Optimal. Leaf size=32 \[ -\frac {\cot (e+f x) \log (\cos (e+f x)) \sqrt {b \tan ^2(e+f x)}}{f} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3739, 3556}
\begin {gather*} -\frac {\cot (e+f x) \sqrt {b \tan ^2(e+f x)} \log (\cos (e+f x))}{f} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3556
Rule 3739
Rubi steps
\begin {align*} \int \sqrt {b \tan ^2(e+f x)} \, dx &=\left (\cot (e+f x) \sqrt {b \tan ^2(e+f x)}\right ) \int \tan (e+f x) \, dx\\ &=-\frac {\cot (e+f x) \log (\cos (e+f x)) \sqrt {b \tan ^2(e+f x)}}{f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 32, normalized size = 1.00 \begin {gather*} -\frac {\cot (e+f x) \log (\cos (e+f x)) \sqrt {b \tan ^2(e+f x)}}{f} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.04, size = 37, normalized size = 1.16
method | result | size |
derivativedivides | \(\frac {\sqrt {b \left (\tan ^{2}\left (f x +e \right )\right )}\, \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f \tan \left (f x +e \right )}\) | \(37\) |
default | \(\frac {\sqrt {b \left (\tan ^{2}\left (f x +e \right )\right )}\, \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f \tan \left (f x +e \right )}\) | \(37\) |
risch | \(\frac {\sqrt {-\frac {b \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{2}}{\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2}}}\, \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) x}{{\mathrm e}^{2 i \left (f x +e \right )}-1}-\frac {2 \sqrt {-\frac {b \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{2}}{\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2}}}\, \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) \left (f x +e \right )}{\left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right ) f}-\frac {i \sqrt {-\frac {b \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{2}}{\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2}}}\, \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}{\left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right ) f}\) | \(197\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.51, size = 20, normalized size = 0.62 \begin {gather*} \frac {\sqrt {b} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{2 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 3.50, size = 41, normalized size = 1.28 \begin {gather*} -\frac {\sqrt {b \tan \left (f x + e\right )^{2}} \log \left (\frac {1}{\tan \left (f x + e\right )^{2} + 1}\right )}{2 \, f \tan \left (f x + e\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {b \tan ^{2}{\left (e + f x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.48, size = 25, normalized size = 0.78 \begin {gather*} -\frac {\sqrt {b} \log \left ({\left | \cos \left (f x + e\right ) \right |}\right ) \mathrm {sgn}\left (\tan \left (f x + e\right )\right )}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________