Optimal. Leaf size=17 \[ \frac {a A \cot ^3(c+d x)}{3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4047, 2687, 30}
\begin {gather*} \frac {a A \cot ^3(c+d x)}{3 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 2687
Rule 4047
Rubi steps
\begin {align*} \int \csc ^2(c+d x) (a-a \csc (c+d x)) (A+A \csc (c+d x)) \, dx &=-\left ((a A) \int \cot ^2(c+d x) \csc ^2(c+d x) \, dx\right )\\ &=-\frac {(a A) \text {Subst}\left (\int x^2 \, dx,x,-\cot (c+d x)\right )}{d}\\ &=\frac {a A \cot ^3(c+d x)}{3 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 17, normalized size = 1.00 \begin {gather*} \frac {a A \cot ^3(c+d x)}{3 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(37\) vs.
\(2(15)=30\).
time = 0.09, size = 38, normalized size = 2.24
method | result | size |
risch | \(-\frac {2 i A a \left (3 \,{\mathrm e}^{4 i \left (d x +c \right )}+1\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}\) | \(35\) |
derivativedivides | \(\frac {-A a \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{3}\right ) \cot \left (d x +c \right )-A a \cot \left (d x +c \right )}{d}\) | \(38\) |
default | \(\frac {-A a \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{3}\right ) \cot \left (d x +c \right )-A a \cot \left (d x +c \right )}{d}\) | \(38\) |
norman | \(\frac {\frac {A a}{24 d}-\frac {A a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {A a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {A a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}\) | \(75\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 42 vs.
\(2 (15) = 30\).
time = 0.27, size = 42, normalized size = 2.47 \begin {gather*} -\frac {\frac {3 \, A a}{\tan \left (d x + c\right )} - \frac {{\left (3 \, \tan \left (d x + c\right )^{2} + 1\right )} A a}{\tan \left (d x + c\right )^{3}}}{3 \, 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. 36 vs.
\(2 (15) = 30\).
time = 3.87, size = 36, normalized size = 2.12 \begin {gather*} -\frac {A a \cos \left (d x + c\right )^{3}}{3 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \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. 54 vs.
\(2 (14) = 28\).
time = 1.30, size = 54, normalized size = 3.18 \begin {gather*} \begin {cases} \frac {- A a \left (- \frac {\cot ^{3}{\left (c + d x \right )}}{3} - \cot {\left (c + d x \right )}\right ) - A a \cot {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (A \csc {\left (c \right )} + A\right ) \left (- a \csc {\left (c \right )} + a\right ) \csc ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.43, size = 15, normalized size = 0.88 \begin {gather*} \frac {A a}{3 \, d \tan \left (d x + c\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.27, size = 15, normalized size = 0.88 \begin {gather*} \frac {A\,a\,{\mathrm {cot}\left (c+d\,x\right )}^3}{3\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________