Optimal. Leaf size=72 \[ \frac {a^3}{d (a-a \sin (c+d x))}+\frac {a^2 \sin ^2(c+d x)}{2 d}+\frac {2 a^2 \sin (c+d x)}{d}+\frac {3 a^2 \log (1-\sin (c+d x))}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2707, 43} \[ \frac {a^2 \sin ^2(c+d x)}{2 d}+\frac {a^3}{d (a-a \sin (c+d x))}+\frac {2 a^2 \sin (c+d x)}{d}+\frac {3 a^2 \log (1-\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 2707
Rubi steps
\begin {align*} \int (a+a \sin (c+d x))^2 \tan ^3(c+d x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^3}{(a-x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (2 a+\frac {a^3}{(a-x)^2}-\frac {3 a^2}{a-x}+x\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {3 a^2 \log (1-\sin (c+d x))}{d}+\frac {2 a^2 \sin (c+d x)}{d}+\frac {a^2 \sin ^2(c+d x)}{2 d}+\frac {a^3}{d (a-a \sin (c+d x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 54, normalized size = 0.75 \[ \frac {a^2 \left (\sin ^2(c+d x)+4 \sin (c+d x)+\frac {2}{1-\sin (c+d x)}+6 \log (1-\sin (c+d x))\right )}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 90, normalized size = 1.25 \[ -\frac {6 \, a^{2} \cos \left (d x + c\right )^{2} - 3 \, a^{2} - 12 \, {\left (a^{2} \sin \left (d x + c\right ) - a^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (2 \, a^{2} \cos \left (d x + c\right )^{2} + 7 \, a^{2}\right )} \sin \left (d x + c\right )}{4 \, {\left (d \sin \left (d x + c\right ) - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.17, size = 162, normalized size = 2.25 \[ \frac {a^{2} \left (\sin ^{6}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{2}}+\frac {a^{2} \left (\sin ^{4}\left (d x +c \right )\right )}{2 d}+\frac {a^{2} \left (\sin ^{2}\left (d x +c \right )\right )}{d}+\frac {3 a^{2} \ln \left (\cos \left (d x +c \right )\right )}{d}+\frac {a^{2} \left (\sin ^{5}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )^{2}}+\frac {a^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{d}+\frac {3 a^{2} \sin \left (d x +c \right )}{d}-\frac {3 a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {a^{2} \left (\tan ^{2}\left (d x +c \right )\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.30, size = 58, normalized size = 0.81 \[ \frac {a^{2} \sin \left (d x + c\right )^{2} + 6 \, a^{2} \log \left (\sin \left (d x + c\right ) - 1\right ) + 4 \, a^{2} \sin \left (d x + c\right ) - \frac {2 \, a^{2}}{\sin \left (d x + c\right ) - 1}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 7.11, size = 204, normalized size = 2.83 \[ \frac {6\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-6\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+8\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-6\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+6\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}+\frac {6\,a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}{d}-\frac {3\,a^2\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a^{2} \left (\int 2 \sin {\left (c + d x \right )} \tan ^{3}{\left (c + d x \right )}\, dx + \int \sin ^{2}{\left (c + d x \right )} \tan ^{3}{\left (c + d x \right )}\, dx + \int \tan ^{3}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________