2 + (24*d^4*Sinh[a + b*x])/b^5 + (12
*d^2*(c + d*x)^2*Sinh[a + b*x])/b^3 + ((c + d*x)^4*Sinh[a + b*x])/b

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {(c+d x)^4 \sinh (a+b x)}{b}-\frac {(4 d) \int (c+d x)^3 \sinh (a+b x) \, dx}{b} \\ & = -\frac {4 d (c+d x)^3 \cosh (a+b x)}{b^2}+\frac {(c+d x)^4 \sinh (a+b x)}{b}+\frac {\left (12 d^2\right ) \int (c+d x)^2 \cosh (a+b x) \, dx}{b^2} \\ & = -\frac {4 d (c+d x)^3 \cosh (a+b x)}{b^2}+\frac {12 d^2 (c+d x)^2 \sinh (a+b x)}{b^3}+\frac {(c+d x)^4 \sinh (a+b x)}{b}-\frac {\left (24 d^3\right ) \int (c+d x) \sinh (a+b x) \, dx}{b^3} \\ & = -\frac {24 d^3 (c+d x) \cosh (a+b x)}{b^4}-\frac {4 d (c+d x)^3 \cosh (a+b x)}{b^2}+\frac {12 d^2 (c+d x)^2 \sinh (a+b x)}{b^3}+\frac {(c+d x)^4 \sinh (a+b x)}{b}+\frac {\left (24 d^4\right ) \int \cosh (a+b x) \, dx}{b^4} \\ & = -\frac {24 d^3 (c+d x) \cosh (a+b x)}{b^4}-\frac {4 d (c+d x)^3 \cosh (a+b x)}{b^2}+\frac {24 d^4 \sinh (a+b x)}{b^5}+\frac {12 d^2 (c+d x)^2 \sinh (a+b x)}{b^3}+\frac {(c+d x)^4 \sinh (a+b x)}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.84 \[ \int (c+d x)^4 \cosh (a+b x) \, dx=\frac {-4 b d (c+d x) \left (6 d^2+b^2 (c+d x)^2\right ) \cosh (a+b x)+\left (24 d^4+12 b^2 d^2 (c+d x)^2+b^4 (c+d x)^4\right ) \sinh (a+b x)}{b^5} \]

[In]

Integrate[(c + d*x)^4*Cosh[a + b*x],x]

[Out]

(-4*b*d*(c + d*x)*(6*d^2 + b^2*(c + d*x)^2)*Cosh[a + b*x] + (24*d^4 + 12*b^2*d^2*(c + d*x)^2 + b^4*(c + d*x)^4
)*Sinh[a + b*x])/b^5

Maple [A] (verified)

Time = 5.56 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.58

method	result	size
parallelrisch	\(\frac {12 d^{2} x b \left (\left (\frac {1}{3} x^{2} d^{2}+c d x +c^{2}\right ) b^{2}+2 d^{2}\right ) \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+2 \left (-\left (d x +c \right )^{4} b^{4}-12 d^{2} \left (d x +c \right )^{2} b^{2}-24 d^{4}\right ) \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )+8 d \left (\frac {d x}{2}+c \right ) \left (\left (x^{2} d^{2}+c d x +c^{2}\right ) b^{2}+6 d^{2}\right ) b}{b^{5} \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1\right )}\)	\(144\)
risch	\(\frac {\left (d^{4} x^{4} b^{4}+4 b^{4} c \,d^{3} x^{3}+6 b^{4} c^{2} d^{2} x^{2}-4 b^{3} d^{4} x^{3}+4 b^{4} c^{3} d x -12 b^{3} c \,d^{3} x^{2}+b^{4} c^{4}-12 b^{3} c^{2} d^{2} x +12 b^{2} d^{4} x^{2}-4 b^{3} c^{3} d +24 b^{2} c \,d^{3} x +12 b^{2} c^{2} d^{2}-24 b \,d^{4} x -24 b c \,d^{3}+24 d^{4}\right ) {\mathrm e}^{b x +a}}{2 b^{5}}-\frac {\left (d^{4} x^{4} b^{4}+4 b^{4} c \,d^{3} x^{3}+6 b^{4} c^{2} d^{2} x^{2}+4 b^{3} d^{4} x^{3}+4 b^{4} c^{3} d x +12 b^{3} c \,d^{3} x^{2}+b^{4} c^{4}+12 b^{3} c^{2} d^{2} x +12 b^{2} d^{4} x^{2}+4 b^{3} c^{3} d +24 b^{2} c \,d^{3} x +12 b^{2} c^{2} d^{2}+24 b \,d^{4} x +24 b c \,d^{3}+24 d^{4}\right ) {\mathrm e}^{-b x -a}}{2 b^{5}}\)	\(325\)
parts	\(\frac {\sinh \left (b x +a \right ) d^{4} x^{4}}{b}+\frac {4 \sinh \left (b x +a \right ) c \,d^{3} x^{3}}{b}+\frac {6 \sinh \left (b x +a \right ) c^{2} d^{2} x^{2}}{b}+\frac {4 \sinh \left (b x +a \right ) c^{3} d x}{b}+\frac {\sinh \left (b x +a \right ) c^{4}}{b}-\frac {4 d \left (\frac {d^{3} \left (\left (b x +a \right )^{3} \cosh \left (b x +a \right )-3 \left (b x +a \right )^{2} \sinh \left (b x +a \right )+6 \left (b x +a \right ) \cosh \left (b x +a \right )-6 \sinh \left (b x +a \right )\right )}{b^{3}}-\frac {3 d^{3} a \left (\left (b x +a \right )^{2} \cosh \left (b x +a \right )-2 \left (b x +a \right ) \sinh \left (b x +a \right )+2 \cosh \left (b x +a \right )\right )}{b^{3}}+\frac {3 d^{2} c \left (\left (b x +a \right )^{2} \cosh \left (b x +a \right )-2 \left (b x +a \right ) \sinh \left (b x +a \right )+2 \cosh \left (b x +a \right )\right )}{b^{2}}+\frac {3 d^{3} a^{2} \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b^{3}}-\frac {6 d^{2} a c \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b^{2}}+\frac {3 d \,c^{2} \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b}-\frac {d^{3} a^{3} \cosh \left (b x +a \right )}{b^{3}}+\frac {3 d^{2} a^{2} c \cosh \left (b x +a \right )}{b^{2}}-\frac {3 d a \,c^{2} \cosh \left (b x +a \right )}{b}+c^{3} \cosh \left (b x +a \right )\right )}{b^{2}}\)	\(394\)
meijerg	\(-\frac {16 i d^{4} \cosh \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 )}{10 \sqrt {\pi }}+\frac {i \left (\frac {5}{8} x^{4} b^{4}+\frac {15}{2} x^{2} b^{2}+15\right ) \sinh \left (b x \right )}{10 \sqrt {\pi }}\right )}{b^{5}}-\frac {16 d^{4} \sinh \left (a \right ) \sqrt {\pi }\, \left (\frac {3}{2 \sqrt {\pi }}-\frac {\left (\frac {3}{8} x^{4} b^{4}+\frac {9}{2} x^{2} b^{2}+9\right ) \cosh \left (b x \right )}{6 \sqrt {\pi }}+\frac {x b \left (\frac {3 x^{2} b^{2}}{2}+9\right ) \sinh \left (b x \right )}{6 \sqrt {\pi }}\right )}{b^{5}}+\frac {32 d^{3} c \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 {32 i d^{3} c \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 {24 i d^{2} c^{2} \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 {24 d^{2} c^{2} \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 {8 d \,c^{3} \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 {4 d \,c^{3} \sinh \left (a \right ) \left (\cosh \left (b x \right ) x b -\sinh \left (b x \right )\right )}{b^{2}}+\frac {c^{4} \cosh \left (a \right ) \sinh \left (b x \right )}{b}-\frac {c^{4} \sinh \left (a \right ) \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cosh \left (b x \right )}{\sqrt {\pi }}\right )}{b}\)	\(458\)
derivativedivides	\(\frac {\frac {d^{4} \left (\left (b x +a \right )^{4} \sinh \left (b x +a \right )-4 \left (b x +a \right )^{3} \cosh \left (b x +a \right )+12 \left (b x +a \right )^{2} \sinh \left (b x +a \right )-24 \left (b x +a \right ) \cosh \left (b x +a \right )+24 \sinh \left (b x +a \right )\right )}{b^{4}}+\frac {d^{4} a^{4} \sinh \left (b x +a \right )}{b^{4}}-\frac {4 d^{4} a \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^{4}}+\frac {4 d^{3} c \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 {6 d^{4} a^{2} \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^{4}}+\frac {6 d^{2} c^{2} \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 {4 d^{4} a^{3} \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b^{4}}+\frac {4 d \,c^{3} \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b}-\frac {4 d^{3} a^{3} c \sinh \left (b x +a \right )}{b^{3}}+\frac {6 d^{2} a^{2} c^{2} \sinh \left (b x +a \right )}{b^{2}}-\frac {4 d a \,c^{3} \sinh \left (b x +a \right )}{b}+c^{4} \sinh \left (b x +a \right )-\frac {12 d^{3} a 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^{3}}+\frac {12 d^{3} a^{2} c \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b^{3}}-\frac {12 d^{2} a \,c^{2} \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b^{2}}}{b}\)	\(547\)
default	\(\frac {\frac {d^{4} \left (\left (b x +a \right )^{4} \sinh \left (b x +a \right )-4 \left (b x +a \right )^{3} \cosh \left (b x +a \right )+12 \left (b x +a \right )^{2} \sinh \left (b x +a \right )-24 \left (b x +a \right ) \cosh \left (b x +a \right )+24 \sinh \left (b x +a \right )\right )}{b^{4}}+\frac {d^{4} a^{4} \sinh \left (b x +a \right )}{b^{4}}-\frac {4 d^{4} a \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^{4}}+\frac {4 d^{3} c \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 {6 d^{4} a^{2} \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^{4}}+\frac {6 d^{2} c^{2} \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 {4 d^{4} a^{3} \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b^{4}}+\frac {4 d \,c^{3} \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b}-\frac {4 d^{3} a^{3} c \sinh \left (b x +a \right )}{b^{3}}+\frac {6 d^{2} a^{2} c^{2} \sinh \left (b x +a \right )}{b^{2}}-\frac {4 d a \,c^{3} \sinh \left (b x +a \right )}{b}+c^{4} \sinh \left (b x +a \right )-\frac {12 d^{3} a 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^{3}}+\frac {12 d^{3} a^{2} c \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b^{3}}-\frac {12 d^{2} a \,c^{2} \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b^{2}}}{b}\)	\(547\)

[In]

int((d*x+c)^4*cosh(b*x+a),x,method=_RETURNVERBOSE)

[Out]

2*(6*d^2*x*b*((1/3*x^2*d^2+c*d*x+c^2)*b^2+2*d^2)*tanh(1/2*b*x+1/2*a)^2+(-(d*x+c)^4*b^4-12*d^2*(d*x+c)^2*b^2-24
*d^4)*tanh(1/2*b*x+1/2*a)+4*d*(1/2*d*x+c)*((d^2*x^2+c*d*x+c^2)*b^2+6*d^2)*b)/b^5/(tanh(1/2*b*x+1/2*a)^2-1)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.88 \[ \int (c+d x)^4 \cosh (a+b x) \, dx=-\frac {4 \, {\left (b^{3} d^{4} x^{3} + 3 \, b^{3} c d^{3} x^{2} + b^{3} c^{3} d + 6 \, b c d^{3} + 3 \, {\left (b^{3} c^{2} d^{2} + 2 \, b d^{4}\right )} x\right )} \cosh \left (b x + a\right ) - {\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + b^{4} c^{4} + 12 \, b^{2} c^{2} d^{2} + 24 \, d^{4} + 6 \, {\left (b^{4} c^{2} d^{2} + 2 \, b^{2} d^{4}\right )} x^{2} + 4 \, {\left (b^{4} c^{3} d + 6 \, b^{2} c d^{3}\right )} x\right )} \sinh \left (b x + a\right )}{b^{5}} \]

[In]

integrate((d*x+c)^4*cosh(b*x+a),x, algorithm="fricas")

[Out]

-(4*(b^3*d^4*x^3 + 3*b^3*c*d^3*x^2 + b^3*c^3*d + 6*b*c*d^3 + 3*(b^3*c^2*d^2 + 2*b*d^4)*x)*cosh(b*x + a) - (b^4
*d^4*x^4 + 4*b^4*c*d^3*x^3 + b^4*c^4 + 12*b^2*c^2*d^2 + 24*d^4 + 6*(b^4*c^2*d^2 + 2*b^2*d^4)*x^2 + 4*(b^4*c^3*
d + 6*b^2*c*d^3)*x)*sinh(b*x + a))/b^5

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (92) = 184\).

Time = 0.36 (sec) , antiderivative size = 311, normalized size of antiderivative = 3.42 \[ \int (c+d x)^4 \cosh (a+b x) \, dx=\begin {cases} \frac {c^{4} \sinh {\left (a + b x \right )}}{b} + \frac {4 c^{3} d x \sinh {\left (a + b x \right )}}{b} + \frac {6 c^{2} d^{2} x^{2} \sinh {\left (a + b x \right )}}{b} + \frac {4 c d^{3} x^{3} \sinh {\left (a + b x \right )}}{b} + \frac {d^{4} x^{4} \sinh {\left (a + b x \right )}}{b} - \frac {4 c^{3} d \cosh {\left (a + b x \right )}}{b^{2}} - \frac {12 c^{2} d^{2} x \cosh {\left (a + b x \right )}}{b^{2}} - \frac {12 c d^{3} x^{2} \cosh {\left (a + b x \right )}}{b^{2}} - \frac {4 d^{4} x^{3} \cosh {\left (a + b x \right )}}{b^{2}} + \frac {12 c^{2} d^{2} \sinh {\left (a + b x \right )}}{b^{3}} + \frac {24 c d^{3} x \sinh {\left (a + b x \right )}}{b^{3}} + \frac {12 d^{4} x^{2} \sinh {\left (a + b x \right )}}{b^{3}} - \frac {24 c d^{3} \cosh {\left (a + b x \right )}}{b^{4}} - \frac {24 d^{4} x \cosh {\left (a + b x \right )}}{b^{4}} + \frac {24 d^{4} \sinh {\left (a + b x \right )}}{b^{5}} & \text {for}\: b \neq 0 \\\left (c^{4} x + 2 c^{3} d x^{2} + 2 c^{2} d^{2} x^{3} + c d^{3} x^{4} + \frac {d^{4} x^{5}}{5}\right ) \cosh {\left (a \right )} & \text {otherwise} \end {cases} \]

[In]

integrate((d*x+c)**4*cosh(b*x+a),x)

[Out]

Piecewise((c**4*sinh(a + b*x)/b + 4*c**3*d*x*sinh(a + b*x)/b + 6*c**2*d**2*x**2*sinh(a + b*x)/b + 4*c*d**3*x**
3*sinh(a + b*x)/b + d**4*x**4*sinh(a + b*x)/b - 4*c**3*d*cosh(a + b*x)/b**2 - 12*c**2*d**2*x*cosh(a + b*x)/b**
2 - 12*c*d**3*x**2*cosh(a + b*x)/b**2 - 4*d**4*x**3*cosh(a + b*x)/b**2 + 12*c**2*d**2*sinh(a + b*x)/b**3 + 24*
c*d**3*x*sinh(a + b*x)/b**3 + 12*d**4*x**2*sinh(a + b*x)/b**3 - 24*c*d**3*cosh(a + b*x)/b**4 - 24*d**4*x*cosh(
a + b*x)/b**4 + 24*d**4*sinh(a + b*x)/b**5, Ne(b, 0)), ((c**4*x + 2*c**3*d*x**2 + 2*c**2*d**2*x**3 + c*d**3*x*
*4 + d**4*x**5/5)*cosh(a), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (91) = 182\).

Time = 0.20 (sec) , antiderivative size = 326, normalized size of antiderivative = 3.58 \[ \int (c+d x)^4 \cosh (a+b x) \, dx=\frac {c^{4} e^{\left (b x + a\right )}}{2 \, b} + \frac {2 \, {\left (b x e^{a} - e^{a}\right )} c^{3} d e^{\left (b x\right )}}{b^{2}} - \frac {c^{4} e^{\left (-b x - a\right )}}{2 \, b} - \frac {2 \, {\left (b x + 1\right )} c^{3} d e^{\left (-b x - a\right )}}{b^{2}} + \frac {3 \, {\left (b^{2} x^{2} e^{a} - 2 \, b x e^{a} + 2 \, e^{a}\right )} c^{2} d^{2} e^{\left (b x\right )}}{b^{3}} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, b x + 2\right )} c^{2} d^{2} e^{\left (-b x - a\right )}}{b^{3}} + \frac {2 \, {\left (b^{3} x^{3} e^{a} - 3 \, b^{2} x^{2} e^{a} + 6 \, b x e^{a} - 6 \, e^{a}\right )} c d^{3} e^{\left (b x\right )}}{b^{4}} - \frac {2 \, {\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} c d^{3} e^{\left (-b x - a\right )}}{b^{4}} + \frac {{\left (b^{4} x^{4} e^{a} - 4 \, b^{3} x^{3} e^{a} + 12 \, b^{2} x^{2} e^{a} - 24 \, b x e^{a} + 24 \, e^{a}\right )} d^{4} e^{\left (b x\right )}}{2 \, b^{5}} - \frac {{\left (b^{4} x^{4} + 4 \, b^{3} x^{3} + 12 \, b^{2} x^{2} + 24 \, b x + 24\right )} d^{4} e^{\left (-b x - a\right )}}{2 \, b^{5}} \]

[In]

integrate((d*x+c)^4*cosh(b*x+a),x, algorithm="maxima")

[Out]

1/2*c^4*e^(b*x + a)/b + 2*(b*x*e^a - e^a)*c^3*d*e^(b*x)/b^2 - 1/2*c^4*e^(-b*x - a)/b - 2*(b*x + 1)*c^3*d*e^(-b
*x - a)/b^2 + 3*(b^2*x^2*e^a - 2*b*x*e^a + 2*e^a)*c^2*d^2*e^(b*x)/b^3 - 3*(b^2*x^2 + 2*b*x + 2)*c^2*d^2*e^(-b*
x - a)/b^3 + 2*(b^3*x^3*e^a - 3*b^2*x^2*e^a + 6*b*x*e^a - 6*e^a)*c*d^3*e^(b*x)/b^4 - 2*(b^3*x^3 + 3*b^2*x^2 +
6*b*x + 6)*c*d^3*e^(-b*x - a)/b^4 + 1/2*(b^4*x^4*e^a - 4*b^3*x^3*e^a + 12*b^2*x^2*e^a - 24*b*x*e^a + 24*e^a)*d
^4*e^(b*x)/b^5 - 1/2*(b^4*x^4 + 4*b^3*x^3 + 12*b^2*x^2 + 24*b*x + 24)*d^4*e^(-b*x - a)/b^5

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 324 vs. \(2 (91) = 182\).

Time = 0.26 (sec) , antiderivative size = 324, normalized size of antiderivative = 3.56 \[ \int (c+d x)^4 \cosh (a+b x) \, dx=\frac {{\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + 6 \, b^{4} c^{2} d^{2} x^{2} - 4 \, b^{3} d^{4} x^{3} + 4 \, b^{4} c^{3} d x - 12 \, b^{3} c d^{3} x^{2} + b^{4} c^{4} - 12 \, b^{3} c^{2} d^{2} x + 12 \, b^{2} d^{4} x^{2} - 4 \, b^{3} c^{3} d + 24 \, b^{2} c d^{3} x + 12 \, b^{2} c^{2} d^{2} - 24 \, b d^{4} x - 24 \, b c d^{3} + 24 \, d^{4}\right )} e^{\left (b x + a\right )}}{2 \, b^{5}} - \frac {{\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + 6 \, b^{4} c^{2} d^{2} x^{2} + 4 \, b^{3} d^{4} x^{3} + 4 \, b^{4} c^{3} d x + 12 \, b^{3} c d^{3} x^{2} + b^{4} c^{4} + 12 \, b^{3} c^{2} d^{2} x + 12 \, b^{2} d^{4} x^{2} + 4 \, b^{3} c^{3} d + 24 \, b^{2} c d^{3} x + 12 \, b^{2} c^{2} d^{2} + 24 \, b d^{4} x + 24 \, b c d^{3} + 24 \, d^{4}\right )} e^{\left (-b x - a\right )}}{2 \, b^{5}} \]

[In]

integrate((d*x+c)^4*cosh(b*x+a),x, algorithm="giac")

[Out]

1/2*(b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 6*b^4*c^2*d^2*x^2 - 4*b^3*d^4*x^3 + 4*b^4*c^3*d*x - 12*b^3*c*d^3*x^2 + b^
4*c^4 - 12*b^3*c^2*d^2*x + 12*b^2*d^4*x^2 - 4*b^3*c^3*d + 24*b^2*c*d^3*x + 12*b^2*c^2*d^2 - 24*b*d^4*x - 24*b*
c*d^3 + 24*d^4)*e^(b*x + a)/b^5 - 1/2*(b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 6*b^4*c^2*d^2*x^2 + 4*b^3*d^4*x^3 + 4*b
^4*c^3*d*x + 12*b^3*c*d^3*x^2 + b^4*c^4 + 12*b^3*c^2*d^2*x + 12*b^2*d^4*x^2 + 4*b^3*c^3*d + 24*b^2*c*d^3*x + 1
2*b^2*c^2*d^2 + 24*b*d^4*x + 24*b*c*d^3 + 24*d^4)*e^(-b*x - a)/b^5

Mupad [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.36 \[ \int (c+d x)^4 \cosh (a+b x) \, dx=\frac {\mathrm {sinh}\left (a+b\,x\right )\,\left (b^4\,c^4+12\,b^2\,c^2\,d^2+24\,d^4\right )}{b^5}-\frac {4\,\mathrm {cosh}\left (a+b\,x\right )\,\left (b^2\,c^3\,d+6\,c\,d^3\right )}{b^4}-\frac {4\,d^4\,x^3\,\mathrm {cosh}\left (a+b\,x\right )}{b^2}-\frac {12\,x\,\mathrm {cosh}\left (a+b\,x\right )\,\left (b^2\,c^2\,d^2+2\,d^4\right )}{b^4}+\frac {d^4\,x^4\,\mathrm {sinh}\left (a+b\,x\right )}{b}+\frac {4\,x\,\mathrm {sinh}\left (a+b\,x\right )\,\left (b^2\,c^3\,d+6\,c\,d^3\right )}{b^3}+\frac {6\,x^2\,\mathrm {sinh}\left (a+b\,x\right )\,\left (b^2\,c^2\,d^2+2\,d^4\right )}{b^3}-\frac {12\,c\,d^3\,x^2\,\mathrm {cosh}\left (a+b\,x\right )}{b^2}+\frac {4\,c\,d^3\,x^3\,\mathrm {sinh}\left (a+b\,x\right )}{b} \]

[In]

int(cosh(a + b*x)*(c + d*x)^4,x)

[Out]

(sinh(a + b*x)*(24*d^4 + b^4*c^4 + 12*b^2*c^2*d^2))/b^5 - (4*cosh(a + b*x)*(6*c*d^3 + b^2*c^3*d))/b^4 - (4*d^4
*x^3*cosh(a + b*x))/b^2 - (12*x*cosh(a + b*x)*(2*d^4 + b^2*c^2*d^2))/b^4 + (d^4*x^4*sinh(a + b*x))/b + (4*x*si
nh(a + b*x)*(6*c*d^3 + b^2*c^3*d))/b^3 + (6*x^2*sinh(a + b*x)*(2*d^4 + b^2*c^2*d^2))/b^3 - (12*c*d^3*x^2*cosh(
a + b*x))/b^2 + (4*c*d^3*x^3*sinh(a + b*x))/b