Optimal. Leaf size=27 \[ A x+\frac {B \tanh ^{-1}(\sin (c+d x))}{d}+\frac {C \tan (c+d x)}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3855, 3852, 8}
\begin {gather*} A x+\frac {B \tanh ^{-1}(\sin (c+d x))}{d}+\frac {C \tan (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 3852
Rule 3855
Rubi steps
\begin {align*} \int \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=A x+B \int \sec (c+d x) \, dx+C \int \sec ^2(c+d x) \, dx\\ &=A x+\frac {B \tanh ^{-1}(\sin (c+d x))}{d}-\frac {C \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=A x+\frac {B \tanh ^{-1}(\sin (c+d x))}{d}+\frac {C \tan (c+d x)}{d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 27, normalized size = 1.00 \begin {gather*} A x+\frac {B \tanh ^{-1}(\sin (c+d x))}{d}+\frac {C \tan (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.38, size = 35, normalized size = 1.30
method | result | size |
default | \(A x +\frac {B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {C \tan \left (d x +c \right )}{d}\) | \(35\) |
derivativedivides | \(\frac {\left (d x +c \right ) A +B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+C \tan \left (d x +c \right )}{d}\) | \(37\) |
risch | \(A x -\frac {B \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {B \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}+\frac {2 i C}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}\) | \(62\) |
norman | \(\frac {A x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-A x -\frac {2 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}}{\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\frac {B \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {B \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(87\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 34, normalized size = 1.26 \begin {gather*} A x + \frac {B \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right )}{d} + \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. 71 vs.
\(2 (27) = 54\).
time = 3.89, size = 71, normalized size = 2.63 \begin {gather*} \frac {2 \, A d x \cos \left (d x + c\right ) + B \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - B \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, C \sin \left (d x + c\right )}{2 \, 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 + B \sec {\left (c + d x \right )} + 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 [B] Leaf count of result is larger than twice the leaf count of optimal. 60 vs.
\(2 (27) = 54\).
time = 0.40, size = 60, normalized size = 2.22 \begin {gather*} A x + \frac {B {\left (\log \left ({\left | \frac {1}{\sin \left (d x + c\right )} + \sin \left (d x + c\right ) + 2 \right |}\right ) - \log \left ({\left | \frac {1}{\sin \left (d x + c\right )} + \sin \left (d x + c\right ) - 2 \right |}\right )\right )}}{4 \, d} + \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 = 2.54, size = 161, normalized size = 5.96 \begin {gather*} \frac {2\,A\,\mathrm {atan}\left (\frac {64\,A^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64\,A^3+64\,A\,B^2}+\frac {64\,A\,B^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64\,A^3+64\,A\,B^2}\right )}{d}+\frac {2\,B\,\mathrm {atanh}\left (\frac {64\,B^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64\,A^2\,B+64\,B^3}+\frac {64\,A^2\,B\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64\,A^2\,B+64\,B^3}\right )}{d}-\frac {2\,C\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________