Integrand size = 16, antiderivative size = 63 \[ \int \log \left (c \left (d+e (f+g x)^2\right )^q\right ) \, dx=-2 q x+\frac {2 \sqrt {d} q \arctan \left (\frac {\sqrt {e} (f+g x)}{\sqrt {d}}\right )}{\sqrt {e} g}+\frac {(f+g x) \log \left (c \left (d+e (f+g x)^2\right )^q\right )}{g} \] Output:
-2*q*x+2*d^(1/2)*q*arctan(e^(1/2)*(g*x+f)/d^(1/2))/e^(1/2)/g+(g*x+f)*ln(c* (d+e*(g*x+f)^2)^q)/g
Time = 0.04 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00 \[ \int \log \left (c \left (d+e (f+g x)^2\right )^q\right ) \, dx=-2 q x+\frac {2 \sqrt {d} q \arctan \left (\frac {\sqrt {e} (f+g x)}{\sqrt {d}}\right )}{\sqrt {e} g}+\frac {(f+g x) \log \left (c \left (d+e (f+g x)^2\right )^q\right )}{g} \] Input:
Integrate[Log[c*(d + e*(f + g*x)^2)^q],x]
Output:
-2*q*x + (2*Sqrt[d]*q*ArcTan[(Sqrt[e]*(f + g*x))/Sqrt[d]])/(Sqrt[e]*g) + ( (f + g*x)*Log[c*(d + e*(f + g*x)^2)^q])/g
Time = 0.36 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.11, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2933, 2898, 262, 218}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \log \left (c \left (d+e (f+g x)^2\right )^q\right ) \, dx\) |
\(\Big \downarrow \) 2933 |
\(\displaystyle \frac {\int \log \left (c \left (e (f+g x)^2+d\right )^q\right )d(f+g x)}{g}\) |
\(\Big \downarrow \) 2898 |
\(\displaystyle \frac {(f+g x) \log \left (c \left (d+e (f+g x)^2\right )^q\right )-2 e q \int \frac {(f+g x)^2}{e (f+g x)^2+d}d(f+g x)}{g}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {(f+g x) \log \left (c \left (d+e (f+g x)^2\right )^q\right )-2 e q \left (\frac {f+g x}{e}-\frac {d \int \frac {1}{e (f+g x)^2+d}d(f+g x)}{e}\right )}{g}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {(f+g x) \log \left (c \left (d+e (f+g x)^2\right )^q\right )-2 e q \left (\frac {f+g x}{e}-\frac {\sqrt {d} \arctan \left (\frac {\sqrt {e} (f+g x)}{\sqrt {d}}\right )}{e^{3/2}}\right )}{g}\) |
Input:
Int[Log[c*(d + e*(f + g*x)^2)^q],x]
Output:
(-2*e*q*((f + g*x)/e - (Sqrt[d]*ArcTan[(Sqrt[e]*(f + g*x))/Sqrt[d]])/e^(3/ 2)) + (f + g*x)*Log[c*(d + e*(f + g*x)^2)^q])/g
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Simp[e*n*p Int[x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*((f_.) + (g_.)*(x_))^(n_))^(p_.)]*(b_. ))^(q_.), x_Symbol] :> Simp[1/g Subst[Int[(a + b*Log[c*(d + e*x^n)^p])^q, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IGtQ[q, 0] && (EqQ[q, 1] || IntegerQ[n])
Time = 1.81 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.67
method | result | size |
parts | \(\ln \left (c \left (d +\left (g x +f \right )^{2} e \right )^{q}\right ) x -2 q e g \left (\frac {x}{g e}+\frac {-\frac {f \ln \left (e \,g^{2} x^{2}+2 e f g x +f^{2} e +d \right )}{2 g}-\frac {d \arctan \left (\frac {2 e \,g^{2} x +2 e f g}{2 g \sqrt {d e}}\right )}{g \sqrt {d e}}}{e g}\right )\) | \(105\) |
default | \(x \ln \left (c \left (e \,g^{2} x^{2}+2 e f g x +f^{2} e +d \right )^{q}\right )-2 q e g \left (\frac {x}{g e}+\frac {-\frac {f \ln \left (e \,g^{2} x^{2}+2 e f g x +f^{2} e +d \right )}{2 g}-\frac {d \arctan \left (\frac {2 e \,g^{2} x +2 e f g}{2 g \sqrt {d e}}\right )}{g \sqrt {d e}}}{e g}\right )\) | \(115\) |
risch | \(x \ln \left (\left (d +\left (g x +f \right )^{2} e \right )^{q}\right )+\frac {i {\operatorname {csgn}\left (i c \left (d +\left (g x +f \right )^{2} e \right )^{q}\right )}^{2} \operatorname {csgn}\left (i \left (d +\left (g x +f \right )^{2} e \right )^{q}\right ) \pi x}{2}-\frac {i \pi x \,\operatorname {csgn}\left (i \left (d +\left (g x +f \right )^{2} e \right )^{q}\right ) \operatorname {csgn}\left (i c \left (d +\left (g x +f \right )^{2} e \right )^{q}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi x {\operatorname {csgn}\left (i c \left (d +\left (g x +f \right )^{2} e \right )^{q}\right )}^{3}}{2}+\frac {i \operatorname {csgn}\left (i c \right ) {\operatorname {csgn}\left (i c \left (d +\left (g x +f \right )^{2} e \right )^{q}\right )}^{2} \pi x}{2}+\frac {\ln \left (-\sqrt {-d e}\, g x -\sqrt {-d e}\, f +d \right ) f q}{g}+\frac {\ln \left (\sqrt {-d e}\, g x +\sqrt {-d e}\, f +d \right ) f q}{g}+\ln \left (c \right ) x -2 q x +\frac {\ln \left (-\sqrt {-d e}\, g x -\sqrt {-d e}\, f +d \right ) \sqrt {-d e}\, q}{e g}-\frac {\sqrt {-d e}\, \ln \left (\sqrt {-d e}\, g x +\sqrt {-d e}\, f +d \right ) q}{e g}\) | \(293\) |
Input:
int(ln(c*(d+(g*x+f)^2*e)^q),x,method=_RETURNVERBOSE)
Output:
ln(c*(d+(g*x+f)^2*e)^q)*x-2*q*e*g*(1/g/e*x+1/e/g*(-1/2*f/g*ln(e*g^2*x^2+2* e*f*g*x+e*f^2+d)-d/g/(d*e)^(1/2)*arctan(1/2*(2*e*g^2*x+2*e*f*g)/g/(d*e)^(1 /2))))
Time = 0.10 (sec) , antiderivative size = 206, normalized size of antiderivative = 3.27 \[ \int \log \left (c \left (d+e (f+g x)^2\right )^q\right ) \, dx=\left [-\frac {2 \, g q x - g x \log \left (c\right ) - q \sqrt {-\frac {d}{e}} \log \left (\frac {e g^{2} x^{2} + 2 \, e f g x + e f^{2} + 2 \, {\left (e g x + e f\right )} \sqrt {-\frac {d}{e}} - d}{e g^{2} x^{2} + 2 \, e f g x + e f^{2} + d}\right ) - {\left (g q x + f q\right )} \log \left (e g^{2} x^{2} + 2 \, e f g x + e f^{2} + d\right )}{g}, -\frac {2 \, g q x - g x \log \left (c\right ) - 2 \, q \sqrt {\frac {d}{e}} \arctan \left (\frac {{\left (e g x + e f\right )} \sqrt {\frac {d}{e}}}{d}\right ) - {\left (g q x + f q\right )} \log \left (e g^{2} x^{2} + 2 \, e f g x + e f^{2} + d\right )}{g}\right ] \] Input:
integrate(log(c*(d+e*(g*x+f)^2)^q),x, algorithm="fricas")
Output:
[-(2*g*q*x - g*x*log(c) - q*sqrt(-d/e)*log((e*g^2*x^2 + 2*e*f*g*x + e*f^2 + 2*(e*g*x + e*f)*sqrt(-d/e) - d)/(e*g^2*x^2 + 2*e*f*g*x + e*f^2 + d)) - ( g*q*x + f*q)*log(e*g^2*x^2 + 2*e*f*g*x + e*f^2 + d))/g, -(2*g*q*x - g*x*lo g(c) - 2*q*sqrt(d/e)*arctan((e*g*x + e*f)*sqrt(d/e)/d) - (g*q*x + f*q)*log (e*g^2*x^2 + 2*e*f*g*x + e*f^2 + d))/g]
Leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (58) = 116\).
Time = 119.94 (sec) , antiderivative size = 235, normalized size of antiderivative = 3.73 \[ \int \log \left (c \left (d+e (f+g x)^2\right )^q\right ) \, dx=\begin {cases} x \log {\left (0^{q} c \right )} & \text {for}\: d = 0 \wedge e = 0 \wedge g = 0 \\x \log {\left (c d^{q} \right )} & \text {for}\: e = 0 \\x \log {\left (c \left (d + e f^{2}\right )^{q} \right )} & \text {for}\: g = 0 \\\frac {f \log {\left (c \left (e f^{2} + 2 e f g x + e g^{2} x^{2}\right )^{q} \right )}}{g} - 2 q x + x \log {\left (c \left (e f^{2} + 2 e f g x + e g^{2} x^{2}\right )^{q} \right )} & \text {for}\: d = 0 \\\frac {2 d q \log {\left (\frac {f}{g} + x - \frac {\sqrt {- d e}}{e g} \right )}}{g \sqrt {- d e}} - \frac {d \log {\left (c \left (d + e f^{2} + 2 e f g x + e g^{2} x^{2}\right )^{q} \right )}}{g \sqrt {- d e}} + \frac {f \log {\left (c \left (d + e f^{2} + 2 e f g x + e g^{2} x^{2}\right )^{q} \right )}}{g} - 2 q x + x \log {\left (c \left (d + e f^{2} + 2 e f g x + e g^{2} x^{2}\right )^{q} \right )} & \text {otherwise} \end {cases} \] Input:
integrate(ln(c*(d+e*(g*x+f)**2)**q),x)
Output:
Piecewise((x*log(0**q*c), Eq(d, 0) & Eq(e, 0) & Eq(g, 0)), (x*log(c*d**q), Eq(e, 0)), (x*log(c*(d + e*f**2)**q), Eq(g, 0)), (f*log(c*(e*f**2 + 2*e*f *g*x + e*g**2*x**2)**q)/g - 2*q*x + x*log(c*(e*f**2 + 2*e*f*g*x + e*g**2*x **2)**q), Eq(d, 0)), (2*d*q*log(f/g + x - sqrt(-d*e)/(e*g))/(g*sqrt(-d*e)) - d*log(c*(d + e*f**2 + 2*e*f*g*x + e*g**2*x**2)**q)/(g*sqrt(-d*e)) + f*l og(c*(d + e*f**2 + 2*e*f*g*x + e*g**2*x**2)**q)/g - 2*q*x + x*log(c*(d + e *f**2 + 2*e*f*g*x + e*g**2*x**2)**q), True))
Exception generated. \[ \int \log \left (c \left (d+e (f+g x)^2\right )^q\right ) \, dx=\text {Exception raised: ValueError} \] Input:
integrate(log(c*(d+e*(g*x+f)^2)^q),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Time = 0.12 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.46 \[ \int \log \left (c \left (d+e (f+g x)^2\right )^q\right ) \, dx=q x \log \left (e g^{2} x^{2} + 2 \, e f g x + e f^{2} + d\right ) - {\left (2 \, q - \log \left (c\right )\right )} x + \frac {f q \log \left (e g^{2} x^{2} + 2 \, e f g x + e f^{2} + d\right )}{g} + \frac {2 \, d q \arctan \left (\frac {e g x + e f}{\sqrt {d e}}\right )}{\sqrt {d e} g} \] Input:
integrate(log(c*(d+e*(g*x+f)^2)^q),x, algorithm="giac")
Output:
q*x*log(e*g^2*x^2 + 2*e*f*g*x + e*f^2 + d) - (2*q - log(c))*x + f*q*log(e* g^2*x^2 + 2*e*f*g*x + e*f^2 + d)/g + 2*d*q*arctan((e*g*x + e*f)/sqrt(d*e)) /(sqrt(d*e)*g)
Time = 25.67 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.30 \[ \int \log \left (c \left (d+e (f+g x)^2\right )^q\right ) \, dx=x\,\ln \left (c\,{\left (d+e\,{\left (f+g\,x\right )}^2\right )}^q\right )-2\,q\,x+\frac {f\,q\,\ln \left (e\,f^2+2\,e\,f\,g\,x+e\,g^2\,x^2+d\right )}{g}+\frac {2\,\sqrt {d}\,q\,\mathrm {atan}\left (\frac {\sqrt {e}\,f}{\sqrt {d}}+\frac {\sqrt {e}\,g\,x}{\sqrt {d}}\right )}{\sqrt {e}\,g} \] Input:
int(log(c*(d + e*(f + g*x)^2)^q),x)
Output:
x*log(c*(d + e*(f + g*x)^2)^q) - 2*q*x + (f*q*log(d + e*f^2 + e*g^2*x^2 + 2*e*f*g*x))/g + (2*d^(1/2)*q*atan((e^(1/2)*f)/d^(1/2) + (e^(1/2)*g*x)/d^(1 /2)))/(e^(1/2)*g)
Time = 0.16 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.56 \[ \int \log \left (c \left (d+e (f+g x)^2\right )^q\right ) \, dx=\frac {2 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e g x +e f}{\sqrt {e}\, \sqrt {d}}\right ) q +\mathrm {log}\left (\left (e \,g^{2} x^{2}+2 e f g x +e \,f^{2}+d \right )^{q} c \right ) e f +\mathrm {log}\left (\left (e \,g^{2} x^{2}+2 e f g x +e \,f^{2}+d \right )^{q} c \right ) e g x -2 e g q x}{e g} \] Input:
int(log(c*(d+e*(g*x+f)^2)^q),x)
Output:
(2*sqrt(e)*sqrt(d)*atan((e*f + e*g*x)/(sqrt(e)*sqrt(d)))*q + log((d + e*f* *2 + 2*e*f*g*x + e*g**2*x**2)**q*c)*e*f + log((d + e*f**2 + 2*e*f*g*x + e* g**2*x**2)**q*c)*e*g*x - 2*e*g*q*x)/(e*g)