\(\int \frac {1}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx\) [18]

Optimal result

Integrand size = 25, antiderivative size = 139 \[ \int \frac {1}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {1-\frac {\sqrt [3]{2} (c-d x)}{\sqrt [3]{-c^3+d^3 x^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} c d}+\frac {\log \left ((c-d x) (c+d x)^2\right )}{4 \sqrt [3]{2} c d}-\frac {3 \log \left (d (c-d x)+2^{2/3} d \sqrt [3]{-c^3+d^3 x^3}\right )}{4 \sqrt [3]{2} c d} \]

[Out]

1/8*ln((-d*x+c)*(d*x+c)^2)*2^(2/3)/c/d-3/8*ln(d*(-d*x+c)+2^(2/3)*d*(d^3*x^3-c^3)^(1/3))*2^(2/3)/c/d+1/4*arctan
(1/3*(1-2^(1/3)*(-d*x+c)/(d^3*x^3-c^3)^(1/3))*3^(1/2))*3^(1/2)*2^(2/3)/c/d

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2174} \[ \int \frac {1}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {1-\frac {\sqrt [3]{2} (c-d x)}{\sqrt [3]{d^3 x^3-c^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} c d}-\frac {3 \log \left (2^{2/3} d \sqrt [3]{d^3 x^3-c^3}+d (c-d x)\right )}{4 \sqrt [3]{2} c d}+\frac {\log \left ((c-d x) (c+d x)^2\right )}{4 \sqrt [3]{2} c d} \]

[In]

Int[1/((c + d*x)*(-c^3 + d^3*x^3)^(1/3)),x]

[Out]

(Sqrt[3]*ArcTan[(1 - (2^(1/3)*(c - d*x))/(-c^3 + d^3*x^3)^(1/3))/Sqrt[3]])/(2*2^(1/3)*c*d) + Log[(c - d*x)*(c
+ d*x)^2]/(4*2^(1/3)*c*d) - (3*Log[d*(c - d*x) + 2^(2/3)*d*(-c^3 + d^3*x^3)^(1/3)])/(4*2^(1/3)*c*d)

Rule 2174

Int[1/(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^3)^(1/3)), x_Symbol] :> Simp[Sqrt[3]*(ArcTan[(1 - 2^(1/3)*Rt[b,
3]*((c - d*x)/(d*(a + b*x^3)^(1/3))))/Sqrt[3]]/(2^(4/3)*Rt[b, 3]*c)), x] + (Simp[Log[(c + d*x)^2*(c - d*x)]/(2
^(7/3)*Rt[b, 3]*c), x] - Simp[(3*Log[Rt[b, 3]*(c - d*x) + 2^(2/3)*d*(a + b*x^3)^(1/3)])/(2^(7/3)*Rt[b, 3]*c),
x]) /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^3 + a*d^3, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {3} \tan ^{-1}\left (\frac {1-\frac {\sqrt [3]{2} (c-d x)}{\sqrt [3]{-c^3+d^3 x^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} c d}+\frac {\log \left ((c-d x) (c+d x)^2\right )}{4 \sqrt [3]{2} c d}-\frac {3 \log \left (d (c-d x)+2^{2/3} d \sqrt [3]{-c^3+d^3 x^3}\right )}{4 \sqrt [3]{2} c d} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.07 (sec) , antiderivative size = 311, normalized size of antiderivative = 2.24 \[ \int \frac {1}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx=\frac {\sqrt [3]{-\frac {1}{2}} \left (2 i \sqrt {3} \text {arctanh}\left (\frac {\sqrt [3]{2} \left (3+i \sqrt {3}\right ) c+\sqrt [3]{2} \left (-3-i \sqrt {3}\right ) d x+2 i \sqrt {3} \sqrt [3]{-c^3+d^3 x^3}}{6 \sqrt [3]{-c^3+d^3 x^3}}\right )+2 \log \left (\sqrt {c} \sqrt {d} \left (-c+i \sqrt {3} c+d x-i \sqrt {3} d x+2\ 2^{2/3} \sqrt [3]{-c^3+d^3 x^3}\right )\right )-\log \left (-c d \left (\left (1+i \sqrt {3}\right ) c^2+\left (1+i \sqrt {3}\right ) d^2 x^2-2 (-2)^{2/3} d x \sqrt [3]{-c^3+d^3 x^3}-4 \sqrt [3]{2} \left (-c^3+d^3 x^3\right )^{2/3}+2 c \left (\left (-1-i \sqrt {3}\right ) d x+(-2)^{2/3} \sqrt [3]{-c^3+d^3 x^3}\right )\right )\right )\right )}{4 c d} \]

[In]

Integrate[1/((c + d*x)*(-c^3 + d^3*x^3)^(1/3)),x]

[Out]

((-1/2)^(1/3)*((2*I)*Sqrt[3]*ArcTanh[(2^(1/3)*(3 + I*Sqrt[3])*c + 2^(1/3)*(-3 - I*Sqrt[3])*d*x + (2*I)*Sqrt[3]
*(-c^3 + d^3*x^3)^(1/3))/(6*(-c^3 + d^3*x^3)^(1/3))] + 2*Log[Sqrt[c]*Sqrt[d]*(-c + I*Sqrt[3]*c + d*x - I*Sqrt[
3]*d*x + 2*2^(2/3)*(-c^3 + d^3*x^3)^(1/3))] - Log[-(c*d*((1 + I*Sqrt[3])*c^2 + (1 + I*Sqrt[3])*d^2*x^2 - 2*(-2
)^(2/3)*d*x*(-c^3 + d^3*x^3)^(1/3) - 4*2^(1/3)*(-c^3 + d^3*x^3)^(2/3) + 2*c*((-1 - I*Sqrt[3])*d*x + (-2)^(2/3)
*(-c^3 + d^3*x^3)^(1/3))))]))/(4*c*d)

Maple [F]

\[\int \frac {1}{\left (d x +c \right ) \left (d^{3} x^{3}-c^{3}\right )^{\frac {1}{3}}}d x\]

[In]

int(1/(d*x+c)/(d^3*x^3-c^3)^(1/3),x)

[Out]

int(1/(d*x+c)/(d^3*x^3-c^3)^(1/3),x)

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx=\text {Timed out} \]

[In]

integrate(1/(d*x+c)/(d^3*x^3-c^3)^(1/3),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {1}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx=\int \frac {1}{\sqrt [3]{\left (- c + d x\right ) \left (c^{2} + c d x + d^{2} x^{2}\right )} \left (c + d x\right )}\, dx \]

[In]

integrate(1/(d*x+c)/(d**3*x**3-c**3)**(1/3),x)

[Out]

Integral(1/(((-c + d*x)*(c**2 + c*d*x + d**2*x**2))**(1/3)*(c + d*x)), x)

Maxima [F]

\[ \int \frac {1}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx=\int { \frac {1}{{\left (d^{3} x^{3} - c^{3}\right )}^{\frac {1}{3}} {\left (d x + c\right )}} \,d x } \]

[In]

integrate(1/(d*x+c)/(d^3*x^3-c^3)^(1/3),x, algorithm="maxima")

[Out]

integrate(1/((d^3*x^3 - c^3)^(1/3)*(d*x + c)), x)

Giac [F]

\[ \int \frac {1}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx=\int { \frac {1}{{\left (d^{3} x^{3} - c^{3}\right )}^{\frac {1}{3}} {\left (d x + c\right )}} \,d x } \]

[In]

integrate(1/(d*x+c)/(d^3*x^3-c^3)^(1/3),x, algorithm="giac")

[Out]

integrate(1/((d^3*x^3 - c^3)^(1/3)*(d*x + c)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx=\int \frac {1}{{\left (d^3\,x^3-c^3\right )}^{1/3}\,\left (c+d\,x\right )} \,d x \]

[In]

int(1/((d^3*x^3 - c^3)^(1/3)*(c + d*x)),x)

[Out]

int(1/((d^3*x^3 - c^3)^(1/3)*(c + d*x)), x)