Optimal. Leaf size=154 \[ \frac {\tan ^{10}(c+d x)}{10 a d}+\frac {63 \tanh ^{-1}(\sin (c+d x))}{256 a d}-\frac {\tan ^9(c+d x) \sec (c+d x)}{10 a d}+\frac {9 \tan ^7(c+d x) \sec (c+d x)}{80 a d}-\frac {21 \tan ^5(c+d x) \sec (c+d x)}{160 a d}+\frac {21 \tan ^3(c+d x) \sec (c+d x)}{128 a d}-\frac {63 \tan (c+d x) \sec (c+d x)}{256 a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.19, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2706, 2607, 30, 2611, 3770} \[ \frac {\tan ^{10}(c+d x)}{10 a d}+\frac {63 \tanh ^{-1}(\sin (c+d x))}{256 a d}-\frac {\tan ^9(c+d x) \sec (c+d x)}{10 a d}+\frac {9 \tan ^7(c+d x) \sec (c+d x)}{80 a d}-\frac {21 \tan ^5(c+d x) \sec (c+d x)}{160 a d}+\frac {21 \tan ^3(c+d x) \sec (c+d x)}{128 a d}-\frac {63 \tan (c+d x) \sec (c+d x)}{256 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 2607
Rule 2611
Rule 2706
Rule 3770
Rubi steps
\begin {align*} \int \frac {\tan ^9(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int \sec ^2(c+d x) \tan ^9(c+d x) \, dx}{a}-\frac {\int \sec (c+d x) \tan ^{10}(c+d x) \, dx}{a}\\ &=-\frac {\sec (c+d x) \tan ^9(c+d x)}{10 a d}+\frac {9 \int \sec (c+d x) \tan ^8(c+d x) \, dx}{10 a}+\frac {\operatorname {Subst}\left (\int x^9 \, dx,x,\tan (c+d x)\right )}{a d}\\ &=\frac {9 \sec (c+d x) \tan ^7(c+d x)}{80 a d}-\frac {\sec (c+d x) \tan ^9(c+d x)}{10 a d}+\frac {\tan ^{10}(c+d x)}{10 a d}-\frac {63 \int \sec (c+d x) \tan ^6(c+d x) \, dx}{80 a}\\ &=-\frac {21 \sec (c+d x) \tan ^5(c+d x)}{160 a d}+\frac {9 \sec (c+d x) \tan ^7(c+d x)}{80 a d}-\frac {\sec (c+d x) \tan ^9(c+d x)}{10 a d}+\frac {\tan ^{10}(c+d x)}{10 a d}+\frac {21 \int \sec (c+d x) \tan ^4(c+d x) \, dx}{32 a}\\ &=\frac {21 \sec (c+d x) \tan ^3(c+d x)}{128 a d}-\frac {21 \sec (c+d x) \tan ^5(c+d x)}{160 a d}+\frac {9 \sec (c+d x) \tan ^7(c+d x)}{80 a d}-\frac {\sec (c+d x) \tan ^9(c+d x)}{10 a d}+\frac {\tan ^{10}(c+d x)}{10 a d}-\frac {63 \int \sec (c+d x) \tan ^2(c+d x) \, dx}{128 a}\\ &=-\frac {63 \sec (c+d x) \tan (c+d x)}{256 a d}+\frac {21 \sec (c+d x) \tan ^3(c+d x)}{128 a d}-\frac {21 \sec (c+d x) \tan ^5(c+d x)}{160 a d}+\frac {9 \sec (c+d x) \tan ^7(c+d x)}{80 a d}-\frac {\sec (c+d x) \tan ^9(c+d x)}{10 a d}+\frac {\tan ^{10}(c+d x)}{10 a d}+\frac {63 \int \sec (c+d x) \, dx}{256 a}\\ &=\frac {63 \tanh ^{-1}(\sin (c+d x))}{256 a d}-\frac {63 \sec (c+d x) \tan (c+d x)}{256 a d}+\frac {21 \sec (c+d x) \tan ^3(c+d x)}{128 a d}-\frac {21 \sec (c+d x) \tan ^5(c+d x)}{160 a d}+\frac {9 \sec (c+d x) \tan ^7(c+d x)}{80 a d}-\frac {\sec (c+d x) \tan ^9(c+d x)}{10 a d}+\frac {\tan ^{10}(c+d x)}{10 a d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 2.42, size = 122, normalized size = 0.79 \[ \frac {\frac {2 \left (965 \sin ^8(c+d x)+325 \sin ^7(c+d x)-2045 \sin ^6(c+d x)-765 \sin ^5(c+d x)+1923 \sin ^4(c+d x)+643 \sin ^3(c+d x)-827 \sin ^2(c+d x)-187 \sin (c+d x)+128\right )}{(\sin (c+d x)-1)^4 (\sin (c+d x)+1)^5}+630 \tanh ^{-1}(\sin (c+d x))}{2560 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.53, size = 187, normalized size = 1.21 \[ \frac {1930 \, \cos \left (d x + c\right )^{8} - 3630 \, \cos \left (d x + c\right )^{6} + 3156 \, \cos \left (d x + c\right )^{4} - 1488 \, \cos \left (d x + c\right )^{2} + 315 \, {\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 315 \, {\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (325 \, \cos \left (d x + c\right )^{6} - 210 \, \cos \left (d x + c\right )^{4} + 88 \, \cos \left (d x + c\right )^{2} - 16\right )} \sin \left (d x + c\right ) + 288}{2560 \, {\left (a d \cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{8}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.37, size = 156, normalized size = 1.01 \[ \frac {\frac {1260 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {1260 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac {5 \, {\left (525 \, \sin \left (d x + c\right )^{4} - 1580 \, \sin \left (d x + c\right )^{3} + 1818 \, \sin \left (d x + c\right )^{2} - 932 \, \sin \left (d x + c\right ) + 177\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{4}} - \frac {2877 \, \sin \left (d x + c\right )^{5} + 9265 \, \sin \left (d x + c\right )^{4} + 12030 \, \sin \left (d x + c\right )^{3} + 7430 \, \sin \left (d x + c\right )^{2} + 1965 \, \sin \left (d x + c\right ) + 113}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{5}}}{10240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.46, size = 198, normalized size = 1.29 \[ \frac {1}{256 a d \left (\sin \left (d x +c \right )-1\right )^{4}}+\frac {1}{32 a d \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {57}{512 a d \left (\sin \left (d x +c \right )-1\right )^{2}}+\frac {65}{256 a d \left (\sin \left (d x +c \right )-1\right )}-\frac {63 \ln \left (\sin \left (d x +c \right )-1\right )}{512 a d}+\frac {1}{160 a d \left (1+\sin \left (d x +c \right )\right )^{5}}-\frac {13}{256 a d \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {23}{128 a d \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {187}{512 a d \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {1}{2 a d \left (1+\sin \left (d x +c \right )\right )}+\frac {63 \ln \left (1+\sin \left (d x +c \right )\right )}{512 a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.52, size = 214, normalized size = 1.39 \[ \frac {\frac {2 \, {\left (965 \, \sin \left (d x + c\right )^{8} + 325 \, \sin \left (d x + c\right )^{7} - 2045 \, \sin \left (d x + c\right )^{6} - 765 \, \sin \left (d x + c\right )^{5} + 1923 \, \sin \left (d x + c\right )^{4} + 643 \, \sin \left (d x + c\right )^{3} - 827 \, \sin \left (d x + c\right )^{2} - 187 \, \sin \left (d x + c\right ) + 128\right )}}{a \sin \left (d x + c\right )^{9} + a \sin \left (d x + c\right )^{8} - 4 \, a \sin \left (d x + c\right )^{7} - 4 \, a \sin \left (d x + c\right )^{6} + 6 \, a \sin \left (d x + c\right )^{5} + 6 \, a \sin \left (d x + c\right )^{4} - 4 \, a \sin \left (d x + c\right )^{3} - 4 \, a \sin \left (d x + c\right )^{2} + a \sin \left (d x + c\right ) + a} + \frac {315 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac {315 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a}}{2560 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 17.05, size = 497, normalized size = 3.23 \[ \frac {63\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{128\,a\,d}-\frac {\frac {63\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{128}+\frac {63\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}}{64}-\frac {105\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{32}-\frac {483\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{64}+\frac {1407\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{160}+\frac {8043\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{320}-\frac {1779\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{160}-\frac {15159\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{320}+\frac {245\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}-\frac {15159\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{320}-\frac {1779\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{160}+\frac {8043\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{320}+\frac {1407\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160}-\frac {483\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64}-\frac {105\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{32}+\frac {63\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64}+\frac {63\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128}}{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}-7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-16\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}+20\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+56\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-28\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-112\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+14\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+140\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+14\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-112\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-28\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+56\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+20\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-16\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-7\,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 )} \]
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]
________________________________________________________________________________________