Optimal. Leaf size=174 \[ \frac {\sec ^{10}(c+d x)}{10 a d}-\frac {\sec ^8(c+d x)}{8 a d}-\frac {3 \tanh ^{-1}(\sin (c+d x))}{256 a d}-\frac {\tan ^3(c+d x) \sec ^7(c+d x)}{10 a d}+\frac {3 \tan (c+d x) \sec ^7(c+d x)}{80 a d}-\frac {\tan (c+d x) \sec ^5(c+d x)}{160 a d}-\frac {\tan (c+d x) \sec ^3(c+d x)}{128 a d}-\frac {3 \tan (c+d x) \sec (c+d x)}{256 a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.24, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2835, 2606, 14, 2611, 3768, 3770} \[ \frac {\sec ^{10}(c+d x)}{10 a d}-\frac {\sec ^8(c+d x)}{8 a d}-\frac {3 \tanh ^{-1}(\sin (c+d x))}{256 a d}-\frac {\tan ^3(c+d x) \sec ^7(c+d x)}{10 a d}+\frac {3 \tan (c+d x) \sec ^7(c+d x)}{80 a d}-\frac {\tan (c+d x) \sec ^5(c+d x)}{160 a d}-\frac {\tan (c+d x) \sec ^3(c+d x)}{128 a d}-\frac {3 \tan (c+d x) \sec (c+d x)}{256 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 2606
Rule 2611
Rule 2835
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\sec ^6(c+d x) \tan ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int \sec ^8(c+d x) \tan ^3(c+d x) \, dx}{a}-\frac {\int \sec ^7(c+d x) \tan ^4(c+d x) \, dx}{a}\\ &=-\frac {\sec ^7(c+d x) \tan ^3(c+d x)}{10 a d}+\frac {3 \int \sec ^7(c+d x) \tan ^2(c+d x) \, dx}{10 a}+\frac {\operatorname {Subst}\left (\int x^7 \left (-1+x^2\right ) \, dx,x,\sec (c+d x)\right )}{a d}\\ &=\frac {3 \sec ^7(c+d x) \tan (c+d x)}{80 a d}-\frac {\sec ^7(c+d x) \tan ^3(c+d x)}{10 a d}-\frac {3 \int \sec ^7(c+d x) \, dx}{80 a}+\frac {\operatorname {Subst}\left (\int \left (-x^7+x^9\right ) \, dx,x,\sec (c+d x)\right )}{a d}\\ &=-\frac {\sec ^8(c+d x)}{8 a d}+\frac {\sec ^{10}(c+d x)}{10 a d}-\frac {\sec ^5(c+d x) \tan (c+d x)}{160 a d}+\frac {3 \sec ^7(c+d x) \tan (c+d x)}{80 a d}-\frac {\sec ^7(c+d x) \tan ^3(c+d x)}{10 a d}-\frac {\int \sec ^5(c+d x) \, dx}{32 a}\\ &=-\frac {\sec ^8(c+d x)}{8 a d}+\frac {\sec ^{10}(c+d x)}{10 a d}-\frac {\sec ^3(c+d x) \tan (c+d x)}{128 a d}-\frac {\sec ^5(c+d x) \tan (c+d x)}{160 a d}+\frac {3 \sec ^7(c+d x) \tan (c+d x)}{80 a d}-\frac {\sec ^7(c+d x) \tan ^3(c+d x)}{10 a d}-\frac {3 \int \sec ^3(c+d x) \, dx}{128 a}\\ &=-\frac {\sec ^8(c+d x)}{8 a d}+\frac {\sec ^{10}(c+d x)}{10 a d}-\frac {3 \sec (c+d x) \tan (c+d x)}{256 a d}-\frac {\sec ^3(c+d x) \tan (c+d x)}{128 a d}-\frac {\sec ^5(c+d x) \tan (c+d x)}{160 a d}+\frac {3 \sec ^7(c+d x) \tan (c+d x)}{80 a d}-\frac {\sec ^7(c+d x) \tan ^3(c+d x)}{10 a d}-\frac {3 \int \sec (c+d x) \, dx}{256 a}\\ &=-\frac {3 \tanh ^{-1}(\sin (c+d x))}{256 a d}-\frac {\sec ^8(c+d x)}{8 a d}+\frac {\sec ^{10}(c+d x)}{10 a d}-\frac {3 \sec (c+d x) \tan (c+d x)}{256 a d}-\frac {\sec ^3(c+d x) \tan (c+d x)}{128 a d}-\frac {\sec ^5(c+d x) \tan (c+d x)}{160 a d}+\frac {3 \sec ^7(c+d x) \tan (c+d x)}{80 a d}-\frac {\sec ^7(c+d x) \tan ^3(c+d x)}{10 a d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 2.81, size = 104, normalized size = 0.60 \[ -\frac {-\frac {30}{\sin (c+d x)-1}+\frac {15}{(\sin (c+d x)-1)^2}+\frac {15}{(\sin (c+d x)+1)^2}+\frac {20}{(\sin (c+d x)+1)^3}-\frac {10}{(\sin (c+d x)-1)^4}+\frac {10}{(\sin (c+d x)+1)^4}-\frac {16}{(\sin (c+d x)+1)^5}+30 \tanh ^{-1}(\sin (c+d x))}{2560 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.51, size = 187, normalized size = 1.07 \[ \frac {30 \, \cos \left (d x + c\right )^{8} - 10 \, \cos \left (d x + c\right )^{6} - 4 \, \cos \left (d x + c\right )^{4} - 368 \, \cos \left (d x + c\right )^{2} - 15 \, {\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, {\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (15 \, \cos \left (d x + c\right )^{6} + 10 \, \cos \left (d x + c\right )^{4} + 8 \, \cos \left (d x + c\right )^{2} - 16\right )} \sin \left (d x + c\right ) + 288}{2560 \, {\left (a d \cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{8}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.31, size = 156, normalized size = 0.90 \[ -\frac {\frac {60 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {60 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac {5 \, {\left (25 \, \sin \left (d x + c\right )^{4} - 124 \, \sin \left (d x + c\right )^{3} + 234 \, \sin \left (d x + c\right )^{2} - 196 \, \sin \left (d x + c\right ) + 53\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{4}} - \frac {137 \, \sin \left (d x + c\right )^{5} + 685 \, \sin \left (d x + c\right )^{4} + 1310 \, \sin \left (d x + c\right )^{3} + 1110 \, \sin \left (d x + c\right )^{2} + 305 \, \sin \left (d x + c\right ) + 21}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{5}}}{10240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.38, size = 162, normalized size = 0.93 \[ \frac {1}{256 a d \left (\sin \left (d x +c \right )-1\right )^{4}}-\frac {3}{512 a d \left (\sin \left (d x +c \right )-1\right )^{2}}+\frac {3}{256 a d \left (\sin \left (d x +c \right )-1\right )}+\frac {3 \ln \left (\sin \left (d x +c \right )-1\right )}{512 a d}+\frac {1}{160 a d \left (1+\sin \left (d x +c \right )\right )^{5}}-\frac {1}{256 a d \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {1}{128 a d \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {3}{512 a d \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {3 \ln \left (1+\sin \left (d x +c \right )\right )}{512 a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.34, size = 214, normalized size = 1.23 \[ \frac {\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{8} + 15 \, \sin \left (d x + c\right )^{7} - 55 \, \sin \left (d x + c\right )^{6} - 55 \, \sin \left (d x + c\right )^{5} + 73 \, \sin \left (d x + c\right )^{4} + 73 \, \sin \left (d x + c\right )^{3} + 143 \, \sin \left (d x + c\right )^{2} - 17 \, \sin \left (d x + c\right ) - 32\right )}}{a \sin \left (d x + c\right )^{9} + a \sin \left (d x + c\right )^{8} - 4 \, a \sin \left (d x + c\right )^{7} - 4 \, a \sin \left (d x + c\right )^{6} + 6 \, a \sin \left (d x + c\right )^{5} + 6 \, a \sin \left (d x + c\right )^{4} - 4 \, a \sin \left (d x + c\right )^{3} - 4 \, a \sin \left (d x + c\right )^{2} + a \sin \left (d x + c\right ) + a} - \frac {15 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac {15 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a}}{2560 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 17.34, size = 496, normalized size = 2.85 \[ \frac {\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{128}+\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}}{64}-\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{32}+\frac {233\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{64}+\frac {323\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{160}+\frac {2687\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{320}-\frac {231\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{160}+\frac {5349\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{320}+\frac {353\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}+\frac {5349\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{320}-\frac {231\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{160}+\frac {2687\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{320}+\frac {323\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160}+\frac {233\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64}-\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{32}+\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64}+\frac {3\,\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 )}-\frac {3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{128\,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]
________________________________________________________________________________________