3.11 \(\int \frac {(a+b \tan (c+d x^2))^2}{x} \, dx\)

Optimal. Leaf size=21 \[ \text {Int}\left (\frac {\left (a+b \tan \left (c+d x^2\right )\right )^2}{x},x\right ) \]

[Out]

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

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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 = {} \[ \int \frac {\left (a+b \tan \left (c+d x^2\right )\right )^2}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*Tan[c + d*x^2])^2/x,x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 8.16, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \tan \left (c+d x^2\right )\right )^2}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \tan \left (d x^{2} + c\right )^{2} + 2 \, a b \tan \left (d x^{2} + c\right ) + a^{2}}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.81, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \tan \left (d \,x^{2}+c \right )\right )^{2}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {{\left (a+b\,\mathrm {tan}\left (d\,x^2+c\right )\right )}^2}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*tan(c + d*x^2))^2/x,x)

[Out]

int((a + b*tan(c + d*x^2))^2/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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