Optimal. Leaf size=236 \[ -\frac {a^3 (A-B)}{80 d (a \sin (c+d x)+a)^5}-\frac {a^2 (2 A-B)}{64 d (a \sin (c+d x)+a)^4}+\frac {3 (7 A+3 B)}{256 d \left (a^2-a^2 \sin (c+d x)\right )}-\frac {5 (7 A+B)}{256 d \left (a^2 \sin (c+d x)+a^2\right )}+\frac {7 (4 A+B) \tanh ^{-1}(\sin (c+d x))}{128 a^2 d}+\frac {a (A+B)}{192 d (a-a \sin (c+d x))^3}-\frac {a (5 A-B)}{96 d (a \sin (c+d x)+a)^3}+\frac {3 A+2 B}{128 d (a-a \sin (c+d x))^2}-\frac {5 A}{64 d (a \sin (c+d x)+a)^2} \]
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Rubi [A] time = 0.28, antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {2836, 77, 206} \[ -\frac {a^3 (A-B)}{80 d (a \sin (c+d x)+a)^5}-\frac {a^2 (2 A-B)}{64 d (a \sin (c+d x)+a)^4}+\frac {3 (7 A+3 B)}{256 d \left (a^2-a^2 \sin (c+d x)\right )}-\frac {5 (7 A+B)}{256 d \left (a^2 \sin (c+d x)+a^2\right )}+\frac {7 (4 A+B) \tanh ^{-1}(\sin (c+d x))}{128 a^2 d}+\frac {a (A+B)}{192 d (a-a \sin (c+d x))^3}-\frac {a (5 A-B)}{96 d (a \sin (c+d x)+a)^3}+\frac {3 A+2 B}{128 d (a-a \sin (c+d x))^2}-\frac {5 A}{64 d (a \sin (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 77
Rule 206
Rule 2836
Rubi steps
\begin {align*} \int \frac {\sec ^7(c+d x) (A+B \sin (c+d x))}{(a+a \sin (c+d x))^2} \, dx &=\frac {a^7 \operatorname {Subst}\left (\int \frac {A+\frac {B x}{a}}{(a-x)^4 (a+x)^6} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^7 \operatorname {Subst}\left (\int \left (\frac {A+B}{64 a^6 (a-x)^4}+\frac {3 A+2 B}{64 a^7 (a-x)^3}+\frac {3 (7 A+3 B)}{256 a^8 (a-x)^2}+\frac {A-B}{16 a^4 (a+x)^6}+\frac {2 A-B}{16 a^5 (a+x)^5}+\frac {5 A-B}{32 a^6 (a+x)^4}+\frac {5 A}{32 a^7 (a+x)^3}+\frac {5 (7 A+B)}{256 a^8 (a+x)^2}+\frac {7 (4 A+B)}{128 a^8 \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a (A+B)}{192 d (a-a \sin (c+d x))^3}+\frac {3 A+2 B}{128 d (a-a \sin (c+d x))^2}-\frac {a^3 (A-B)}{80 d (a+a \sin (c+d x))^5}-\frac {a^2 (2 A-B)}{64 d (a+a \sin (c+d x))^4}-\frac {a (5 A-B)}{96 d (a+a \sin (c+d x))^3}-\frac {5 A}{64 d (a+a \sin (c+d x))^2}+\frac {3 (7 A+3 B)}{256 d \left (a^2-a^2 \sin (c+d x)\right )}-\frac {5 (7 A+B)}{256 d \left (a^2+a^2 \sin (c+d x)\right )}+\frac {(7 (4 A+B)) \operatorname {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{128 a d}\\ &=\frac {7 (4 A+B) \tanh ^{-1}(\sin (c+d x))}{128 a^2 d}+\frac {a (A+B)}{192 d (a-a \sin (c+d x))^3}+\frac {3 A+2 B}{128 d (a-a \sin (c+d x))^2}-\frac {a^3 (A-B)}{80 d (a+a \sin (c+d x))^5}-\frac {a^2 (2 A-B)}{64 d (a+a \sin (c+d x))^4}-\frac {a (5 A-B)}{96 d (a+a \sin (c+d x))^3}-\frac {5 A}{64 d (a+a \sin (c+d x))^2}+\frac {3 (7 A+3 B)}{256 d \left (a^2-a^2 \sin (c+d x)\right )}-\frac {5 (7 A+B)}{256 d \left (a^2+a^2 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 1.48, size = 160, normalized size = 0.68 \[ \frac {210 (4 A+B) \tanh ^{-1}(\sin (c+d x))-\frac {2 \left (105 (4 A+B) \sin ^7(c+d x)+210 (4 A+B) \sin ^6(c+d x)-175 (4 A+B) \sin ^5(c+d x)-560 (4 A+B) \sin ^4(c+d x)-49 (4 A+B) \sin ^3(c+d x)+462 (4 A+B) \sin ^2(c+d x)+183 (4 A+B) \sin (c+d x)+48 (3 B-8 A)\right )}{(\sin (c+d x)-1)^3 (\sin (c+d x)+1)^5}}{3840 a^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 290, normalized size = 1.23 \[ \frac {420 \, {\left (4 \, A + B\right )} \cos \left (d x + c\right )^{6} - 140 \, {\left (4 \, A + B\right )} \cos \left (d x + c\right )^{4} - 56 \, {\left (4 \, A + B\right )} \cos \left (d x + c\right )^{2} + 105 \, {\left ({\left (4 \, A + B\right )} \cos \left (d x + c\right )^{8} - 2 \, {\left (4 \, A + B\right )} \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) - 2 \, {\left (4 \, A + B\right )} \cos \left (d x + c\right )^{6}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left ({\left (4 \, A + B\right )} \cos \left (d x + c\right )^{8} - 2 \, {\left (4 \, A + B\right )} \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) - 2 \, {\left (4 \, A + B\right )} \cos \left (d x + c\right )^{6}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (105 \, {\left (4 \, A + B\right )} \cos \left (d x + c\right )^{6} - 140 \, {\left (4 \, A + B\right )} \cos \left (d x + c\right )^{4} - 84 \, {\left (4 \, A + B\right )} \cos \left (d x + c\right )^{2} - 256 \, A - 64 \, B\right )} \sin \left (d x + c\right ) - 128 \, A - 512 \, B}{3840 \, {\left (a^{2} d \cos \left (d x + c\right )^{8} - 2 \, a^{2} d \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) - 2 \, a^{2} d \cos \left (d x + c\right )^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 258, normalized size = 1.09 \[ \frac {\frac {420 \, {\left (4 \, A + B\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{2}} - \frac {420 \, {\left (4 \, A + B\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{2}} + \frac {10 \, {\left (308 \, A \sin \left (d x + c\right )^{3} + 77 \, B \sin \left (d x + c\right )^{3} - 1050 \, A \sin \left (d x + c\right )^{2} - 285 \, B \sin \left (d x + c\right )^{2} + 1212 \, A \sin \left (d x + c\right ) + 363 \, B \sin \left (d x + c\right ) - 478 \, A - 163 \, B\right )}}{a^{2} {\left (\sin \left (d x + c\right ) - 1\right )}^{3}} - \frac {3836 \, A \sin \left (d x + c\right )^{5} + 959 \, B \sin \left (d x + c\right )^{5} + 21280 \, A \sin \left (d x + c\right )^{4} + 5095 \, B \sin \left (d x + c\right )^{4} + 47960 \, A \sin \left (d x + c\right )^{3} + 10790 \, B \sin \left (d x + c\right )^{3} + 55360 \, A \sin \left (d x + c\right )^{2} + 11230 \, B \sin \left (d x + c\right )^{2} + 33260 \, A \sin \left (d x + c\right ) + 5435 \, B \sin \left (d x + c\right ) + 8608 \, A + 667 \, B}{a^{2} {\left (\sin \left (d x + c\right ) + 1\right )}^{5}}}{15360 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.76, size = 359, normalized size = 1.52 \[ -\frac {7 \ln \left (\sin \left (d x +c \right )-1\right ) A}{64 d \,a^{2}}-\frac {7 \ln \left (\sin \left (d x +c \right )-1\right ) B}{256 d \,a^{2}}+\frac {3 A}{128 d \,a^{2} \left (\sin \left (d x +c \right )-1\right )^{2}}+\frac {B}{64 d \,a^{2} \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {A}{192 d \,a^{2} \left (\sin \left (d x +c \right )-1\right )^{3}}-\frac {B}{192 d \,a^{2} \left (\sin \left (d x +c \right )-1\right )^{3}}-\frac {21 A}{256 d \,a^{2} \left (\sin \left (d x +c \right )-1\right )}-\frac {9 B}{256 d \,a^{2} \left (\sin \left (d x +c \right )-1\right )}-\frac {5 A}{64 d \,a^{2} \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {A}{80 d \,a^{2} \left (1+\sin \left (d x +c \right )\right )^{5}}+\frac {B}{80 d \,a^{2} \left (1+\sin \left (d x +c \right )\right )^{5}}-\frac {A}{32 d \,a^{2} \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {B}{64 d \,a^{2} \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {5 A}{96 d \,a^{2} \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {B}{96 d \,a^{2} \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {7 \ln \left (1+\sin \left (d x +c \right )\right ) A}{64 d \,a^{2}}+\frac {7 B \ln \left (1+\sin \left (d x +c \right )\right )}{256 a^{2} d}-\frac {35 A}{256 d \,a^{2} \left (1+\sin \left (d x +c \right )\right )}-\frac {5 B}{256 d \,a^{2} \left (1+\sin \left (d x +c \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 252, normalized size = 1.07 \[ -\frac {\frac {2 \, {\left (105 \, {\left (4 \, A + B\right )} \sin \left (d x + c\right )^{7} + 210 \, {\left (4 \, A + B\right )} \sin \left (d x + c\right )^{6} - 175 \, {\left (4 \, A + B\right )} \sin \left (d x + c\right )^{5} - 560 \, {\left (4 \, A + B\right )} \sin \left (d x + c\right )^{4} - 49 \, {\left (4 \, A + B\right )} \sin \left (d x + c\right )^{3} + 462 \, {\left (4 \, A + B\right )} \sin \left (d x + c\right )^{2} + 183 \, {\left (4 \, A + B\right )} \sin \left (d x + c\right ) - 384 \, A + 144 \, B\right )}}{a^{2} \sin \left (d x + c\right )^{8} + 2 \, a^{2} \sin \left (d x + c\right )^{7} - 2 \, a^{2} \sin \left (d x + c\right )^{6} - 6 \, a^{2} \sin \left (d x + c\right )^{5} + 6 \, a^{2} \sin \left (d x + c\right )^{3} + 2 \, a^{2} \sin \left (d x + c\right )^{2} - 2 \, a^{2} \sin \left (d x + c\right ) - a^{2}} - \frac {105 \, {\left (4 \, A + B\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{2}} + \frac {105 \, {\left (4 \, A + B\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{2}}}{3840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.60, size = 240, normalized size = 1.02 \[ \frac {\left (\frac {7\,A}{32}+\frac {7\,B}{128}\right )\,{\sin \left (c+d\,x\right )}^7+\left (\frac {7\,A}{16}+\frac {7\,B}{64}\right )\,{\sin \left (c+d\,x\right )}^6+\left (-\frac {35\,A}{96}-\frac {35\,B}{384}\right )\,{\sin \left (c+d\,x\right )}^5+\left (-\frac {7\,A}{6}-\frac {7\,B}{24}\right )\,{\sin \left (c+d\,x\right )}^4+\left (-\frac {49\,A}{480}-\frac {49\,B}{1920}\right )\,{\sin \left (c+d\,x\right )}^3+\left (\frac {77\,A}{80}+\frac {77\,B}{320}\right )\,{\sin \left (c+d\,x\right )}^2+\left (\frac {61\,A}{160}+\frac {61\,B}{640}\right )\,\sin \left (c+d\,x\right )-\frac {A}{5}+\frac {3\,B}{40}}{d\,\left (-a^2\,{\sin \left (c+d\,x\right )}^8-2\,a^2\,{\sin \left (c+d\,x\right )}^7+2\,a^2\,{\sin \left (c+d\,x\right )}^6+6\,a^2\,{\sin \left (c+d\,x\right )}^5-6\,a^2\,{\sin \left (c+d\,x\right )}^3-2\,a^2\,{\sin \left (c+d\,x\right )}^2+2\,a^2\,\sin \left (c+d\,x\right )+a^2\right )}+\frac {7\,\mathrm {atanh}\left (\sin \left (c+d\,x\right )\right )\,\left (4\,A+B\right )}{128\,a^2\,d} \]
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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