Optimal. Leaf size=23 \[ \text {Int}\left (\frac {1}{(c+d x)^2 (a+b \cosh (e+f x))^2},x\right ) \]
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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 = {} \[ \int \frac {1}{(c+d x)^2 (a+b \cosh (e+f x))^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{(c+d x)^2 (a+b \cosh (e+f x))^2} \, dx &=\int \frac {1}{(c+d x)^2 (a+b \cosh (e+f x))^2} \, dx\\ \end {align*}
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Mathematica [A] time = 54.96, size = 0, normalized size = 0.00 \[ \int \frac {1}{(c+d x)^2 (a+b \cosh (e+f x))^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{a^{2} d^{2} x^{2} + 2 \, a^{2} c d x + a^{2} c^{2} + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (a b d^{2} x^{2} + 2 \, a b c d x + a b c^{2}\right )} \cosh \left (f x + e\right )}, x\right ) \]
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 \[ \int \frac {1}{{\left (d x + c\right )}^{2} {\left (b \cosh \left (f x + e\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.78, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d x +c \right )^{2} \left (a +b \cosh \left (f x +e \right )\right )^{2}}\, dx \]
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 \[ \frac {2 \, {\left (a e^{\left (f x + e\right )} + b\right )}}{a^{2} b c^{2} f - b^{3} c^{2} f + {\left (a^{2} b d^{2} f - b^{3} d^{2} f\right )} x^{2} + 2 \, {\left (a^{2} b c d f - b^{3} c d f\right )} x + {\left (a^{2} b c^{2} f e^{\left (2 \, e\right )} - b^{3} c^{2} f e^{\left (2 \, e\right )} + {\left (a^{2} b d^{2} f e^{\left (2 \, e\right )} - b^{3} d^{2} f e^{\left (2 \, e\right )}\right )} x^{2} + 2 \, {\left (a^{2} b c d f e^{\left (2 \, e\right )} - b^{3} c d f e^{\left (2 \, e\right )}\right )} x\right )} e^{\left (2 \, f x\right )} + 2 \, {\left (a^{3} c^{2} f e^{e} - a b^{2} c^{2} f e^{e} + {\left (a^{3} d^{2} f e^{e} - a b^{2} d^{2} f e^{e}\right )} x^{2} + 2 \, {\left (a^{3} c d f e^{e} - a b^{2} c d f e^{e}\right )} x\right )} e^{\left (f x\right )}} + \int \frac {2 \, {\left (2 \, b d + {\left (a d f x e^{e} + {\left (c f e^{e} + 2 \, d e^{e}\right )} a\right )} e^{\left (f x\right )}\right )}}{a^{2} b c^{3} f - b^{3} c^{3} f + {\left (a^{2} b d^{3} f - b^{3} d^{3} f\right )} x^{3} + 3 \, {\left (a^{2} b c d^{2} f - b^{3} c d^{2} f\right )} x^{2} + 3 \, {\left (a^{2} b c^{2} d f - b^{3} c^{2} d f\right )} x + {\left (a^{2} b c^{3} f e^{\left (2 \, e\right )} - b^{3} c^{3} f e^{\left (2 \, e\right )} + {\left (a^{2} b d^{3} f e^{\left (2 \, e\right )} - b^{3} d^{3} f e^{\left (2 \, e\right )}\right )} x^{3} + 3 \, {\left (a^{2} b c d^{2} f e^{\left (2 \, e\right )} - b^{3} c d^{2} f e^{\left (2 \, e\right )}\right )} x^{2} + 3 \, {\left (a^{2} b c^{2} d f e^{\left (2 \, e\right )} - b^{3} c^{2} d f e^{\left (2 \, e\right )}\right )} x\right )} e^{\left (2 \, f x\right )} + 2 \, {\left (a^{3} c^{3} f e^{e} - a b^{2} c^{3} f e^{e} + {\left (a^{3} d^{3} f e^{e} - a b^{2} d^{3} f e^{e}\right )} x^{3} + 3 \, {\left (a^{3} c d^{2} f e^{e} - a b^{2} c d^{2} f e^{e}\right )} x^{2} + 3 \, {\left (a^{3} c^{2} d f e^{e} - a b^{2} c^{2} d f e^{e}\right )} x\right )} e^{\left (f x\right )}}\,{d x} \]
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 \[ \int \frac {1}{{\left (a+b\,\mathrm {cosh}\left (e+f\,x\right )\right )}^2\,{\left (c+d\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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