Optimal. Leaf size=48 \[ \frac {a^3 \sin (c+d x)}{d}+\frac {a^3 \tan (c+d x)}{d}+\frac {3 a^3 \tanh ^{-1}(\sin (c+d x))}{d}+3 a^3 x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2757, 2637, 3770, 3767, 8} \[ \frac {a^3 \sin (c+d x)}{d}+\frac {a^3 \tan (c+d x)}{d}+\frac {3 a^3 \tanh ^{-1}(\sin (c+d x))}{d}+3 a^3 x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2637
Rule 2757
Rule 3767
Rule 3770
Rubi steps
\begin {align*} \int (a+a \cos (c+d x))^3 \sec ^2(c+d x) \, dx &=\int \left (3 a^3+a^3 \cos (c+d x)+3 a^3 \sec (c+d x)+a^3 \sec ^2(c+d x)\right ) \, dx\\ &=3 a^3 x+a^3 \int \cos (c+d x) \, dx+a^3 \int \sec ^2(c+d x) \, dx+\left (3 a^3\right ) \int \sec (c+d x) \, dx\\ &=3 a^3 x+\frac {3 a^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a^3 \sin (c+d x)}{d}-\frac {a^3 \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=3 a^3 x+\frac {3 a^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a^3 \sin (c+d x)}{d}+\frac {a^3 \tan (c+d x)}{d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.73, size = 211, normalized size = 4.40 \[ \frac {1}{8} a^3 (\cos (c+d x)+1)^3 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \left (\frac {\sin (c) \cos (d x)}{d}+\frac {\cos (c) \sin (d x)}{d}+\frac {\sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {\sin \left (\frac {d x}{2}\right )}{d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}-\frac {3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{d}+3 x\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.80, size = 91, normalized size = 1.90 \[ \frac {6 \, a^{3} d x \cos \left (d x + c\right ) + 3 \, a^{3} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, a^{3} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (a^{3} \cos \left (d x + c\right ) + a^{3}\right )} \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.61, size = 80, normalized size = 1.67 \[ \frac {3 \, {\left (d x + c\right )} a^{3} + 3 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {4 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.14, size = 65, normalized size = 1.35 \[ 3 a^{3} x +\frac {3 a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {a^{3} \sin \left (d x +c \right )}{d}+\frac {a^{3} \tan \left (d x +c \right )}{d}+\frac {3 a^{3} c}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.00, size = 64, normalized size = 1.33 \[ \frac {6 \, {\left (d x + c\right )} a^{3} + 3 \, a^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, a^{3} \sin \left (d x + c\right ) + 2 \, a^{3} \tan \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.41, size = 57, normalized size = 1.19 \[ 3\,a^3\,x+\frac {6\,a^3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {4\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a^{3} \left (\int 3 \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 3 \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \cos ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sec ^{2}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________