\(\int x^2 (a+b x^2)^2 \cosh (c+d x) \, dx\) [49]

Optimal result

Integrand size = 19, antiderivative size = 234 \[ \int x^2 \left (a+b x^2\right )^2 \cosh (c+d x) \, dx=-\frac {720 b^2 x \cosh (c+d x)}{d^6}-\frac {48 a b x \cosh (c+d x)}{d^4}-\frac {2 a^2 x \cosh (c+d x)}{d^2}-\frac {120 b^2 x^3 \cosh (c+d x)}{d^4}-\frac {8 a b x^3 \cosh (c+d x)}{d^2}-\frac {6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac {720 b^2 \sinh (c+d x)}{d^7}+\frac {48 a b \sinh (c+d x)}{d^5}+\frac {2 a^2 \sinh (c+d x)}{d^3}+\frac {360 b^2 x^2 \sinh (c+d x)}{d^5}+\frac {24 a b x^2 \sinh (c+d x)}{d^3}+\frac {a^2 x^2 \sinh (c+d x)}{d}+\frac {30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac {2 a b x^4 \sinh (c+d x)}{d}+\frac {b^2 x^6 \sinh (c+d x)}{d} \]

[Out]

-720*b^2*x*cosh(d*x+c)/d^6-48*a*b*x*cosh(d*x+c)/d^4-2*a^2*x*cosh(d*x+c)/d^2-120*b^2*x^3*cosh(d*x+c)/d^4-8*a*b*
x^3*cosh(d*x+c)/d^2-6*b^2*x^5*cosh(d*x+c)/d^2+720*b^2*sinh(d*x+c)/d^7+48*a*b*sinh(d*x+c)/d^5+2*a^2*sinh(d*x+c)
/d^3+360*b^2*x^2*sinh(d*x+c)/d^5+24*a*b*x^2*sinh(d*x+c)/d^3+a^2*x^2*sinh(d*x+c)/d+30*b^2*x^4*sinh(d*x+c)/d^3+2
*a*b*x^4*sinh(d*x+c)/d+b^2*x^6*sinh(d*x+c)/d

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {5395, 3377, 2717} \[ \int x^2 \left (a+b x^2\right )^2 \cosh (c+d x) \, dx=\frac {2 a^2 \sinh (c+d x)}{d^3}-\frac {2 a^2 x \cosh (c+d x)}{d^2}+\frac {a^2 x^2 \sinh (c+d x)}{d}+\frac {48 a b \sinh (c+d x)}{d^5}-\frac {48 a b x \cosh (c+d x)}{d^4}+\frac {24 a b x^2 \sinh (c+d x)}{d^3}-\frac {8 a b x^3 \cosh (c+d x)}{d^2}+\frac {2 a b x^4 \sinh (c+d x)}{d}+\frac {720 b^2 \sinh (c+d x)}{d^7}-\frac {720 b^2 x \cosh (c+d x)}{d^6}+\frac {360 b^2 x^2 \sinh (c+d x)}{d^5}-\frac {120 b^2 x^3 \cosh (c+d x)}{d^4}+\frac {30 b^2 x^4 \sinh (c+d x)}{d^3}-\frac {6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac {b^2 x^6 \sinh (c+d x)}{d} \]

[In]

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

[Out]

(-720*b^2*x*Cosh[c + d*x])/d^6 - (48*a*b*x*Cosh[c + d*x])/d^4 - (2*a^2*x*Cosh[c + d*x])/d^2 - (120*b^2*x^3*Cos
h[c + d*x])/d^4 - (8*a*b*x^3*Cosh[c + d*x])/d^2 - (6*b^2*x^5*Cosh[c + d*x])/d^2 + (720*b^2*Sinh[c + d*x])/d^7
+ (48*a*b*Sinh[c + d*x])/d^5 + (2*a^2*Sinh[c + d*x])/d^3 + (360*b^2*x^2*Sinh[c + d*x])/d^5 + (24*a*b*x^2*Sinh[
c + d*x])/d^3 + (a^2*x^2*Sinh[c + d*x])/d + (30*b^2*x^4*Sinh[c + d*x])/d^3 + (2*a*b*x^4*Sinh[c + d*x])/d + (b^
2*x^6*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 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^2 \cosh (c+d x)+2 a b x^4 \cosh (c+d x)+b^2 x^6 \cosh (c+d x)\right ) \, dx \\ & = a^2 \int x^2 \cosh (c+d x) \, dx+(2 a b) \int x^4 \cosh (c+d x) \, dx+b^2 \int x^6 \cosh (c+d x) \, dx \\ & = \frac {a^2 x^2 \sinh (c+d x)}{d}+\frac {2 a b x^4 \sinh (c+d x)}{d}+\frac {b^2 x^6 \sinh (c+d x)}{d}-\frac {\left (2 a^2\right ) \int x \sinh (c+d x) \, dx}{d}-\frac {(8 a b) \int x^3 \sinh (c+d x) \, dx}{d}-\frac {\left (6 b^2\right ) \int x^5 \sinh (c+d x) \, dx}{d} \\ & = -\frac {2 a^2 x \cosh (c+d x)}{d^2}-\frac {8 a b x^3 \cosh (c+d x)}{d^2}-\frac {6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac {a^2 x^2 \sinh (c+d x)}{d}+\frac {2 a b x^4 \sinh (c+d x)}{d}+\frac {b^2 x^6 \sinh (c+d x)}{d}+\frac {\left (2 a^2\right ) \int \cosh (c+d x) \, dx}{d^2}+\frac {(24 a b) \int x^2 \cosh (c+d x) \, dx}{d^2}+\frac {\left (30 b^2\right ) \int x^4 \cosh (c+d x) \, dx}{d^2} \\ & = -\frac {2 a^2 x \cosh (c+d x)}{d^2}-\frac {8 a b x^3 \cosh (c+d x)}{d^2}-\frac {6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac {2 a^2 \sinh (c+d x)}{d^3}+\frac {24 a b x^2 \sinh (c+d x)}{d^3}+\frac {a^2 x^2 \sinh (c+d x)}{d}+\frac {30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac {2 a b x^4 \sinh (c+d x)}{d}+\frac {b^2 x^6 \sinh (c+d x)}{d}-\frac {(48 a b) \int x \sinh (c+d x) \, dx}{d^3}-\frac {\left (120 b^2\right ) \int x^3 \sinh (c+d x) \, dx}{d^3} \\ & = -\frac {48 a b x \cosh (c+d x)}{d^4}-\frac {2 a^2 x \cosh (c+d x)}{d^2}-\frac {120 b^2 x^3 \cosh (c+d x)}{d^4}-\frac {8 a b x^3 \cosh (c+d x)}{d^2}-\frac {6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac {2 a^2 \sinh (c+d x)}{d^3}+\frac {24 a b x^2 \sinh (c+d x)}{d^3}+\frac {a^2 x^2 \sinh (c+d x)}{d}+\frac {30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac {2 a b x^4 \sinh (c+d x)}{d}+\frac {b^2 x^6 \sinh (c+d x)}{d}+\frac {(48 a b) \int \cosh (c+d x) \, dx}{d^4}+\frac {\left (360 b^2\right ) \int x^2 \cosh (c+d x) \, dx}{d^4} \\ & = -\frac {48 a b x \cosh (c+d x)}{d^4}-\frac {2 a^2 x \cosh (c+d x)}{d^2}-\frac {120 b^2 x^3 \cosh (c+d x)}{d^4}-\frac {8 a b x^3 \cosh (c+d x)}{d^2}-\frac {6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac {48 a b \sinh (c+d x)}{d^5}+\frac {2 a^2 \sinh (c+d x)}{d^3}+\frac {360 b^2 x^2 \sinh (c+d x)}{d^5}+\frac {24 a b x^2 \sinh (c+d x)}{d^3}+\frac {a^2 x^2 \sinh (c+d x)}{d}+\frac {30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac {2 a b x^4 \sinh (c+d x)}{d}+\frac {b^2 x^6 \sinh (c+d x)}{d}-\frac {\left (720 b^2\right ) \int x \sinh (c+d x) \, dx}{d^5} \\ & = -\frac {720 b^2 x \cosh (c+d x)}{d^6}-\frac {48 a b x \cosh (c+d x)}{d^4}-\frac {2 a^2 x \cosh (c+d x)}{d^2}-\frac {120 b^2 x^3 \cosh (c+d x)}{d^4}-\frac {8 a b x^3 \cosh (c+d x)}{d^2}-\frac {6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac {48 a b \sinh (c+d x)}{d^5}+\frac {2 a^2 \sinh (c+d x)}{d^3}+\frac {360 b^2 x^2 \sinh (c+d x)}{d^5}+\frac {24 a b x^2 \sinh (c+d x)}{d^3}+\frac {a^2 x^2 \sinh (c+d x)}{d}+\frac {30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac {2 a b x^4 \sinh (c+d x)}{d}+\frac {b^2 x^6 \sinh (c+d x)}{d}+\frac {\left (720 b^2\right ) \int \cosh (c+d x) \, dx}{d^6} \\ & = -\frac {720 b^2 x \cosh (c+d x)}{d^6}-\frac {48 a b x \cosh (c+d x)}{d^4}-\frac {2 a^2 x \cosh (c+d x)}{d^2}-\frac {120 b^2 x^3 \cosh (c+d x)}{d^4}-\frac {8 a b x^3 \cosh (c+d x)}{d^2}-\frac {6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac {720 b^2 \sinh (c+d x)}{d^7}+\frac {48 a b \sinh (c+d x)}{d^5}+\frac {2 a^2 \sinh (c+d x)}{d^3}+\frac {360 b^2 x^2 \sinh (c+d x)}{d^5}+\frac {24 a b x^2 \sinh (c+d x)}{d^3}+\frac {a^2 x^2 \sinh (c+d x)}{d}+\frac {30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac {2 a b x^4 \sinh (c+d x)}{d}+\frac {b^2 x^6 \sinh (c+d x)}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.59 \[ \int x^2 \left (a+b x^2\right )^2 \cosh (c+d x) \, dx=\frac {-2 d x \left (a^2 d^4+4 a b d^2 \left (6+d^2 x^2\right )+3 b^2 \left (120+20 d^2 x^2+d^4 x^4\right )\right ) \cosh (c+d x)+\left (a^2 d^4 \left (2+d^2 x^2\right )+2 a b d^2 \left (24+12 d^2 x^2+d^4 x^4\right )+b^2 \left (720+360 d^2 x^2+30 d^4 x^4+d^6 x^6\right )\right ) \sinh (c+d x)}{d^7} \]

[In]

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

[Out]

(-2*d*x*(a^2*d^4 + 4*a*b*d^2*(6 + d^2*x^2) + 3*b^2*(120 + 20*d^2*x^2 + d^4*x^4))*Cosh[c + d*x] + (a^2*d^4*(2 +
d^2*x^2) + 2*a*b*d^2*(24 + 12*d^2*x^2 + d^4*x^4) + b^2*(720 + 360*d^2*x^2 + 30*d^4*x^4 + d^6*x^6))*Sinh[c + d
*x])/d^7

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.86

method	result	size
parallelrisch	\(\frac {2 \left (\left (3 b \,x^{2}+a \right ) \left (b \,x^{2}+a \right ) d^{4}+12 \left (5 x^{2} b^{2}+2 a b \right ) d^{2}+360 b^{2}\right ) d x \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \left (-x^{2} \left (b \,x^{2}+a \right )^{2} d^{6}+2 \left (-15 b^{2} x^{4}-12 a b \,x^{2}-a^{2}\right ) d^{4}+24 \left (-15 x^{2} b^{2}-2 a b \right ) d^{2}-720 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \left (\left (3 b \,x^{2}+a \right ) \left (b \,x^{2}+a \right ) d^{4}+12 \left (5 x^{2} b^{2}+2 a b \right ) d^{2}+360 b^{2}\right ) d x}{d^{7} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\)	\(202\)
risch	\(\frac {\left (b^{2} x^{6} d^{6}+2 a b \,d^{6} x^{4}-6 b^{2} x^{5} d^{5}+a^{2} d^{6} x^{2}-8 a b \,d^{5} x^{3}+30 b^{2} x^{4} d^{4}-2 a^{2} d^{5} x +24 a b \,d^{4} x^{2}-120 b^{2} d^{3} x^{3}+2 a^{2} d^{4}-48 a b \,d^{3} x +360 x^{2} d^{2} b^{2}+48 a \,d^{2} b -720 b^{2} d x +720 b^{2}\right ) {\mathrm e}^{d x +c}}{2 d^{7}}-\frac {\left (b^{2} x^{6} d^{6}+2 a b \,d^{6} x^{4}+6 b^{2} x^{5} d^{5}+a^{2} d^{6} x^{2}+8 a b \,d^{5} x^{3}+30 b^{2} x^{4} d^{4}+2 a^{2} d^{5} x +24 a b \,d^{4} x^{2}+120 b^{2} d^{3} x^{3}+2 a^{2} d^{4}+48 a b \,d^{3} x +360 x^{2} d^{2} b^{2}+48 a \,d^{2} b +720 b^{2} d x +720 b^{2}\right ) {\mathrm e}^{-d x -c}}{2 d^{7}}\)	\(305\)
meijerg	\(\frac {64 i b^{2} \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {i x d \left (\frac {21}{8} d^{4} x^{4}+\frac {105}{2} x^{2} d^{2}+315\right ) \cosh \left (d x \right )}{28 \sqrt {\pi }}-\frac {i \left (\frac {7}{16} x^{6} d^{6}+\frac {105}{8} d^{4} x^{4}+\frac {315}{2} x^{2} d^{2}+315\right ) \sinh \left (d x \right )}{28 \sqrt {\pi }}\right )}{d^{7}}+\frac {64 b^{2} \sinh \left (c \right ) \sqrt {\pi }\, \left (-\frac {45}{4 \sqrt {\pi }}+\frac {\left (\frac {1}{16} x^{6} d^{6}+\frac {15}{8} d^{4} x^{4}+\frac {45}{2} x^{2} d^{2}+45\right ) \cosh \left (d x \right )}{4 \sqrt {\pi }}-\frac {x d \left (\frac {3}{8} d^{4} x^{4}+\frac {15}{2} x^{2} d^{2}+45\right ) \sinh \left (d x \right )}{4 \sqrt {\pi }}\right )}{d^{7}}-\frac {32 i a b \cosh \left (c \right ) \sqrt {\pi }\, \left (-\frac {i x d \left (\frac {5 x^{2} d^{2}}{2}+15\right ) \cosh \left (d x \right )}{10 \sqrt {\pi }}+\frac {i \left (\frac {5}{8} d^{4} x^{4}+\frac {15}{2} x^{2} d^{2}+15\right ) \sinh \left (d x \right )}{10 \sqrt {\pi }}\right )}{d^{5}}-\frac {32 b a \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {3}{2 \sqrt {\pi }}-\frac {\left (\frac {3}{8} d^{4} x^{4}+\frac {9}{2} x^{2} d^{2}+9\right ) \cosh \left (d x \right )}{6 \sqrt {\pi }}+\frac {x d \left (\frac {3 x^{2} d^{2}}{2}+9\right ) \sinh \left (d x \right )}{6 \sqrt {\pi }}\right )}{d^{5}}+\frac {4 i a^{2} \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {i x d \cosh \left (d x \right )}{2 \sqrt {\pi }}-\frac {i \left (\frac {3 x^{2} d^{2}}{2}+3\right ) \sinh \left (d x \right )}{6 \sqrt {\pi }}\right )}{d^{3}}+\frac {4 a^{2} \sinh \left (c \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\left (\frac {x^{2} d^{2}}{2}+1\right ) \cosh \left (d x \right )}{2 \sqrt {\pi }}-\frac {d x \sinh \left (d x \right )}{2 \sqrt {\pi }}\right )}{d^{3}}\)	\(392\)
parts	\(\frac {b^{2} x^{6} \sinh \left (d x +c \right )}{d}+\frac {2 a b \,x^{4} \sinh \left (d x +c \right )}{d}+\frac {a^{2} x^{2} \sinh \left (d x +c \right )}{d}-\frac {2 \left (\frac {15 b^{2} c^{4} \left (\left (d x +c \right ) \cosh \left (d x +c \right )-\sinh \left (d x +c \right )\right )}{d^{5}}-\frac {30 b^{2} c^{3} \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^{5}}+\frac {30 b^{2} c^{2} \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^{5}}+\frac {12 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 {15 b^{2} c \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^{5}}-\frac {12 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}}+\frac {3 b^{2} \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^{5}}+\frac {4 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 {a^{2} \left (\left (d x +c \right ) \cosh \left (d x +c \right )-\sinh \left (d x +c \right )\right )}{d}-\frac {3 b^{2} c^{5} \cosh \left (d x +c \right )}{d^{5}}-\frac {4 b \,c^{3} a \cosh \left (d x +c \right )}{d^{3}}-\frac {a^{2} c \cosh \left (d x +c \right )}{d}\right )}{d^{2}}\)	\(570\)
derivativedivides	\(\frac {a^{2} c^{2} \sinh \left (d x +c \right )+a^{2} \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )+\frac {b^{2} c^{6} \sinh \left (d x +c \right )}{d^{4}}+\frac {b^{2} \left (\left (d x +c \right )^{6} \sinh \left (d x +c \right )-6 \left (d x +c \right )^{5} \cosh \left (d x +c \right )+30 \left (d x +c \right )^{4} \sinh \left (d x +c \right )-120 \left (d x +c \right )^{3} \cosh \left (d x +c \right )+360 \left (d x +c \right )^{2} \sinh \left (d x +c \right )-720 \left (d x +c \right ) \cosh \left (d x +c \right )+720 \sinh \left (d x +c \right )\right )}{d^{4}}-2 a^{2} c \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )-\frac {6 b^{2} c^{5} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{4}}+\frac {15 b^{2} c^{4} \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )}{d^{4}}-\frac {20 b^{2} c^{3} \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )}{d^{4}}+\frac {15 b^{2} c^{2} \left (\left (d x +c \right )^{4} \sinh \left (d x +c \right )-4 \left (d x +c \right )^{3} \cosh \left (d x +c \right )+12 \left (d x +c \right )^{2} \sinh \left (d x +c \right )-24 \left (d x +c \right ) \cosh \left (d x +c \right )+24 \sinh \left (d x +c \right )\right )}{d^{4}}+\frac {2 b \,c^{4} a \sinh \left (d x +c \right )}{d^{2}}-\frac {6 b^{2} c \left (\left (d x +c \right )^{5} \sinh \left (d x +c \right )-5 \left (d x +c \right )^{4} \cosh \left (d x +c \right )+20 \left (d x +c \right )^{3} \sinh \left (d x +c \right )-60 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+120 \left (d x +c \right ) \sinh \left (d x +c \right )-120 \cosh \left (d x +c \right )\right )}{d^{4}}+\frac {2 b a \left (\left (d x +c \right )^{4} \sinh \left (d x +c \right )-4 \left (d x +c \right )^{3} \cosh \left (d x +c \right )+12 \left (d x +c \right )^{2} \sinh \left (d x +c \right )-24 \left (d x +c \right ) \cosh \left (d x +c \right )+24 \sinh \left (d x +c \right )\right )}{d^{2}}+\frac {12 b \,c^{2} a \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )}{d^{2}}-\frac {8 b c a \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )}{d^{2}}-\frac {8 b \,c^{3} a \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{2}}}{d^{3}}\)	\(738\)
default	\(\frac {a^{2} c^{2} \sinh \left (d x +c \right )+a^{2} \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )+\frac {b^{2} c^{6} \sinh \left (d x +c \right )}{d^{4}}+\frac {b^{2} \left (\left (d x +c \right )^{6} \sinh \left (d x +c \right )-6 \left (d x +c \right )^{5} \cosh \left (d x +c \right )+30 \left (d x +c \right )^{4} \sinh \left (d x +c \right )-120 \left (d x +c \right )^{3} \cosh \left (d x +c \right )+360 \left (d x +c \right )^{2} \sinh \left (d x +c \right )-720 \left (d x +c \right ) \cosh \left (d x +c \right )+720 \sinh \left (d x +c \right )\right )}{d^{4}}-2 a^{2} c \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )-\frac {6 b^{2} c^{5} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{4}}+\frac {15 b^{2} c^{4} \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )}{d^{4}}-\frac {20 b^{2} c^{3} \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )}{d^{4}}+\frac {15 b^{2} c^{2} \left (\left (d x +c \right )^{4} \sinh \left (d x +c \right )-4 \left (d x +c \right )^{3} \cosh \left (d x +c \right )+12 \left (d x +c \right )^{2} \sinh \left (d x +c \right )-24 \left (d x +c \right ) \cosh \left (d x +c \right )+24 \sinh \left (d x +c \right )\right )}{d^{4}}+\frac {2 b \,c^{4} a \sinh \left (d x +c \right )}{d^{2}}-\frac {6 b^{2} c \left (\left (d x +c \right )^{5} \sinh \left (d x +c \right )-5 \left (d x +c \right )^{4} \cosh \left (d x +c \right )+20 \left (d x +c \right )^{3} \sinh \left (d x +c \right )-60 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+120 \left (d x +c \right ) \sinh \left (d x +c \right )-120 \cosh \left (d x +c \right )\right )}{d^{4}}+\frac {2 b a \left (\left (d x +c \right )^{4} \sinh \left (d x +c \right )-4 \left (d x +c \right )^{3} \cosh \left (d x +c \right )+12 \left (d x +c \right )^{2} \sinh \left (d x +c \right )-24 \left (d x +c \right ) \cosh \left (d x +c \right )+24 \sinh \left (d x +c \right )\right )}{d^{2}}+\frac {12 b \,c^{2} a \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )}{d^{2}}-\frac {8 b c a \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )}{d^{2}}-\frac {8 b \,c^{3} a \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{2}}}{d^{3}}\)	\(738\)

[In]

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

[Out]

2*(((3*b*x^2+a)*(b*x^2+a)*d^4+12*(5*b^2*x^2+2*a*b)*d^2+360*b^2)*d*x*tanh(1/2*d*x+1/2*c)^2+(-x^2*(b*x^2+a)^2*d^
6+2*(-15*b^2*x^4-12*a*b*x^2-a^2)*d^4+24*(-15*b^2*x^2-2*a*b)*d^2-720*b^2)*tanh(1/2*d*x+1/2*c)+((3*b*x^2+a)*(b*x
^2+a)*d^4+12*(5*b^2*x^2+2*a*b)*d^2+360*b^2)*d*x)/d^7/(tanh(1/2*d*x+1/2*c)^2-1)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.66 \[ \int x^2 \left (a+b x^2\right )^2 \cosh (c+d x) \, dx=-\frac {2 \, {\left (3 \, b^{2} d^{5} x^{5} + 4 \, {\left (a b d^{5} + 15 \, b^{2} d^{3}\right )} x^{3} + {\left (a^{2} d^{5} + 24 \, a b d^{3} + 360 \, b^{2} d\right )} x\right )} \cosh \left (d x + c\right ) - {\left (b^{2} d^{6} x^{6} + 2 \, a^{2} d^{4} + 2 \, {\left (a b d^{6} + 15 \, b^{2} d^{4}\right )} x^{4} + 48 \, a b d^{2} + {\left (a^{2} d^{6} + 24 \, a b d^{4} + 360 \, b^{2} d^{2}\right )} x^{2} + 720 \, b^{2}\right )} \sinh \left (d x + c\right )}{d^{7}} \]

[In]

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

[Out]

-(2*(3*b^2*d^5*x^5 + 4*(a*b*d^5 + 15*b^2*d^3)*x^3 + (a^2*d^5 + 24*a*b*d^3 + 360*b^2*d)*x)*cosh(d*x + c) - (b^2
*d^6*x^6 + 2*a^2*d^4 + 2*(a*b*d^6 + 15*b^2*d^4)*x^4 + 48*a*b*d^2 + (a^2*d^6 + 24*a*b*d^4 + 360*b^2*d^2)*x^2 +
720*b^2)*sinh(d*x + c))/d^7

Sympy [A] (verification not implemented)

Time = 0.57 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.22 \[ \int x^2 \left (a+b x^2\right )^2 \cosh (c+d x) \, dx=\begin {cases} \frac {a^{2} x^{2} \sinh {\left (c + d x \right )}}{d} - \frac {2 a^{2} x \cosh {\left (c + d x \right )}}{d^{2}} + \frac {2 a^{2} \sinh {\left (c + d x \right )}}{d^{3}} + \frac {2 a b x^{4} \sinh {\left (c + d x \right )}}{d} - \frac {8 a b x^{3} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {24 a b x^{2} \sinh {\left (c + d x \right )}}{d^{3}} - \frac {48 a b x \cosh {\left (c + d x \right )}}{d^{4}} + \frac {48 a b \sinh {\left (c + d x \right )}}{d^{5}} + \frac {b^{2} x^{6} \sinh {\left (c + d x \right )}}{d} - \frac {6 b^{2} x^{5} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {30 b^{2} x^{4} \sinh {\left (c + d x \right )}}{d^{3}} - \frac {120 b^{2} x^{3} \cosh {\left (c + d x \right )}}{d^{4}} + \frac {360 b^{2} x^{2} \sinh {\left (c + d x \right )}}{d^{5}} - \frac {720 b^{2} x \cosh {\left (c + d x \right )}}{d^{6}} + \frac {720 b^{2} \sinh {\left (c + d x \right )}}{d^{7}} & \text {for}\: d \neq 0 \\\left (\frac {a^{2} x^{3}}{3} + \frac {2 a b x^{5}}{5} + \frac {b^{2} x^{7}}{7}\right ) \cosh {\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((a**2*x**2*sinh(c + d*x)/d - 2*a**2*x*cosh(c + d*x)/d**2 + 2*a**2*sinh(c + d*x)/d**3 + 2*a*b*x**4*si
nh(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**6*sinh(c + d*x)/d - 6*b**2*x**5*cosh(c + d*x)/d**2 + 30*b**2*x**4*sinh(c
+ d*x)/d**3 - 120*b**2*x**3*cosh(c + d*x)/d**4 + 360*b**2*x**2*sinh(c + d*x)/d**5 - 720*b**2*x*cosh(c + d*x)/d
**6 + 720*b**2*sinh(c + d*x)/d**7, Ne(d, 0)), ((a**2*x**3/3 + 2*a*b*x**5/5 + b**2*x**7/7)*cosh(c), True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.64 \[ \int x^2 \left (a+b x^2\right )^2 \cosh (c+d x) \, dx=-\frac {1}{210} \, d {\left (\frac {35 \, {\left (d^{3} x^{3} e^{c} - 3 \, d^{2} x^{2} e^{c} + 6 \, d x e^{c} - 6 \, e^{c}\right )} a^{2} e^{\left (d x\right )}}{d^{4}} + \frac {35 \, {\left (d^{3} x^{3} + 3 \, d^{2} x^{2} + 6 \, d x + 6\right )} a^{2} e^{\left (-d x - c\right )}}{d^{4}} + \frac {42 \, {\left (d^{5} x^{5} e^{c} - 5 \, d^{4} x^{4} e^{c} + 20 \, d^{3} x^{3} e^{c} - 60 \, d^{2} x^{2} e^{c} + 120 \, d x e^{c} - 120 \, e^{c}\right )} a b e^{\left (d x\right )}}{d^{6}} + \frac {42 \, {\left (d^{5} x^{5} + 5 \, d^{4} x^{4} + 20 \, d^{3} x^{3} + 60 \, d^{2} x^{2} + 120 \, d x + 120\right )} a b e^{\left (-d x - c\right )}}{d^{6}} + \frac {15 \, {\left (d^{7} x^{7} e^{c} - 7 \, d^{6} x^{6} e^{c} + 42 \, d^{5} x^{5} e^{c} - 210 \, d^{4} x^{4} e^{c} + 840 \, d^{3} x^{3} e^{c} - 2520 \, d^{2} x^{2} e^{c} + 5040 \, d x e^{c} - 5040 \, e^{c}\right )} b^{2} e^{\left (d x\right )}}{d^{8}} + \frac {15 \, {\left (d^{7} x^{7} + 7 \, d^{6} x^{6} + 42 \, d^{5} x^{5} + 210 \, d^{4} x^{4} + 840 \, d^{3} x^{3} + 2520 \, d^{2} x^{2} + 5040 \, d x + 5040\right )} b^{2} e^{\left (-d x - c\right )}}{d^{8}}\right )} + \frac {1}{105} \, {\left (15 \, b^{2} x^{7} + 42 \, a b x^{5} + 35 \, a^{2} x^{3}\right )} \cosh \left (d x + c\right ) \]

[In]

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

[Out]

-1/210*d*(35*(d^3*x^3*e^c - 3*d^2*x^2*e^c + 6*d*x*e^c - 6*e^c)*a^2*e^(d*x)/d^4 + 35*(d^3*x^3 + 3*d^2*x^2 + 6*d
*x + 6)*a^2*e^(-d*x - c)/d^4 + 42*(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 + 42*(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 + 15*(d^7*x^7*e^c - 7*d^6*x^6*e^c + 42*d^5*x^5*e^c - 210*d^4*x^4*e^c + 840*d^3*x^3*e^c - 2520*d^2*x^2*
e^c + 5040*d*x*e^c - 5040*e^c)*b^2*e^(d*x)/d^8 + 15*(d^7*x^7 + 7*d^6*x^6 + 42*d^5*x^5 + 210*d^4*x^4 + 840*d^3*
x^3 + 2520*d^2*x^2 + 5040*d*x + 5040)*b^2*e^(-d*x - c)/d^8) + 1/105*(15*b^2*x^7 + 42*a*b*x^5 + 35*a^2*x^3)*cos
h(d*x + c)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.30 \[ \int x^2 \left (a+b x^2\right )^2 \cosh (c+d x) \, dx=\frac {{\left (b^{2} d^{6} x^{6} + 2 \, a b d^{6} x^{4} - 6 \, b^{2} d^{5} x^{5} + a^{2} d^{6} x^{2} - 8 \, a b d^{5} x^{3} + 30 \, b^{2} d^{4} x^{4} - 2 \, a^{2} d^{5} x + 24 \, a b d^{4} x^{2} - 120 \, b^{2} d^{3} x^{3} + 2 \, a^{2} d^{4} - 48 \, a b d^{3} x + 360 \, b^{2} d^{2} x^{2} + 48 \, a b d^{2} - 720 \, b^{2} d x + 720 \, b^{2}\right )} e^{\left (d x + c\right )}}{2 \, d^{7}} - \frac {{\left (b^{2} d^{6} x^{6} + 2 \, a b d^{6} x^{4} + 6 \, b^{2} d^{5} x^{5} + a^{2} d^{6} x^{2} + 8 \, a b d^{5} x^{3} + 30 \, b^{2} d^{4} x^{4} + 2 \, a^{2} d^{5} x + 24 \, a b d^{4} x^{2} + 120 \, b^{2} d^{3} x^{3} + 2 \, a^{2} d^{4} + 48 \, a b d^{3} x + 360 \, b^{2} d^{2} x^{2} + 48 \, a b d^{2} + 720 \, b^{2} d x + 720 \, b^{2}\right )} e^{\left (-d x - c\right )}}{2 \, d^{7}} \]

[In]

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

[Out]

1/2*(b^2*d^6*x^6 + 2*a*b*d^6*x^4 - 6*b^2*d^5*x^5 + a^2*d^6*x^2 - 8*a*b*d^5*x^3 + 30*b^2*d^4*x^4 - 2*a^2*d^5*x
+ 24*a*b*d^4*x^2 - 120*b^2*d^3*x^3 + 2*a^2*d^4 - 48*a*b*d^3*x + 360*b^2*d^2*x^2 + 48*a*b*d^2 - 720*b^2*d*x + 7
20*b^2)*e^(d*x + c)/d^7 - 1/2*(b^2*d^6*x^6 + 2*a*b*d^6*x^4 + 6*b^2*d^5*x^5 + a^2*d^6*x^2 + 8*a*b*d^5*x^3 + 30*
b^2*d^4*x^4 + 2*a^2*d^5*x + 24*a*b*d^4*x^2 + 120*b^2*d^3*x^3 + 2*a^2*d^4 + 48*a*b*d^3*x + 360*b^2*d^2*x^2 + 48
*a*b*d^2 + 720*b^2*d*x + 720*b^2)*e^(-d*x - c)/d^7

Mupad [B] (verification not implemented)

Time = 1.79 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.78 \[ \int x^2 \left (a+b x^2\right )^2 \cosh (c+d x) \, dx=\frac {2\,\mathrm {sinh}\left (c+d\,x\right )\,\left (a^2\,d^4+24\,a\,b\,d^2+360\,b^2\right )}{d^7}-\frac {6\,b^2\,x^5\,\mathrm {cosh}\left (c+d\,x\right )}{d^2}+\frac {b^2\,x^6\,\mathrm {sinh}\left (c+d\,x\right )}{d}-\frac {2\,x\,\mathrm {cosh}\left (c+d\,x\right )\,\left (a^2\,d^4+24\,a\,b\,d^2+360\,b^2\right )}{d^6}+\frac {x^2\,\mathrm {sinh}\left (c+d\,x\right )\,\left (a^2\,d^4+24\,a\,b\,d^2+360\,b^2\right )}{d^5}-\frac {8\,x^3\,\mathrm {cosh}\left (c+d\,x\right )\,\left (15\,b^2+a\,b\,d^2\right )}{d^4}+\frac {2\,x^4\,\mathrm {sinh}\left (c+d\,x\right )\,\left (15\,b^2+a\,b\,d^2\right )}{d^3} \]

[In]

int(x^2*cosh(c + d*x)*(a + b*x^2)^2,x)

[Out]

(2*sinh(c + d*x)*(360*b^2 + a^2*d^4 + 24*a*b*d^2))/d^7 - (6*b^2*x^5*cosh(c + d*x))/d^2 + (b^2*x^6*sinh(c + d*x
))/d - (2*x*cosh(c + d*x)*(360*b^2 + a^2*d^4 + 24*a*b*d^2))/d^6 + (x^2*sinh(c + d*x)*(360*b^2 + a^2*d^4 + 24*a
*b*d^2))/d^5 - (8*x^3*cosh(c + d*x)*(15*b^2 + a*b*d^2))/d^4 + (2*x^4*sinh(c + d*x)*(15*b^2 + a*b*d^2))/d^3