\(\int (c+d x)^3 \cosh ^3(a+b x) \, dx\) [17]

Optimal result

Integrand size = 16, antiderivative size = 175 \[ \int (c+d x)^3 \cosh ^3(a+b x) \, dx=-\frac {40 d^3 \cosh (a+b x)}{9 b^4}-\frac {2 d (c+d x)^2 \cosh (a+b x)}{b^2}-\frac {2 d^3 \cosh ^3(a+b x)}{27 b^4}-\frac {d (c+d x)^2 \cosh ^3(a+b x)}{3 b^2}+\frac {40 d^2 (c+d x) \sinh (a+b x)}{9 b^3}+\frac {2 (c+d x)^3 \sinh (a+b x)}{3 b}+\frac {2 d^2 (c+d x) \cosh ^2(a+b x) \sinh (a+b x)}{9 b^3}+\frac {(c+d x)^3 \cosh ^2(a+b x) \sinh (a+b x)}{3 b} \]

[Out]

-40/9*d^3*cosh(b*x+a)/b^4-2*d*(d*x+c)^2*cosh(b*x+a)/b^2-2/27*d^3*cosh(b*x+a)^3/b^4-1/3*d*(d*x+c)^2*cosh(b*x+a)
^3/b^2+40/9*d^2*(d*x+c)*sinh(b*x+a)/b^3+2/3*(d*x+c)^3*sinh(b*x+a)/b+2/9*d^2*(d*x+c)*cosh(b*x+a)^2*sinh(b*x+a)/
b^3+1/3*(d*x+c)^3*cosh(b*x+a)^2*sinh(b*x+a)/b

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3392, 3377, 2718, 3391} \[ \int (c+d x)^3 \cosh ^3(a+b x) \, dx=-\frac {2 d^3 \cosh ^3(a+b x)}{27 b^4}-\frac {40 d^3 \cosh (a+b x)}{9 b^4}+\frac {40 d^2 (c+d x) \sinh (a+b x)}{9 b^3}+\frac {2 d^2 (c+d x) \sinh (a+b x) \cosh ^2(a+b x)}{9 b^3}-\frac {d (c+d x)^2 \cosh ^3(a+b x)}{3 b^2}-\frac {2 d (c+d x)^2 \cosh (a+b x)}{b^2}+\frac {2 (c+d x)^3 \sinh (a+b x)}{3 b}+\frac {(c+d x)^3 \sinh (a+b x) \cosh ^2(a+b x)}{3 b} \]

[In]

Int[(c + d*x)^3*Cosh[a + b*x]^3,x]

[Out]

(-40*d^3*Cosh[a + b*x])/(9*b^4) - (2*d*(c + d*x)^2*Cosh[a + b*x])/b^2 - (2*d^3*Cosh[a + b*x]^3)/(27*b^4) - (d*
(c + d*x)^2*Cosh[a + b*x]^3)/(3*b^2) + (40*d^2*(c + d*x)*Sinh[a + b*x])/(9*b^3) + (2*(c + d*x)^3*Sinh[a + b*x]
)/(3*b) + (2*d^2*(c + d*x)*Cosh[a + b*x]^2*Sinh[a + b*x])/(9*b^3) + ((c + d*x)^3*Cosh[a + b*x]^2*Sinh[a + b*x]
)/(3*b)

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[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]

Rule 3391

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] + (Dist[b^2*((n - 1)/n), Int[(c + d*x)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[b*(c + d*x)*Cos[e + f*x
]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]

Rule 3392

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*m*(c + d*x)^(m - 1)*((
b*Sin[e + f*x])^n/(f^2*n^2)), x] + (Dist[b^2*((n - 1)/n), Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[d^2*m*((m - 1)/(f^2*n^2)), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[b*(c + d*x)^m*Cos[e + f
*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]

Rubi steps \begin{align*} \text {integral}& = -\frac {d (c+d x)^2 \cosh ^3(a+b x)}{3 b^2}+\frac {(c+d x)^3 \cosh ^2(a+b x) \sinh (a+b x)}{3 b}+\frac {2}{3} \int (c+d x)^3 \cosh (a+b x) \, dx+\frac {\left (2 d^2\right ) \int (c+d x) \cosh ^3(a+b x) \, dx}{3 b^2} \\ & = -\frac {2 d^3 \cosh ^3(a+b x)}{27 b^4}-\frac {d (c+d x)^2 \cosh ^3(a+b x)}{3 b^2}+\frac {2 (c+d x)^3 \sinh (a+b x)}{3 b}+\frac {2 d^2 (c+d x) \cosh ^2(a+b x) \sinh (a+b x)}{9 b^3}+\frac {(c+d x)^3 \cosh ^2(a+b x) \sinh (a+b x)}{3 b}-\frac {(2 d) \int (c+d x)^2 \sinh (a+b x) \, dx}{b}+\frac {\left (4 d^2\right ) \int (c+d x) \cosh (a+b x) \, dx}{9 b^2} \\ & = -\frac {2 d (c+d x)^2 \cosh (a+b x)}{b^2}-\frac {2 d^3 \cosh ^3(a+b x)}{27 b^4}-\frac {d (c+d x)^2 \cosh ^3(a+b x)}{3 b^2}+\frac {4 d^2 (c+d x) \sinh (a+b x)}{9 b^3}+\frac {2 (c+d x)^3 \sinh (a+b x)}{3 b}+\frac {2 d^2 (c+d x) \cosh ^2(a+b x) \sinh (a+b x)}{9 b^3}+\frac {(c+d x)^3 \cosh ^2(a+b x) \sinh (a+b x)}{3 b}+\frac {\left (4 d^2\right ) \int (c+d x) \cosh (a+b x) \, dx}{b^2}-\frac {\left (4 d^3\right ) \int \sinh (a+b x) \, dx}{9 b^3} \\ & = -\frac {4 d^3 \cosh (a+b x)}{9 b^4}-\frac {2 d (c+d x)^2 \cosh (a+b x)}{b^2}-\frac {2 d^3 \cosh ^3(a+b x)}{27 b^4}-\frac {d (c+d x)^2 \cosh ^3(a+b x)}{3 b^2}+\frac {40 d^2 (c+d x) \sinh (a+b x)}{9 b^3}+\frac {2 (c+d x)^3 \sinh (a+b x)}{3 b}+\frac {2 d^2 (c+d x) \cosh ^2(a+b x) \sinh (a+b x)}{9 b^3}+\frac {(c+d x)^3 \cosh ^2(a+b x) \sinh (a+b x)}{3 b}-\frac {\left (4 d^3\right ) \int \sinh (a+b x) \, dx}{b^3} \\ & = -\frac {40 d^3 \cosh (a+b x)}{9 b^4}-\frac {2 d (c+d x)^2 \cosh (a+b x)}{b^2}-\frac {2 d^3 \cosh ^3(a+b x)}{27 b^4}-\frac {d (c+d x)^2 \cosh ^3(a+b x)}{3 b^2}+\frac {40 d^2 (c+d x) \sinh (a+b x)}{9 b^3}+\frac {2 (c+d x)^3 \sinh (a+b x)}{3 b}+\frac {2 d^2 (c+d x) \cosh ^2(a+b x) \sinh (a+b x)}{9 b^3}+\frac {(c+d x)^3 \cosh ^2(a+b x) \sinh (a+b x)}{3 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.66 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.70 \[ \int (c+d x)^3 \cosh ^3(a+b x) \, dx=\frac {-486 d \left (2 d^2+b^2 (c+d x)^2\right ) \cosh (a+b x)-2 d \left (2 d^2+9 b^2 (c+d x)^2\right ) \cosh (3 (a+b x))+12 b (c+d x) \left (82 d^2+15 b^2 (c+d x)^2+\left (2 d^2+3 b^2 (c+d x)^2\right ) \cosh (2 (a+b x))\right ) \sinh (a+b x)}{216 b^4} \]

[In]

Integrate[(c + d*x)^3*Cosh[a + b*x]^3,x]

[Out]

(-486*d*(2*d^2 + b^2*(c + d*x)^2)*Cosh[a + b*x] - 2*d*(2*d^2 + 9*b^2*(c + d*x)^2)*Cosh[3*(a + b*x)] + 12*b*(c
+ d*x)*(82*d^2 + 15*b^2*(c + d*x)^2 + (2*d^2 + 3*b^2*(c + d*x)^2)*Cosh[2*(a + b*x)])*Sinh[a + b*x])/(216*b^4)

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.62

method	result	size
parallelrisch	\(\frac {126 d^{2} \left (\frac {d x}{2}+c \right ) x \,b^{2} \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{6}-54 \left (d x +c \right ) \left (\left (d x +c \right )^{2} b^{2}+\frac {14 d^{2}}{3}\right ) b \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{5}+\left (\left (-27 d^{3} x^{2}-54 c \,d^{2} x +162 d \,c^{2}\right ) b^{2}+252 d^{3}\right ) \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}+36 \left (d x +c \right ) \left (\left (d x +c \right )^{2} b^{2}+\frac {38 d^{2}}{3}\right ) b \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}+\left (\left (-27 d^{3} x^{2}-54 c \,d^{2} x -216 d \,c^{2}\right ) b^{2}-480 d^{3}\right ) \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-54 \left (d x +c \right ) \left (\left (d x +c \right )^{2} b^{2}+\frac {14 d^{2}}{3}\right ) b \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )+\left (63 d^{3} x^{2}+126 c \,d^{2} x +126 d \,c^{2}\right ) b^{2}+244 d^{3}}{27 b^{4} \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )^{3} \left (1+\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{3}}\)	\(284\)
risch	\(\frac {\left (9 d^{3} x^{3} b^{3}+27 b^{3} c \,d^{2} x^{2}+27 b^{3} c^{2} d x -9 b^{2} d^{3} x^{2}+9 b^{3} c^{3}-18 b^{2} c \,d^{2} x -9 b^{2} c^{2} d +6 b \,d^{3} x +6 b c \,d^{2}-2 d^{3}\right ) {\mathrm e}^{3 b x +3 a}}{216 b^{4}}+\frac {3 \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x -3 b^{2} d^{3} x^{2}+b^{3} c^{3}-6 b^{2} c \,d^{2} x -3 b^{2} c^{2} d +6 b \,d^{3} x +6 b c \,d^{2}-6 d^{3}\right ) {\mathrm e}^{b x +a}}{8 b^{4}}-\frac {3 \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +3 b^{2} d^{3} x^{2}+b^{3} c^{3}+6 b^{2} c \,d^{2} x +3 b^{2} c^{2} d +6 b \,d^{3} x +6 b c \,d^{2}+6 d^{3}\right ) {\mathrm e}^{-b x -a}}{8 b^{4}}-\frac {\left (9 d^{3} x^{3} b^{3}+27 b^{3} c \,d^{2} x^{2}+27 b^{3} c^{2} d x +9 b^{2} d^{3} x^{2}+9 b^{3} c^{3}+18 b^{2} c \,d^{2} x +9 b^{2} c^{2} d +6 b \,d^{3} x +6 b c \,d^{2}+2 d^{3}\right ) {\mathrm e}^{-3 b x -3 a}}{216 b^{4}}\)	\(415\)
derivativedivides	\(\frac {\frac {d^{3} \left (\frac {2 \left (b x +a \right )^{3} \sinh \left (b x +a \right )}{3}+\frac {\left (b x +a \right )^{3} \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{3}-2 \left (b x +a \right )^{2} \cosh \left (b x +a \right )+\frac {40 \left (b x +a \right ) \sinh \left (b x +a \right )}{9}-\frac {40 \cosh \left (b x +a \right )}{9}-\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right )^{3}}{3}+\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{9}-\frac {2 \cosh \left (b x +a \right )^{3}}{27}\right )}{b^{3}}-\frac {3 d^{3} a \left (\frac {2 \left (b x +a \right )^{2} \sinh \left (b x +a \right )}{3}+\frac {\left (b x +a \right )^{2} \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{3}-\frac {4 \left (b x +a \right ) \cosh \left (b x +a \right )}{3}+\frac {40 \sinh \left (b x +a \right )}{27}-\frac {2 \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{9}+\frac {2 \cosh \left (b x +a \right )^{2} \sinh \left (b x +a \right )}{27}\right )}{b^{3}}+\frac {3 d^{2} c \left (\frac {2 \left (b x +a \right )^{2} \sinh \left (b x +a \right )}{3}+\frac {\left (b x +a \right )^{2} \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{3}-\frac {4 \left (b x +a \right ) \cosh \left (b x +a \right )}{3}+\frac {40 \sinh \left (b x +a \right )}{27}-\frac {2 \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{9}+\frac {2 \cosh \left (b x +a \right )^{2} \sinh \left (b x +a \right )}{27}\right )}{b^{2}}+\frac {3 d^{3} a^{2} \left (\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right )}{3}+\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{3}-\frac {2 \cosh \left (b x +a \right )}{3}-\frac {\cosh \left (b x +a \right )^{3}}{9}\right )}{b^{3}}-\frac {6 d^{2} a c \left (\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right )}{3}+\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{3}-\frac {2 \cosh \left (b x +a \right )}{3}-\frac {\cosh \left (b x +a \right )^{3}}{9}\right )}{b^{2}}+\frac {3 d \,c^{2} \left (\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right )}{3}+\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{3}-\frac {2 \cosh \left (b x +a \right )}{3}-\frac {\cosh \left (b x +a \right )^{3}}{9}\right )}{b}-\frac {d^{3} a^{3} \left (\frac {2}{3}+\frac {\cosh \left (b x +a \right )^{2}}{3}\right ) \sinh \left (b x +a \right )}{b^{3}}+\frac {3 d^{2} a^{2} c \left (\frac {2}{3}+\frac {\cosh \left (b x +a \right )^{2}}{3}\right ) \sinh \left (b x +a \right )}{b^{2}}-\frac {3 d a \,c^{2} \left (\frac {2}{3}+\frac {\cosh \left (b x +a \right )^{2}}{3}\right ) \sinh \left (b x +a \right )}{b}+c^{3} \left (\frac {2}{3}+\frac {\cosh \left (b x +a \right )^{2}}{3}\right ) \sinh \left (b x +a \right )}{b}\)	\(634\)
default	\(\frac {\frac {d^{3} \left (\frac {2 \left (b x +a \right )^{3} \sinh \left (b x +a \right )}{3}+\frac {\left (b x +a \right )^{3} \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{3}-2 \left (b x +a \right )^{2} \cosh \left (b x +a \right )+\frac {40 \left (b x +a \right ) \sinh \left (b x +a \right )}{9}-\frac {40 \cosh \left (b x +a \right )}{9}-\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right )^{3}}{3}+\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{9}-\frac {2 \cosh \left (b x +a \right )^{3}}{27}\right )}{b^{3}}-\frac {3 d^{3} a \left (\frac {2 \left (b x +a \right )^{2} \sinh \left (b x +a \right )}{3}+\frac {\left (b x +a \right )^{2} \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{3}-\frac {4 \left (b x +a \right ) \cosh \left (b x +a \right )}{3}+\frac {40 \sinh \left (b x +a \right )}{27}-\frac {2 \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{9}+\frac {2 \cosh \left (b x +a \right )^{2} \sinh \left (b x +a \right )}{27}\right )}{b^{3}}+\frac {3 d^{2} c \left (\frac {2 \left (b x +a \right )^{2} \sinh \left (b x +a \right )}{3}+\frac {\left (b x +a \right )^{2} \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{3}-\frac {4 \left (b x +a \right ) \cosh \left (b x +a \right )}{3}+\frac {40 \sinh \left (b x +a \right )}{27}-\frac {2 \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{9}+\frac {2 \cosh \left (b x +a \right )^{2} \sinh \left (b x +a \right )}{27}\right )}{b^{2}}+\frac {3 d^{3} a^{2} \left (\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right )}{3}+\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{3}-\frac {2 \cosh \left (b x +a \right )}{3}-\frac {\cosh \left (b x +a \right )^{3}}{9}\right )}{b^{3}}-\frac {6 d^{2} a c \left (\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right )}{3}+\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{3}-\frac {2 \cosh \left (b x +a \right )}{3}-\frac {\cosh \left (b x +a \right )^{3}}{9}\right )}{b^{2}}+\frac {3 d \,c^{2} \left (\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right )}{3}+\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{3}-\frac {2 \cosh \left (b x +a \right )}{3}-\frac {\cosh \left (b x +a \right )^{3}}{9}\right )}{b}-\frac {d^{3} a^{3} \left (\frac {2}{3}+\frac {\cosh \left (b x +a \right )^{2}}{3}\right ) \sinh \left (b x +a \right )}{b^{3}}+\frac {3 d^{2} a^{2} c \left (\frac {2}{3}+\frac {\cosh \left (b x +a \right )^{2}}{3}\right ) \sinh \left (b x +a \right )}{b^{2}}-\frac {3 d a \,c^{2} \left (\frac {2}{3}+\frac {\cosh \left (b x +a \right )^{2}}{3}\right ) \sinh \left (b x +a \right )}{b}+c^{3} \left (\frac {2}{3}+\frac {\cosh \left (b x +a \right )^{2}}{3}\right ) \sinh \left (b x +a \right )}{b}\)	\(634\)

[In]

int((d*x+c)^3*cosh(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

1/27*(126*d^2*(1/2*d*x+c)*x*b^2*tanh(1/2*b*x+1/2*a)^6-54*(d*x+c)*((d*x+c)^2*b^2+14/3*d^2)*b*tanh(1/2*b*x+1/2*a
)^5+((-27*d^3*x^2-54*c*d^2*x+162*c^2*d)*b^2+252*d^3)*tanh(1/2*b*x+1/2*a)^4+36*(d*x+c)*((d*x+c)^2*b^2+38/3*d^2)
*b*tanh(1/2*b*x+1/2*a)^3+((-27*d^3*x^2-54*c*d^2*x-216*c^2*d)*b^2-480*d^3)*tanh(1/2*b*x+1/2*a)^2-54*(d*x+c)*((d
*x+c)^2*b^2+14/3*d^2)*b*tanh(1/2*b*x+1/2*a)+(63*d^3*x^2+126*c*d^2*x+126*c^2*d)*b^2+244*d^3)/b^4/(tanh(1/2*b*x+
1/2*a)-1)^3/(1+tanh(1/2*b*x+1/2*a))^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (161) = 322\).

Time = 0.25 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.96 \[ \int (c+d x)^3 \cosh ^3(a+b x) \, dx=-\frac {{\left (9 \, b^{2} d^{3} x^{2} + 18 \, b^{2} c d^{2} x + 9 \, b^{2} c^{2} d + 2 \, d^{3}\right )} \cosh \left (b x + a\right )^{3} + 3 \, {\left (9 \, b^{2} d^{3} x^{2} + 18 \, b^{2} c d^{2} x + 9 \, b^{2} c^{2} d + 2 \, d^{3}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 3 \, {\left (3 \, b^{3} d^{3} x^{3} + 9 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{3} + 2 \, b c d^{2} + {\left (9 \, b^{3} c^{2} d + 2 \, b d^{3}\right )} x\right )} \sinh \left (b x + a\right )^{3} + 243 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d + 2 \, d^{3}\right )} \cosh \left (b x + a\right ) - 9 \, {\left (9 \, b^{3} d^{3} x^{3} + 27 \, b^{3} c d^{2} x^{2} + 9 \, b^{3} c^{3} + 54 \, b c d^{2} + {\left (3 \, b^{3} d^{3} x^{3} + 9 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{3} + 2 \, b c d^{2} + {\left (9 \, b^{3} c^{2} d + 2 \, b d^{3}\right )} x\right )} \cosh \left (b x + a\right )^{2} + 27 \, {\left (b^{3} c^{2} d + 2 \, b d^{3}\right )} x\right )} \sinh \left (b x + a\right )}{108 \, b^{4}} \]

[In]

integrate((d*x+c)^3*cosh(b*x+a)^3,x, algorithm="fricas")

[Out]

-1/108*((9*b^2*d^3*x^2 + 18*b^2*c*d^2*x + 9*b^2*c^2*d + 2*d^3)*cosh(b*x + a)^3 + 3*(9*b^2*d^3*x^2 + 18*b^2*c*d
^2*x + 9*b^2*c^2*d + 2*d^3)*cosh(b*x + a)*sinh(b*x + a)^2 - 3*(3*b^3*d^3*x^3 + 9*b^3*c*d^2*x^2 + 3*b^3*c^3 + 2
*b*c*d^2 + (9*b^3*c^2*d + 2*b*d^3)*x)*sinh(b*x + a)^3 + 243*(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) - 9*(9*b^3*d^3*x^3 + 27*b^3*c*d^2*x^2 + 9*b^3*c^3 + 54*b*c*d^2 + (3*b^3*d^3*x^3 + 9*b^3*c*d^2*x^
2 + 3*b^3*c^3 + 2*b*c*d^2 + (9*b^3*c^2*d + 2*b*d^3)*x)*cosh(b*x + a)^2 + 27*(b^3*c^2*d + 2*b*d^3)*x)*sinh(b*x
+ a))/b^4

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (173) = 346\).

Time = 0.50 (sec) , antiderivative size = 495, normalized size of antiderivative = 2.83 \[ \int (c+d x)^3 \cosh ^3(a+b x) \, dx=\begin {cases} - \frac {2 c^{3} \sinh ^{3}{\left (a + b x \right )}}{3 b} + \frac {c^{3} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b} - \frac {2 c^{2} d x \sinh ^{3}{\left (a + b x \right )}}{b} + \frac {3 c^{2} d x \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b} - \frac {2 c d^{2} x^{2} \sinh ^{3}{\left (a + b x \right )}}{b} + \frac {3 c d^{2} x^{2} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b} - \frac {2 d^{3} x^{3} \sinh ^{3}{\left (a + b x \right )}}{3 b} + \frac {d^{3} x^{3} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b} + \frac {2 c^{2} d \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b^{2}} - \frac {7 c^{2} d \cosh ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {4 c d^{2} x \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b^{2}} - \frac {14 c d^{2} x \cosh ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {2 d^{3} x^{2} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b^{2}} - \frac {7 d^{3} x^{2} \cosh ^{3}{\left (a + b x \right )}}{3 b^{2}} - \frac {40 c d^{2} \sinh ^{3}{\left (a + b x \right )}}{9 b^{3}} + \frac {14 c d^{2} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3 b^{3}} - \frac {40 d^{3} x \sinh ^{3}{\left (a + b x \right )}}{9 b^{3}} + \frac {14 d^{3} x \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3 b^{3}} + \frac {40 d^{3} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{9 b^{4}} - \frac {122 d^{3} \cosh ^{3}{\left (a + b x \right )}}{27 b^{4}} & \text {for}\: b \neq 0 \\\left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) \cosh ^{3}{\left (a \right )} & \text {otherwise} \end {cases} \]

[In]

integrate((d*x+c)**3*cosh(b*x+a)**3,x)

[Out]

Piecewise((-2*c**3*sinh(a + b*x)**3/(3*b) + c**3*sinh(a + b*x)*cosh(a + b*x)**2/b - 2*c**2*d*x*sinh(a + b*x)**
3/b + 3*c**2*d*x*sinh(a + b*x)*cosh(a + b*x)**2/b - 2*c*d**2*x**2*sinh(a + b*x)**3/b + 3*c*d**2*x**2*sinh(a +
b*x)*cosh(a + b*x)**2/b - 2*d**3*x**3*sinh(a + b*x)**3/(3*b) + d**3*x**3*sinh(a + b*x)*cosh(a + b*x)**2/b + 2*
c**2*d*sinh(a + b*x)**2*cosh(a + b*x)/b**2 - 7*c**2*d*cosh(a + b*x)**3/(3*b**2) + 4*c*d**2*x*sinh(a + b*x)**2*
cosh(a + b*x)/b**2 - 14*c*d**2*x*cosh(a + b*x)**3/(3*b**2) + 2*d**3*x**2*sinh(a + b*x)**2*cosh(a + b*x)/b**2 -
7*d**3*x**2*cosh(a + b*x)**3/(3*b**2) - 40*c*d**2*sinh(a + b*x)**3/(9*b**3) + 14*c*d**2*sinh(a + b*x)*cosh(a
+ b*x)**2/(3*b**3) - 40*d**3*x*sinh(a + b*x)**3/(9*b**3) + 14*d**3*x*sinh(a + b*x)*cosh(a + b*x)**2/(3*b**3) +
40*d**3*sinh(a + b*x)**2*cosh(a + b*x)/(9*b**4) - 122*d**3*cosh(a + b*x)**3/(27*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)**3, True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 439 vs. \(2 (161) = 322\).

Time = 0.20 (sec) , antiderivative size = 439, normalized size of antiderivative = 2.51 \[ \int (c+d x)^3 \cosh ^3(a+b x) \, dx=\frac {1}{24} \, c^{2} d {\left (\frac {{\left (3 \, b x e^{\left (3 \, a\right )} - e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{b^{2}} + \frac {27 \, {\left (b x e^{a} - e^{a}\right )} e^{\left (b x\right )}}{b^{2}} - \frac {27 \, {\left (b x + 1\right )} e^{\left (-b x - a\right )}}{b^{2}} - \frac {{\left (3 \, b x + 1\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{b^{2}}\right )} + \frac {1}{24} \, c^{3} {\left (\frac {e^{\left (3 \, b x + 3 \, a\right )}}{b} + \frac {9 \, e^{\left (b x + a\right )}}{b} - \frac {9 \, e^{\left (-b x - a\right )}}{b} - \frac {e^{\left (-3 \, b x - 3 \, a\right )}}{b}\right )} + \frac {1}{72} \, c d^{2} {\left (\frac {{\left (9 \, b^{2} x^{2} e^{\left (3 \, a\right )} - 6 \, b x e^{\left (3 \, a\right )} + 2 \, e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{b^{3}} + \frac {81 \, {\left (b^{2} x^{2} e^{a} - 2 \, b x e^{a} + 2 \, e^{a}\right )} e^{\left (b x\right )}}{b^{3}} - \frac {81 \, {\left (b^{2} x^{2} + 2 \, b x + 2\right )} e^{\left (-b x - a\right )}}{b^{3}} - \frac {{\left (9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{b^{3}}\right )} + \frac {1}{216} \, d^{3} {\left (\frac {{\left (9 \, b^{3} x^{3} e^{\left (3 \, a\right )} - 9 \, b^{2} x^{2} e^{\left (3 \, a\right )} + 6 \, b x e^{\left (3 \, a\right )} - 2 \, e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{b^{4}} + \frac {81 \, {\left (b^{3} x^{3} e^{a} - 3 \, b^{2} x^{2} e^{a} + 6 \, b x e^{a} - 6 \, e^{a}\right )} e^{\left (b x\right )}}{b^{4}} - \frac {81 \, {\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} e^{\left (-b x - a\right )}}{b^{4}} - \frac {{\left (9 \, b^{3} x^{3} + 9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{b^{4}}\right )} \]

[In]

integrate((d*x+c)^3*cosh(b*x+a)^3,x, algorithm="maxima")

[Out]

1/24*c^2*d*((3*b*x*e^(3*a) - e^(3*a))*e^(3*b*x)/b^2 + 27*(b*x*e^a - e^a)*e^(b*x)/b^2 - 27*(b*x + 1)*e^(-b*x -
a)/b^2 - (3*b*x + 1)*e^(-3*b*x - 3*a)/b^2) + 1/24*c^3*(e^(3*b*x + 3*a)/b + 9*e^(b*x + a)/b - 9*e^(-b*x - a)/b
- e^(-3*b*x - 3*a)/b) + 1/72*c*d^2*((9*b^2*x^2*e^(3*a) - 6*b*x*e^(3*a) + 2*e^(3*a))*e^(3*b*x)/b^3 + 81*(b^2*x^
2*e^a - 2*b*x*e^a + 2*e^a)*e^(b*x)/b^3 - 81*(b^2*x^2 + 2*b*x + 2)*e^(-b*x - a)/b^3 - (9*b^2*x^2 + 6*b*x + 2)*e
^(-3*b*x - 3*a)/b^3) + 1/216*d^3*((9*b^3*x^3*e^(3*a) - 9*b^2*x^2*e^(3*a) + 6*b*x*e^(3*a) - 2*e^(3*a))*e^(3*b*x
)/b^4 + 81*(b^3*x^3*e^a - 3*b^2*x^2*e^a + 6*b*x*e^a - 6*e^a)*e^(b*x)/b^4 - 81*(b^3*x^3 + 3*b^2*x^2 + 6*b*x + 6
)*e^(-b*x - a)/b^4 - (9*b^3*x^3 + 9*b^2*x^2 + 6*b*x + 2)*e^(-3*b*x - 3*a)/b^4)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 414 vs. \(2 (161) = 322\).

Time = 0.28 (sec) , antiderivative size = 414, normalized size of antiderivative = 2.37 \[ \int (c+d x)^3 \cosh ^3(a+b x) \, dx=\frac {{\left (9 \, b^{3} d^{3} x^{3} + 27 \, b^{3} c d^{2} x^{2} + 27 \, b^{3} c^{2} d x - 9 \, b^{2} d^{3} x^{2} + 9 \, b^{3} c^{3} - 18 \, b^{2} c d^{2} x - 9 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} - 2 \, d^{3}\right )} e^{\left (3 \, b x + 3 \, a\right )}}{216 \, b^{4}} + \frac {3 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x - 3 \, b^{2} d^{3} x^{2} + b^{3} c^{3} - 6 \, b^{2} c d^{2} x - 3 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} - 6 \, d^{3}\right )} e^{\left (b x + a\right )}}{8 \, b^{4}} - \frac {3 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + 3 \, b^{2} d^{3} x^{2} + b^{3} c^{3} + 6 \, b^{2} c d^{2} x + 3 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} + 6 \, d^{3}\right )} e^{\left (-b x - a\right )}}{8 \, b^{4}} - \frac {{\left (9 \, b^{3} d^{3} x^{3} + 27 \, b^{3} c d^{2} x^{2} + 27 \, b^{3} c^{2} d x + 9 \, b^{2} d^{3} x^{2} + 9 \, b^{3} c^{3} + 18 \, b^{2} c d^{2} x + 9 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} + 2 \, d^{3}\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{216 \, b^{4}} \]

[In]

integrate((d*x+c)^3*cosh(b*x+a)^3,x, algorithm="giac")

[Out]

1/216*(9*b^3*d^3*x^3 + 27*b^3*c*d^2*x^2 + 27*b^3*c^2*d*x - 9*b^2*d^3*x^2 + 9*b^3*c^3 - 18*b^2*c*d^2*x - 9*b^2*
c^2*d + 6*b*d^3*x + 6*b*c*d^2 - 2*d^3)*e^(3*b*x + 3*a)/b^4 + 3/8*(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 - 3
/8*(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 - 1/216*(9*b^3*d^3*x^3 + 27*b^3*c*d^2*x^2 + 27*b^3*c^2*d*x + 9*b
^2*d^3*x^2 + 9*b^3*c^3 + 18*b^2*c*d^2*x + 9*b^2*c^2*d + 6*b*d^3*x + 6*b*c*d^2 + 2*d^3)*e^(-3*b*x - 3*a)/b^4

Mupad [B] (verification not implemented)

Time = 2.04 (sec) , antiderivative size = 364, normalized size of antiderivative = 2.08 \[ \int (c+d x)^3 \cosh ^3(a+b x) \, dx=\frac {{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )\,\left (3\,b^2\,c^3+14\,c\,d^2\right )}{3\,b^3}-\frac {2\,{\mathrm {sinh}\left (a+b\,x\right )}^3\,\left (3\,b^2\,c^3+20\,c\,d^2\right )}{9\,b^3}-\frac {{\mathrm {cosh}\left (a+b\,x\right )}^3\,\left (63\,b^2\,c^2\,d+122\,d^3\right )}{27\,b^4}+\frac {2\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2\,\left (9\,b^2\,c^2\,d+20\,d^3\right )}{9\,b^4}-\frac {2\,x\,{\mathrm {sinh}\left (a+b\,x\right )}^3\,\left (9\,b^2\,c^2\,d+20\,d^3\right )}{9\,b^3}-\frac {7\,d^3\,x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{3\,b^2}-\frac {2\,d^3\,x^3\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{3\,b}-\frac {14\,c\,d^2\,x\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{3\,b^2}+\frac {x\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )\,\left (9\,b^2\,c^2\,d+14\,d^3\right )}{3\,b^3}+\frac {d^3\,x^3\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )}{b}+\frac {2\,d^3\,x^2\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2}{b^2}-\frac {2\,c\,d^2\,x^2\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{b}+\frac {3\,c\,d^2\,x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )}{b}+\frac {4\,c\,d^2\,x\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2}{b^2} \]

[In]

int(cosh(a + b*x)^3*(c + d*x)^3,x)

[Out]

(cosh(a + b*x)^2*sinh(a + b*x)*(14*c*d^2 + 3*b^2*c^3))/(3*b^3) - (2*sinh(a + b*x)^3*(20*c*d^2 + 3*b^2*c^3))/(9
*b^3) - (cosh(a + b*x)^3*(122*d^3 + 63*b^2*c^2*d))/(27*b^4) + (2*cosh(a + b*x)*sinh(a + b*x)^2*(20*d^3 + 9*b^2
*c^2*d))/(9*b^4) - (2*x*sinh(a + b*x)^3*(20*d^3 + 9*b^2*c^2*d))/(9*b^3) - (7*d^3*x^2*cosh(a + b*x)^3)/(3*b^2)
- (2*d^3*x^3*sinh(a + b*x)^3)/(3*b) - (14*c*d^2*x*cosh(a + b*x)^3)/(3*b^2) + (x*cosh(a + b*x)^2*sinh(a + b*x)*
(14*d^3 + 9*b^2*c^2*d))/(3*b^3) + (d^3*x^3*cosh(a + b*x)^2*sinh(a + b*x))/b + (2*d^3*x^2*cosh(a + b*x)*sinh(a
+ b*x)^2)/b^2 - (2*c*d^2*x^2*sinh(a + b*x)^3)/b + (3*c*d^2*x^2*cosh(a + b*x)^2*sinh(a + b*x))/b + (4*c*d^2*x*c
osh(a + b*x)*sinh(a + b*x)^2)/b^2