Optimal. Leaf size=58 \[ \frac {2 b \sqrt {d \sec (e+f x)}}{f}+\frac {2 a \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {d \sec (e+f x)}}{f} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3567, 3856,
2720} \begin {gather*} \frac {2 a \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {d \sec (e+f x)}}{f}+\frac {2 b \sqrt {d \sec (e+f x)}}{f} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2720
Rule 3567
Rule 3856
Rubi steps
\begin {align*} \int \sqrt {d \sec (e+f x)} (a+b \tan (e+f x)) \, dx &=\frac {2 b \sqrt {d \sec (e+f x)}}{f}+a \int \sqrt {d \sec (e+f x)} \, dx\\ &=\frac {2 b \sqrt {d \sec (e+f x)}}{f}+\left (a \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)}} \, dx\\ &=\frac {2 b \sqrt {d \sec (e+f x)}}{f}+\frac {2 a \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {d \sec (e+f x)}}{f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.23, size = 42, normalized size = 0.72 \begin {gather*} \frac {2 \left (b+a \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right )\right ) \sqrt {d \sec (e+f x)}}{f} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains complex when optimal does not.
time = 0.51, size = 168, normalized size = 2.90
method | result | size |
default | \(\frac {2 \sqrt {\frac {d}{\cos \left (f x +e \right )}}\, \left (\cos \left (f x +e \right )+1\right )^{2} \left (\cos \left (f x +e \right )-1\right )^{2} \left (i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (\cos \left (f x +e \right )-1\right )}{\sin \left (f x +e \right )}, i\right ) \cos \left (f x +e \right ) a +i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (\cos \left (f x +e \right )-1\right )}{\sin \left (f x +e \right )}, i\right ) a +b \right )}{f \sin \left (f x +e \right )^{4}}\) | \(168\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
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]
[Out]
________________________________________________________________________________________
Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.12, size = 79, normalized size = 1.36 \begin {gather*} \frac {-i \, \sqrt {2} a \sqrt {d} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + i \, \sqrt {2} a \sqrt {d} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) + 2 \, b \sqrt {\frac {d}{\cos \left (f x + e\right )}}}{f} \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 {d \sec {\left (e + f x \right )}} \left (a + b \tan {\left (e + f x \right )}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
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]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.36, size = 39, normalized size = 0.67 \begin {gather*} \frac {2\,\left (b+a\,\sqrt {\cos \left (e+f\,x\right )}\,\mathrm {F}\left (\frac {e}{2}+\frac {f\,x}{2}\middle |2\right )\right )\,\sqrt {\frac {d}{\cos \left (e+f\,x\right )}}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________