Optimal. Leaf size=26 \[ \frac {(a+b \sin (c+d x))^{1+m}}{b d (1+m)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2747, 32}
\begin {gather*} \frac {(a+b \sin (c+d x))^{m+1}}{b d (m+1)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 32
Rule 2747
Rubi steps
\begin {align*} \int \cos (c+d x) (a+b \sin (c+d x))^m \, dx &=\frac {\text {Subst}\left (\int (a+x)^m \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac {(a+b \sin (c+d x))^{1+m}}{b d (1+m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 26, normalized size = 1.00 \begin {gather*} \frac {(a+b \sin (c+d x))^{1+m}}{b d (1+m)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.12, size = 27, normalized size = 1.04
method | result | size |
derivativedivides | \(\frac {\left (a +b \sin \left (d x +c \right )\right )^{1+m}}{b d \left (1+m \right )}\) | \(27\) |
default | \(\frac {\left (a +b \sin \left (d x +c \right )\right )^{1+m}}{b d \left (1+m \right )}\) | \(27\) |
norman | \(\frac {\frac {a \,{\mathrm e}^{m \ln \left (a +\frac {2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\right )}}{b d \left (1+m \right )}+\frac {a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) {\mathrm e}^{m \ln \left (a +\frac {2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\right )}}{b d \left (1+m \right )}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) {\mathrm e}^{m \ln \left (a +\frac {2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\right )}}{d \left (1+m \right )}}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\) | \(173\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 26, normalized size = 1.00 \begin {gather*} \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{m + 1}}{b d {\left (m + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.35, size = 33, normalized size = 1.27 \begin {gather*} \frac {{\left (b \sin \left (d x + c\right ) + a\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{m}}{b d m + b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 99 vs.
\(2 (19) = 38\).
time = 0.53, size = 99, normalized size = 3.81 \begin {gather*} \begin {cases} \frac {x \cos {\left (c \right )}}{a} & \text {for}\: b = 0 \wedge d = 0 \wedge m = -1 \\\frac {a^{m} \sin {\left (c + d x \right )}}{d} & \text {for}\: b = 0 \\x \left (a + b \sin {\left (c \right )}\right )^{m} \cos {\left (c \right )} & \text {for}\: d = 0 \\\frac {\log {\left (\frac {a}{b} + \sin {\left (c + d x \right )} \right )}}{b d} & \text {for}\: m = -1 \\\frac {a \left (a + b \sin {\left (c + d x \right )}\right )^{m}}{b d m + b d} + \frac {b \left (a + b \sin {\left (c + d x \right )}\right )^{m} \sin {\left (c + d x \right )}}{b d m + b d} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 3.94, size = 26, normalized size = 1.00 \begin {gather*} \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{m + 1}}{b d {\left (m + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 6.32, size = 26, normalized size = 1.00 \begin {gather*} \frac {{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{m+1}}{b\,d\,\left (m+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________