Optimal. Leaf size=179 \[ -\frac {a^2 (A-B)}{32 d (a \sin (c+d x)+a)^4}+\frac {5 A+3 B}{64 d \left (a^2-a^2 \sin (c+d x)\right )}-\frac {5 A+B}{32 d \left (a^2 \sin (c+d x)+a^2\right )}+\frac {5 (3 A+B) \tanh ^{-1}(\sin (c+d x))}{64 a^2 d}-\frac {a (3 A-B)}{48 d (a \sin (c+d x)+a)^3}+\frac {A+B}{64 d (a-a \sin (c+d x))^2}-\frac {3 A}{32 d (a \sin (c+d x)+a)^2} \]
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Rubi [A] time = 0.21, antiderivative size = 179, 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} \[ -\frac {a^2 (A-B)}{32 d (a \sin (c+d x)+a)^4}+\frac {5 A+3 B}{64 d \left (a^2-a^2 \sin (c+d x)\right )}-\frac {5 A+B}{32 d \left (a^2 \sin (c+d x)+a^2\right )}+\frac {5 (3 A+B) \tanh ^{-1}(\sin (c+d x))}{64 a^2 d}-\frac {a (3 A-B)}{48 d (a \sin (c+d x)+a)^3}+\frac {A+B}{64 d (a-a \sin (c+d x))^2}-\frac {3 A}{32 d (a \sin (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 77
Rule 206
Rule 2836
Rubi steps
\begin {align*} \int \frac {\sec ^5(c+d x) (A+B \sin (c+d x))}{(a+a \sin (c+d x))^2} \, dx &=\frac {a^5 \operatorname {Subst}\left (\int \frac {A+\frac {B x}{a}}{(a-x)^3 (a+x)^5} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^5 \operatorname {Subst}\left (\int \left (\frac {A+B}{32 a^5 (a-x)^3}+\frac {5 A+3 B}{64 a^6 (a-x)^2}+\frac {A-B}{8 a^3 (a+x)^5}+\frac {3 A-B}{16 a^4 (a+x)^4}+\frac {3 A}{16 a^5 (a+x)^3}+\frac {5 A+B}{32 a^6 (a+x)^2}+\frac {5 (3 A+B)}{64 a^6 \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {A+B}{64 d (a-a \sin (c+d x))^2}-\frac {a^2 (A-B)}{32 d (a+a \sin (c+d x))^4}-\frac {a (3 A-B)}{48 d (a+a \sin (c+d x))^3}-\frac {3 A}{32 d (a+a \sin (c+d x))^2}+\frac {5 A+3 B}{64 d \left (a^2-a^2 \sin (c+d x)\right )}-\frac {5 A+B}{32 d \left (a^2+a^2 \sin (c+d x)\right )}+\frac {(5 (3 A+B)) \operatorname {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{64 a d}\\ &=\frac {5 (3 A+B) \tanh ^{-1}(\sin (c+d x))}{64 a^2 d}+\frac {A+B}{64 d (a-a \sin (c+d x))^2}-\frac {a^2 (A-B)}{32 d (a+a \sin (c+d x))^4}-\frac {a (3 A-B)}{48 d (a+a \sin (c+d x))^3}-\frac {3 A}{32 d (a+a \sin (c+d x))^2}+\frac {5 A+3 B}{64 d \left (a^2-a^2 \sin (c+d x)\right )}-\frac {5 A+B}{32 d \left (a^2+a^2 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.69, size = 123, normalized size = 0.69 \[ \frac {-\frac {3 (5 A+3 B)}{\sin (c+d x)-1}-\frac {6 (5 A+B)}{\sin (c+d x)+1}+\frac {3 (A+B)}{(\sin (c+d x)-1)^2}+\frac {4 (B-3 A)}{(\sin (c+d x)+1)^3}-\frac {6 (A-B)}{(\sin (c+d x)+1)^4}+15 (3 A+B) \tanh ^{-1}(\sin (c+d x))-\frac {18 A}{(\sin (c+d x)+1)^2}}{192 a^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 260, normalized size = 1.45 \[ \frac {60 \, {\left (3 \, A + B\right )} \cos \left (d x + c\right )^{4} - 20 \, {\left (3 \, A + B\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left ({\left (3 \, A + B\right )} \cos \left (d x + c\right )^{6} - 2 \, {\left (3 \, A + B\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) - 2 \, {\left (3 \, A + B\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left ({\left (3 \, A + B\right )} \cos \left (d x + c\right )^{6} - 2 \, {\left (3 \, A + B\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) - 2 \, {\left (3 \, A + B\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (15 \, {\left (3 \, A + B\right )} \cos \left (d x + c\right )^{4} - 20 \, {\left (3 \, A + B\right )} \cos \left (d x + c\right )^{2} - 36 \, A - 12 \, B\right )} \sin \left (d x + c\right ) - 24 \, A - 72 \, B}{384 \, {\left (a^{2} d \cos \left (d x + c\right )^{6} - 2 \, a^{2} d \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) - 2 \, a^{2} d \cos \left (d x + c\right )^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 214, normalized size = 1.20 \[ \frac {\frac {60 \, {\left (3 \, A + B\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{2}} - \frac {60 \, {\left (3 \, A + B\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{2}} + \frac {6 \, {\left (45 \, A \sin \left (d x + c\right )^{2} + 15 \, B \sin \left (d x + c\right )^{2} - 110 \, A \sin \left (d x + c\right ) - 42 \, B \sin \left (d x + c\right ) + 69 \, A + 31 \, B\right )}}{a^{2} {\left (\sin \left (d x + c\right ) - 1\right )}^{2}} - \frac {375 \, A \sin \left (d x + c\right )^{4} + 125 \, B \sin \left (d x + c\right )^{4} + 1740 \, A \sin \left (d x + c\right )^{3} + 548 \, B \sin \left (d x + c\right )^{3} + 3114 \, A \sin \left (d x + c\right )^{2} + 894 \, B \sin \left (d x + c\right )^{2} + 2604 \, A \sin \left (d x + c\right ) + 612 \, B \sin \left (d x + c\right ) + 903 \, A + 93 \, B}{a^{2} {\left (\sin \left (d x + c\right ) + 1\right )}^{4}}}{1536 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.76, size = 283, normalized size = 1.58 \[ -\frac {15 \ln \left (\sin \left (d x +c \right )-1\right ) A}{128 d \,a^{2}}-\frac {5 \ln \left (\sin \left (d x +c \right )-1\right ) B}{128 d \,a^{2}}+\frac {A}{64 d \,a^{2} \left (\sin \left (d x +c \right )-1\right )^{2}}+\frac {B}{64 d \,a^{2} \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {5 A}{64 d \,a^{2} \left (\sin \left (d x +c \right )-1\right )}-\frac {3 B}{64 d \,a^{2} \left (\sin \left (d x +c \right )-1\right )}-\frac {3 A}{32 d \,a^{2} \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {A}{32 d \,a^{2} \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {B}{32 d \,a^{2} \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {A}{16 d \,a^{2} \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {B}{48 d \,a^{2} \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {5 A}{32 d \,a^{2} \left (1+\sin \left (d x +c \right )\right )}-\frac {B}{32 d \,a^{2} \left (1+\sin \left (d x +c \right )\right )}+\frac {15 \ln \left (1+\sin \left (d x +c \right )\right ) A}{128 d \,a^{2}}+\frac {5 B \ln \left (1+\sin \left (d x +c \right )\right )}{128 a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 207, normalized size = 1.16 \[ -\frac {\frac {2 \, {\left (15 \, {\left (3 \, A + B\right )} \sin \left (d x + c\right )^{5} + 30 \, {\left (3 \, A + B\right )} \sin \left (d x + c\right )^{4} - 10 \, {\left (3 \, A + B\right )} \sin \left (d x + c\right )^{3} - 50 \, {\left (3 \, A + B\right )} \sin \left (d x + c\right )^{2} - 17 \, {\left (3 \, A + B\right )} \sin \left (d x + c\right ) + 48 \, A - 16 \, B\right )}}{a^{2} \sin \left (d x + c\right )^{6} + 2 \, a^{2} \sin \left (d x + c\right )^{5} - a^{2} \sin \left (d x + c\right )^{4} - 4 \, a^{2} \sin \left (d x + c\right )^{3} - a^{2} \sin \left (d x + c\right )^{2} + 2 \, a^{2} \sin \left (d x + c\right ) + a^{2}} - \frac {15 \, {\left (3 \, A + B\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{2}} + \frac {15 \, {\left (3 \, A + B\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{2}}}{384 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.50, size = 193, normalized size = 1.08 \[ \frac {\left (-\frac {15\,A}{64}-\frac {5\,B}{64}\right )\,{\sin \left (c+d\,x\right )}^5+\left (-\frac {15\,A}{32}-\frac {5\,B}{32}\right )\,{\sin \left (c+d\,x\right )}^4+\left (\frac {5\,A}{32}+\frac {5\,B}{96}\right )\,{\sin \left (c+d\,x\right )}^3+\left (\frac {25\,A}{32}+\frac {25\,B}{96}\right )\,{\sin \left (c+d\,x\right )}^2+\left (\frac {17\,A}{64}+\frac {17\,B}{192}\right )\,\sin \left (c+d\,x\right )-\frac {A}{4}+\frac {B}{12}}{d\,\left (a^2\,{\sin \left (c+d\,x\right )}^6+2\,a^2\,{\sin \left (c+d\,x\right )}^5-a^2\,{\sin \left (c+d\,x\right )}^4-4\,a^2\,{\sin \left (c+d\,x\right )}^3-a^2\,{\sin \left (c+d\,x\right )}^2+2\,a^2\,\sin \left (c+d\,x\right )+a^2\right )}+\frac {5\,\mathrm {atanh}\left (\sin \left (c+d\,x\right )\right )\,\left (3\,A+B\right )}{64\,a^2\,d} \]
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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