\(\int \frac {1}{(c+d x) \sqrt [3]{2 c^3+d^3 x^3}} \, dx\) [139]

Optimal result

Integrand size = 25, antiderivative size = 186 \[ \int \frac {1}{(c+d x) \sqrt [3]{2 c^3+d^3 x^3}} \, dx=\frac {\arctan \left (\frac {1+\frac {2 d x}{\sqrt [3]{2 c^3+d^3 x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3} c d}-\frac {\sqrt {3} \arctan \left (\frac {1+\frac {2 (2 c+d x)}{\sqrt [3]{2 c^3+d^3 x^3}}}{\sqrt {3}}\right )}{2 c d}-\frac {\log (c+d x)}{2 c d}-\frac {\log \left (-d x+\sqrt [3]{2 c^3+d^3 x^3}\right )}{4 c d}+\frac {3 \log \left (d (2 c+d x)-d \sqrt [3]{2 c^3+d^3 x^3}\right )}{4 c d} \] Output:

1/6*arctan(1/3*(1+2*d*x/(d^3*x^3+2*c^3)^(1/3))*3^(1/2))*3^(1/2)/c/d-1/2*3^ 
(1/2)*arctan(1/3*(1+2*(d*x+2*c)/(d^3*x^3+2*c^3)^(1/3))*3^(1/2))/c/d-1/2*ln 
(d*x+c)/c/d-1/4*ln(-d*x+(d^3*x^3+2*c^3)^(1/3))/c/d+3/4*ln(d*(d*x+2*c)-d*(d 
^3*x^3+2*c^3)^(1/3))/c/d
                                                                                    
                                                                                    
 

Mathematica [F]

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

Integrate[1/((c + d*x)*(2*c^3 + d^3*x^3)^(1/3)),x]
 

Output:

Integrate[1/((c + d*x)*(2*c^3 + d^3*x^3)^(1/3)), x]
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2575, 769, 2576}

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.

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

\(\Big \downarrow \) 2575

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

\(\Big \downarrow \) 769

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

\(\Big \downarrow \) 2576

\(\displaystyle \frac {\frac {\arctan \left (\frac {\frac {2 d x}{\sqrt [3]{2 c^3+d^3 x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} d}-\frac {\log \left (\sqrt [3]{2 c^3+d^3 x^3}-d x\right )}{2 d}}{2 c}+\frac {-\frac {\sqrt {3} \arctan \left (\frac {\frac {2 (2 c+d x)}{\sqrt [3]{2 c^3+d^3 x^3}}+1}{\sqrt {3}}\right )}{d}+\frac {3 \log \left (d (2 c+d x)-d \sqrt [3]{2 c^3+d^3 x^3}\right )}{2 d}-\frac {\log (c+d x)}{d}}{2 c}\)

Input:

Int[1/((c + d*x)*(2*c^3 + d^3*x^3)^(1/3)),x]
 

Output:

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

Defintions of rubi rules used

rule 769

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]* 
(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^ 
3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]
 

rule 2575

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

rule 2576

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

Maple [F]

\[\int \frac {1}{\left (d x +c \right ) \left (d^{3} x^{3}+2 c^{3}\right )^{\frac {1}{3}}}d x\]

Input:

int(1/(d*x+c)/(d^3*x^3+2*c^3)^(1/3),x)
 

Output:

int(1/(d*x+c)/(d^3*x^3+2*c^3)^(1/3),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(c+d x) \sqrt [3]{2 c^3+d^3 x^3}} \, dx=\text {Timed out} \] Input:

integrate(1/(d*x+c)/(d^3*x^3+2*c^3)^(1/3),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(1/(d*x+c)/(d^3*x^3+2*c^3)^(1/3),x, algorithm="maxima")
 

Output:

integrate(1/((d^3*x^3 + 2*c^3)^(1/3)*(d*x + c)), x)
 

Giac [F]

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

integrate(1/(d*x+c)/(d^3*x^3+2*c^3)^(1/3),x, algorithm="giac")
 

Output:

integrate(1/((d^3*x^3 + 2*c^3)^(1/3)*(d*x + c)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(c+d x) \sqrt [3]{2 c^3+d^3 x^3}} \, dx=\int \frac {1}{{\left (2\,c^3+d^3\,x^3\right )}^{1/3}\,\left (c+d\,x\right )} \,d x \] Input:

int(1/((2*c^3 + d^3*x^3)^(1/3)*(c + d*x)),x)
 

Output:

int(1/((2*c^3 + d^3*x^3)^(1/3)*(c + d*x)), x)
 

Reduce [F]

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

int(1/(d*x+c)/(d^3*x^3+2*c^3)^(1/3),x)
 

Output:

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