Optimal. Leaf size=15 \[ A x+\frac {C \tan (c+d x)}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3852, 8}
\begin {gather*} A x+\frac {C \tan (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 3852
Rubi steps
\begin {align*} \int \left (A+C \sec ^2(c+d x)\right ) \, dx &=A x+C \int \sec ^2(c+d x) \, dx\\ &=A x-\frac {C \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=A x+\frac {C \tan (c+d x)}{d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} A x+\frac {C \tan (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.36, size = 16, normalized size = 1.07
method | result | size |
default | \(A x +\frac {C \tan \left (d x +c \right )}{d}\) | \(16\) |
derivativedivides | \(\frac {\left (d x +c \right ) A +C \tan \left (d x +c \right )}{d}\) | \(21\) |
risch | \(A x +\frac {2 i C}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}\) | \(25\) |
norman | \(\frac {A x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-A x -\frac {2 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}}{\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1}\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 15, normalized size = 1.00 \begin {gather*} A x + \frac {C \tan \left (d x + c\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 31 vs.
\(2 (15) = 30\).
time = 1.50, size = 31, normalized size = 2.07 \begin {gather*} \frac {A d x \cos \left (d x + c\right ) + C \sin \left (d x + c\right )}{d \cos \left (d x + c\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 \left (A + C \sec ^{2}{\left (c + d x \right )}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.41, size = 15, normalized size = 1.00 \begin {gather*} A x + \frac {C \tan \left (d x + c\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 2.34, size = 17, normalized size = 1.13 \begin {gather*} \frac {C\,\mathrm {tan}\left (c+d\,x\right )+A\,d\,x}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________