\(\int (d x)^m (a+b \text {arccosh}(c x))^2 \, dx\) [164]

Optimal result

Integrand size = 16, antiderivative size = 181 \[ \int (d x)^m (a+b \text {arccosh}(c x))^2 \, dx=\frac {(d x)^{1+m} (a+b \text {arccosh}(c x))^2}{d (1+m)}-\frac {2 b c (d x)^{2+m} \sqrt {1-c x} (a+b \text {arccosh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{d^2 (1+m) (2+m) \sqrt {-1+c x}}-\frac {2 b^2 c^2 (d x)^{3+m} \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};c^2 x^2\right )}{d^3 (1+m) (2+m) (3+m)} \]

[Out]

(d*x)^(1+m)*(a+b*arccosh(c*x))^2/d/(1+m)-2*b^2*c^2*(d*x)^(3+m)*hypergeom([1, 3/2+1/2*m, 3/2+1/2*m],[2+1/2*m, 5
/2+1/2*m],c^2*x^2)/d^3/(3+m)/(m^2+3*m+2)-2*b*c*(d*x)^(2+m)*(a+b*arccosh(c*x))*hypergeom([1/2, 1+1/2*m],[2+1/2*
m],c^2*x^2)*(-c*x+1)^(1/2)/d^2/(1+m)/(2+m)/(c*x-1)^(1/2)

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5883, 5949} \[ \int (d x)^m (a+b \text {arccosh}(c x))^2 \, dx=-\frac {2 b^2 c^2 (d x)^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};c^2 x^2\right )}{d^3 (m+1) (m+2) (m+3)}-\frac {2 b c \sqrt {1-c x} (d x)^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right ) (a+b \text {arccosh}(c x))}{d^2 (m+1) (m+2) \sqrt {c x-1}}+\frac {(d x)^{m+1} (a+b \text {arccosh}(c x))^2}{d (m+1)} \]

[In]

Int[(d*x)^m*(a + b*ArcCosh[c*x])^2,x]

[Out]

((d*x)^(1 + m)*(a + b*ArcCosh[c*x])^2)/(d*(1 + m)) - (2*b*c*(d*x)^(2 + m)*Sqrt[1 - c*x]*(a + b*ArcCosh[c*x])*H
ypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(d^2*(1 + m)*(2 + m)*Sqrt[-1 + c*x]) - (2*b^2*c^2*(d*x)^
(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, c^2*x^2])/(d^3*(1 + m)*(2 + m)*(3 +
m))

Rule 5883

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcC
osh[c*x])^n/(d*(m + 1))), x] - Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt
[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5949

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_
)]), x_Symbol] :> Simp[((f*x)^(m + 1)/(f*(m + 1)))*Simp[Sqrt[1 - c^2*x^2]/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])]*(
a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2], x] + Simp[b*c*((f*x)^(m + 2)/(f^2*(
m + 1)*(m + 2)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*HypergeometricPFQ[{1
, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, m}, x] && EqQ[e1
, c*d1] && EqQ[e2, (-c)*d2] &&  !IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = \frac {(d x)^{1+m} (a+b \text {arccosh}(c x))^2}{d (1+m)}-\frac {(2 b c) \int \frac {(d x)^{1+m} (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{d (1+m)} \\ & = \frac {(d x)^{1+m} (a+b \text {arccosh}(c x))^2}{d (1+m)}-\frac {2 b c (d x)^{2+m} \sqrt {1-c x} (a+b \text {arccosh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{d^2 (1+m) (2+m) \sqrt {-1+c x}}-\frac {2 b^2 c^2 (d x)^{3+m} \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};c^2 x^2\right )}{d^3 (1+m) (2+m) (3+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.91 \[ \int (d x)^m (a+b \text {arccosh}(c x))^2 \, dx=\frac {x (d x)^m \left ((a+b \text {arccosh}(c x))^2-\frac {2 b c x \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{(2+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b^2 c^2 x^2 \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};c^2 x^2\right )}{6+5 m+m^2}\right )}{1+m} \]

[In]

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

[Out]

(x*(d*x)^m*((a + b*ArcCosh[c*x])^2 - (2*b*c*x*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (2
+ m)/2, (4 + m)/2, c^2*x^2])/((2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b^2*c^2*x^2*HypergeometricPFQ[{1, 3/
2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, c^2*x^2])/(6 + 5*m + m^2)))/(1 + m)

Maple [F]

\[\int \left (d x \right )^{m} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int (d x)^m (a+b \text {arccosh}(c x))^2 \, dx=\int { {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} \left (d x\right )^{m} \,d x } \]

[In]

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

[Out]

integral((b^2*arccosh(c*x)^2 + 2*a*b*arccosh(c*x) + a^2)*(d*x)^m, x)

Sympy [F]

\[ \int (d x)^m (a+b \text {arccosh}(c x))^2 \, dx=\int \left (d x\right )^{m} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}\, dx \]

[In]

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

[Out]

Integral((d*x)**m*(a + b*acosh(c*x))**2, x)

Maxima [F]

\[ \int (d x)^m (a+b \text {arccosh}(c x))^2 \, dx=\int { {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} \left (d x\right )^{m} \,d x } \]

[In]

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

[Out]

b^2*d^m*x*x^m*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2/(m + 1) + (d*x)^(m + 1)*a^2/(d*(m + 1)) + integrate(-2*
((a*b*d^m*(m + 1) - (a*b*c^2*d^m*(m + 1) - b^2*c^2*d^m)*x^2)*sqrt(c*x + 1)*sqrt(c*x - 1)*x^m - ((a*b*c^3*d^m*(
m + 1) - b^2*c^3*d^m)*x^3 - (a*b*c*d^m*(m + 1) - b^2*c*d^m)*x)*x^m)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(c^
3*(m + 1)*x^3 - c*(m + 1)*x + (c^2*(m + 1)*x^2 - m - 1)*sqrt(c*x + 1)*sqrt(c*x - 1)), x)

Giac [F]

\[ \int (d x)^m (a+b \text {arccosh}(c x))^2 \, dx=\int { {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} \left (d x\right )^{m} \,d x } \]

[In]

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

[Out]

integrate((b*arccosh(c*x) + a)^2*(d*x)^m, x)

Mupad [F(-1)]

Timed out. \[ \int (d x)^m (a+b \text {arccosh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d\,x\right )}^m \,d x \]

[In]

int((a + b*acosh(c*x))^2*(d*x)^m,x)

[Out]

int((a + b*acosh(c*x))^2*(d*x)^m, x)