3.633 \(\int \log (c (d+\frac {e}{(f+g x)^2})^q) \, dx\)

Optimal. Leaf size=59 \[ \frac {(f+g x) \log \left (c \left (d+\frac {e}{(f+g x)^2}\right )^q\right )}{g}+\frac {2 \sqrt {e} q \tan ^{-1}\left (\frac {\sqrt {d} (f+g x)}{\sqrt {e}}\right )}{\sqrt {d} g} \]

[Out]

(g*x+f)*ln(c*(d+e/(g*x+f)^2)^q)/g+2*q*arctan((g*x+f)*d^(1/2)/e^(1/2))*e^(1/2)/g/d^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2483, 2448, 263, 205} \[ \frac {(f+g x) \log \left (c \left (d+\frac {e}{(f+g x)^2}\right )^q\right )}{g}+\frac {2 \sqrt {e} q \tan ^{-1}\left (\frac {\sqrt {d} (f+g x)}{\sqrt {e}}\right )}{\sqrt {d} g} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(d + e/(f + g*x)^2)^q],x]

[Out]

(2*Sqrt[e]*q*ArcTan[(Sqrt[d]*(f + g*x))/Sqrt[e]])/(Sqrt[d]*g) + ((f + g*x)*Log[c*(d + e/(f + g*x)^2)^q])/g

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 2448

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

Rule 2483

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*((f_.) + (g_.)*(x_))^(n_))^(p_.)]*(b_.))^(q_.), x_Symbol] :> Dist[1/g, Su
bst[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])

Rubi steps

\begin {align*} \int \log \left (c \left (d+\frac {e}{(f+g x)^2}\right )^q\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \log \left (c \left (d+\frac {e}{x^2}\right )^q\right ) \, dx,x,f+g x\right )}{g}\\ &=\frac {(f+g x) \log \left (c \left (d+\frac {e}{(f+g x)^2}\right )^q\right )}{g}+\frac {(2 e q) \operatorname {Subst}\left (\int \frac {1}{\left (d+\frac {e}{x^2}\right ) x^2} \, dx,x,f+g x\right )}{g}\\ &=\frac {(f+g x) \log \left (c \left (d+\frac {e}{(f+g x)^2}\right )^q\right )}{g}+\frac {(2 e q) \operatorname {Subst}\left (\int \frac {1}{e+d x^2} \, dx,x,f+g x\right )}{g}\\ &=\frac {2 \sqrt {e} q \tan ^{-1}\left (\frac {\sqrt {d} (f+g x)}{\sqrt {e}}\right )}{\sqrt {d} g}+\frac {(f+g x) \log \left (c \left (d+\frac {e}{(f+g x)^2}\right )^q\right )}{g}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 61, normalized size = 1.03 \[ \frac {(f+g x) \log \left (c \left (d+\frac {e}{(f+g x)^2}\right )^q\right )}{g}-\frac {2 \sqrt {e} q \tan ^{-1}\left (\frac {\sqrt {e}}{\sqrt {d} (f+g x)}\right )}{\sqrt {d} g} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(d + e/(f + g*x)^2)^q],x]

[Out]

(-2*Sqrt[e]*q*ArcTan[Sqrt[e]/(Sqrt[d]*(f + g*x))])/(Sqrt[d]*g) + ((f + g*x)*Log[c*(d + e/(f + g*x)^2)^q])/g

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fricas [B]  time = 0.85, size = 287, normalized size = 4.86 \[ \left [\frac {g q x \log \left (\frac {d g^{2} x^{2} + 2 \, d f g x + d f^{2} + e}{g^{2} x^{2} + 2 \, f g x + f^{2}}\right ) + f q \log \left (d g^{2} x^{2} + 2 \, d f g x + d f^{2} + e\right ) - 2 \, f q \log \left (g x + f\right ) + g x \log \relax (c) + q \sqrt {-\frac {e}{d}} \log \left (\frac {d g^{2} x^{2} + 2 \, d f g x + d f^{2} + 2 \, {\left (d g x + d f\right )} \sqrt {-\frac {e}{d}} - e}{d g^{2} x^{2} + 2 \, d f g x + d f^{2} + e}\right )}{g}, \frac {g q x \log \left (\frac {d g^{2} x^{2} + 2 \, d f g x + d f^{2} + e}{g^{2} x^{2} + 2 \, f g x + f^{2}}\right ) + f q \log \left (d g^{2} x^{2} + 2 \, d f g x + d f^{2} + e\right ) - 2 \, f q \log \left (g x + f\right ) + g x \log \relax (c) + 2 \, q \sqrt {\frac {e}{d}} \arctan \left (\frac {{\left (d g x + d f\right )} \sqrt {\frac {e}{d}}}{e}\right )}{g}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(d+e/(g*x+f)^2)^q),x, algorithm="fricas")

[Out]

[(g*q*x*log((d*g^2*x^2 + 2*d*f*g*x + d*f^2 + e)/(g^2*x^2 + 2*f*g*x + f^2)) + f*q*log(d*g^2*x^2 + 2*d*f*g*x + d
*f^2 + e) - 2*f*q*log(g*x + f) + g*x*log(c) + q*sqrt(-e/d)*log((d*g^2*x^2 + 2*d*f*g*x + d*f^2 + 2*(d*g*x + d*f
)*sqrt(-e/d) - e)/(d*g^2*x^2 + 2*d*f*g*x + d*f^2 + e)))/g, (g*q*x*log((d*g^2*x^2 + 2*d*f*g*x + d*f^2 + e)/(g^2
*x^2 + 2*f*g*x + f^2)) + f*q*log(d*g^2*x^2 + 2*d*f*g*x + d*f^2 + e) - 2*f*q*log(g*x + f) + g*x*log(c) + 2*q*sq
rt(e/d)*arctan((d*g*x + d*f)*sqrt(e/d)/e))/g]

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giac [B]  time = 0.47, size = 137, normalized size = 2.32 \[ d g^{4} q {\left (\frac {f e^{\left (-1\right )} \log \left (d g^{2} x^{2} + 2 \, d f g x + d f^{2} + e\right )}{d g^{5}} - \frac {2 \, f e^{\left (-1\right )} \log \left ({\left | g x + f \right |}\right )}{d g^{5}} + \frac {2 \, \arctan \left (\frac {{\left (d g x + d f\right )} e^{\left (-\frac {1}{2}\right )}}{\sqrt {d}}\right ) e^{\left (-\frac {1}{2}\right )}}{d^{\frac {3}{2}} g^{5}}\right )} e + q x \log \left (d g^{2} x^{2} + 2 \, d f g x + d f^{2} + e\right ) - q x \log \left (g^{2} x^{2} + 2 \, f g x + f^{2}\right ) + x \log \relax (c) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(d+e/(g*x+f)^2)^q),x, algorithm="giac")

[Out]

d*g^4*q*(f*e^(-1)*log(d*g^2*x^2 + 2*d*f*g*x + d*f^2 + e)/(d*g^5) - 2*f*e^(-1)*log(abs(g*x + f))/(d*g^5) + 2*ar
ctan((d*g*x + d*f)*e^(-1/2)/sqrt(d))*e^(-1/2)/(d^(3/2)*g^5))*e + q*x*log(d*g^2*x^2 + 2*d*f*g*x + d*f^2 + e) -
q*x*log(g^2*x^2 + 2*f*g*x + f^2) + x*log(c)

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maple [B]  time = 0.13, size = 115, normalized size = 1.95 \[ \frac {2 e q \arctan \left (\frac {2 d \,g^{2} x +2 d f g}{2 \sqrt {d e}\, g}\right )}{\sqrt {d e}\, g}-\frac {2 f q \ln \left (g x +f \right )}{g}+\frac {f q \ln \left (d \,g^{2} x^{2}+2 d f g x +d \,f^{2}+e \right )}{g}+x \ln \left (c \left (\frac {d \,g^{2} x^{2}+2 d f g x +d \,f^{2}+e}{\left (g x +f \right )^{2}}\right )^{q}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(d+e/(g*x+f)^2)^q),x)

[Out]

x*ln(c*((d*g^2*x^2+2*d*f*g*x+d*f^2+e)/(g*x+f)^2)^q)+1/g*q*f*ln(d*g^2*x^2+2*d*f*g*x+d*f^2+e)+2/g*e*q/(d*e)^(1/2
)*arctan(1/2*(2*d*g^2*x+2*d*f*g)/g/(d*e)^(1/2))-2*f/g*q*ln(g*x+f)

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maxima [A]  time = 1.64, size = 100, normalized size = 1.69 \[ e g q {\left (\frac {f \log \left (d g^{2} x^{2} + 2 \, d f g x + d f^{2} + e\right )}{e g^{2}} - \frac {2 \, f \log \left (g x + f\right )}{e g^{2}} + \frac {2 \, \arctan \left (\frac {d g^{2} x + d f g}{\sqrt {d e} g}\right )}{\sqrt {d e} g^{2}}\right )} + x \log \left (c {\left (d + \frac {e}{{\left (g x + f\right )}^{2}}\right )}^{q}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(d+e/(g*x+f)^2)^q),x, algorithm="maxima")

[Out]

e*g*q*(f*log(d*g^2*x^2 + 2*d*f*g*x + d*f^2 + e)/(e*g^2) - 2*f*log(g*x + f)/(e*g^2) + 2*arctan((d*g^2*x + d*f*g
)/(sqrt(d*e)*g))/(sqrt(d*e)*g^2)) + x*log(c*(d + e/(g*x + f)^2)^q)

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mupad [B]  time = 0.52, size = 163, normalized size = 2.76 \[ x\,\ln \left (c\,{\left (d+\frac {e}{{\left (f+g\,x\right )}^2}\right )}^q\right )-\frac {2\,f\,q\,\ln \left (f+g\,x\right )}{g}+\frac {\ln \left (e\,\sqrt {-d\,e}-3\,d\,f^2\,\sqrt {-d\,e}+4\,d\,e\,f+d\,e\,g\,x-3\,d\,f\,g\,x\,\sqrt {-d\,e}\right )\,\left (q\,\sqrt {-d\,e}+d\,f\,q\right )}{d\,g}-\frac {\ln \left (3\,d\,f^2\,\sqrt {-d\,e}-e\,\sqrt {-d\,e}+4\,d\,e\,f+d\,e\,g\,x+3\,d\,f\,g\,x\,\sqrt {-d\,e}\right )\,\left (q\,\sqrt {-d\,e}-d\,f\,q\right )}{d\,g} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*(d + e/(f + g*x)^2)^q),x)

[Out]

x*log(c*(d + e/(f + g*x)^2)^q) - (2*f*q*log(f + g*x))/g + (log(e*(-d*e)^(1/2) - 3*d*f^2*(-d*e)^(1/2) + 4*d*e*f
+ d*e*g*x - 3*d*f*g*x*(-d*e)^(1/2))*(q*(-d*e)^(1/2) + d*f*q))/(d*g) - (log(3*d*f^2*(-d*e)^(1/2) - e*(-d*e)^(1
/2) + 4*d*e*f + d*e*g*x + 3*d*f*g*x*(-d*e)^(1/2))*(q*(-d*e)^(1/2) - d*f*q))/(d*g)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*(d+e/(g*x+f)**2)**q),x)

[Out]

Timed out

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