3.59 \(\int x^2 \cosh (a+b \sqrt {c+d x}) \, dx\)

Optimal. Leaf size=346 \[ -\frac {240 \cosh \left (a+b \sqrt {c+d x}\right )}{b^6 d^3}+\frac {240 \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b^5 d^3}-\frac {120 (c+d x) \cosh \left (a+b \sqrt {c+d x}\right )}{b^4 d^3}+\frac {24 c \cosh \left (a+b \sqrt {c+d x}\right )}{b^4 d^3}+\frac {40 (c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b^3 d^3}-\frac {24 c \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b^3 d^3}-\frac {2 c^2 \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}-\frac {10 (c+d x)^2 \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}+\frac {12 c (c+d x) \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}+\frac {2 c^2 \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}+\frac {2 (c+d x)^{5/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}-\frac {4 c (c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3} \]

[Out]

-240*cosh(a+b*(d*x+c)^(1/2))/b^6/d^3+24*c*cosh(a+b*(d*x+c)^(1/2))/b^4/d^3-2*c^2*cosh(a+b*(d*x+c)^(1/2))/b^2/d^
3-120*(d*x+c)*cosh(a+b*(d*x+c)^(1/2))/b^4/d^3+12*c*(d*x+c)*cosh(a+b*(d*x+c)^(1/2))/b^2/d^3-10*(d*x+c)^2*cosh(a
+b*(d*x+c)^(1/2))/b^2/d^3+40*(d*x+c)^(3/2)*sinh(a+b*(d*x+c)^(1/2))/b^3/d^3-4*c*(d*x+c)^(3/2)*sinh(a+b*(d*x+c)^
(1/2))/b/d^3+2*(d*x+c)^(5/2)*sinh(a+b*(d*x+c)^(1/2))/b/d^3+240*sinh(a+b*(d*x+c)^(1/2))*(d*x+c)^(1/2)/b^5/d^3-2
4*c*sinh(a+b*(d*x+c)^(1/2))*(d*x+c)^(1/2)/b^3/d^3+2*c^2*sinh(a+b*(d*x+c)^(1/2))*(d*x+c)^(1/2)/b/d^3

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Rubi [A]  time = 0.42, antiderivative size = 346, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5365, 5287, 3296, 2638} \[ -\frac {2 c^2 \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}+\frac {40 (c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b^3 d^3}-\frac {24 c \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b^3 d^3}+\frac {240 \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b^5 d^3}-\frac {10 (c+d x)^2 \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}+\frac {12 c (c+d x) \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}-\frac {120 (c+d x) \cosh \left (a+b \sqrt {c+d x}\right )}{b^4 d^3}+\frac {24 c \cosh \left (a+b \sqrt {c+d x}\right )}{b^4 d^3}-\frac {240 \cosh \left (a+b \sqrt {c+d x}\right )}{b^6 d^3}+\frac {2 c^2 \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}+\frac {2 (c+d x)^{5/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}-\frac {4 c (c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2*Cosh[a + b*Sqrt[c + d*x]],x]

[Out]

(-240*Cosh[a + b*Sqrt[c + d*x]])/(b^6*d^3) + (24*c*Cosh[a + b*Sqrt[c + d*x]])/(b^4*d^3) - (2*c^2*Cosh[a + b*Sq
rt[c + d*x]])/(b^2*d^3) - (120*(c + d*x)*Cosh[a + b*Sqrt[c + d*x]])/(b^4*d^3) + (12*c*(c + d*x)*Cosh[a + b*Sqr
t[c + d*x]])/(b^2*d^3) - (10*(c + d*x)^2*Cosh[a + b*Sqrt[c + d*x]])/(b^2*d^3) + (240*Sqrt[c + d*x]*Sinh[a + b*
Sqrt[c + d*x]])/(b^5*d^3) - (24*c*Sqrt[c + d*x]*Sinh[a + b*Sqrt[c + d*x]])/(b^3*d^3) + (2*c^2*Sqrt[c + d*x]*Si
nh[a + b*Sqrt[c + d*x]])/(b*d^3) + (40*(c + d*x)^(3/2)*Sinh[a + b*Sqrt[c + d*x]])/(b^3*d^3) - (4*c*(c + d*x)^(
3/2)*Sinh[a + b*Sqrt[c + d*x]])/(b*d^3) + (2*(c + d*x)^(5/2)*Sinh[a + b*Sqrt[c + d*x]])/(b*d^3)

Rule 2638

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

Rule 3296

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 5287

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]

Rule 5365

Int[((a_.) + Cosh[(c_.) + (d_.)*(u_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/Coefficient[u, x, 1]^(
m + 1), Subst[Int[(x - Coefficient[u, x, 0])^m*(a + b*Cosh[c + d*x^n])^p, x], x, u], x] /; FreeQ[{a, b, c, d,
n, p}, x] && LinearQ[u, x] && NeQ[u, x] && IntegerQ[m]

Rubi steps

\begin {align*} \int x^2 \cosh \left (a+b \sqrt {c+d x}\right ) \, dx &=\frac {\operatorname {Subst}\left (\int (-c+x)^2 \cosh \left (a+b \sqrt {x}\right ) \, dx,x,c+d x\right )}{d^3}\\ &=\frac {2 \operatorname {Subst}\left (\int x \left (c-x^2\right )^2 \cosh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{d^3}\\ &=\frac {2 \operatorname {Subst}\left (\int \left (c^2 x \cosh (a+b x)-2 c x^3 \cosh (a+b x)+x^5 \cosh (a+b x)\right ) \, dx,x,\sqrt {c+d x}\right )}{d^3}\\ &=\frac {2 \operatorname {Subst}\left (\int x^5 \cosh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{d^3}-\frac {(4 c) \operatorname {Subst}\left (\int x^3 \cosh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{d^3}+\frac {\left (2 c^2\right ) \operatorname {Subst}\left (\int x \cosh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{d^3}\\ &=\frac {2 c^2 \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}-\frac {4 c (c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}+\frac {2 (c+d x)^{5/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}-\frac {10 \operatorname {Subst}\left (\int x^4 \sinh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{b d^3}+\frac {(12 c) \operatorname {Subst}\left (\int x^2 \sinh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{b d^3}-\frac {\left (2 c^2\right ) \operatorname {Subst}\left (\int \sinh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{b d^3}\\ &=-\frac {2 c^2 \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}+\frac {12 c (c+d x) \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}-\frac {10 (c+d x)^2 \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}+\frac {2 c^2 \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}-\frac {4 c (c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}+\frac {2 (c+d x)^{5/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}+\frac {40 \operatorname {Subst}\left (\int x^3 \cosh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{b^2 d^3}-\frac {(24 c) \operatorname {Subst}\left (\int x \cosh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{b^2 d^3}\\ &=-\frac {2 c^2 \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}+\frac {12 c (c+d x) \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}-\frac {10 (c+d x)^2 \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}-\frac {24 c \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b^3 d^3}+\frac {2 c^2 \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}+\frac {40 (c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b^3 d^3}-\frac {4 c (c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}+\frac {2 (c+d x)^{5/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}-\frac {120 \operatorname {Subst}\left (\int x^2 \sinh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{b^3 d^3}+\frac {(24 c) \operatorname {Subst}\left (\int \sinh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{b^3 d^3}\\ &=\frac {24 c \cosh \left (a+b \sqrt {c+d x}\right )}{b^4 d^3}-\frac {2 c^2 \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}-\frac {120 (c+d x) \cosh \left (a+b \sqrt {c+d x}\right )}{b^4 d^3}+\frac {12 c (c+d x) \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}-\frac {10 (c+d x)^2 \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}-\frac {24 c \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b^3 d^3}+\frac {2 c^2 \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}+\frac {40 (c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b^3 d^3}-\frac {4 c (c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}+\frac {2 (c+d x)^{5/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}+\frac {240 \operatorname {Subst}\left (\int x \cosh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{b^4 d^3}\\ &=\frac {24 c \cosh \left (a+b \sqrt {c+d x}\right )}{b^4 d^3}-\frac {2 c^2 \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}-\frac {120 (c+d x) \cosh \left (a+b \sqrt {c+d x}\right )}{b^4 d^3}+\frac {12 c (c+d x) \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}-\frac {10 (c+d x)^2 \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}+\frac {240 \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b^5 d^3}-\frac {24 c \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b^3 d^3}+\frac {2 c^2 \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}+\frac {40 (c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b^3 d^3}-\frac {4 c (c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}+\frac {2 (c+d x)^{5/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}-\frac {240 \operatorname {Subst}\left (\int \sinh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{b^5 d^3}\\ &=-\frac {240 \cosh \left (a+b \sqrt {c+d x}\right )}{b^6 d^3}+\frac {24 c \cosh \left (a+b \sqrt {c+d x}\right )}{b^4 d^3}-\frac {2 c^2 \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}-\frac {120 (c+d x) \cosh \left (a+b \sqrt {c+d x}\right )}{b^4 d^3}+\frac {12 c (c+d x) \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}-\frac {10 (c+d x)^2 \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}+\frac {240 \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b^5 d^3}-\frac {24 c \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b^3 d^3}+\frac {2 c^2 \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}+\frac {40 (c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b^3 d^3}-\frac {4 c (c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}+\frac {2 (c+d x)^{5/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}\\ \end {align*}

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Mathematica [A]  time = 0.54, size = 104, normalized size = 0.30 \[ \frac {2 b \sqrt {c+d x} \left (b^4 d^2 x^2+4 b^2 (2 c+5 d x)+120\right ) \sinh \left (a+b \sqrt {c+d x}\right )-2 \left (b^4 d x (4 c+5 d x)+12 b^2 (4 c+5 d x)+120\right ) \cosh \left (a+b \sqrt {c+d x}\right )}{b^6 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(120 + 12*b^2*(4*c + 5*d*x) + b^4*d*x*(4*c + 5*d*x))*Cosh[a + b*Sqrt[c + d*x]] + 2*b*Sqrt[c + d*x]*(120 +
b^4*d^2*x^2 + 4*b^2*(2*c + 5*d*x))*Sinh[a + b*Sqrt[c + d*x]])/(b^6*d^3)

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fricas [A]  time = 0.50, size = 104, normalized size = 0.30 \[ \frac {2 \, {\left ({\left (b^{5} d^{2} x^{2} + 20 \, b^{3} d x + 8 \, b^{3} c + 120 \, b\right )} \sqrt {d x + c} \sinh \left (\sqrt {d x + c} b + a\right ) - {\left (5 \, b^{4} d^{2} x^{2} + 48 \, b^{2} c + 4 \, {\left (b^{4} c + 15 \, b^{2}\right )} d x + 120\right )} \cosh \left (\sqrt {d x + c} b + a\right )\right )}}{b^{6} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*cosh(a+b*(d*x+c)^(1/2)),x, algorithm="fricas")

[Out]

2*((b^5*d^2*x^2 + 20*b^3*d*x + 8*b^3*c + 120*b)*sqrt(d*x + c)*sinh(sqrt(d*x + c)*b + a) - (5*b^4*d^2*x^2 + 48*
b^2*c + 4*(b^4*c + 15*b^2)*d*x + 120)*cosh(sqrt(d*x + c)*b + a))/(b^6*d^3)

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giac [B]  time = 0.16, size = 915, normalized size = 2.64 \[ \frac {\frac {{\left ({\left (\sqrt {d x + c} b + a\right )} b^{4} c^{2} - a b^{4} c^{2} - 2 \, {\left (\sqrt {d x + c} b + a\right )}^{3} b^{2} c + 6 \, {\left (\sqrt {d x + c} b + a\right )}^{2} a b^{2} c - 6 \, {\left (\sqrt {d x + c} b + a\right )} a^{2} b^{2} c + 2 \, a^{3} b^{2} c - b^{4} c^{2} + {\left (\sqrt {d x + c} b + a\right )}^{5} - 5 \, {\left (\sqrt {d x + c} b + a\right )}^{4} a + 10 \, {\left (\sqrt {d x + c} b + a\right )}^{3} a^{2} - 10 \, {\left (\sqrt {d x + c} b + a\right )}^{2} a^{3} + 5 \, {\left (\sqrt {d x + c} b + a\right )} a^{4} - a^{5} + 6 \, {\left (\sqrt {d x + c} b + a\right )}^{2} b^{2} c - 12 \, {\left (\sqrt {d x + c} b + a\right )} a b^{2} c + 6 \, a^{2} b^{2} c - 5 \, {\left (\sqrt {d x + c} b + a\right )}^{4} + 20 \, {\left (\sqrt {d x + c} b + a\right )}^{3} a - 30 \, {\left (\sqrt {d x + c} b + a\right )}^{2} a^{2} + 20 \, {\left (\sqrt {d x + c} b + a\right )} a^{3} - 5 \, a^{4} - 12 \, {\left (\sqrt {d x + c} b + a\right )} b^{2} c + 12 \, a b^{2} c + 20 \, {\left (\sqrt {d x + c} b + a\right )}^{3} - 60 \, {\left (\sqrt {d x + c} b + a\right )}^{2} a + 60 \, {\left (\sqrt {d x + c} b + a\right )} a^{2} - 20 \, a^{3} + 12 \, b^{2} c - 60 \, {\left (\sqrt {d x + c} b + a\right )}^{2} + 120 \, {\left (\sqrt {d x + c} b + a\right )} a - 60 \, a^{2} + 120 \, \sqrt {d x + c} b - 120\right )} e^{\left (\sqrt {d x + c} b + a\right )}}{b^{5} d^{2}} - \frac {{\left ({\left (\sqrt {d x + c} b + a\right )} b^{4} c^{2} - a b^{4} c^{2} - 2 \, {\left (\sqrt {d x + c} b + a\right )}^{3} b^{2} c + 6 \, {\left (\sqrt {d x + c} b + a\right )}^{2} a b^{2} c - 6 \, {\left (\sqrt {d x + c} b + a\right )} a^{2} b^{2} c + 2 \, a^{3} b^{2} c + b^{4} c^{2} + {\left (\sqrt {d x + c} b + a\right )}^{5} - 5 \, {\left (\sqrt {d x + c} b + a\right )}^{4} a + 10 \, {\left (\sqrt {d x + c} b + a\right )}^{3} a^{2} - 10 \, {\left (\sqrt {d x + c} b + a\right )}^{2} a^{3} + 5 \, {\left (\sqrt {d x + c} b + a\right )} a^{4} - a^{5} - 6 \, {\left (\sqrt {d x + c} b + a\right )}^{2} b^{2} c + 12 \, {\left (\sqrt {d x + c} b + a\right )} a b^{2} c - 6 \, a^{2} b^{2} c + 5 \, {\left (\sqrt {d x + c} b + a\right )}^{4} - 20 \, {\left (\sqrt {d x + c} b + a\right )}^{3} a + 30 \, {\left (\sqrt {d x + c} b + a\right )}^{2} a^{2} - 20 \, {\left (\sqrt {d x + c} b + a\right )} a^{3} + 5 \, a^{4} - 12 \, {\left (\sqrt {d x + c} b + a\right )} b^{2} c + 12 \, a b^{2} c + 20 \, {\left (\sqrt {d x + c} b + a\right )}^{3} - 60 \, {\left (\sqrt {d x + c} b + a\right )}^{2} a + 60 \, {\left (\sqrt {d x + c} b + a\right )} a^{2} - 20 \, a^{3} - 12 \, b^{2} c + 60 \, {\left (\sqrt {d x + c} b + a\right )}^{2} - 120 \, {\left (\sqrt {d x + c} b + a\right )} a + 60 \, a^{2} + 120 \, \sqrt {d x + c} b + 120\right )} e^{\left (-\sqrt {d x + c} b - a\right )}}{b^{5} d^{2}}}{b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*cosh(a+b*(d*x+c)^(1/2)),x, algorithm="giac")

[Out]

(((sqrt(d*x + c)*b + a)*b^4*c^2 - a*b^4*c^2 - 2*(sqrt(d*x + c)*b + a)^3*b^2*c + 6*(sqrt(d*x + c)*b + a)^2*a*b^
2*c - 6*(sqrt(d*x + c)*b + a)*a^2*b^2*c + 2*a^3*b^2*c - b^4*c^2 + (sqrt(d*x + c)*b + a)^5 - 5*(sqrt(d*x + c)*b
+ a)^4*a + 10*(sqrt(d*x + c)*b + a)^3*a^2 - 10*(sqrt(d*x + c)*b + a)^2*a^3 + 5*(sqrt(d*x + c)*b + a)*a^4 - a^
5 + 6*(sqrt(d*x + c)*b + a)^2*b^2*c - 12*(sqrt(d*x + c)*b + a)*a*b^2*c + 6*a^2*b^2*c - 5*(sqrt(d*x + c)*b + a)
^4 + 20*(sqrt(d*x + c)*b + a)^3*a - 30*(sqrt(d*x + c)*b + a)^2*a^2 + 20*(sqrt(d*x + c)*b + a)*a^3 - 5*a^4 - 12
*(sqrt(d*x + c)*b + a)*b^2*c + 12*a*b^2*c + 20*(sqrt(d*x + c)*b + a)^3 - 60*(sqrt(d*x + c)*b + a)^2*a + 60*(sq
rt(d*x + c)*b + a)*a^2 - 20*a^3 + 12*b^2*c - 60*(sqrt(d*x + c)*b + a)^2 + 120*(sqrt(d*x + c)*b + a)*a - 60*a^2
+ 120*sqrt(d*x + c)*b - 120)*e^(sqrt(d*x + c)*b + a)/(b^5*d^2) - ((sqrt(d*x + c)*b + a)*b^4*c^2 - a*b^4*c^2 -
2*(sqrt(d*x + c)*b + a)^3*b^2*c + 6*(sqrt(d*x + c)*b + a)^2*a*b^2*c - 6*(sqrt(d*x + c)*b + a)*a^2*b^2*c + 2*a
^3*b^2*c + b^4*c^2 + (sqrt(d*x + c)*b + a)^5 - 5*(sqrt(d*x + c)*b + a)^4*a + 10*(sqrt(d*x + c)*b + a)^3*a^2 -
10*(sqrt(d*x + c)*b + a)^2*a^3 + 5*(sqrt(d*x + c)*b + a)*a^4 - a^5 - 6*(sqrt(d*x + c)*b + a)^2*b^2*c + 12*(sqr
t(d*x + c)*b + a)*a*b^2*c - 6*a^2*b^2*c + 5*(sqrt(d*x + c)*b + a)^4 - 20*(sqrt(d*x + c)*b + a)^3*a + 30*(sqrt(
d*x + c)*b + a)^2*a^2 - 20*(sqrt(d*x + c)*b + a)*a^3 + 5*a^4 - 12*(sqrt(d*x + c)*b + a)*b^2*c + 12*a*b^2*c + 2
0*(sqrt(d*x + c)*b + a)^3 - 60*(sqrt(d*x + c)*b + a)^2*a + 60*(sqrt(d*x + c)*b + a)*a^2 - 20*a^3 - 12*b^2*c +
60*(sqrt(d*x + c)*b + a)^2 - 120*(sqrt(d*x + c)*b + a)*a + 60*a^2 + 120*sqrt(d*x + c)*b + 120)*e^(-sqrt(d*x +
c)*b - a)/(b^5*d^2))/(b*d)

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maple [B]  time = 0.10, size = 831, normalized size = 2.40 \[ \frac {\frac {2 \left (\left (a +b \sqrt {d x +c}\right )^{5} \sinh \left (a +b \sqrt {d x +c}\right )-5 \left (a +b \sqrt {d x +c}\right )^{4} \cosh \left (a +b \sqrt {d x +c}\right )+20 \left (a +b \sqrt {d x +c}\right )^{3} \sinh \left (a +b \sqrt {d x +c}\right )-60 \left (a +b \sqrt {d x +c}\right )^{2} \cosh \left (a +b \sqrt {d x +c}\right )+120 \sinh \left (a +b \sqrt {d x +c}\right ) \left (a +b \sqrt {d x +c}\right )-120 \cosh \left (a +b \sqrt {d x +c}\right )\right )}{b^{4}}-\frac {10 a \left (\left (a +b \sqrt {d x +c}\right )^{4} \sinh \left (a +b \sqrt {d x +c}\right )-4 \left (a +b \sqrt {d x +c}\right )^{3} \cosh \left (a +b \sqrt {d x +c}\right )+12 \sinh \left (a +b \sqrt {d x +c}\right ) \left (a +b \sqrt {d x +c}\right )^{2}-24 \left (a +b \sqrt {d x +c}\right ) \cosh \left (a +b \sqrt {d x +c}\right )+24 \sinh \left (a +b \sqrt {d x +c}\right )\right )}{b^{4}}+\frac {20 a^{2} \left (\left (a +b \sqrt {d x +c}\right )^{3} \sinh \left (a +b \sqrt {d x +c}\right )-3 \left (a +b \sqrt {d x +c}\right )^{2} \cosh \left (a +b \sqrt {d x +c}\right )+6 \sinh \left (a +b \sqrt {d x +c}\right ) \left (a +b \sqrt {d x +c}\right )-6 \cosh \left (a +b \sqrt {d x +c}\right )\right )}{b^{4}}-\frac {20 a^{3} \left (\sinh \left (a +b \sqrt {d x +c}\right ) \left (a +b \sqrt {d x +c}\right )^{2}-2 \left (a +b \sqrt {d x +c}\right ) \cosh \left (a +b \sqrt {d x +c}\right )+2 \sinh \left (a +b \sqrt {d x +c}\right )\right )}{b^{4}}-\frac {4 c \left (\left (a +b \sqrt {d x +c}\right )^{3} \sinh \left (a +b \sqrt {d x +c}\right )-3 \left (a +b \sqrt {d x +c}\right )^{2} \cosh \left (a +b \sqrt {d x +c}\right )+6 \sinh \left (a +b \sqrt {d x +c}\right ) \left (a +b \sqrt {d x +c}\right )-6 \cosh \left (a +b \sqrt {d x +c}\right )\right )}{b^{2}}+\frac {12 c a \left (\sinh \left (a +b \sqrt {d x +c}\right ) \left (a +b \sqrt {d x +c}\right )^{2}-2 \left (a +b \sqrt {d x +c}\right ) \cosh \left (a +b \sqrt {d x +c}\right )+2 \sinh \left (a +b \sqrt {d x +c}\right )\right )}{b^{2}}+\frac {10 a^{4} \left (\sinh \left (a +b \sqrt {d x +c}\right ) \left (a +b \sqrt {d x +c}\right )-\cosh \left (a +b \sqrt {d x +c}\right )\right )}{b^{4}}-\frac {12 a^{2} c \left (\sinh \left (a +b \sqrt {d x +c}\right ) \left (a +b \sqrt {d x +c}\right )-\cosh \left (a +b \sqrt {d x +c}\right )\right )}{b^{2}}-\frac {2 a^{5} \sinh \left (a +b \sqrt {d x +c}\right )}{b^{4}}+\frac {4 a^{3} c \sinh \left (a +b \sqrt {d x +c}\right )}{b^{2}}+2 c^{2} \left (\sinh \left (a +b \sqrt {d x +c}\right ) \left (a +b \sqrt {d x +c}\right )-\cosh \left (a +b \sqrt {d x +c}\right )\right )-2 c^{2} a \sinh \left (a +b \sqrt {d x +c}\right )}{d^{3} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*cosh(a+b*(d*x+c)^(1/2)),x)

[Out]

2/d^3/b^2*(1/b^4*((a+b*(d*x+c)^(1/2))^5*sinh(a+b*(d*x+c)^(1/2))-5*(a+b*(d*x+c)^(1/2))^4*cosh(a+b*(d*x+c)^(1/2)
)+20*(a+b*(d*x+c)^(1/2))^3*sinh(a+b*(d*x+c)^(1/2))-60*(a+b*(d*x+c)^(1/2))^2*cosh(a+b*(d*x+c)^(1/2))+120*sinh(a
+b*(d*x+c)^(1/2))*(a+b*(d*x+c)^(1/2))-120*cosh(a+b*(d*x+c)^(1/2)))-5/b^4*a*((a+b*(d*x+c)^(1/2))^4*sinh(a+b*(d*
x+c)^(1/2))-4*(a+b*(d*x+c)^(1/2))^3*cosh(a+b*(d*x+c)^(1/2))+12*sinh(a+b*(d*x+c)^(1/2))*(a+b*(d*x+c)^(1/2))^2-2
4*(a+b*(d*x+c)^(1/2))*cosh(a+b*(d*x+c)^(1/2))+24*sinh(a+b*(d*x+c)^(1/2)))+10/b^4*a^2*((a+b*(d*x+c)^(1/2))^3*si
nh(a+b*(d*x+c)^(1/2))-3*(a+b*(d*x+c)^(1/2))^2*cosh(a+b*(d*x+c)^(1/2))+6*sinh(a+b*(d*x+c)^(1/2))*(a+b*(d*x+c)^(
1/2))-6*cosh(a+b*(d*x+c)^(1/2)))-10/b^4*a^3*(sinh(a+b*(d*x+c)^(1/2))*(a+b*(d*x+c)^(1/2))^2-2*(a+b*(d*x+c)^(1/2
))*cosh(a+b*(d*x+c)^(1/2))+2*sinh(a+b*(d*x+c)^(1/2)))-2/b^2*c*((a+b*(d*x+c)^(1/2))^3*sinh(a+b*(d*x+c)^(1/2))-3
*(a+b*(d*x+c)^(1/2))^2*cosh(a+b*(d*x+c)^(1/2))+6*sinh(a+b*(d*x+c)^(1/2))*(a+b*(d*x+c)^(1/2))-6*cosh(a+b*(d*x+c
)^(1/2)))+6/b^2*c*a*(sinh(a+b*(d*x+c)^(1/2))*(a+b*(d*x+c)^(1/2))^2-2*(a+b*(d*x+c)^(1/2))*cosh(a+b*(d*x+c)^(1/2
))+2*sinh(a+b*(d*x+c)^(1/2)))+5*a^4/b^4*(sinh(a+b*(d*x+c)^(1/2))*(a+b*(d*x+c)^(1/2))-cosh(a+b*(d*x+c)^(1/2)))-
6*a^2/b^2*c*(sinh(a+b*(d*x+c)^(1/2))*(a+b*(d*x+c)^(1/2))-cosh(a+b*(d*x+c)^(1/2)))-a^5/b^4*sinh(a+b*(d*x+c)^(1/
2))+2*a^3/b^2*c*sinh(a+b*(d*x+c)^(1/2))+c^2*(sinh(a+b*(d*x+c)^(1/2))*(a+b*(d*x+c)^(1/2))-cosh(a+b*(d*x+c)^(1/2
)))-c^2*a*sinh(a+b*(d*x+c)^(1/2)))

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maxima [A]  time = 0.34, size = 486, normalized size = 1.40 \[ \frac {2 \, d^{3} x^{3} \cosh \left (\sqrt {d x + c} b + a\right ) + {\left (\frac {c^{3} e^{\left (\sqrt {d x + c} b + a\right )}}{b} + \frac {c^{3} e^{\left (-\sqrt {d x + c} b - a\right )}}{b} - \frac {3 \, {\left ({\left (d x + c\right )} b^{2} e^{a} - 2 \, \sqrt {d x + c} b e^{a} + 2 \, e^{a}\right )} c^{2} e^{\left (\sqrt {d x + c} b\right )}}{b^{3}} - \frac {3 \, {\left ({\left (d x + c\right )} b^{2} + 2 \, \sqrt {d x + c} b + 2\right )} c^{2} e^{\left (-\sqrt {d x + c} b - a\right )}}{b^{3}} + \frac {3 \, {\left ({\left (d x + c\right )}^{2} b^{4} e^{a} - 4 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} e^{a} + 12 \, {\left (d x + c\right )} b^{2} e^{a} - 24 \, \sqrt {d x + c} b e^{a} + 24 \, e^{a}\right )} c e^{\left (\sqrt {d x + c} b\right )}}{b^{5}} + \frac {3 \, {\left ({\left (d x + c\right )}^{2} b^{4} + 4 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} + 12 \, {\left (d x + c\right )} b^{2} + 24 \, \sqrt {d x + c} b + 24\right )} c e^{\left (-\sqrt {d x + c} b - a\right )}}{b^{5}} - \frac {{\left ({\left (d x + c\right )}^{3} b^{6} e^{a} - 6 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{5} e^{a} + 30 \, {\left (d x + c\right )}^{2} b^{4} e^{a} - 120 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} e^{a} + 360 \, {\left (d x + c\right )} b^{2} e^{a} - 720 \, \sqrt {d x + c} b e^{a} + 720 \, e^{a}\right )} e^{\left (\sqrt {d x + c} b\right )}}{b^{7}} - \frac {{\left ({\left (d x + c\right )}^{3} b^{6} + 6 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{5} + 30 \, {\left (d x + c\right )}^{2} b^{4} + 120 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} + 360 \, {\left (d x + c\right )} b^{2} + 720 \, \sqrt {d x + c} b + 720\right )} e^{\left (-\sqrt {d x + c} b - a\right )}}{b^{7}}\right )} b}{6 \, d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*cosh(a+b*(d*x+c)^(1/2)),x, algorithm="maxima")

[Out]

1/6*(2*d^3*x^3*cosh(sqrt(d*x + c)*b + a) + (c^3*e^(sqrt(d*x + c)*b + a)/b + c^3*e^(-sqrt(d*x + c)*b - a)/b - 3
*((d*x + c)*b^2*e^a - 2*sqrt(d*x + c)*b*e^a + 2*e^a)*c^2*e^(sqrt(d*x + c)*b)/b^3 - 3*((d*x + c)*b^2 + 2*sqrt(d
*x + c)*b + 2)*c^2*e^(-sqrt(d*x + c)*b - a)/b^3 + 3*((d*x + c)^2*b^4*e^a - 4*(d*x + c)^(3/2)*b^3*e^a + 12*(d*x
+ c)*b^2*e^a - 24*sqrt(d*x + c)*b*e^a + 24*e^a)*c*e^(sqrt(d*x + c)*b)/b^5 + 3*((d*x + c)^2*b^4 + 4*(d*x + c)^
(3/2)*b^3 + 12*(d*x + c)*b^2 + 24*sqrt(d*x + c)*b + 24)*c*e^(-sqrt(d*x + c)*b - a)/b^5 - ((d*x + c)^3*b^6*e^a
- 6*(d*x + c)^(5/2)*b^5*e^a + 30*(d*x + c)^2*b^4*e^a - 120*(d*x + c)^(3/2)*b^3*e^a + 360*(d*x + c)*b^2*e^a - 7
20*sqrt(d*x + c)*b*e^a + 720*e^a)*e^(sqrt(d*x + c)*b)/b^7 - ((d*x + c)^3*b^6 + 6*(d*x + c)^(5/2)*b^5 + 30*(d*x
+ c)^2*b^4 + 120*(d*x + c)^(3/2)*b^3 + 360*(d*x + c)*b^2 + 720*sqrt(d*x + c)*b + 720)*e^(-sqrt(d*x + c)*b - a
)/b^7)*b)/d^3

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,\mathrm {cosh}\left (a+b\,\sqrt {c+d\,x}\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*cosh(a + b*(c + d*x)^(1/2)),x)

[Out]

int(x^2*cosh(a + b*(c + d*x)^(1/2)), x)

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sympy [A]  time = 1.73, size = 269, normalized size = 0.78 \[ \begin {cases} \frac {x^{3} \cosh {\relax (a )}}{3} & \text {for}\: b = 0 \wedge \left (b = 0 \vee d = 0\right ) \\\frac {x^{3} \cosh {\left (a + b \sqrt {c} \right )}}{3} & \text {for}\: d = 0 \\\frac {2 x^{2} \sqrt {c + d x} \sinh {\left (a + b \sqrt {c + d x} \right )}}{b d} - \frac {8 c x \cosh {\left (a + b \sqrt {c + d x} \right )}}{b^{2} d^{2}} - \frac {10 x^{2} \cosh {\left (a + b \sqrt {c + d x} \right )}}{b^{2} d} + \frac {16 c \sqrt {c + d x} \sinh {\left (a + b \sqrt {c + d x} \right )}}{b^{3} d^{3}} + \frac {40 x \sqrt {c + d x} \sinh {\left (a + b \sqrt {c + d x} \right )}}{b^{3} d^{2}} - \frac {96 c \cosh {\left (a + b \sqrt {c + d x} \right )}}{b^{4} d^{3}} - \frac {120 x \cosh {\left (a + b \sqrt {c + d x} \right )}}{b^{4} d^{2}} + \frac {240 \sqrt {c + d x} \sinh {\left (a + b \sqrt {c + d x} \right )}}{b^{5} d^{3}} - \frac {240 \cosh {\left (a + b \sqrt {c + d x} \right )}}{b^{6} d^{3}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*cosh(a+b*(d*x+c)**(1/2)),x)

[Out]

Piecewise((x**3*cosh(a)/3, Eq(b, 0) & (Eq(b, 0) | Eq(d, 0))), (x**3*cosh(a + b*sqrt(c))/3, Eq(d, 0)), (2*x**2*
sqrt(c + d*x)*sinh(a + b*sqrt(c + d*x))/(b*d) - 8*c*x*cosh(a + b*sqrt(c + d*x))/(b**2*d**2) - 10*x**2*cosh(a +
b*sqrt(c + d*x))/(b**2*d) + 16*c*sqrt(c + d*x)*sinh(a + b*sqrt(c + d*x))/(b**3*d**3) + 40*x*sqrt(c + d*x)*sin
h(a + b*sqrt(c + d*x))/(b**3*d**2) - 96*c*cosh(a + b*sqrt(c + d*x))/(b**4*d**3) - 120*x*cosh(a + b*sqrt(c + d*
x))/(b**4*d**2) + 240*sqrt(c + d*x)*sinh(a + b*sqrt(c + d*x))/(b**5*d**3) - 240*cosh(a + b*sqrt(c + d*x))/(b**
6*d**3), True))

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