Optimal. Leaf size=206 \[ -\frac {\left (a^2-b^2\right )^2 (A b-a B)}{b^6 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \left (-5 a^2 B+4 a A b+b^2 B\right ) \log (a+b \sin (c+d x))}{b^6 d}-\frac {\left (-3 a^2 B+2 a A b+2 b^2 B\right ) \sin ^2(c+d x)}{2 b^4 d}+\frac {\left (-4 a^3 B+3 a^2 A b+4 a b^2 B-2 A b^3\right ) \sin (c+d x)}{b^5 d}+\frac {(A b-2 a B) \sin ^3(c+d x)}{3 b^3 d}+\frac {B \sin ^4(c+d x)}{4 b^2 d} \]
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Rubi [A] time = 0.27, antiderivative size = 206, 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} \[ -\frac {\left (-3 a^2 B+2 a A b+2 b^2 B\right ) \sin ^2(c+d x)}{2 b^4 d}+\frac {\left (3 a^2 A b-4 a^3 B+4 a b^2 B-2 A b^3\right ) \sin (c+d x)}{b^5 d}-\frac {\left (a^2-b^2\right )^2 (A b-a B)}{b^6 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \left (-5 a^2 B+4 a A b+b^2 B\right ) \log (a+b \sin (c+d x))}{b^6 d}+\frac {(A b-2 a B) \sin ^3(c+d x)}{3 b^3 d}+\frac {B \sin ^4(c+d x)}{4 b^2 d} \]
Antiderivative was successfully verified.
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Rule 772
Rule 2837
Rubi steps
\begin {align*} \int \frac {\cos ^5(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (A+\frac {B x}{b}\right ) \left (b^2-x^2\right )^2}{(a+x)^2} \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {3 a^2 A b-2 A b^3-4 a^3 B+4 a b^2 B}{b}+\frac {\left (-2 a A b+3 a^2 B-2 b^2 B\right ) x}{b}+\frac {(A b-2 a B) x^2}{b}+\frac {B x^3}{b}+\frac {\left (-a^2+b^2\right )^2 (A b-a B)}{b (a+x)^2}+\frac {\left (-a^2+b^2\right ) \left (4 a A b-5 a^2 B+b^2 B\right )}{b (a+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=-\frac {\left (a^2-b^2\right ) \left (4 a A b-5 a^2 B+b^2 B\right ) \log (a+b \sin (c+d x))}{b^6 d}+\frac {\left (3 a^2 A b-2 A b^3-4 a^3 B+4 a b^2 B\right ) \sin (c+d x)}{b^5 d}-\frac {\left (2 a A b-3 a^2 B+2 b^2 B\right ) \sin ^2(c+d x)}{2 b^4 d}+\frac {(A b-2 a B) \sin ^3(c+d x)}{3 b^3 d}+\frac {B \sin ^4(c+d x)}{4 b^2 d}-\frac {\left (a^2-b^2\right )^2 (A b-a B)}{b^6 d (a+b \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 2.16, size = 234, normalized size = 1.14 \[ \frac {4 \left (A-\frac {a B}{b}\right ) \left (\left (8 a^2 b-4 b^3\right ) \sin (c+d x)+\frac {b^4 \cos ^4(c+d x)-4 \left (a^2-b^2\right ) \left (3 a^2 \log (a+b \sin (c+d x))+a^2+3 a b \sin (c+d x) \log (a+b \sin (c+d x))-b^2\right )}{a+b \sin (c+d x)}-2 a b^2 \sin ^2(c+d x)\right )+B \left (6 b \left (a^2-b^2\right ) \sin ^2(c+d x)-12 a \left (a^2-2 b^2\right ) \sin (c+d x)+\frac {12 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{b}-4 a b^2 \sin ^3(c+d x)+3 b^3 \cos ^4(c+d x)\right )}{12 b^5 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 322, normalized size = 1.56 \[ \frac {96 \, B a^{5} - 96 \, A a^{4} b - 504 \, B a^{3} b^{2} + 432 \, A a^{2} b^{3} + 383 \, B a b^{4} - 256 \, A b^{5} - 8 \, {\left (5 \, B a b^{4} - 4 \, A b^{5}\right )} \cos \left (d x + c\right )^{4} + 16 \, {\left (15 \, B a^{3} b^{2} - 12 \, A a^{2} b^{3} - 13 \, B a b^{4} + 8 \, A b^{5}\right )} \cos \left (d x + c\right )^{2} + 96 \, {\left (5 \, B a^{5} - 4 \, A a^{4} b - 6 \, B a^{3} b^{2} + 4 \, A a^{2} b^{3} + B a b^{4} + {\left (5 \, B a^{4} b - 4 \, A a^{3} b^{2} - 6 \, B a^{2} b^{3} + 4 \, A a b^{4} + B b^{5}\right )} \sin \left (d x + c\right )\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) + {\left (24 \, B b^{5} \cos \left (d x + c\right )^{4} - 384 \, B a^{4} b + 288 \, A a^{3} b^{2} + 392 \, B a^{2} b^{3} - 208 \, A a b^{4} - 33 \, B b^{5} - 16 \, {\left (5 \, B a^{2} b^{3} - 4 \, A a b^{4} - 3 \, B b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{96 \, {\left (b^{7} d \sin \left (d x + c\right ) + a b^{6} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 328, normalized size = 1.59 \[ \frac {\frac {12 \, {\left (5 \, B a^{4} - 4 \, A a^{3} b - 6 \, B a^{2} b^{2} + 4 \, A a b^{3} + B b^{4}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{6}} - \frac {12 \, {\left (5 \, B a^{4} b \sin \left (d x + c\right ) - 4 \, A a^{3} b^{2} \sin \left (d x + c\right ) - 6 \, B a^{2} b^{3} \sin \left (d x + c\right ) + 4 \, A a b^{4} \sin \left (d x + c\right ) + B b^{5} \sin \left (d x + c\right ) + 4 \, B a^{5} - 3 \, A a^{4} b - 4 \, B a^{3} b^{2} + 2 \, A a^{2} b^{3} + A b^{5}\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )} b^{6}} + \frac {3 \, B b^{6} \sin \left (d x + c\right )^{4} - 8 \, B a b^{5} \sin \left (d x + c\right )^{3} + 4 \, A b^{6} \sin \left (d x + c\right )^{3} + 18 \, B a^{2} b^{4} \sin \left (d x + c\right )^{2} - 12 \, A a b^{5} \sin \left (d x + c\right )^{2} - 12 \, B b^{6} \sin \left (d x + c\right )^{2} - 48 \, B a^{3} b^{3} \sin \left (d x + c\right ) + 36 \, A a^{2} b^{4} \sin \left (d x + c\right ) + 48 \, B a b^{5} \sin \left (d x + c\right ) - 24 \, A b^{6} \sin \left (d x + c\right )}{b^{8}}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.66, size = 422, normalized size = 2.05 \[ \frac {B \left (\sin ^{4}\left (d x +c \right )\right )}{4 b^{2} d}+\frac {A \left (\sin ^{3}\left (d x +c \right )\right )}{3 d \,b^{2}}-\frac {2 B \left (\sin ^{3}\left (d x +c \right )\right ) a}{3 d \,b^{3}}-\frac {A \left (\sin ^{2}\left (d x +c \right )\right ) a}{d \,b^{3}}+\frac {3 B \left (\sin ^{2}\left (d x +c \right )\right ) a^{2}}{2 d \,b^{4}}-\frac {B \left (\sin ^{2}\left (d x +c \right )\right )}{b^{2} d}+\frac {3 A \,a^{2} \sin \left (d x +c \right )}{d \,b^{4}}-\frac {2 A \sin \left (d x +c \right )}{d \,b^{2}}-\frac {4 B \,a^{3} \sin \left (d x +c \right )}{d \,b^{5}}+\frac {4 B a \sin \left (d x +c \right )}{d \,b^{3}}-\frac {4 \ln \left (a +b \sin \left (d x +c \right )\right ) A \,a^{3}}{d \,b^{5}}+\frac {4 \ln \left (a +b \sin \left (d x +c \right )\right ) A a}{d \,b^{3}}+\frac {5 \ln \left (a +b \sin \left (d x +c \right )\right ) B \,a^{4}}{d \,b^{6}}-\frac {6 \ln \left (a +b \sin \left (d x +c \right )\right ) B \,a^{2}}{d \,b^{4}}+\frac {B \ln \left (a +b \sin \left (d x +c \right )\right )}{b^{2} d}-\frac {A \,a^{4}}{d \,b^{5} \left (a +b \sin \left (d x +c \right )\right )}+\frac {2 A \,a^{2}}{d \,b^{3} \left (a +b \sin \left (d x +c \right )\right )}-\frac {A}{d b \left (a +b \sin \left (d x +c \right )\right )}+\frac {B \,a^{5}}{d \,b^{6} \left (a +b \sin \left (d x +c \right )\right )}-\frac {2 B \,a^{3}}{d \,b^{4} \left (a +b \sin \left (d x +c \right )\right )}+\frac {a B}{d \,b^{2} \left (a +b \sin \left (d x +c \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.39, size = 229, normalized size = 1.11 \[ \frac {\frac {12 \, {\left (B a^{5} - A a^{4} b - 2 \, B a^{3} b^{2} + 2 \, A a^{2} b^{3} + B a b^{4} - A b^{5}\right )}}{b^{7} \sin \left (d x + c\right ) + a b^{6}} + \frac {3 \, B b^{3} \sin \left (d x + c\right )^{4} - 4 \, {\left (2 \, B a b^{2} - A b^{3}\right )} \sin \left (d x + c\right )^{3} + 6 \, {\left (3 \, B a^{2} b - 2 \, A a b^{2} - 2 \, B b^{3}\right )} \sin \left (d x + c\right )^{2} - 12 \, {\left (4 \, B a^{3} - 3 \, A a^{2} b - 4 \, B a b^{2} + 2 \, A b^{3}\right )} \sin \left (d x + c\right )}{b^{5}} + \frac {12 \, {\left (5 \, B a^{4} - 4 \, A a^{3} b - 6 \, B a^{2} b^{2} + 4 \, A a b^{3} + B b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{6}}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 290, normalized size = 1.41 \[ \frac {{\sin \left (c+d\,x\right )}^3\,\left (\frac {A}{3\,b^2}-\frac {2\,B\,a}{3\,b^3}\right )}{d}-\frac {\sin \left (c+d\,x\right )\,\left (\frac {2\,A}{b^2}+\frac {a^2\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{b^2}-\frac {2\,a\,\left (\frac {2\,B}{b^2}+\frac {2\,a\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{b}+\frac {B\,a^2}{b^4}\right )}{b}\right )}{d}-\frac {{\sin \left (c+d\,x\right )}^2\,\left (\frac {B}{b^2}+\frac {a\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{b}+\frac {B\,a^2}{2\,b^4}\right )}{d}+\frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )\,\left (5\,B\,a^4-4\,A\,a^3\,b-6\,B\,a^2\,b^2+4\,A\,a\,b^3+B\,b^4\right )}{b^6\,d}-\frac {-B\,a^5+A\,a^4\,b+2\,B\,a^3\,b^2-2\,A\,a^2\,b^3-B\,a\,b^4+A\,b^5}{b\,d\,\left (\sin \left (c+d\,x\right )\,b^6+a\,b^5\right )}+\frac {B\,{\sin \left (c+d\,x\right )}^4}{4\,b^2\,d} \]
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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