3.56 \(\int \frac {(a+b \tan (c+d \sqrt [3]{x}))^2}{x^2} \, dx\)

Optimal. Leaf size=23 \[ \text {Int}\left (\frac {\left (a+b \tan \left (c+d \sqrt [3]{x}\right )\right )^2}{x^2},x\right ) \]

[Out]

Unintegrable((a+b*tan(c+d*x^(1/3)))^2/x^2,x)

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\left (a+b \tan \left (c+d \sqrt [3]{x}\right )\right )^2}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {\left (a+b \tan \left (c+d \sqrt [3]{x}\right )\right )^2}{x^2} \, dx &=\int \frac {\left (a+b \tan \left (c+d \sqrt [3]{x}\right )\right )^2}{x^2} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 8.80, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \tan \left (c+d \sqrt [3]{x}\right )\right )^2}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \tan \left (d x^{\frac {1}{3}} + c\right )^{2} + 2 \, a b \tan \left (d x^{\frac {1}{3}} + c\right ) + a^{2}}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^2*tan(d*x^(1/3) + c)^2 + 2*a*b*tan(d*x^(1/3) + c) + a^2)/x^2, x)

________________________________________________________________________________________

giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \tan \left (d x^{\frac {1}{3}} + c\right ) + a\right )}^{2}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 1.46, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \tan \left (c +d \,x^{\frac {1}{3}}\right )\right )^{2}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {4 \, {\left (d \cos \left (2 \, d x^{\frac {1}{3}} + 2 \, c\right )^{2} + d \sin \left (2 \, d x^{\frac {1}{3}} + 2 \, c\right )^{2} + 2 \, d \cos \left (2 \, d x^{\frac {1}{3}} + 2 \, c\right ) + d\right )} x^{2} \int \frac {a b d x \sin \left (2 \, d x^{\frac {1}{3}} + 2 \, c\right ) + 2 \, b^{2} x^{\frac {2}{3}} \sin \left (2 \, d x^{\frac {1}{3}} + 2 \, c\right )}{{\left (d \cos \left (2 \, d x^{\frac {1}{3}} + 2 \, c\right )^{2} + d \sin \left (2 \, d x^{\frac {1}{3}} + 2 \, c\right )^{2} + 2 \, d \cos \left (2 \, d x^{\frac {1}{3}} + 2 \, c\right ) + d\right )} x^{3}}\,{d x} + 6 \, b^{2} x^{\frac {2}{3}} \sin \left (2 \, d x^{\frac {1}{3}} + 2 \, c\right ) - {\left ({\left (a^{2} - b^{2}\right )} d \cos \left (2 \, d x^{\frac {1}{3}} + 2 \, c\right )^{2} + {\left (a^{2} - b^{2}\right )} d \sin \left (2 \, d x^{\frac {1}{3}} + 2 \, c\right )^{2} + 2 \, {\left (a^{2} - b^{2}\right )} d \cos \left (2 \, d x^{\frac {1}{3}} + 2 \, c\right ) + {\left (a^{2} - b^{2}\right )} d\right )} x}{{\left (d \cos \left (2 \, d x^{\frac {1}{3}} + 2 \, c\right )^{2} + d \sin \left (2 \, d x^{\frac {1}{3}} + 2 \, c\right )^{2} + 2 \, d \cos \left (2 \, d x^{\frac {1}{3}} + 2 \, c\right ) + d\right )} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d*cos(2*d*x^(1/3) + 2*c)^2 + d*sin(2*d*x^(1/3) + 2*c)^2 + 2*d*cos(2*d*x^(1/3) + 2*c) + d)*x^2*integrate(4*(a
*b*d*x*sin(2*d*x^(1/3) + 2*c) + 2*b^2*x^(2/3)*sin(2*d*x^(1/3) + 2*c))/((d*cos(2*d*x^(1/3) + 2*c)^2 + d*sin(2*d
*x^(1/3) + 2*c)^2 + 2*d*cos(2*d*x^(1/3) + 2*c) + d)*x^3), x) + 6*b^2*x^(2/3)*sin(2*d*x^(1/3) + 2*c) - ((a^2 -
b^2)*d*cos(2*d*x^(1/3) + 2*c)^2 + (a^2 - b^2)*d*sin(2*d*x^(1/3) + 2*c)^2 + 2*(a^2 - b^2)*d*cos(2*d*x^(1/3) + 2
*c) + (a^2 - b^2)*d)*x)/((d*cos(2*d*x^(1/3) + 2*c)^2 + d*sin(2*d*x^(1/3) + 2*c)^2 + 2*d*cos(2*d*x^(1/3) + 2*c)
+ d)*x^2)

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\left (a+b\,\mathrm {tan}\left (c+d\,x^{1/3}\right )\right )}^2}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \tan {\left (c + d \sqrt [3]{x} \right )}\right )^{2}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________