3.1.26 \(\int \frac {(a+b \text {ArcTan}(c x^2))^2}{d+e x} \, dx\) [26]

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

[Out]

Unintegrable((a+b*arctan(c*x^2))^2/(e*x+d),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 = {} \begin {gather*} \int \frac {\left (a+b \text {ArcTan}\left (c x^2\right )\right )^2}{d+e x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {\left (a+b \tan ^{-1}\left (c x^2\right )\right )^2}{d+e x} \, dx &=\int \left (\frac {a^2}{d+e x}+\frac {2 a b \tan ^{-1}\left (c x^2\right )}{d+e x}+\frac {b^2 \tan ^{-1}\left (c x^2\right )^2}{d+e x}\right ) \, dx\\ &=\frac {a^2 \log (d+e x)}{e}+(2 a b) \int \frac {\tan ^{-1}\left (c x^2\right )}{d+e x} \, dx+b^2 \int \frac {\tan ^{-1}\left (c x^2\right )^2}{d+e x} \, dx\\ \end {align*}

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Mathematica [A]
time = 47.02, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b \text {ArcTan}\left (c x^2\right )\right )^2}{d+e x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arctan \left (c \,x^{2}\right )\right )^{2}}{e x +d}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*e^(-1)*log(x*e + d) + integrate((b^2*arctan(c*x^2)^2 + 2*a*b*arctan(c*x^2))/(x*e + d), x)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x^2\right )\right )}^2}{d+e\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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