Optimal. Leaf size=315 \[ -\frac {\left (a^2-b^2\right )^3 (A b-a B) \log (a+b \sin (c+d x))}{b^8 d}-\frac {\left (a^2-3 b^2\right ) (A b-a B) \sin ^4(c+d x)}{4 b^4 d}+\frac {\left (a^2 (-B)+a A b+3 b^2 B\right ) \sin ^5(c+d x)}{5 b^3 d}-\frac {\left (a^4-3 a^2 b^2+3 b^4\right ) (A b-a B) \sin ^2(c+d x)}{2 b^6 d}+\frac {\left (a^4 (-B)+a^3 A b+3 a^2 b^2 B-3 a A b^3-3 b^4 B\right ) \sin ^3(c+d x)}{3 b^5 d}+\frac {\left (a^6 (-B)+a^5 A b+3 a^4 b^2 B-3 a^3 A b^3-3 a^2 b^4 B+3 a A b^5+b^6 B\right ) \sin (c+d x)}{b^7 d}-\frac {(A b-a B) \sin ^6(c+d x)}{6 b^2 d}-\frac {B \sin ^7(c+d x)}{7 b d} \]
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Rubi [A] time = 0.36, antiderivative size = 315, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2837, 772} \[ \frac {\left (a^2 (-B)+a A b+3 b^2 B\right ) \sin ^5(c+d x)}{5 b^3 d}-\frac {\left (a^2-3 b^2\right ) (A b-a B) \sin ^4(c+d x)}{4 b^4 d}+\frac {\left (a^3 A b+3 a^2 b^2 B+a^4 (-B)-3 a A b^3-3 b^4 B\right ) \sin ^3(c+d x)}{3 b^5 d}-\frac {\left (-3 a^2 b^2+a^4+3 b^4\right ) (A b-a B) \sin ^2(c+d x)}{2 b^6 d}+\frac {\left (-3 a^3 A b^3+a^5 A b+3 a^4 b^2 B-3 a^2 b^4 B+a^6 (-B)+3 a A b^5+b^6 B\right ) \sin (c+d x)}{b^7 d}-\frac {\left (a^2-b^2\right )^3 (A b-a B) \log (a+b \sin (c+d x))}{b^8 d}-\frac {(A b-a B) \sin ^6(c+d x)}{6 b^2 d}-\frac {B \sin ^7(c+d x)}{7 b d} \]
Antiderivative was successfully verified.
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Rule 772
Rule 2837
Rubi steps
\begin {align*} \int \frac {\cos ^7(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (A+\frac {B x}{b}\right ) \left (b^2-x^2\right )^3}{a+x} \, dx,x,b \sin (c+d x)\right )}{b^7 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a^5 A b-3 a^3 A b^3+3 a A b^5-a^6 B+3 a^4 b^2 B-3 a^2 b^4 B+b^6 B}{b}-\frac {\left (a^4-3 a^2 b^2+3 b^4\right ) (A b-a B) x}{b}-\frac {\left (-a^3 A b+3 a A b^3+a^4 B-3 a^2 b^2 B+3 b^4 B\right ) x^2}{b}+\frac {\left (-a^2+3 b^2\right ) (A b-a B) x^3}{b}+\frac {\left (a A b-a^2 B+3 b^2 B\right ) x^4}{b}-\frac {(A b-a B) x^5}{b}-\frac {B x^6}{b}+\frac {\left (-a^2+b^2\right )^3 (A b-a B)}{b (a+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{b^7 d}\\ &=-\frac {\left (a^2-b^2\right )^3 (A b-a B) \log (a+b \sin (c+d x))}{b^8 d}+\frac {\left (a^5 A b-3 a^3 A b^3+3 a A b^5-a^6 B+3 a^4 b^2 B-3 a^2 b^4 B+b^6 B\right ) \sin (c+d x)}{b^7 d}-\frac {\left (a^4-3 a^2 b^2+3 b^4\right ) (A b-a B) \sin ^2(c+d x)}{2 b^6 d}+\frac {\left (a^3 A b-3 a A b^3-a^4 B+3 a^2 b^2 B-3 b^4 B\right ) \sin ^3(c+d x)}{3 b^5 d}-\frac {\left (a^2-3 b^2\right ) (A b-a B) \sin ^4(c+d x)}{4 b^4 d}+\frac {\left (a A b-a^2 B+3 b^2 B\right ) \sin ^5(c+d x)}{5 b^3 d}-\frac {(A b-a B) \sin ^6(c+d x)}{6 b^2 d}-\frac {B \sin ^7(c+d x)}{7 b d}\\ \end {align*}
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Mathematica [A] time = 0.87, size = 218, normalized size = 0.69 \[ \frac {\frac {(A b-a B) \left (-30 b^2 \left (a^2-b^2\right )^2 \sin ^2(c+d x)-60 \left (a^2-b^2\right )^3 \log (a+b \sin (c+d x))+15 b^4 \left (b^2-a^2\right ) \cos ^4(c+d x)+20 a b^3 \left (a^2-3 b^2\right ) \sin ^3(c+d x)+60 a b \left (a^4-3 a^2 b^2+3 b^4\right ) \sin (c+d x)+12 a b^5 \sin ^5(c+d x)+10 b^6 \cos ^6(c+d x)\right )}{60 b}+\frac {b^6 B (1225 \sin (c+d x)+245 \sin (3 (c+d x))+49 \sin (5 (c+d x))+5 \sin (7 (c+d x)))}{2240}}{b^7 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 366, normalized size = 1.16 \[ -\frac {70 \, {\left (B a b^{6} - A b^{7}\right )} \cos \left (d x + c\right )^{6} - 105 \, {\left (B a^{3} b^{4} - A a^{2} b^{5} - B a b^{6} + A b^{7}\right )} \cos \left (d x + c\right )^{4} + 210 \, {\left (B a^{5} b^{2} - A a^{4} b^{3} - 2 \, B a^{3} b^{4} + 2 \, A a^{2} b^{5} + B a b^{6} - A b^{7}\right )} \cos \left (d x + c\right )^{2} - 420 \, {\left (B a^{7} - A a^{6} b - 3 \, B a^{5} b^{2} + 3 \, A a^{4} b^{3} + 3 \, B a^{3} b^{4} - 3 \, A a^{2} b^{5} - B a b^{6} + A b^{7}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) - 4 \, {\left (15 \, B b^{7} \cos \left (d x + c\right )^{6} - 105 \, B a^{6} b + 105 \, A a^{5} b^{2} + 280 \, B a^{4} b^{3} - 280 \, A a^{3} b^{4} - 231 \, B a^{2} b^{5} + 231 \, A a b^{6} + 48 \, B b^{7} - 3 \, {\left (7 \, B a^{2} b^{5} - 7 \, A a b^{6} - 6 \, B b^{7}\right )} \cos \left (d x + c\right )^{4} + {\left (35 \, B a^{4} b^{3} - 35 \, A a^{3} b^{4} - 63 \, B a^{2} b^{5} + 63 \, A a b^{6} + 24 \, B b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{420 \, b^{8} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 511, normalized size = 1.62 \[ -\frac {\frac {60 \, B b^{6} \sin \left (d x + c\right )^{7} - 70 \, B a b^{5} \sin \left (d x + c\right )^{6} + 70 \, A b^{6} \sin \left (d x + c\right )^{6} + 84 \, B a^{2} b^{4} \sin \left (d x + c\right )^{5} - 84 \, A a b^{5} \sin \left (d x + c\right )^{5} - 252 \, B b^{6} \sin \left (d x + c\right )^{5} - 105 \, B a^{3} b^{3} \sin \left (d x + c\right )^{4} + 105 \, A a^{2} b^{4} \sin \left (d x + c\right )^{4} + 315 \, B a b^{5} \sin \left (d x + c\right )^{4} - 315 \, A b^{6} \sin \left (d x + c\right )^{4} + 140 \, B a^{4} b^{2} \sin \left (d x + c\right )^{3} - 140 \, A a^{3} b^{3} \sin \left (d x + c\right )^{3} - 420 \, B a^{2} b^{4} \sin \left (d x + c\right )^{3} + 420 \, A a b^{5} \sin \left (d x + c\right )^{3} + 420 \, B b^{6} \sin \left (d x + c\right )^{3} - 210 \, B a^{5} b \sin \left (d x + c\right )^{2} + 210 \, A a^{4} b^{2} \sin \left (d x + c\right )^{2} + 630 \, B a^{3} b^{3} \sin \left (d x + c\right )^{2} - 630 \, A a^{2} b^{4} \sin \left (d x + c\right )^{2} - 630 \, B a b^{5} \sin \left (d x + c\right )^{2} + 630 \, A b^{6} \sin \left (d x + c\right )^{2} + 420 \, B a^{6} \sin \left (d x + c\right ) - 420 \, A a^{5} b \sin \left (d x + c\right ) - 1260 \, B a^{4} b^{2} \sin \left (d x + c\right ) + 1260 \, A a^{3} b^{3} \sin \left (d x + c\right ) + 1260 \, B a^{2} b^{4} \sin \left (d x + c\right ) - 1260 \, A a b^{5} \sin \left (d x + c\right ) - 420 \, B b^{6} \sin \left (d x + c\right )}{b^{7}} - \frac {420 \, {\left (B a^{7} - A a^{6} b - 3 \, B a^{5} b^{2} + 3 \, A a^{4} b^{3} + 3 \, B a^{3} b^{4} - 3 \, A a^{2} b^{5} - B a b^{6} + A b^{7}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{8}}}{420 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.40, size = 689, normalized size = 2.19 \[ -\frac {3 A \left (\sin ^{2}\left (d x +c \right )\right )}{2 d b}-\frac {A \left (\sin ^{6}\left (d x +c \right )\right )}{6 d b}+\frac {3 A \sin \left (d x +c \right ) a}{d \,b^{2}}+\frac {3 B \sin \left (d x +c \right ) a^{4}}{d \,b^{5}}+\frac {\ln \left (a +b \sin \left (d x +c \right )\right ) A}{d b}-\frac {3 B \sin \left (d x +c \right ) a^{2}}{d \,b^{3}}+\frac {A \left (\sin ^{3}\left (d x +c \right )\right ) a^{3}}{3 d \,b^{4}}-\frac {B \left (\sin ^{3}\left (d x +c \right )\right ) a^{4}}{3 d \,b^{5}}-\frac {B \left (\sin ^{3}\left (d x +c \right )\right )}{b d}-\frac {B \left (\sin ^{7}\left (d x +c \right )\right )}{7 b d}+\frac {3 B \left (\sin ^{5}\left (d x +c \right )\right )}{5 b d}+\frac {B \left (\sin ^{6}\left (d x +c \right )\right ) a}{6 d \,b^{2}}+\frac {B \left (\sin ^{3}\left (d x +c \right )\right ) a^{2}}{d \,b^{3}}+\frac {B \left (\sin ^{4}\left (d x +c \right )\right ) a^{3}}{4 d \,b^{4}}+\frac {3 B \left (\sin ^{2}\left (d x +c \right )\right ) a}{2 d \,b^{2}}-\frac {B \left (\sin ^{5}\left (d x +c \right )\right ) a^{2}}{5 d \,b^{3}}-\frac {A \left (\sin ^{4}\left (d x +c \right )\right ) a^{2}}{4 d \,b^{3}}-\frac {3 B \left (\sin ^{4}\left (d x +c \right )\right ) a}{4 d \,b^{2}}-\frac {A \left (\sin ^{2}\left (d x +c \right )\right ) a^{4}}{2 d \,b^{5}}+\frac {3 A \left (\sin ^{2}\left (d x +c \right )\right ) a^{2}}{2 d \,b^{3}}+\frac {B \left (\sin ^{2}\left (d x +c \right )\right ) a^{5}}{2 d \,b^{6}}-\frac {3 B \left (\sin ^{2}\left (d x +c \right )\right ) a^{3}}{2 d \,b^{4}}+\frac {3 A \left (\sin ^{4}\left (d x +c \right )\right )}{4 d b}-\frac {3 A \sin \left (d x +c \right ) a^{3}}{d \,b^{4}}-\frac {3 \ln \left (a +b \sin \left (d x +c \right )\right ) B \,a^{5}}{d \,b^{6}}+\frac {3 \ln \left (a +b \sin \left (d x +c \right )\right ) B \,a^{3}}{d \,b^{4}}-\frac {\ln \left (a +b \sin \left (d x +c \right )\right ) B a}{d \,b^{2}}-\frac {\ln \left (a +b \sin \left (d x +c \right )\right ) A \,a^{6}}{d \,b^{7}}-\frac {3 \ln \left (a +b \sin \left (d x +c \right )\right ) A \,a^{2}}{d \,b^{3}}+\frac {B \sin \left (d x +c \right )}{b d}+\frac {A \left (\sin ^{5}\left (d x +c \right )\right ) a}{5 d \,b^{2}}+\frac {3 \ln \left (a +b \sin \left (d x +c \right )\right ) A \,a^{4}}{d \,b^{5}}-\frac {B \sin \left (d x +c \right ) a^{6}}{d \,b^{7}}+\frac {A \sin \left (d x +c \right ) a^{5}}{d \,b^{6}}+\frac {\ln \left (a +b \sin \left (d x +c \right )\right ) B \,a^{7}}{d \,b^{8}}-\frac {A \left (\sin ^{3}\left (d x +c \right )\right ) a}{d \,b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 366, normalized size = 1.16 \[ -\frac {\frac {60 \, B b^{6} \sin \left (d x + c\right )^{7} - 70 \, {\left (B a b^{5} - A b^{6}\right )} \sin \left (d x + c\right )^{6} + 84 \, {\left (B a^{2} b^{4} - A a b^{5} - 3 \, B b^{6}\right )} \sin \left (d x + c\right )^{5} - 105 \, {\left (B a^{3} b^{3} - A a^{2} b^{4} - 3 \, B a b^{5} + 3 \, A b^{6}\right )} \sin \left (d x + c\right )^{4} + 140 \, {\left (B a^{4} b^{2} - A a^{3} b^{3} - 3 \, B a^{2} b^{4} + 3 \, A a b^{5} + 3 \, B b^{6}\right )} \sin \left (d x + c\right )^{3} - 210 \, {\left (B a^{5} b - A a^{4} b^{2} - 3 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4} + 3 \, B a b^{5} - 3 \, A b^{6}\right )} \sin \left (d x + c\right )^{2} + 420 \, {\left (B a^{6} - A a^{5} b - 3 \, B a^{4} b^{2} + 3 \, A a^{3} b^{3} + 3 \, B a^{2} b^{4} - 3 \, A a b^{5} - B b^{6}\right )} \sin \left (d x + c\right )}{b^{7}} - \frac {420 \, {\left (B a^{7} - A a^{6} b - 3 \, B a^{5} b^{2} + 3 \, A a^{4} b^{3} + 3 \, B a^{3} b^{4} - 3 \, A a^{2} b^{5} - B a b^{6} + A b^{7}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{8}}}{420 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.37, size = 435, normalized size = 1.38 \[ \frac {{\sin \left (c+d\,x\right )}^4\,\left (\frac {3\,A}{4\,b}-\frac {a\,\left (\frac {3\,B}{b}+\frac {a\,\left (\frac {A}{b}-\frac {B\,a}{b^2}\right )}{b}\right )}{4\,b}\right )}{d}-\frac {{\sin \left (c+d\,x\right )}^3\,\left (\frac {B}{b}+\frac {a\,\left (\frac {3\,A}{b}-\frac {a\,\left (\frac {3\,B}{b}+\frac {a\,\left (\frac {A}{b}-\frac {B\,a}{b^2}\right )}{b}\right )}{b}\right )}{3\,b}\right )}{d}+\frac {\sin \left (c+d\,x\right )\,\left (\frac {B}{b}+\frac {a\,\left (\frac {3\,A}{b}-\frac {a\,\left (\frac {3\,B}{b}+\frac {a\,\left (\frac {3\,A}{b}-\frac {a\,\left (\frac {3\,B}{b}+\frac {a\,\left (\frac {A}{b}-\frac {B\,a}{b^2}\right )}{b}\right )}{b}\right )}{b}\right )}{b}\right )}{b}\right )}{d}-\frac {{\sin \left (c+d\,x\right )}^6\,\left (\frac {A}{6\,b}-\frac {B\,a}{6\,b^2}\right )}{d}-\frac {{\sin \left (c+d\,x\right )}^2\,\left (\frac {3\,A}{2\,b}-\frac {a\,\left (\frac {3\,B}{b}+\frac {a\,\left (\frac {3\,A}{b}-\frac {a\,\left (\frac {3\,B}{b}+\frac {a\,\left (\frac {A}{b}-\frac {B\,a}{b^2}\right )}{b}\right )}{b}\right )}{b}\right )}{2\,b}\right )}{d}+\frac {{\sin \left (c+d\,x\right )}^5\,\left (\frac {3\,B}{5\,b}+\frac {a\,\left (\frac {A}{b}-\frac {B\,a}{b^2}\right )}{5\,b}\right )}{d}+\frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )\,\left (B\,a^7-A\,a^6\,b-3\,B\,a^5\,b^2+3\,A\,a^4\,b^3+3\,B\,a^3\,b^4-3\,A\,a^2\,b^5-B\,a\,b^6+A\,b^7\right )}{b^8\,d}-\frac {B\,{\sin \left (c+d\,x\right )}^7}{7\,b\,d} \]
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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