Optimal. Leaf size=194 \[ \frac {\sec ^{10}(c+d x)}{10 a d}-\frac {\sec ^8(c+d x)}{4 a d}+\frac {\sec ^6(c+d x)}{6 a d}+\frac {3 \tanh ^{-1}(\sin (c+d x))}{256 a d}-\frac {\tan ^5(c+d x) \sec ^5(c+d x)}{10 a d}+\frac {\tan ^3(c+d x) \sec ^5(c+d x)}{16 a d}-\frac {\tan (c+d x) \sec ^5(c+d x)}{32 a d}+\frac {\tan (c+d x) \sec ^3(c+d x)}{128 a d}+\frac {3 \tan (c+d x) \sec (c+d x)}{256 a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.27, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2835, 2606, 266, 43, 2611, 3768, 3770} \[ \frac {\sec ^{10}(c+d x)}{10 a d}-\frac {\sec ^8(c+d x)}{4 a d}+\frac {\sec ^6(c+d x)}{6 a d}+\frac {3 \tanh ^{-1}(\sin (c+d x))}{256 a d}-\frac {\tan ^5(c+d x) \sec ^5(c+d x)}{10 a d}+\frac {\tan ^3(c+d x) \sec ^5(c+d x)}{16 a d}-\frac {\tan (c+d x) \sec ^5(c+d x)}{32 a d}+\frac {\tan (c+d x) \sec ^3(c+d x)}{128 a d}+\frac {3 \tan (c+d x) \sec (c+d x)}{256 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 266
Rule 2606
Rule 2611
Rule 2835
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\sec ^4(c+d x) \tan ^5(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int \sec ^6(c+d x) \tan ^5(c+d x) \, dx}{a}-\frac {\int \sec ^5(c+d x) \tan ^6(c+d x) \, dx}{a}\\ &=-\frac {\sec ^5(c+d x) \tan ^5(c+d x)}{10 a d}+\frac {\int \sec ^5(c+d x) \tan ^4(c+d x) \, dx}{2 a}+\frac {\operatorname {Subst}\left (\int x^5 \left (-1+x^2\right )^2 \, dx,x,\sec (c+d x)\right )}{a d}\\ &=\frac {\sec ^5(c+d x) \tan ^3(c+d x)}{16 a d}-\frac {\sec ^5(c+d x) \tan ^5(c+d x)}{10 a d}-\frac {3 \int \sec ^5(c+d x) \tan ^2(c+d x) \, dx}{16 a}+\frac {\operatorname {Subst}\left (\int (-1+x)^2 x^2 \, dx,x,\sec ^2(c+d x)\right )}{2 a d}\\ &=-\frac {\sec ^5(c+d x) \tan (c+d x)}{32 a d}+\frac {\sec ^5(c+d x) \tan ^3(c+d x)}{16 a d}-\frac {\sec ^5(c+d x) \tan ^5(c+d x)}{10 a d}+\frac {\int \sec ^5(c+d x) \, dx}{32 a}+\frac {\operatorname {Subst}\left (\int \left (x^2-2 x^3+x^4\right ) \, dx,x,\sec ^2(c+d x)\right )}{2 a d}\\ &=\frac {\sec ^6(c+d x)}{6 a d}-\frac {\sec ^8(c+d x)}{4 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)}{32 a d}+\frac {\sec ^5(c+d x) \tan ^3(c+d x)}{16 a d}-\frac {\sec ^5(c+d x) \tan ^5(c+d x)}{10 a d}+\frac {3 \int \sec ^3(c+d x) \, dx}{128 a}\\ &=\frac {\sec ^6(c+d x)}{6 a d}-\frac {\sec ^8(c+d x)}{4 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)}{32 a d}+\frac {\sec ^5(c+d x) \tan ^3(c+d x)}{16 a d}-\frac {\sec ^5(c+d x) \tan ^5(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 ^6(c+d x)}{6 a d}-\frac {\sec ^8(c+d x)}{4 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)}{32 a d}+\frac {\sec ^5(c+d x) \tan ^3(c+d x)}{16 a d}-\frac {\sec ^5(c+d x) \tan ^5(c+d x)}{10 a d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 5.49, size = 116, normalized size = 0.60 \[ \frac {-\frac {90}{\sin (c+d x)-1}+\frac {15}{(\sin (c+d x)-1)^2}+\frac {75}{(\sin (c+d x)+1)^2}+\frac {80}{(\sin (c+d x)-1)^3}+\frac {100}{(\sin (c+d x)+1)^3}+\frac {30}{(\sin (c+d x)-1)^4}-\frac {150}{(\sin (c+d x)+1)^4}+\frac {48}{(\sin (c+d x)+1)^5}+90 \tanh ^{-1}(\sin (c+d x))}{7680 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.51, size = 187, normalized size = 0.96 \[ -\frac {90 \, \cos \left (d x + c\right )^{8} - 30 \, \cos \left (d x + c\right )^{6} - 1548 \, \cos \left (d x + c\right )^{4} + 2224 \, \cos \left (d x + c\right )^{2} - 45 \, {\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 45 \, {\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (45 \, \cos \left (d x + c\right )^{6} + 30 \, \cos \left (d x + c\right )^{4} - 104 \, \cos \left (d x + c\right )^{2} + 48\right )} \sin \left (d x + c\right ) - 864}{7680 \, {\left (a d \cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{8}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.32, size = 156, normalized size = 0.80 \[ \frac {\frac {180 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {180 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac {5 \, {\left (75 \, \sin \left (d x + c\right )^{4} - 372 \, \sin \left (d x + c\right )^{3} + 678 \, \sin \left (d x + c\right )^{2} - 476 \, \sin \left (d x + c\right ) + 119\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{4}} - \frac {411 \, \sin \left (d x + c\right )^{5} + 2055 \, \sin \left (d x + c\right )^{4} + 3810 \, \sin \left (d x + c\right )^{3} + 2810 \, \sin \left (d x + c\right )^{2} + 955 \, \sin \left (d x + c\right ) + 119}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{5}}}{30720 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.40, size = 180, normalized size = 0.93 \[ \frac {1}{256 a d \left (\sin \left (d x +c \right )-1\right )^{4}}+\frac {1}{96 a d \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {1}{512 a d \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {3}{256 a d \left (\sin \left (d x +c \right )-1\right )}-\frac {3 \ln \left (\sin \left (d x +c \right )-1\right )}{512 a d}+\frac {1}{160 a d \left (1+\sin \left (d x +c \right )\right )^{5}}-\frac {5}{256 a d \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {5}{384 a d \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {5}{512 a d \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {3 \ln \left (1+\sin \left (d x +c \right )\right )}{512 a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.48, size = 214, normalized size = 1.10 \[ -\frac {\frac {2 \, {\left (45 \, \sin \left (d x + c\right )^{8} + 45 \, \sin \left (d x + c\right )^{7} - 165 \, \sin \left (d x + c\right )^{6} - 165 \, \sin \left (d x + c\right )^{5} - 549 \, \sin \left (d x + c\right )^{4} + 91 \, \sin \left (d x + c\right )^{3} + 301 \, \sin \left (d x + c\right )^{2} - 19 \, \sin \left (d x + c\right ) - 64\right )}}{a \sin \left (d x + c\right )^{9} + a \sin \left (d x + c\right )^{8} - 4 \, a \sin \left (d x + c\right )^{7} - 4 \, a \sin \left (d x + c\right )^{6} + 6 \, a \sin \left (d x + c\right )^{5} + 6 \, a \sin \left (d x + c\right )^{4} - 4 \, a \sin \left (d x + c\right )^{3} - 4 \, a \sin \left (d x + c\right )^{2} + a \sin \left (d x + c\right ) + a} - \frac {45 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac {45 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a}}{7680 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 16.62, size = 496, normalized size = 2.56 \[ \frac {3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{128\,a\,d}+\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 {23\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{64}-\frac {67\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{160}+\frac {9091\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{960}+\frac {1717\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{480}+\frac {18257\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{960}-\frac {35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{192}+\frac {18257\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{960}+\frac {1717\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{480}+\frac {9091\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{960}-\frac {67\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160}+\frac {23\,{\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 )} \]
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]
________________________________________________________________________________________