Optimal. Leaf size=35 \[ (A b+a B) x+\frac {b B \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a A \sin (c+d x)}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {4081, 3855}
\begin {gather*} x (a B+A b)+\frac {a A \sin (c+d x)}{d}+\frac {b B \tanh ^{-1}(\sin (c+d x))}{d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3855
Rule 4081
Rubi steps
\begin {align*} \int \cos (c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx &=\frac {a A \sin (c+d x)}{d}-\int (-A b-a B-b B \sec (c+d x)) \, dx\\ &=(A b+a B) x+\frac {a A \sin (c+d x)}{d}+(b B) \int \sec (c+d x) \, dx\\ &=(A b+a B) x+\frac {b B \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a A \sin (c+d x)}{d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 46, normalized size = 1.31 \begin {gather*} A b x+a B x+\frac {b B \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a A \cos (d x) \sin (c)}{d}+\frac {a A \cos (c) \sin (d x)}{d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.21, size = 48, normalized size = 1.37
method | result | size |
derivativedivides | \(\frac {A \sin \left (d x +c \right ) a +B a \left (d x +c \right )+A b \left (d x +c \right )+B b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(48\) |
default | \(\frac {A \sin \left (d x +c \right ) a +B a \left (d x +c \right )+A b \left (d x +c \right )+B b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(48\) |
risch | \(A b x +B a x -\frac {i A a \,{\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {i A a \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B b}{d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B b}{d}\) | \(83\) |
norman | \(\frac {\left (-A b -B a \right ) x +\left (A b +B a \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {2 A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 A a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {B b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {B b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(136\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.30, size = 58, normalized size = 1.66 \begin {gather*} \frac {2 \, {\left (d x + c\right )} B a + 2 \, {\left (d x + c\right )} A b + B b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, A a \sin \left (d x + c\right )}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 3.72, size = 54, normalized size = 1.54 \begin {gather*} \frac {2 \, {\left (B a + A b\right )} d x + B b \log \left (\sin \left (d x + c\right ) + 1\right ) - B b \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, A a \sin \left (d x + c\right )}{2 \, d} \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 )}\right ) \left (a + b \sec {\left (c + d x \right )}\right ) \cos {\left (c + d x \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. 79 vs.
\(2 (35) = 70\).
time = 0.46, size = 79, normalized size = 2.26 \begin {gather*} \frac {B b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - B b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + {\left (B a + A b\right )} {\left (d x + c\right )} + \frac {2 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 2.19, size = 100, normalized size = 2.86 \begin {gather*} \frac {A\,a\,\sin \left (c+d\,x\right )}{d}+\frac {2\,A\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,B\,a\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,B\,b\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________