Optimal. Leaf size=23 \[ \text {Int}\left (\frac {(a+b \tanh (e+f x))^2}{(c+d x)^2},x\right ) \]
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Rubi [A] time = 0.05, 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 = {} \[ \int \frac {(a+b \tanh (e+f x))^2}{(c+d x)^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(a+b \tanh (e+f x))^2}{(c+d x)^2} \, dx &=\int \frac {(a+b \tanh (e+f x))^2}{(c+d x)^2} \, dx\\ \end {align*}
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Mathematica [A] time = 27.17, size = 0, normalized size = 0.00 \[ \int \frac {(a+b \tanh (e+f x))^2}{(c+d x)^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \tanh \left (f x + e\right )^{2} + 2 \, a b \tanh \left (f x + e\right ) + a^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \tanh \left (f x + e\right ) + a\right )}^{2}}{{\left (d x + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.94, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \tanh \left (f x +e \right )\right )^{2}}{\left (d x +c \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {a^{2}}{d^{2} x + c d} - \frac {2 \, a b c f + {\left (c f - 2 \, d\right )} b^{2} + {\left (2 \, a b d f + b^{2} d f\right )} x + {\left (2 \, a b c f e^{\left (2 \, e\right )} + b^{2} c f e^{\left (2 \, e\right )} + {\left (2 \, a b d f e^{\left (2 \, e\right )} + b^{2} d f e^{\left (2 \, e\right )}\right )} x\right )} e^{\left (2 \, f x\right )}}{d^{3} f x^{2} + 2 \, c d^{2} f x + c^{2} d f + {\left (d^{3} f x^{2} e^{\left (2 \, e\right )} + 2 \, c d^{2} f x e^{\left (2 \, e\right )} + c^{2} d f e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}} - \int \frac {4 \, {\left (a b d f x + a b c f - b^{2} d\right )}}{d^{3} f x^{3} + 3 \, c d^{2} f x^{2} + 3 \, c^{2} d f x + c^{3} f + {\left (d^{3} f x^{3} e^{\left (2 \, e\right )} + 3 \, c d^{2} f x^{2} e^{\left (2 \, e\right )} + 3 \, c^{2} d f x e^{\left (2 \, e\right )} + c^{3} f e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\left (a+b\,\mathrm {tanh}\left (e+f\,x\right )\right )}^2}{{\left (c+d\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \tanh {\left (e + f x \right )}\right )^{2}}{\left (c + d x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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