Optimal. Leaf size=205 \[ -\frac {a^3 (A-B)}{64 d (a \sin (c+d x)+a)^4}+\frac {a^2 (A+B)}{96 d (a-a \sin (c+d x))^3}-\frac {a^2 (2 A-B)}{48 d (a \sin (c+d x)+a)^3}+\frac {a (5 A+3 B)}{128 d (a-a \sin (c+d x))^2}-\frac {a (5 A-B)}{64 d (a \sin (c+d x)+a)^2}+\frac {5 (3 A+B)}{128 d (a-a \sin (c+d x))}+\frac {5 (7 A+B) \tanh ^{-1}(\sin (c+d x))}{128 a d}-\frac {5 A}{32 d (a \sin (c+d x)+a)} \]
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Rubi [A] time = 0.25, antiderivative size = 205, 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} \[ -\frac {a^3 (A-B)}{64 d (a \sin (c+d x)+a)^4}+\frac {a^2 (A+B)}{96 d (a-a \sin (c+d x))^3}-\frac {a^2 (2 A-B)}{48 d (a \sin (c+d x)+a)^3}+\frac {a (5 A+3 B)}{128 d (a-a \sin (c+d x))^2}-\frac {a (5 A-B)}{64 d (a \sin (c+d x)+a)^2}+\frac {5 (3 A+B)}{128 d (a-a \sin (c+d x))}+\frac {5 (7 A+B) \tanh ^{-1}(\sin (c+d x))}{128 a d}-\frac {5 A}{32 d (a \sin (c+d x)+a)} \]
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)} \, dx &=\frac {a^7 \operatorname {Subst}\left (\int \frac {A+\frac {B x}{a}}{(a-x)^4 (a+x)^5} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^7 \operatorname {Subst}\left (\int \left (\frac {A+B}{32 a^5 (a-x)^4}+\frac {5 A+3 B}{64 a^6 (a-x)^3}+\frac {5 (3 A+B)}{128 a^7 (a-x)^2}+\frac {A-B}{16 a^4 (a+x)^5}+\frac {2 A-B}{16 a^5 (a+x)^4}+\frac {5 A-B}{32 a^6 (a+x)^3}+\frac {5 A}{32 a^7 (a+x)^2}+\frac {5 (7 A+B)}{128 a^7 \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^2 (A+B)}{96 d (a-a \sin (c+d x))^3}+\frac {a (5 A+3 B)}{128 d (a-a \sin (c+d x))^2}+\frac {5 (3 A+B)}{128 d (a-a \sin (c+d x))}-\frac {a^3 (A-B)}{64 d (a+a \sin (c+d x))^4}-\frac {a^2 (2 A-B)}{48 d (a+a \sin (c+d x))^3}-\frac {a (5 A-B)}{64 d (a+a \sin (c+d x))^2}-\frac {5 A}{32 d (a+a \sin (c+d x))}+\frac {(5 (7 A+B)) \operatorname {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{128 d}\\ &=\frac {5 (7 A+B) \tanh ^{-1}(\sin (c+d x))}{128 a d}+\frac {a^2 (A+B)}{96 d (a-a \sin (c+d x))^3}+\frac {a (5 A+3 B)}{128 d (a-a \sin (c+d x))^2}+\frac {5 (3 A+B)}{128 d (a-a \sin (c+d x))}-\frac {a^3 (A-B)}{64 d (a+a \sin (c+d x))^4}-\frac {a^2 (2 A-B)}{48 d (a+a \sin (c+d x))^3}-\frac {a (5 A-B)}{64 d (a+a \sin (c+d x))^2}-\frac {5 A}{32 d (a+a \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.88, size = 142, normalized size = 0.69 \[ \frac {\frac {-15 (7 A+B) \sin ^6(c+d x)-15 (7 A+B) \sin ^5(c+d x)+40 (7 A+B) \sin ^4(c+d x)+40 (7 A+B) \sin ^3(c+d x)-33 (7 A+B) \sin ^2(c+d x)-33 (7 A+B) \sin (c+d x)+48 (A-B)}{(\sin (c+d x)-1)^3 (\sin (c+d x)+1)^4}+15 (7 A+B) \tanh ^{-1}(\sin (c+d x))}{384 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 224, normalized size = 1.09 \[ -\frac {30 \, {\left (7 \, A + B\right )} \cos \left (d x + c\right )^{6} - 10 \, {\left (7 \, A + B\right )} \cos \left (d x + c\right )^{4} - 4 \, {\left (7 \, A + B\right )} \cos \left (d x + c\right )^{2} - 15 \, {\left ({\left (7 \, A + B\right )} \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + {\left (7 \, A + B\right )} \cos \left (d x + c\right )^{6}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, {\left ({\left (7 \, A + B\right )} \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + {\left (7 \, A + B\right )} \cos \left (d x + c\right )^{6}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (15 \, {\left (7 \, A + B\right )} \cos \left (d x + c\right )^{4} + 10 \, {\left (7 \, A + B\right )} \cos \left (d x + c\right )^{2} + 56 \, A + 8 \, B\right )} \sin \left (d x + c\right ) - 16 \, A - 112 \, B}{768 \, {\left (a d \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 236, normalized size = 1.15 \[ \frac {\frac {60 \, {\left (7 \, A + B\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {60 \, {\left (7 \, A + B\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac {2 \, {\left (385 \, A \sin \left (d x + c\right )^{3} + 55 \, B \sin \left (d x + c\right )^{3} - 1335 \, A \sin \left (d x + c\right )^{2} - 225 \, B \sin \left (d x + c\right )^{2} + 1575 \, A \sin \left (d x + c\right ) + 321 \, B \sin \left (d x + c\right ) - 641 \, A - 167 \, B\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{3}} - \frac {875 \, A \sin \left (d x + c\right )^{4} + 125 \, B \sin \left (d x + c\right )^{4} + 3980 \, A \sin \left (d x + c\right )^{3} + 500 \, B \sin \left (d x + c\right )^{3} + 6930 \, A \sin \left (d x + c\right )^{2} + 702 \, B \sin \left (d x + c\right )^{2} + 5548 \, A \sin \left (d x + c\right ) + 340 \, B \sin \left (d x + c\right ) + 1771 \, A - 35 \, B}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{4}}}{3072 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.50, size = 321, normalized size = 1.57 \[ -\frac {35 \ln \left (\sin \left (d x +c \right )-1\right ) A}{256 a d}-\frac {5 \ln \left (\sin \left (d x +c \right )-1\right ) B}{256 a d}+\frac {5 A}{128 a d \left (\sin \left (d x +c \right )-1\right )^{2}}+\frac {3 B}{128 a d \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {A}{96 a d \left (\sin \left (d x +c \right )-1\right )^{3}}-\frac {B}{96 a d \left (\sin \left (d x +c \right )-1\right )^{3}}-\frac {15 A}{128 a d \left (\sin \left (d x +c \right )-1\right )}-\frac {5 B}{128 a d \left (\sin \left (d x +c \right )-1\right )}-\frac {5 A}{32 a d \left (1+\sin \left (d x +c \right )\right )}-\frac {A}{64 a d \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {B}{64 a d \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {A}{24 a d \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {B}{48 a d \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {5 A}{64 a d \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {B}{64 a d \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {35 \ln \left (1+\sin \left (d x +c \right )\right ) A}{256 d a}+\frac {5 \ln \left (1+\sin \left (d x +c \right )\right ) B}{256 d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 220, normalized size = 1.07 \[ \frac {\frac {15 \, {\left (7 \, A + B\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac {15 \, {\left (7 \, A + B\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a} - \frac {2 \, {\left (15 \, {\left (7 \, A + B\right )} \sin \left (d x + c\right )^{6} + 15 \, {\left (7 \, A + B\right )} \sin \left (d x + c\right )^{5} - 40 \, {\left (7 \, A + B\right )} \sin \left (d x + c\right )^{4} - 40 \, {\left (7 \, A + B\right )} \sin \left (d x + c\right )^{3} + 33 \, {\left (7 \, A + B\right )} \sin \left (d x + c\right )^{2} + 33 \, {\left (7 \, A + B\right )} \sin \left (d x + c\right ) - 48 \, A + 48 \, B\right )}}{a \sin \left (d x + c\right )^{7} + a \sin \left (d x + c\right )^{6} - 3 \, a \sin \left (d x + c\right )^{5} - 3 \, a \sin \left (d x + c\right )^{4} + 3 \, a \sin \left (d x + c\right )^{3} + 3 \, a \sin \left (d x + c\right )^{2} - a \sin \left (d x + c\right ) - a}}{768 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.35, size = 206, normalized size = 1.00 \[ \frac {\left (\frac {35\,A}{128}+\frac {5\,B}{128}\right )\,{\sin \left (c+d\,x\right )}^6+\left (\frac {35\,A}{128}+\frac {5\,B}{128}\right )\,{\sin \left (c+d\,x\right )}^5+\left (-\frac {35\,A}{48}-\frac {5\,B}{48}\right )\,{\sin \left (c+d\,x\right )}^4+\left (-\frac {35\,A}{48}-\frac {5\,B}{48}\right )\,{\sin \left (c+d\,x\right )}^3+\left (\frac {77\,A}{128}+\frac {11\,B}{128}\right )\,{\sin \left (c+d\,x\right )}^2+\left (\frac {77\,A}{128}+\frac {11\,B}{128}\right )\,\sin \left (c+d\,x\right )-\frac {A}{8}+\frac {B}{8}}{d\,\left (-a\,{\sin \left (c+d\,x\right )}^7-a\,{\sin \left (c+d\,x\right )}^6+3\,a\,{\sin \left (c+d\,x\right )}^5+3\,a\,{\sin \left (c+d\,x\right )}^4-3\,a\,{\sin \left (c+d\,x\right )}^3-3\,a\,{\sin \left (c+d\,x\right )}^2+a\,\sin \left (c+d\,x\right )+a\right )}+\frac {5\,\mathrm {atanh}\left (\sin \left (c+d\,x\right )\right )\,\left (7\,A+B\right )}{128\,a\,d} \]
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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