\(\int x (a+b x^3)^2 \cosh (c+d x) \, dx\) [87]

Optimal result

Integrand size = 17, antiderivative size = 234 \[ \int x \left (a+b x^3\right )^2 \cosh (c+d x) \, dx=-\frac {5040 b^2 \cosh (c+d x)}{d^8}-\frac {a^2 \cosh (c+d x)}{d^2}-\frac {48 a b x \cosh (c+d x)}{d^4}-\frac {2520 b^2 x^2 \cosh (c+d x)}{d^6}-\frac {8 a b x^3 \cosh (c+d x)}{d^2}-\frac {210 b^2 x^4 \cosh (c+d x)}{d^4}-\frac {7 b^2 x^6 \cosh (c+d x)}{d^2}+\frac {48 a b \sinh (c+d x)}{d^5}+\frac {5040 b^2 x \sinh (c+d x)}{d^7}+\frac {a^2 x \sinh (c+d x)}{d}+\frac {24 a b x^2 \sinh (c+d x)}{d^3}+\frac {840 b^2 x^3 \sinh (c+d x)}{d^5}+\frac {2 a b x^4 \sinh (c+d x)}{d}+\frac {42 b^2 x^5 \sinh (c+d x)}{d^3}+\frac {b^2 x^7 \sinh (c+d x)}{d} \]

[Out]

-5040*b^2*cosh(d*x+c)/d^8-a^2*cosh(d*x+c)/d^2-48*a*b*x*cosh(d*x+c)/d^4-2520*b^2*x^2*cosh(d*x+c)/d^6-8*a*b*x^3*
cosh(d*x+c)/d^2-210*b^2*x^4*cosh(d*x+c)/d^4-7*b^2*x^6*cosh(d*x+c)/d^2+48*a*b*sinh(d*x+c)/d^5+5040*b^2*x*sinh(d
*x+c)/d^7+a^2*x*sinh(d*x+c)/d+24*a*b*x^2*sinh(d*x+c)/d^3+840*b^2*x^3*sinh(d*x+c)/d^5+2*a*b*x^4*sinh(d*x+c)/d+4
2*b^2*x^5*sinh(d*x+c)/d^3+b^2*x^7*sinh(d*x+c)/d

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {5395, 3377, 2718, 2717} \[ \int x \left (a+b x^3\right )^2 \cosh (c+d x) \, dx=-\frac {a^2 \cosh (c+d x)}{d^2}+\frac {a^2 x \sinh (c+d x)}{d}+\frac {48 a b \sinh (c+d x)}{d^5}-\frac {48 a b x \cosh (c+d x)}{d^4}+\frac {24 a b x^2 \sinh (c+d x)}{d^3}-\frac {8 a b x^3 \cosh (c+d x)}{d^2}+\frac {2 a b x^4 \sinh (c+d x)}{d}-\frac {5040 b^2 \cosh (c+d x)}{d^8}+\frac {5040 b^2 x \sinh (c+d x)}{d^7}-\frac {2520 b^2 x^2 \cosh (c+d x)}{d^6}+\frac {840 b^2 x^3 \sinh (c+d x)}{d^5}-\frac {210 b^2 x^4 \cosh (c+d x)}{d^4}+\frac {42 b^2 x^5 \sinh (c+d x)}{d^3}-\frac {7 b^2 x^6 \cosh (c+d x)}{d^2}+\frac {b^2 x^7 \sinh (c+d x)}{d} \]

[In]

Int[x*(a + b*x^3)^2*Cosh[c + d*x],x]

[Out]

(-5040*b^2*Cosh[c + d*x])/d^8 - (a^2*Cosh[c + d*x])/d^2 - (48*a*b*x*Cosh[c + d*x])/d^4 - (2520*b^2*x^2*Cosh[c
+ d*x])/d^6 - (8*a*b*x^3*Cosh[c + d*x])/d^2 - (210*b^2*x^4*Cosh[c + d*x])/d^4 - (7*b^2*x^6*Cosh[c + d*x])/d^2
+ (48*a*b*Sinh[c + d*x])/d^5 + (5040*b^2*x*Sinh[c + d*x])/d^7 + (a^2*x*Sinh[c + d*x])/d + (24*a*b*x^2*Sinh[c +
d*x])/d^3 + (840*b^2*x^3*Sinh[c + d*x])/d^5 + (2*a*b*x^4*Sinh[c + d*x])/d + (42*b^2*x^5*Sinh[c + d*x])/d^3 +
(b^2*x^7*Sinh[c + d*x])/d

Rule 2717

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

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 5395

Int[Cosh[(c_.) + (d_.)*(x_)]*((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[Cosh[c + d*x], (e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 x \cosh (c+d x)+2 a b x^4 \cosh (c+d x)+b^2 x^7 \cosh (c+d x)\right ) \, dx \\ & = a^2 \int x \cosh (c+d x) \, dx+(2 a b) \int x^4 \cosh (c+d x) \, dx+b^2 \int x^7 \cosh (c+d x) \, dx \\ & = \frac {a^2 x \sinh (c+d x)}{d}+\frac {2 a b x^4 \sinh (c+d x)}{d}+\frac {b^2 x^7 \sinh (c+d x)}{d}-\frac {a^2 \int \sinh (c+d x) \, dx}{d}-\frac {(8 a b) \int x^3 \sinh (c+d x) \, dx}{d}-\frac {\left (7 b^2\right ) \int x^6 \sinh (c+d x) \, dx}{d} \\ & = -\frac {a^2 \cosh (c+d x)}{d^2}-\frac {8 a b x^3 \cosh (c+d x)}{d^2}-\frac {7 b^2 x^6 \cosh (c+d x)}{d^2}+\frac {a^2 x \sinh (c+d x)}{d}+\frac {2 a b x^4 \sinh (c+d x)}{d}+\frac {b^2 x^7 \sinh (c+d x)}{d}+\frac {(24 a b) \int x^2 \cosh (c+d x) \, dx}{d^2}+\frac {\left (42 b^2\right ) \int x^5 \cosh (c+d x) \, dx}{d^2} \\ & = -\frac {a^2 \cosh (c+d x)}{d^2}-\frac {8 a b x^3 \cosh (c+d x)}{d^2}-\frac {7 b^2 x^6 \cosh (c+d x)}{d^2}+\frac {a^2 x \sinh (c+d x)}{d}+\frac {24 a b x^2 \sinh (c+d x)}{d^3}+\frac {2 a b x^4 \sinh (c+d x)}{d}+\frac {42 b^2 x^5 \sinh (c+d x)}{d^3}+\frac {b^2 x^7 \sinh (c+d x)}{d}-\frac {(48 a b) \int x \sinh (c+d x) \, dx}{d^3}-\frac {\left (210 b^2\right ) \int x^4 \sinh (c+d x) \, dx}{d^3} \\ & = -\frac {a^2 \cosh (c+d x)}{d^2}-\frac {48 a b x \cosh (c+d x)}{d^4}-\frac {8 a b x^3 \cosh (c+d x)}{d^2}-\frac {210 b^2 x^4 \cosh (c+d x)}{d^4}-\frac {7 b^2 x^6 \cosh (c+d x)}{d^2}+\frac {a^2 x \sinh (c+d x)}{d}+\frac {24 a b x^2 \sinh (c+d x)}{d^3}+\frac {2 a b x^4 \sinh (c+d x)}{d}+\frac {42 b^2 x^5 \sinh (c+d x)}{d^3}+\frac {b^2 x^7 \sinh (c+d x)}{d}+\frac {(48 a b) \int \cosh (c+d x) \, dx}{d^4}+\frac {\left (840 b^2\right ) \int x^3 \cosh (c+d x) \, dx}{d^4} \\ & = -\frac {a^2 \cosh (c+d x)}{d^2}-\frac {48 a b x \cosh (c+d x)}{d^4}-\frac {8 a b x^3 \cosh (c+d x)}{d^2}-\frac {210 b^2 x^4 \cosh (c+d x)}{d^4}-\frac {7 b^2 x^6 \cosh (c+d x)}{d^2}+\frac {48 a b \sinh (c+d x)}{d^5}+\frac {a^2 x \sinh (c+d x)}{d}+\frac {24 a b x^2 \sinh (c+d x)}{d^3}+\frac {840 b^2 x^3 \sinh (c+d x)}{d^5}+\frac {2 a b x^4 \sinh (c+d x)}{d}+\frac {42 b^2 x^5 \sinh (c+d x)}{d^3}+\frac {b^2 x^7 \sinh (c+d x)}{d}-\frac {\left (2520 b^2\right ) \int x^2 \sinh (c+d x) \, dx}{d^5} \\ & = -\frac {a^2 \cosh (c+d x)}{d^2}-\frac {48 a b x \cosh (c+d x)}{d^4}-\frac {2520 b^2 x^2 \cosh (c+d x)}{d^6}-\frac {8 a b x^3 \cosh (c+d x)}{d^2}-\frac {210 b^2 x^4 \cosh (c+d x)}{d^4}-\frac {7 b^2 x^6 \cosh (c+d x)}{d^2}+\frac {48 a b \sinh (c+d x)}{d^5}+\frac {a^2 x \sinh (c+d x)}{d}+\frac {24 a b x^2 \sinh (c+d x)}{d^3}+\frac {840 b^2 x^3 \sinh (c+d x)}{d^5}+\frac {2 a b x^4 \sinh (c+d x)}{d}+\frac {42 b^2 x^5 \sinh (c+d x)}{d^3}+\frac {b^2 x^7 \sinh (c+d x)}{d}+\frac {\left (5040 b^2\right ) \int x \cosh (c+d x) \, dx}{d^6} \\ & = -\frac {a^2 \cosh (c+d x)}{d^2}-\frac {48 a b x \cosh (c+d x)}{d^4}-\frac {2520 b^2 x^2 \cosh (c+d x)}{d^6}-\frac {8 a b x^3 \cosh (c+d x)}{d^2}-\frac {210 b^2 x^4 \cosh (c+d x)}{d^4}-\frac {7 b^2 x^6 \cosh (c+d x)}{d^2}+\frac {48 a b \sinh (c+d x)}{d^5}+\frac {5040 b^2 x \sinh (c+d x)}{d^7}+\frac {a^2 x \sinh (c+d x)}{d}+\frac {24 a b x^2 \sinh (c+d x)}{d^3}+\frac {840 b^2 x^3 \sinh (c+d x)}{d^5}+\frac {2 a b x^4 \sinh (c+d x)}{d}+\frac {42 b^2 x^5 \sinh (c+d x)}{d^3}+\frac {b^2 x^7 \sinh (c+d x)}{d}-\frac {\left (5040 b^2\right ) \int \sinh (c+d x) \, dx}{d^7} \\ & = -\frac {5040 b^2 \cosh (c+d x)}{d^8}-\frac {a^2 \cosh (c+d x)}{d^2}-\frac {48 a b x \cosh (c+d x)}{d^4}-\frac {2520 b^2 x^2 \cosh (c+d x)}{d^6}-\frac {8 a b x^3 \cosh (c+d x)}{d^2}-\frac {210 b^2 x^4 \cosh (c+d x)}{d^4}-\frac {7 b^2 x^6 \cosh (c+d x)}{d^2}+\frac {48 a b \sinh (c+d x)}{d^5}+\frac {5040 b^2 x \sinh (c+d x)}{d^7}+\frac {a^2 x \sinh (c+d x)}{d}+\frac {24 a b x^2 \sinh (c+d x)}{d^3}+\frac {840 b^2 x^3 \sinh (c+d x)}{d^5}+\frac {2 a b x^4 \sinh (c+d x)}{d}+\frac {42 b^2 x^5 \sinh (c+d x)}{d^3}+\frac {b^2 x^7 \sinh (c+d x)}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.59 \[ \int x \left (a+b x^3\right )^2 \cosh (c+d x) \, dx=\frac {-\left (\left (a^2 d^6+8 a b d^4 x \left (6+d^2 x^2\right )+7 b^2 \left (720+360 d^2 x^2+30 d^4 x^4+d^6 x^6\right )\right ) \cosh (c+d x)\right )+d \left (a^2 d^6 x+2 a b d^2 \left (24+12 d^2 x^2+d^4 x^4\right )+b^2 x \left (5040+840 d^2 x^2+42 d^4 x^4+d^6 x^6\right )\right ) \sinh (c+d x)}{d^8} \]

[In]

Integrate[x*(a + b*x^3)^2*Cosh[c + d*x],x]

[Out]

(-((a^2*d^6 + 8*a*b*d^4*x*(6 + d^2*x^2) + 7*b^2*(720 + 360*d^2*x^2 + 30*d^4*x^4 + d^6*x^6))*Cosh[c + d*x]) + d
*(a^2*d^6*x + 2*a*b*d^2*(24 + 12*d^2*x^2 + d^4*x^4) + b^2*x*(5040 + 840*d^2*x^2 + 42*d^4*x^4 + d^6*x^6))*Sinh[
c + d*x])/d^8

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.88

method	result	size
parallelrisch	\(\frac {8 d^{2} \left (x^{2} \left (\frac {7 b \,x^{3}}{8}+a \right ) d^{4}+3 \left (\frac {35 b \,x^{3}}{4}+2 a \right ) d^{2}+315 b x \right ) x b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 d \left (x \left (b \,x^{3}+a \right )^{2} d^{6}+6 \left (7 b^{2} x^{5}+4 a b \,x^{2}\right ) d^{4}+24 \left (35 b^{2} x^{3}+2 a b \right ) d^{2}+5040 b^{2} x \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (7 b^{2} x^{6}+8 a b \,x^{3}+2 a^{2}\right ) d^{6}+6 \left (35 b^{2} x^{4}+8 a b x \right ) d^{4}+2520 x^{2} d^{2} b^{2}+10080 b^{2}}{d^{8} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\)	\(206\)
risch	\(\frac {\left (b^{2} x^{7} d^{7}-7 b^{2} x^{6} d^{6}+2 a b \,d^{7} x^{4}+42 b^{2} x^{5} d^{5}-8 a b \,d^{6} x^{3}+a^{2} d^{7} x -210 b^{2} x^{4} d^{4}+24 a b \,d^{5} x^{2}-a^{2} d^{6}+840 b^{2} d^{3} x^{3}-48 a b \,d^{4} x -2520 x^{2} d^{2} b^{2}+48 a b \,d^{3}+5040 b^{2} d x -5040 b^{2}\right ) {\mathrm e}^{d x +c}}{2 d^{8}}-\frac {\left (b^{2} x^{7} d^{7}+7 b^{2} x^{6} d^{6}+2 a b \,d^{7} x^{4}+42 b^{2} x^{5} d^{5}+8 a b \,d^{6} x^{3}+a^{2} d^{7} x +210 b^{2} x^{4} d^{4}+24 a b \,d^{5} x^{2}+a^{2} d^{6}+840 b^{2} d^{3} x^{3}+48 a b \,d^{4} x +2520 x^{2} d^{2} b^{2}+48 a b \,d^{3}+5040 b^{2} d x +5040 b^{2}\right ) {\mathrm e}^{-d x -c}}{2 d^{8}}\)	\(304\)
meijerg	\(\frac {128 b^{2} \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {315}{8 \sqrt {\pi }}-\frac {\left (\frac {7}{16} x^{6} d^{6}+\frac {105}{8} d^{4} x^{4}+\frac {315}{2} x^{2} d^{2}+315\right ) \cosh \left (d x \right )}{8 \sqrt {\pi }}+\frac {x d \left (\frac {1}{16} x^{6} d^{6}+\frac {21}{8} d^{4} x^{4}+\frac {105}{2} x^{2} d^{2}+315\right ) \sinh \left (d x \right )}{8 \sqrt {\pi }}\right )}{d^{8}}-\frac {128 i b^{2} \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {i x d \left (\frac {9}{16} x^{6} d^{6}+\frac {189}{8} d^{4} x^{4}+\frac {945}{2} x^{2} d^{2}+2835\right ) \cosh \left (d x \right )}{72 \sqrt {\pi }}-\frac {i \left (\frac {63}{16} x^{6} d^{6}+\frac {945}{8} d^{4} x^{4}+\frac {2835}{2} x^{2} d^{2}+2835\right ) \sinh \left (d x \right )}{72 \sqrt {\pi }}\right )}{d^{8}}-\frac {32 i a b \cosh \left (c \right ) \sqrt {\pi }\, \left (-\frac {i x d \left (\frac {5 x^{2} d^{2}}{2}+15\right ) \cosh \left (d x \right )}{10 \sqrt {\pi }}+\frac {i \left (\frac {5}{8} d^{4} x^{4}+\frac {15}{2} x^{2} d^{2}+15\right ) \sinh \left (d x \right )}{10 \sqrt {\pi }}\right )}{d^{5}}-\frac {32 b a \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {3}{2 \sqrt {\pi }}-\frac {\left (\frac {3}{8} d^{4} x^{4}+\frac {9}{2} x^{2} d^{2}+9\right ) \cosh \left (d x \right )}{6 \sqrt {\pi }}+\frac {x d \left (\frac {3 x^{2} d^{2}}{2}+9\right ) \sinh \left (d x \right )}{6 \sqrt {\pi }}\right )}{d^{5}}-\frac {2 a^{2} \cosh \left (c \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\cosh \left (d x \right )}{2 \sqrt {\pi }}-\frac {d x \sinh \left (d x \right )}{2 \sqrt {\pi }}\right )}{d^{2}}+\frac {a^{2} \sinh \left (c \right ) \left (\cosh \left (d x \right ) x d -\sinh \left (d x \right )\right )}{d^{2}}\)	\(374\)
parts	\(\frac {b^{2} x^{7} \sinh \left (d x +c \right )}{d}+\frac {2 a b \,x^{4} \sinh \left (d x +c \right )}{d}+\frac {a^{2} x \sinh \left (d x +c \right )}{d}-\frac {\frac {7 b^{2} c^{6} \cosh \left (d x +c \right )}{d^{6}}+\frac {7 b^{2} \left (\left (d x +c \right )^{6} \cosh \left (d x +c \right )-6 \left (d x +c \right )^{5} \sinh \left (d x +c \right )+30 \left (d x +c \right )^{4} \cosh \left (d x +c \right )-120 \left (d x +c \right )^{3} \sinh \left (d x +c \right )+360 \left (d x +c \right )^{2} \cosh \left (d x +c \right )-720 \left (d x +c \right ) \sinh \left (d x +c \right )+720 \cosh \left (d x +c \right )\right )}{d^{6}}+\cosh \left (d x +c \right ) a^{2}-\frac {42 b^{2} c^{5} \left (\left (d x +c \right ) \cosh \left (d x +c \right )-\sinh \left (d x +c \right )\right )}{d^{6}}+\frac {105 b^{2} c^{4} \left (\left (d x +c \right )^{2} \cosh \left (d x +c \right )-2 \left (d x +c \right ) \sinh \left (d x +c \right )+2 \cosh \left (d x +c \right )\right )}{d^{6}}-\frac {140 b^{2} c^{3} \left (\left (d x +c \right )^{3} \cosh \left (d x +c \right )-3 \left (d x +c \right )^{2} \sinh \left (d x +c \right )+6 \left (d x +c \right ) \cosh \left (d x +c \right )-6 \sinh \left (d x +c \right )\right )}{d^{6}}-\frac {8 b \,c^{3} a \cosh \left (d x +c \right )}{d^{3}}+\frac {105 b^{2} c^{2} \left (\left (d x +c \right )^{4} \cosh \left (d x +c \right )-4 \left (d x +c \right )^{3} \sinh \left (d x +c \right )+12 \left (d x +c \right )^{2} \cosh \left (d x +c \right )-24 \left (d x +c \right ) \sinh \left (d x +c \right )+24 \cosh \left (d x +c \right )\right )}{d^{6}}-\frac {42 b^{2} c \left (\left (d x +c \right )^{5} \cosh \left (d x +c \right )-5 \left (d x +c \right )^{4} \sinh \left (d x +c \right )+20 \left (d x +c \right )^{3} \cosh \left (d x +c \right )-60 \left (d x +c \right )^{2} \sinh \left (d x +c \right )+120 \left (d x +c \right ) \cosh \left (d x +c \right )-120 \sinh \left (d x +c \right )\right )}{d^{6}}+\frac {8 b a \left (\left (d x +c \right )^{3} \cosh \left (d x +c \right )-3 \left (d x +c \right )^{2} \sinh \left (d x +c \right )+6 \left (d x +c \right ) \cosh \left (d x +c \right )-6 \sinh \left (d x +c \right )\right )}{d^{3}}+\frac {24 b \,c^{2} a \left (\left (d x +c \right ) \cosh \left (d x +c \right )-\sinh \left (d x +c \right )\right )}{d^{3}}-\frac {24 b c a \left (\left (d x +c \right )^{2} \cosh \left (d x +c \right )-2 \left (d x +c \right ) \sinh \left (d x +c \right )+2 \cosh \left (d x +c \right )\right )}{d^{3}}}{d^{2}}\)	\(642\)
derivativedivides	\(\text {Expression too large to display}\)	\(818\)
default	\(\text {Expression too large to display}\)	\(818\)

[In]

int(x*(b*x^3+a)^2*cosh(d*x+c),x,method=_RETURNVERBOSE)

[Out]

(8*d^2*(x^2*(7/8*b*x^3+a)*d^4+3*(35/4*b*x^3+2*a)*d^2+315*b*x)*x*b*tanh(1/2*d*x+1/2*c)^2-2*d*(x*(b*x^3+a)^2*d^6
+6*(7*b^2*x^5+4*a*b*x^2)*d^4+24*(35*b^2*x^3+2*a*b)*d^2+5040*b^2*x)*tanh(1/2*d*x+1/2*c)+(7*b^2*x^6+8*a*b*x^3+2*
a^2)*d^6+6*(35*b^2*x^4+8*a*b*x)*d^4+2520*x^2*d^2*b^2+10080*b^2)/d^8/(tanh(1/2*d*x+1/2*c)^2-1)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.69 \[ \int x \left (a+b x^3\right )^2 \cosh (c+d x) \, dx=-\frac {{\left (7 \, b^{2} d^{6} x^{6} + 8 \, a b d^{6} x^{3} + 210 \, b^{2} d^{4} x^{4} + a^{2} d^{6} + 48 \, a b d^{4} x + 2520 \, b^{2} d^{2} x^{2} + 5040 \, b^{2}\right )} \cosh \left (d x + c\right ) - {\left (b^{2} d^{7} x^{7} + 2 \, a b d^{7} x^{4} + 42 \, b^{2} d^{5} x^{5} + 24 \, a b d^{5} x^{2} + 840 \, b^{2} d^{3} x^{3} + 48 \, a b d^{3} + {\left (a^{2} d^{7} + 5040 \, b^{2} d\right )} x\right )} \sinh \left (d x + c\right )}{d^{8}} \]

[In]

integrate(x*(b*x^3+a)^2*cosh(d*x+c),x, algorithm="fricas")

[Out]

-((7*b^2*d^6*x^6 + 8*a*b*d^6*x^3 + 210*b^2*d^4*x^4 + a^2*d^6 + 48*a*b*d^4*x + 2520*b^2*d^2*x^2 + 5040*b^2)*cos
h(d*x + c) - (b^2*d^7*x^7 + 2*a*b*d^7*x^4 + 42*b^2*d^5*x^5 + 24*a*b*d^5*x^2 + 840*b^2*d^3*x^3 + 48*a*b*d^3 + (
a^2*d^7 + 5040*b^2*d)*x)*sinh(d*x + c))/d^8

Sympy [A] (verification not implemented)

Time = 0.79 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.21 \[ \int x \left (a+b x^3\right )^2 \cosh (c+d x) \, dx=\begin {cases} \frac {a^{2} x \sinh {\left (c + d x \right )}}{d} - \frac {a^{2} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {2 a b x^{4} \sinh {\left (c + d x \right )}}{d} - \frac {8 a b x^{3} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {24 a b x^{2} \sinh {\left (c + d x \right )}}{d^{3}} - \frac {48 a b x \cosh {\left (c + d x \right )}}{d^{4}} + \frac {48 a b \sinh {\left (c + d x \right )}}{d^{5}} + \frac {b^{2} x^{7} \sinh {\left (c + d x \right )}}{d} - \frac {7 b^{2} x^{6} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {42 b^{2} x^{5} \sinh {\left (c + d x \right )}}{d^{3}} - \frac {210 b^{2} x^{4} \cosh {\left (c + d x \right )}}{d^{4}} + \frac {840 b^{2} x^{3} \sinh {\left (c + d x \right )}}{d^{5}} - \frac {2520 b^{2} x^{2} \cosh {\left (c + d x \right )}}{d^{6}} + \frac {5040 b^{2} x \sinh {\left (c + d x \right )}}{d^{7}} - \frac {5040 b^{2} \cosh {\left (c + d x \right )}}{d^{8}} & \text {for}\: d \neq 0 \\\left (\frac {a^{2} x^{2}}{2} + \frac {2 a b x^{5}}{5} + \frac {b^{2} x^{8}}{8}\right ) \cosh {\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(x*(b*x**3+a)**2*cosh(d*x+c),x)

[Out]

Piecewise((a**2*x*sinh(c + d*x)/d - a**2*cosh(c + d*x)/d**2 + 2*a*b*x**4*sinh(c + d*x)/d - 8*a*b*x**3*cosh(c +
d*x)/d**2 + 24*a*b*x**2*sinh(c + d*x)/d**3 - 48*a*b*x*cosh(c + d*x)/d**4 + 48*a*b*sinh(c + d*x)/d**5 + b**2*x
**7*sinh(c + d*x)/d - 7*b**2*x**6*cosh(c + d*x)/d**2 + 42*b**2*x**5*sinh(c + d*x)/d**3 - 210*b**2*x**4*cosh(c
+ d*x)/d**4 + 840*b**2*x**3*sinh(c + d*x)/d**5 - 2520*b**2*x**2*cosh(c + d*x)/d**6 + 5040*b**2*x*sinh(c + d*x)
/d**7 - 5040*b**2*cosh(c + d*x)/d**8, Ne(d, 0)), ((a**2*x**2/2 + 2*a*b*x**5/5 + b**2*x**8/8)*cosh(c), True))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.64 \[ \int x \left (a+b x^3\right )^2 \cosh (c+d x) \, dx=-\frac {1}{80} \, d {\left (\frac {20 \, {\left (d^{2} x^{2} e^{c} - 2 \, d x e^{c} + 2 \, e^{c}\right )} a^{2} e^{\left (d x\right )}}{d^{3}} + \frac {20 \, {\left (d^{2} x^{2} + 2 \, d x + 2\right )} a^{2} e^{\left (-d x - c\right )}}{d^{3}} + \frac {16 \, {\left (d^{5} x^{5} e^{c} - 5 \, d^{4} x^{4} e^{c} + 20 \, d^{3} x^{3} e^{c} - 60 \, d^{2} x^{2} e^{c} + 120 \, d x e^{c} - 120 \, e^{c}\right )} a b e^{\left (d x\right )}}{d^{6}} + \frac {16 \, {\left (d^{5} x^{5} + 5 \, d^{4} x^{4} + 20 \, d^{3} x^{3} + 60 \, d^{2} x^{2} + 120 \, d x + 120\right )} a b e^{\left (-d x - c\right )}}{d^{6}} + \frac {5 \, {\left (d^{8} x^{8} e^{c} - 8 \, d^{7} x^{7} e^{c} + 56 \, d^{6} x^{6} e^{c} - 336 \, d^{5} x^{5} e^{c} + 1680 \, d^{4} x^{4} e^{c} - 6720 \, d^{3} x^{3} e^{c} + 20160 \, d^{2} x^{2} e^{c} - 40320 \, d x e^{c} + 40320 \, e^{c}\right )} b^{2} e^{\left (d x\right )}}{d^{9}} + \frac {5 \, {\left (d^{8} x^{8} + 8 \, d^{7} x^{7} + 56 \, d^{6} x^{6} + 336 \, d^{5} x^{5} + 1680 \, d^{4} x^{4} + 6720 \, d^{3} x^{3} + 20160 \, d^{2} x^{2} + 40320 \, d x + 40320\right )} b^{2} e^{\left (-d x - c\right )}}{d^{9}}\right )} + \frac {1}{40} \, {\left (5 \, b^{2} x^{8} + 16 \, a b x^{5} + 20 \, a^{2} x^{2}\right )} \cosh \left (d x + c\right ) \]

[In]

integrate(x*(b*x^3+a)^2*cosh(d*x+c),x, algorithm="maxima")

[Out]

-1/80*d*(20*(d^2*x^2*e^c - 2*d*x*e^c + 2*e^c)*a^2*e^(d*x)/d^3 + 20*(d^2*x^2 + 2*d*x + 2)*a^2*e^(-d*x - c)/d^3
+ 16*(d^5*x^5*e^c - 5*d^4*x^4*e^c + 20*d^3*x^3*e^c - 60*d^2*x^2*e^c + 120*d*x*e^c - 120*e^c)*a*b*e^(d*x)/d^6 +
16*(d^5*x^5 + 5*d^4*x^4 + 20*d^3*x^3 + 60*d^2*x^2 + 120*d*x + 120)*a*b*e^(-d*x - c)/d^6 + 5*(d^8*x^8*e^c - 8*
d^7*x^7*e^c + 56*d^6*x^6*e^c - 336*d^5*x^5*e^c + 1680*d^4*x^4*e^c - 6720*d^3*x^3*e^c + 20160*d^2*x^2*e^c - 403
20*d*x*e^c + 40320*e^c)*b^2*e^(d*x)/d^9 + 5*(d^8*x^8 + 8*d^7*x^7 + 56*d^6*x^6 + 336*d^5*x^5 + 1680*d^4*x^4 + 6
720*d^3*x^3 + 20160*d^2*x^2 + 40320*d*x + 40320)*b^2*e^(-d*x - c)/d^9) + 1/40*(5*b^2*x^8 + 16*a*b*x^5 + 20*a^2
*x^2)*cosh(d*x + c)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.29 \[ \int x \left (a+b x^3\right )^2 \cosh (c+d x) \, dx=\frac {{\left (b^{2} d^{7} x^{7} - 7 \, b^{2} d^{6} x^{6} + 2 \, a b d^{7} x^{4} + 42 \, b^{2} d^{5} x^{5} - 8 \, a b d^{6} x^{3} + a^{2} d^{7} x - 210 \, b^{2} d^{4} x^{4} + 24 \, a b d^{5} x^{2} - a^{2} d^{6} + 840 \, b^{2} d^{3} x^{3} - 48 \, a b d^{4} x - 2520 \, b^{2} d^{2} x^{2} + 48 \, a b d^{3} + 5040 \, b^{2} d x - 5040 \, b^{2}\right )} e^{\left (d x + c\right )}}{2 \, d^{8}} - \frac {{\left (b^{2} d^{7} x^{7} + 7 \, b^{2} d^{6} x^{6} + 2 \, a b d^{7} x^{4} + 42 \, b^{2} d^{5} x^{5} + 8 \, a b d^{6} x^{3} + a^{2} d^{7} x + 210 \, b^{2} d^{4} x^{4} + 24 \, a b d^{5} x^{2} + a^{2} d^{6} + 840 \, b^{2} d^{3} x^{3} + 48 \, a b d^{4} x + 2520 \, b^{2} d^{2} x^{2} + 48 \, a b d^{3} + 5040 \, b^{2} d x + 5040 \, b^{2}\right )} e^{\left (-d x - c\right )}}{2 \, d^{8}} \]

[In]

integrate(x*(b*x^3+a)^2*cosh(d*x+c),x, algorithm="giac")

[Out]

1/2*(b^2*d^7*x^7 - 7*b^2*d^6*x^6 + 2*a*b*d^7*x^4 + 42*b^2*d^5*x^5 - 8*a*b*d^6*x^3 + a^2*d^7*x - 210*b^2*d^4*x^
4 + 24*a*b*d^5*x^2 - a^2*d^6 + 840*b^2*d^3*x^3 - 48*a*b*d^4*x - 2520*b^2*d^2*x^2 + 48*a*b*d^3 + 5040*b^2*d*x -
5040*b^2)*e^(d*x + c)/d^8 - 1/2*(b^2*d^7*x^7 + 7*b^2*d^6*x^6 + 2*a*b*d^7*x^4 + 42*b^2*d^5*x^5 + 8*a*b*d^6*x^3
+ a^2*d^7*x + 210*b^2*d^4*x^4 + 24*a*b*d^5*x^2 + a^2*d^6 + 840*b^2*d^3*x^3 + 48*a*b*d^4*x + 2520*b^2*d^2*x^2
+ 48*a*b*d^3 + 5040*b^2*d*x + 5040*b^2)*e^(-d*x - c)/d^8

Mupad [B] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.29 \[ \int x \left (a+b x^3\right )^2 \cosh (c+d x) \, dx=-{\mathrm {e}}^{c+d\,x}\,\left (\frac {a^2\,d^6-48\,a\,b\,d^3+5040\,b^2}{2\,d^8}-\frac {x\,\left (a^2\,d^7-48\,a\,b\,d^4+5040\,b^2\,d\right )}{2\,d^8}-\frac {b^2\,x^7}{2\,d}+\frac {7\,b^2\,x^6}{2\,d^2}-\frac {21\,b^2\,x^5}{d^3}+\frac {b\,x^4\,\left (105\,b-a\,d^3\right )}{d^4}-\frac {4\,b\,x^3\,\left (105\,b-a\,d^3\right )}{d^5}+\frac {12\,b\,x^2\,\left (105\,b-a\,d^3\right )}{d^6}\right )-{\mathrm {e}}^{-c-d\,x}\,\left (\frac {a^2\,d^6+48\,a\,b\,d^3+5040\,b^2}{2\,d^8}+\frac {x\,\left (a^2\,d^7+48\,a\,b\,d^4+5040\,b^2\,d\right )}{2\,d^8}+\frac {b^2\,x^7}{2\,d}+\frac {7\,b^2\,x^6}{2\,d^2}+\frac {21\,b^2\,x^5}{d^3}+\frac {b\,x^4\,\left (a\,d^3+105\,b\right )}{d^4}+\frac {4\,b\,x^3\,\left (a\,d^3+105\,b\right )}{d^5}+\frac {12\,b\,x^2\,\left (a\,d^3+105\,b\right )}{d^6}\right ) \]

[In]

int(x*cosh(c + d*x)*(a + b*x^3)^2,x)

[Out]

- exp(c + d*x)*((5040*b^2 + a^2*d^6 - 48*a*b*d^3)/(2*d^8) - (x*(5040*b^2*d + a^2*d^7 - 48*a*b*d^4))/(2*d^8) -
(b^2*x^7)/(2*d) + (7*b^2*x^6)/(2*d^2) - (21*b^2*x^5)/d^3 + (b*x^4*(105*b - a*d^3))/d^4 - (4*b*x^3*(105*b - a*d
^3))/d^5 + (12*b*x^2*(105*b - a*d^3))/d^6) - exp(- c - d*x)*((5040*b^2 + a^2*d^6 + 48*a*b*d^3)/(2*d^8) + (x*(5
040*b^2*d + a^2*d^7 + 48*a*b*d^4))/(2*d^8) + (b^2*x^7)/(2*d) + (7*b^2*x^6)/(2*d^2) + (21*b^2*x^5)/d^3 + (b*x^4
*(105*b + a*d^3))/d^4 + (4*b*x^3*(105*b + a*d^3))/d^5 + (12*b*x^2*(105*b + a*d^3))/d^6)