Optimal. Leaf size=154 \[ \frac {\sec ^{10}(c+d x)}{10 a d}+\frac {7 \tanh ^{-1}(\sin (c+d x))}{256 a d}-\frac {\tan (c+d x) \sec ^9(c+d x)}{10 a d}+\frac {\tan (c+d x) \sec ^7(c+d x)}{80 a d}+\frac {7 \tan (c+d x) \sec ^5(c+d x)}{480 a d}+\frac {7 \tan (c+d x) \sec ^3(c+d x)}{384 a d}+\frac {7 \tan (c+d x) \sec (c+d x)}{256 a d} \]
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Rubi [A] time = 0.17, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2835, 2606, 30, 2611, 3768, 3770} \[ \frac {\sec ^{10}(c+d x)}{10 a d}+\frac {7 \tanh ^{-1}(\sin (c+d x))}{256 a d}-\frac {\tan (c+d x) \sec ^9(c+d x)}{10 a d}+\frac {\tan (c+d x) \sec ^7(c+d x)}{80 a d}+\frac {7 \tan (c+d x) \sec ^5(c+d x)}{480 a d}+\frac {7 \tan (c+d x) \sec ^3(c+d x)}{384 a d}+\frac {7 \tan (c+d x) \sec (c+d x)}{256 a d} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2606
Rule 2611
Rule 2835
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\sec ^8(c+d x) \tan (c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int \sec ^{10}(c+d x) \tan (c+d x) \, dx}{a}-\frac {\int \sec ^9(c+d x) \tan ^2(c+d x) \, dx}{a}\\ &=-\frac {\sec ^9(c+d x) \tan (c+d x)}{10 a d}+\frac {\int \sec ^9(c+d x) \, dx}{10 a}+\frac {\operatorname {Subst}\left (\int x^9 \, dx,x,\sec (c+d x)\right )}{a d}\\ &=\frac {\sec ^{10}(c+d x)}{10 a d}+\frac {\sec ^7(c+d x) \tan (c+d x)}{80 a d}-\frac {\sec ^9(c+d x) \tan (c+d x)}{10 a d}+\frac {7 \int \sec ^7(c+d x) \, dx}{80 a}\\ &=\frac {\sec ^{10}(c+d x)}{10 a d}+\frac {7 \sec ^5(c+d x) \tan (c+d x)}{480 a d}+\frac {\sec ^7(c+d x) \tan (c+d x)}{80 a d}-\frac {\sec ^9(c+d x) \tan (c+d x)}{10 a d}+\frac {7 \int \sec ^5(c+d x) \, dx}{96 a}\\ &=\frac {\sec ^{10}(c+d x)}{10 a d}+\frac {7 \sec ^3(c+d x) \tan (c+d x)}{384 a d}+\frac {7 \sec ^5(c+d x) \tan (c+d x)}{480 a d}+\frac {\sec ^7(c+d x) \tan (c+d x)}{80 a d}-\frac {\sec ^9(c+d x) \tan (c+d x)}{10 a d}+\frac {7 \int \sec ^3(c+d x) \, dx}{128 a}\\ &=\frac {\sec ^{10}(c+d x)}{10 a d}+\frac {7 \sec (c+d x) \tan (c+d x)}{256 a d}+\frac {7 \sec ^3(c+d x) \tan (c+d x)}{384 a d}+\frac {7 \sec ^5(c+d x) \tan (c+d x)}{480 a d}+\frac {\sec ^7(c+d x) \tan (c+d x)}{80 a d}-\frac {\sec ^9(c+d x) \tan (c+d x)}{10 a d}+\frac {7 \int \sec (c+d x) \, dx}{256 a}\\ &=\frac {7 \tanh ^{-1}(\sin (c+d x))}{256 a d}+\frac {\sec ^{10}(c+d x)}{10 a d}+\frac {7 \sec (c+d x) \tan (c+d x)}{256 a d}+\frac {7 \sec ^3(c+d x) \tan (c+d x)}{384 a d}+\frac {7 \sec ^5(c+d x) \tan (c+d x)}{480 a d}+\frac {\sec ^7(c+d x) \tan (c+d x)}{80 a d}-\frac {\sec ^9(c+d x) \tan (c+d x)}{10 a d}\\ \end {align*}
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Mathematica [A] time = 5.53, size = 116, normalized size = 0.75 \[ \frac {-\frac {210}{\sin (c+d x)-1}+\frac {135}{(\sin (c+d x)-1)^2}+\frac {75}{(\sin (c+d x)+1)^2}-\frac {80}{(\sin (c+d x)-1)^3}+\frac {100}{(\sin (c+d x)+1)^3}+\frac {30}{(\sin (c+d x)-1)^4}+\frac {90}{(\sin (c+d x)+1)^4}+\frac {48}{(\sin (c+d x)+1)^5}+210 \tanh ^{-1}(\sin (c+d x))}{7680 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 187, normalized size = 1.21 \[ -\frac {210 \, \cos \left (d x + c\right )^{8} - 70 \, \cos \left (d x + c\right )^{6} - 28 \, \cos \left (d x + c\right )^{4} - 16 \, \cos \left (d x + c\right )^{2} - 105 \, {\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, {\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (105 \, \cos \left (d x + c\right )^{6} + 70 \, \cos \left (d x + c\right )^{4} + 56 \, \cos \left (d x + c\right )^{2} + 48\right )} \sin \left (d x + c\right ) - 864}{7680 \, {\left (a d \cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{8}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 156, normalized size = 1.01 \[ \frac {\frac {420 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {420 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac {5 \, {\left (175 \, \sin \left (d x + c\right )^{4} - 868 \, \sin \left (d x + c\right )^{3} + 1662 \, \sin \left (d x + c\right )^{2} - 1484 \, \sin \left (d x + c\right ) + 539\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{4}} - \frac {959 \, \sin \left (d x + c\right )^{5} + 4795 \, \sin \left (d x + c\right )^{4} + 9290 \, \sin \left (d x + c\right )^{3} + 8290 \, \sin \left (d x + c\right )^{2} + 2735 \, \sin \left (d x + c\right ) - 293}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{5}}}{30720 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 180, normalized size = 1.17 \[ \frac {1}{256 a d \left (\sin \left (d x +c \right )-1\right )^{4}}-\frac {1}{96 a d \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {9}{512 a d \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {7}{256 a d \left (\sin \left (d x +c \right )-1\right )}-\frac {7 \ln \left (\sin \left (d x +c \right )-1\right )}{512 a d}+\frac {1}{160 a d \left (1+\sin \left (d x +c \right )\right )^{5}}+\frac {3}{256 a d \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {5}{384 a d \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {5}{512 a d \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {7 \ln \left (1+\sin \left (d x +c \right )\right )}{512 a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 214, normalized size = 1.39 \[ -\frac {\frac {2 \, {\left (105 \, \sin \left (d x + c\right )^{8} + 105 \, \sin \left (d x + c\right )^{7} - 385 \, \sin \left (d x + c\right )^{6} - 385 \, \sin \left (d x + c\right )^{5} + 511 \, \sin \left (d x + c\right )^{4} + 511 \, \sin \left (d x + c\right )^{3} - 279 \, \sin \left (d x + c\right )^{2} - 279 \, \sin \left (d x + c\right ) - 384\right )}}{a \sin \left (d x + c\right )^{9} + a \sin \left (d x + c\right )^{8} - 4 \, a \sin \left (d x + c\right )^{7} - 4 \, a \sin \left (d x + c\right )^{6} + 6 \, a \sin \left (d x + c\right )^{5} + 6 \, a \sin \left (d x + c\right )^{4} - 4 \, a \sin \left (d x + c\right )^{3} - 4 \, a \sin \left (d x + c\right )^{2} + a \sin \left (d x + c\right ) + a} - \frac {105 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac {105 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a}}{7680 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 16.75, size = 496, normalized size = 3.22 \[ \frac {7\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{128\,a\,d}+\frac {-\frac {7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{128}+\frac {121\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}}{64}+\frac {163\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{96}+\frac {289\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{192}-\frac {2261\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{480}+\frac {12551\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{960}+\frac {6097\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{480}+\frac {11837\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{960}-\frac {2471\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{192}+\frac {11837\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{960}+\frac {6097\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{480}+\frac {12551\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{960}-\frac {2261\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{480}+\frac {289\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{192}+\frac {163\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96}+\frac {121\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64}-\frac {7\,\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.
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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.
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