Optimal. Leaf size=220 \[ -\frac {a^3}{64 d (a \sin (c+d x)+a)^4}+\frac {a^2}{96 d (a-a \sin (c+d x))^3}+\frac {a^2}{6 d (a \sin (c+d x)+a)^3}+\frac {\sin ^2(c+d x)}{2 a d}-\frac {15 a}{128 d (a-a \sin (c+d x))^2}-\frac {55 a}{64 d (a \sin (c+d x)+a)^2}+\frac {95}{128 d (a-a \sin (c+d x))}+\frac {105}{32 d (a \sin (c+d x)+a)}-\frac {\sin (c+d x)}{a d}+\frac {325 \log (1-\sin (c+d x))}{256 a d}+\frac {955 \log (\sin (c+d x)+1)}{256 a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.23, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2836, 12, 88} \[ -\frac {a^3}{64 d (a \sin (c+d x)+a)^4}+\frac {a^2}{96 d (a-a \sin (c+d x))^3}+\frac {a^2}{6 d (a \sin (c+d x)+a)^3}+\frac {\sin ^2(c+d x)}{2 a d}-\frac {15 a}{128 d (a-a \sin (c+d x))^2}-\frac {55 a}{64 d (a \sin (c+d x)+a)^2}+\frac {95}{128 d (a-a \sin (c+d x))}+\frac {105}{32 d (a \sin (c+d x)+a)}-\frac {\sin (c+d x)}{a d}+\frac {325 \log (1-\sin (c+d x))}{256 a d}+\frac {955 \log (\sin (c+d x)+1)}{256 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 88
Rule 2836
Rubi steps
\begin {align*} \int \frac {\sin ^3(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {a^7 \operatorname {Subst}\left (\int \frac {x^{10}}{a^{10} (a-x)^4 (a+x)^5} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^{10}}{(a-x)^4 (a+x)^5} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-a+\frac {a^5}{32 (a-x)^4}-\frac {15 a^4}{64 (a-x)^3}+\frac {95 a^3}{128 (a-x)^2}-\frac {325 a^2}{256 (a-x)}+x+\frac {a^6}{16 (a+x)^5}-\frac {a^5}{2 (a+x)^4}+\frac {55 a^4}{32 (a+x)^3}-\frac {105 a^3}{32 (a+x)^2}+\frac {955 a^2}{256 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {325 \log (1-\sin (c+d x))}{256 a d}+\frac {955 \log (1+\sin (c+d x))}{256 a d}-\frac {\sin (c+d x)}{a d}+\frac {\sin ^2(c+d x)}{2 a d}+\frac {a^2}{96 d (a-a \sin (c+d x))^3}-\frac {15 a}{128 d (a-a \sin (c+d x))^2}+\frac {95}{128 d (a-a \sin (c+d x))}-\frac {a^3}{64 d (a+a \sin (c+d x))^4}+\frac {a^2}{6 d (a+a \sin (c+d x))^3}-\frac {55 a}{64 d (a+a \sin (c+d x))^2}+\frac {105}{32 d (a+a \sin (c+d x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 6.15, size = 143, normalized size = 0.65 \[ \frac {384 \sin ^2(c+d x)-768 \sin (c+d x)+\frac {570}{1-\sin (c+d x)}+\frac {2520}{\sin (c+d x)+1}-\frac {90}{(1-\sin (c+d x))^2}-\frac {660}{(\sin (c+d x)+1)^2}+\frac {8}{(1-\sin (c+d x))^3}+\frac {128}{(\sin (c+d x)+1)^3}-\frac {12}{(\sin (c+d x)+1)^4}+975 \log (1-\sin (c+d x))+2865 \log (\sin (c+d x)+1)}{768 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.55, size = 197, normalized size = 0.90 \[ \frac {384 \, \cos \left (d x + c\right )^{8} + 1374 \, \cos \left (d x + c\right )^{6} + 630 \, \cos \left (d x + c\right )^{4} - 132 \, \cos \left (d x + c\right )^{2} + 2865 \, {\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 975 \, {\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (192 \, \cos \left (d x + c\right )^{8} + 288 \, \cos \left (d x + c\right )^{6} - 945 \, \cos \left (d x + c\right )^{4} + 330 \, \cos \left (d x + c\right )^{2} - 56\right )} \sin \left (d x + c\right ) + 16}{768 \, {\left (a d \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.36, size = 161, normalized size = 0.73 \[ \frac {\frac {11460 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} + \frac {3900 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac {1536 \, {\left (a \sin \left (d x + c\right )^{2} - 2 \, a \sin \left (d x + c\right )\right )}}{a^{2}} - \frac {2 \, {\left (3575 \, \sin \left (d x + c\right )^{3} - 9585 \, \sin \left (d x + c\right )^{2} + 8625 \, \sin \left (d x + c\right ) - 2599\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{3}} - \frac {23875 \, \sin \left (d x + c\right )^{4} + 85420 \, \sin \left (d x + c\right )^{3} + 115650 \, \sin \left (d x + c\right )^{2} + 70028 \, \sin \left (d x + c\right ) + 15971}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{4}}}{3072 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.47, size = 192, normalized size = 0.87 \[ -\frac {1}{96 a d \left (\sin \left (d x +c \right )-1\right )^{3}}-\frac {15}{128 a d \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {95}{128 a d \left (\sin \left (d x +c \right )-1\right )}+\frac {325 \ln \left (\sin \left (d x +c \right )-1\right )}{256 a d}+\frac {\sin ^{2}\left (d x +c \right )}{2 a d}-\frac {\sin \left (d x +c \right )}{a d}-\frac {1}{64 a d \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {1}{6 a d \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {55}{64 a d \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {105}{32 a d \left (1+\sin \left (d x +c \right )\right )}+\frac {955 \ln \left (1+\sin \left (d x +c \right )\right )}{256 a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.40, size = 197, normalized size = 0.90 \[ \frac {\frac {2 \, {\left (975 \, \sin \left (d x + c\right )^{6} - 945 \, \sin \left (d x + c\right )^{5} - 3240 \, \sin \left (d x + c\right )^{4} + 1560 \, \sin \left (d x + c\right )^{3} + 3489 \, \sin \left (d x + c\right )^{2} - 671 \, \sin \left (d x + c\right ) - 1232\right )}}{a \sin \left (d x + c\right )^{7} + a \sin \left (d x + c\right )^{6} - 3 \, a \sin \left (d x + c\right )^{5} - 3 \, a \sin \left (d x + c\right )^{4} + 3 \, a \sin \left (d x + c\right )^{3} + 3 \, a \sin \left (d x + c\right )^{2} - a \sin \left (d x + c\right ) - a} + \frac {384 \, {\left (\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right )\right )}}{a} + \frac {2865 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac {975 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a}}{768 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 10.34, size = 512, normalized size = 2.33 \[ \frac {325\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}{128\,a\,d}+\frac {955\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{128\,a\,d}+\frac {-\frac {315\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{64}+\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}}{32}+\frac {265\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{8}+\frac {195\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{32}-\frac {1217\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{16}-\frac {2389\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{96}+\frac {1189\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{24}+\frac {767\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{32}+\frac {6845\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{96}+\frac {767\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{32}+\frac {1189\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{24}-\frac {2389\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{96}-\frac {1217\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{16}+\frac {195\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{32}+\frac {265\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8}+\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{32}-\frac {315\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}}{d\,\left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}+2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}-3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}+8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-6\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-20\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-6\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )}-\frac {5\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [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]
________________________________________________________________________________________