3.1.31 \(\int \frac {(a+b \tanh ^{-1}(c x^2))^2}{(d+e x)^2} \, dx\) [31]

Optimal. Leaf size=23 \[ \text {Int}\left (\frac {\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{(d+e x)^2},x\right ) \]

[Out]

Unintegrable((a+b*arctanh(c*x^2))^2/(e*x+d)^2,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 \tanh ^{-1}\left (c x^2\right )\right )^2}{(d+e x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{(d+e x)^2} \, dx &=\int \left (\frac {a^2}{(d+e x)^2}+\frac {2 a b \tanh ^{-1}\left (c x^2\right )}{(d+e x)^2}+\frac {b^2 \tanh ^{-1}\left (c x^2\right )^2}{(d+e x)^2}\right ) \, dx\\ &=-\frac {a^2}{e (d+e x)}+(2 a b) \int \frac {\tanh ^{-1}\left (c x^2\right )}{(d+e x)^2} \, dx+b^2 \int \frac {\tanh ^{-1}\left (c x^2\right )^2}{(d+e x)^2} \, dx\\ &=-\frac {a^2}{e (d+e x)}-\frac {2 a b \tanh ^{-1}\left (c x^2\right )}{e (d+e x)}+b^2 \int \frac {\tanh ^{-1}\left (c x^2\right )^2}{(d+e x)^2} \, dx+\frac {(2 a b) \int \frac {2 c x}{(d+e x) \left (1-c^2 x^4\right )} \, dx}{e}\\ &=-\frac {a^2}{e (d+e x)}-\frac {2 a b \tanh ^{-1}\left (c x^2\right )}{e (d+e x)}+b^2 \int \frac {\tanh ^{-1}\left (c x^2\right )^2}{(d+e x)^2} \, dx+\frac {(4 a b c) \int \frac {x}{(d+e x) \left (1-c^2 x^4\right )} \, dx}{e}\\ &=-\frac {a^2}{e (d+e x)}-\frac {2 a b \tanh ^{-1}\left (c x^2\right )}{e (d+e x)}+b^2 \int \frac {\tanh ^{-1}\left (c x^2\right )^2}{(d+e x)^2} \, dx+\frac {(4 a b c) \int \left (-\frac {d e^3}{\left (-c^2 d^4+e^4\right ) (d+e x)}+\frac {e-c d x}{2 \left (c d^2-e^2\right ) \left (-1+c x^2\right )}+\frac {e+c d x}{2 \left (c d^2+e^2\right ) \left (1+c x^2\right )}\right ) \, dx}{e}\\ &=-\frac {a^2}{e (d+e x)}-\frac {2 a b \tanh ^{-1}\left (c x^2\right )}{e (d+e x)}+\frac {4 a b c d e \log (d+e x)}{c^2 d^4-e^4}+b^2 \int \frac {\tanh ^{-1}\left (c x^2\right )^2}{(d+e x)^2} \, dx+\frac {(2 a b c) \int \frac {e-c d x}{-1+c x^2} \, dx}{e \left (c d^2-e^2\right )}+\frac {(2 a b c) \int \frac {e+c d x}{1+c x^2} \, dx}{e \left (c d^2+e^2\right )}\\ &=-\frac {a^2}{e (d+e x)}-\frac {2 a b \tanh ^{-1}\left (c x^2\right )}{e (d+e x)}+\frac {4 a b c d e \log (d+e x)}{c^2 d^4-e^4}+b^2 \int \frac {\tanh ^{-1}\left (c x^2\right )^2}{(d+e x)^2} \, dx+\frac {(2 a b c) \int \frac {1}{-1+c x^2} \, dx}{c d^2-e^2}-\frac {\left (2 a b c^2 d\right ) \int \frac {x}{-1+c x^2} \, dx}{e \left (c d^2-e^2\right )}+\frac {(2 a b c) \int \frac {1}{1+c x^2} \, dx}{c d^2+e^2}+\frac {\left (2 a b c^2 d\right ) \int \frac {x}{1+c x^2} \, dx}{e \left (c d^2+e^2\right )}\\ &=-\frac {a^2}{e (d+e x)}+\frac {2 a b \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right )}{c d^2+e^2}-\frac {2 a b \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right )}{c d^2-e^2}-\frac {2 a b \tanh ^{-1}\left (c x^2\right )}{e (d+e x)}+\frac {4 a b c d e \log (d+e x)}{c^2 d^4-e^4}-\frac {a b c d \log \left (1-c x^2\right )}{e \left (c d^2-e^2\right )}+\frac {a b c d \log \left (1+c x^2\right )}{e \left (c d^2+e^2\right )}+b^2 \int \frac {\tanh ^{-1}\left (c x^2\right )^2}{(d+e x)^2} \, dx\\ \end {align*}

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Mathematica [A]
time = 25.99, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{(d+e x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.10, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arctanh \left (c \,x^{2}\right )\right )^{2}}{\left (e x +d \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

((4*d*e*log(x*e + d)/(c^2*d^4 - e^4) + d*log(c*x^2 + 1)/(c*d^2*e + e^3) - d*log(c*x^2 - 1)/(c*d^2*e - e^3) + 2
*arctan(sqrt(c)*x)/((c*d^2 + e^2)*sqrt(c)) + log((c*x - sqrt(c))/(c*x + sqrt(c)))/((c*d^2 - e^2)*sqrt(c)))*c -
2*arctanh(c*x^2)/(x*e^2 + d*e))*a*b - 1/4*b^2*(log(-c*x^2 + 1)^2/(x*e^2 + d*e) + integrate(-((c*x^2*e - e)*lo
g(c*x^2 + 1)^2 + 2*(2*c*x^2*e + 2*c*d*x - (c*x^2*e - e)*log(c*x^2 + 1))*log(-c*x^2 + 1))/(c*x^4*e^3 + 2*c*d*x^
3*e^2 + (c*d^2*e - e^3)*x^2 - 2*d*x*e^2 - d^2*e), x)) - a^2/(x*e^2 + d*e)

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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*arctanh(c*x^2))^2/(e*x+d)^2,x, algorithm="fricas")

[Out]

integral((b^2*arctanh(c*x^2)^2 + 2*a*b*arctanh(c*x^2) + a^2)/(x^2*e^2 + 2*d*x*e + d^2), 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*atanh(c*x**2))**2/(e*x+d)**2,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*arctanh(c*x^2))^2/(e*x+d)^2,x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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