Optimal. Leaf size=23 \[ \text {Int}\left (\frac {(a \sec (e+f x)+a)^2}{c+d x},x\right ) \]
[Out]
________________________________________________________________________________________
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 = {} \[ \int \frac {(a+a \sec (e+f x))^2}{c+d x} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {(a+a \sec (e+f x))^2}{c+d x} \, dx &=\int \frac {(a+a \sec (e+f x))^2}{c+d x} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 0.00, size = 0, normalized size = 0.00 \[ \text {\$Aborted} \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{2} \sec \left (f x + e\right )^{2} + 2 \, a^{2} \sec \left (f x + e\right ) + a^{2}}{d x + c}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a \sec \left (f x + e\right ) + a\right )}^{2}}{d x + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 4.23, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +a \sec \left (f x +e \right )\right )^{2}}{d x +c}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (a^{2} d f x + a^{2} c f\right )} \cos \left (2 \, f x + 2 \, e\right )^{2} \log \left (d x + c\right ) + 2 \, a^{2} d \sin \left (2 \, f x + 2 \, e\right ) + {\left (a^{2} d f x + a^{2} c f\right )} \log \left (d x + c\right ) \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \, {\left (a^{2} d f x + a^{2} c f\right )} \cos \left (2 \, f x + 2 \, e\right ) \log \left (d x + c\right ) + 2 \, {\left (d^{2} f x + c d f + {\left (d^{2} f x + c d f\right )} \cos \left (2 \, f x + 2 \, e\right )^{2} + {\left (d^{2} f x + c d f\right )} \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \, {\left (d^{2} f x + c d f\right )} \cos \left (2 \, f x + 2 \, e\right )\right )} \int \frac {2 \, {\left (a^{2} d f x + a^{2} c f\right )} \cos \left (2 \, f x + 2 \, e\right ) \cos \left (f x + e\right ) + 2 \, {\left (a^{2} d f x + a^{2} c f\right )} \cos \left (f x + e\right ) + {\left (a^{2} d + 2 \, {\left (a^{2} d f x + a^{2} c f\right )} \sin \left (f x + e\right )\right )} \sin \left (2 \, f x + 2 \, e\right )}{d^{2} f x^{2} + 2 \, c d f x + c^{2} f + {\left (d^{2} f x^{2} + 2 \, c d f x + c^{2} f\right )} \cos \left (2 \, f x + 2 \, e\right )^{2} + {\left (d^{2} f x^{2} + 2 \, c d f x + c^{2} f\right )} \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \, {\left (d^{2} f x^{2} + 2 \, c d f x + c^{2} f\right )} \cos \left (2 \, f x + 2 \, e\right )}\,{d x} + {\left (a^{2} d f x + a^{2} c f\right )} \log \left (d x + c\right )}{d^{2} f x + c d f + {\left (d^{2} f x + c d f\right )} \cos \left (2 \, f x + 2 \, e\right )^{2} + {\left (d^{2} f x + c d f\right )} \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \, {\left (d^{2} f x + c d f\right )} \cos \left (2 \, f x + 2 \, e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^2}{c+d\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ a^{2} \left (\int \frac {2 \sec {\left (e + f x \right )}}{c + d x}\, dx + \int \frac {\sec ^{2}{\left (e + f x \right )}}{c + d x}\, dx + \int \frac {1}{c + d x}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________