Optimal. Leaf size=160 \[ \frac {209 a^3 \log (1-\sin (c+d x))}{16 d}-\frac {a^3 \log (1+\sin (c+d x))}{16 d}+\frac {7 a^3 \sin (c+d x)}{d}+\frac {3 a^3 \sin ^2(c+d x)}{2 d}+\frac {a^3 \sin ^3(c+d x)}{3 d}+\frac {a^6}{6 d (a-a \sin (c+d x))^3}-\frac {13 a^5}{8 d (a-a \sin (c+d x))^2}+\frac {71 a^4}{8 d (a-a \sin (c+d x))} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.08, antiderivative size = 160, 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 = {2786, 90}
\begin {gather*} \frac {a^6}{6 d (a-a \sin (c+d x))^3}-\frac {13 a^5}{8 d (a-a \sin (c+d x))^2}+\frac {71 a^4}{8 d (a-a \sin (c+d x))}+\frac {a^3 \sin ^3(c+d x)}{3 d}+\frac {3 a^3 \sin ^2(c+d x)}{2 d}+\frac {7 a^3 \sin (c+d x)}{d}+\frac {209 a^3 \log (1-\sin (c+d x))}{16 d}-\frac {a^3 \log (\sin (c+d x)+1)}{16 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 90
Rule 2786
Rubi steps
\begin {align*} \int (a+a \sin (c+d x))^3 \tan ^7(c+d x) \, dx &=\frac {\text {Subst}\left (\int \frac {x^7}{(a-x)^4 (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (7 a^2+\frac {a^6}{2 (a-x)^4}-\frac {13 a^5}{4 (a-x)^3}+\frac {71 a^4}{8 (a-x)^2}-\frac {209 a^3}{16 (a-x)}+3 a x+x^2-\frac {a^3}{16 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {209 a^3 \log (1-\sin (c+d x))}{16 d}-\frac {a^3 \log (1+\sin (c+d x))}{16 d}+\frac {7 a^3 \sin (c+d x)}{d}+\frac {3 a^3 \sin ^2(c+d x)}{2 d}+\frac {a^3 \sin ^3(c+d x)}{3 d}+\frac {a^6}{6 d (a-a \sin (c+d x))^3}-\frac {13 a^5}{8 d (a-a \sin (c+d x))^2}+\frac {71 a^4}{8 d (a-a \sin (c+d x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.36, size = 99, normalized size = 0.62 \begin {gather*} \frac {a^3 \left (627 \log (1-\sin (c+d x))-3 \log (1+\sin (c+d x))-\frac {8}{(-1+\sin (c+d x))^3}-\frac {78}{(-1+\sin (c+d x))^2}-\frac {426}{-1+\sin (c+d x)}+336 \sin (c+d x)+72 \sin ^2(c+d x)+16 \sin ^3(c+d x)\right )}{48 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(391\) vs.
\(2(146)=292\).
time = 0.23, size = 392, normalized size = 2.45
method | result | size |
risch | \(-13 i a^{3} x +\frac {i a^{3} {\mathrm e}^{3 i \left (d x +c \right )}}{24 d}-\frac {3 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {29 i a^{3} {\mathrm e}^{i \left (d x +c \right )}}{8 d}+\frac {29 i a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{8 d}-\frac {3 a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {i a^{3} {\mathrm e}^{-3 i \left (d x +c \right )}}{24 d}-\frac {26 i a^{3} c}{d}-\frac {i \left (213 a^{3} {\mathrm e}^{i \left (d x +c \right )}+774 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-1138 a^{3} {\mathrm e}^{3 i \left (d x +c \right )}-774 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}+213 a^{3} {\mathrm e}^{5 i \left (d x +c \right )}\right )}{12 \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{6} d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}+\frac {209 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}\) | \(260\) |
derivativedivides | \(\frac {a^{3} \left (\frac {\sin ^{11}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{6}}-\frac {5 \left (\sin ^{11}\left (d x +c \right )\right )}{24 \cos \left (d x +c \right )^{4}}+\frac {35 \left (\sin ^{11}\left (d x +c \right )\right )}{48 \cos \left (d x +c \right )^{2}}+\frac {35 \left (\sin ^{9}\left (d x +c \right )\right )}{48}+\frac {15 \left (\sin ^{7}\left (d x +c \right )\right )}{16}+\frac {21 \left (\sin ^{5}\left (d x +c \right )\right )}{16}+\frac {35 \left (\sin ^{3}\left (d x +c \right )\right )}{16}+\frac {105 \sin \left (d x +c \right )}{16}-\frac {105 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+3 a^{3} \left (\frac {\sin ^{10}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{6}}-\frac {\sin ^{10}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{4}}+\frac {\sin ^{10}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{8}\left (d x +c \right )\right )}{2}+\frac {2 \left (\sin ^{6}\left (d x +c \right )\right )}{3}+\sin ^{4}\left (d x +c \right )+2 \left (\sin ^{2}\left (d x +c \right )\right )+4 \ln \left (\cos \left (d x +c \right )\right )\right )+3 a^{3} \left (\frac {\sin ^{9}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{6}}-\frac {\sin ^{9}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{4}}+\frac {5 \left (\sin ^{9}\left (d x +c \right )\right )}{16 \cos \left (d x +c \right )^{2}}+\frac {5 \left (\sin ^{7}\left (d x +c \right )\right )}{16}+\frac {7 \left (\sin ^{5}\left (d x +c \right )\right )}{16}+\frac {35 \left (\sin ^{3}\left (d x +c \right )\right )}{48}+\frac {35 \sin \left (d x +c \right )}{16}-\frac {35 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+a^{3} \left (\frac {\left (\tan ^{6}\left (d x +c \right )\right )}{6}-\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) | \(392\) |
default | \(\frac {a^{3} \left (\frac {\sin ^{11}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{6}}-\frac {5 \left (\sin ^{11}\left (d x +c \right )\right )}{24 \cos \left (d x +c \right )^{4}}+\frac {35 \left (\sin ^{11}\left (d x +c \right )\right )}{48 \cos \left (d x +c \right )^{2}}+\frac {35 \left (\sin ^{9}\left (d x +c \right )\right )}{48}+\frac {15 \left (\sin ^{7}\left (d x +c \right )\right )}{16}+\frac {21 \left (\sin ^{5}\left (d x +c \right )\right )}{16}+\frac {35 \left (\sin ^{3}\left (d x +c \right )\right )}{16}+\frac {105 \sin \left (d x +c \right )}{16}-\frac {105 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+3 a^{3} \left (\frac {\sin ^{10}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{6}}-\frac {\sin ^{10}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{4}}+\frac {\sin ^{10}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{8}\left (d x +c \right )\right )}{2}+\frac {2 \left (\sin ^{6}\left (d x +c \right )\right )}{3}+\sin ^{4}\left (d x +c \right )+2 \left (\sin ^{2}\left (d x +c \right )\right )+4 \ln \left (\cos \left (d x +c \right )\right )\right )+3 a^{3} \left (\frac {\sin ^{9}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{6}}-\frac {\sin ^{9}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{4}}+\frac {5 \left (\sin ^{9}\left (d x +c \right )\right )}{16 \cos \left (d x +c \right )^{2}}+\frac {5 \left (\sin ^{7}\left (d x +c \right )\right )}{16}+\frac {7 \left (\sin ^{5}\left (d x +c \right )\right )}{16}+\frac {35 \left (\sin ^{3}\left (d x +c \right )\right )}{48}+\frac {35 \sin \left (d x +c \right )}{16}-\frac {35 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+a^{3} \left (\frac {\left (\tan ^{6}\left (d x +c \right )\right )}{6}-\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) | \(392\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 133, normalized size = 0.83 \begin {gather*} \frac {16 \, a^{3} \sin \left (d x + c\right )^{3} + 72 \, a^{3} \sin \left (d x + c\right )^{2} - 3 \, a^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) + 627 \, a^{3} \log \left (\sin \left (d x + c\right ) - 1\right ) + 336 \, a^{3} \sin \left (d x + c\right ) - \frac {2 \, {\left (213 \, a^{3} \sin \left (d x + c\right )^{2} - 387 \, a^{3} \sin \left (d x + c\right ) + 178 \, a^{3}\right )}}{\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )^{2} + 3 \, \sin \left (d x + c\right ) - 1}}{48 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.40, size = 240, normalized size = 1.50 \begin {gather*} -\frac {16 \, a^{3} \cos \left (d x + c\right )^{6} - 216 \, a^{3} \cos \left (d x + c\right )^{4} + 1002 \, a^{3} \cos \left (d x + c\right )^{2} - 482 \, a^{3} + 3 \, {\left (3 \, a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3} - {\left (a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 627 \, {\left (3 \, a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3} - {\left (a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (12 \, a^{3} \cos \left (d x + c\right )^{4} + 398 \, a^{3} \cos \left (d x + c\right )^{2} - 245 \, a^{3}\right )} \sin \left (d x + c\right )}{48 \, {\left (3 \, d \cos \left (d x + c\right )^{2} - {\left (d \cos \left (d x + c\right )^{2} - 4 \, d\right )} \sin \left (d x + c\right ) - 4 \, d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} a^{3} \left (\int 3 \sin {\left (c + d x \right )} \tan ^{7}{\left (c + d x \right )}\, dx + \int 3 \sin ^{2}{\left (c + d x \right )} \tan ^{7}{\left (c + d x \right )}\, dx + \int \sin ^{3}{\left (c + d x \right )} \tan ^{7}{\left (c + d x \right )}\, dx + \int \tan ^{7}{\left (c + d x \right )}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 6.47, size = 398, normalized size = 2.49 \begin {gather*} \frac {\frac {105\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{4}-\frac {263\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{2}+\frac {1301\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4}-582\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {1657\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{2}-\frac {2767\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {1657\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{2}-582\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {1301\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{4}-\frac {263\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}+\frac {105\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+18\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-38\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+63\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-84\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+92\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-84\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+63\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-38\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+18\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}+\frac {209\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}{8\,d}-\frac {a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{8\,d}-\frac {13\,a^3\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________