Optimal. Leaf size=36 \[ \frac {a \tan (c+d x)}{d}+\frac {a \sec (c+d x)}{d}-\frac {a \tanh ^{-1}(\cos (c+d x))}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2838, 2622, 321, 207, 3767, 8} \[ \frac {a \tan (c+d x)}{d}+\frac {a \sec (c+d x)}{d}-\frac {a \tanh ^{-1}(\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 207
Rule 321
Rule 2622
Rule 2838
Rule 3767
Rubi steps
\begin {align*} \int \csc (c+d x) \sec ^2(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \sec ^2(c+d x) \, dx+a \int \csc (c+d x) \sec ^2(c+d x) \, dx\\ &=-\frac {a \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}+\frac {a \operatorname {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac {a \sec (c+d x)}{d}+\frac {a \tan (c+d x)}{d}+\frac {a \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac {a \tanh ^{-1}(\cos (c+d x))}{d}+\frac {a \sec (c+d x)}{d}+\frac {a \tan (c+d x)}{d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 56, normalized size = 1.56 \[ \frac {a \tan (c+d x)}{d}+\frac {a \sec (c+d x)}{d}+\frac {a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}-\frac {a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.46, size = 108, normalized size = 3.00 \[ \frac {2 \, a \cos \left (d x + c\right ) - {\left (a \cos \left (d x + c\right ) - a \sin \left (d x + c\right ) + a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left (a \cos \left (d x + c\right ) - a \sin \left (d x + c\right ) + a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, a \sin \left (d x + c\right ) + 2 \, a}{2 \, {\left (d \cos \left (d x + c\right ) - d \sin \left (d x + c\right ) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 34, normalized size = 0.94 \[ \frac {a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {2 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.41, size = 47, normalized size = 1.31 \[ \frac {a \tan \left (d x +c \right )}{d}+\frac {a}{d \cos \left (d x +c \right )}+\frac {a \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.34, size = 48, normalized size = 1.33 \[ \frac {a {\left (\frac {2}{\cos \left (d x + c\right )} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 2 \, a \tan \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 8.95, size = 35, normalized size = 0.97 \[ \frac {a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {2\,a}{d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a \left (\int \csc {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sin {\left (c + d x \right )} \csc {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________