Optimal. Leaf size=163 \[ -\frac {a x \left (4 a^2-3 b^2\right )}{b^5}-\frac {\cos (c+d x) \left (4 a^2-2 a b \sin (c+d x)-b^2\right )}{b^4 d}+\frac {2 \left (4 a^4-5 a^2 b^2+b^4\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^5 d \sqrt {a^2-b^2}}+\frac {\cos ^3(c+d x) (4 a+b \sin (c+d x))}{3 b^2 d (a+b \sin (c+d x))} \]
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Rubi [A] time = 0.31, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2863, 2865, 2735, 2660, 618, 204} \[ \frac {2 \left (-5 a^2 b^2+4 a^4+b^4\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^5 d \sqrt {a^2-b^2}}-\frac {\cos (c+d x) \left (4 a^2-2 a b \sin (c+d x)-b^2\right )}{b^4 d}-\frac {a x \left (4 a^2-3 b^2\right )}{b^5}+\frac {\cos ^3(c+d x) (4 a+b \sin (c+d x))}{3 b^2 d (a+b \sin (c+d x))} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 2660
Rule 2735
Rule 2863
Rule 2865
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx &=\frac {\cos ^3(c+d x) (4 a+b \sin (c+d x))}{3 b^2 d (a+b \sin (c+d x))}-\frac {\int \frac {\cos ^2(c+d x) (-b-4 a \sin (c+d x))}{a+b \sin (c+d x)} \, dx}{b^2}\\ &=\frac {\cos ^3(c+d x) (4 a+b \sin (c+d x))}{3 b^2 d (a+b \sin (c+d x))}-\frac {\cos (c+d x) \left (4 a^2-b^2-2 a b \sin (c+d x)\right )}{b^4 d}-\frac {\int \frac {2 b \left (2 a^2-b^2\right )+2 a \left (4 a^2-3 b^2\right ) \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{2 b^4}\\ &=-\frac {a \left (4 a^2-3 b^2\right ) x}{b^5}+\frac {\cos ^3(c+d x) (4 a+b \sin (c+d x))}{3 b^2 d (a+b \sin (c+d x))}-\frac {\cos (c+d x) \left (4 a^2-b^2-2 a b \sin (c+d x)\right )}{b^4 d}+\frac {\left (4 a^4-5 a^2 b^2+b^4\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{b^5}\\ &=-\frac {a \left (4 a^2-3 b^2\right ) x}{b^5}+\frac {\cos ^3(c+d x) (4 a+b \sin (c+d x))}{3 b^2 d (a+b \sin (c+d x))}-\frac {\cos (c+d x) \left (4 a^2-b^2-2 a b \sin (c+d x)\right )}{b^4 d}+\frac {\left (2 \left (4 a^4-5 a^2 b^2+b^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^5 d}\\ &=-\frac {a \left (4 a^2-3 b^2\right ) x}{b^5}+\frac {\cos ^3(c+d x) (4 a+b \sin (c+d x))}{3 b^2 d (a+b \sin (c+d x))}-\frac {\cos (c+d x) \left (4 a^2-b^2-2 a b \sin (c+d x)\right )}{b^4 d}-\frac {\left (4 \left (4 a^4-5 a^2 b^2+b^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^5 d}\\ &=-\frac {a \left (4 a^2-3 b^2\right ) x}{b^5}+\frac {2 \left (4 a^4-5 a^2 b^2+b^4\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^5 \sqrt {a^2-b^2} d}+\frac {\cos ^3(c+d x) (4 a+b \sin (c+d x))}{3 b^2 d (a+b \sin (c+d x))}-\frac {\cos (c+d x) \left (4 a^2-b^2-2 a b \sin (c+d x)\right )}{b^4 d}\\ \end {align*}
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Mathematica [A] time = 2.51, size = 247, normalized size = 1.52 \[ \frac {\frac {48 \left (4 a^4-5 a^2 b^2+b^4\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+\frac {-96 a^4 c-96 a^4 d x+\left (60 a b^3-96 a^3 b\right ) \cos (c+d x)-96 a^3 b c \sin (c+d x)-96 a^3 b d x \sin (c+d x)-24 a^2 b^2 \sin (2 (c+d x))+72 a^2 b^2 c+72 a^2 b^2 d x+72 a b^3 c \sin (c+d x)+72 a b^3 d x \sin (c+d x)-4 a b^3 \cos (3 (c+d x))+14 b^4 \sin (2 (c+d x))+b^4 \sin (4 (c+d x))}{a+b \sin (c+d x)}}{24 b^5 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 507, normalized size = 3.11 \[ \left [-\frac {4 \, a b^{3} \cos \left (d x + c\right )^{3} + 6 \, {\left (4 \, a^{4} - 3 \, a^{2} b^{2}\right )} d x + 3 \, {\left (4 \, a^{3} - a b^{2} + {\left (4 \, a^{2} b - b^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} + 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) + 6 \, {\left (4 \, a^{3} b - 3 \, a b^{3}\right )} \cos \left (d x + c\right ) - 2 \, {\left (b^{4} \cos \left (d x + c\right )^{3} - 3 \, {\left (4 \, a^{3} b - 3 \, a b^{3}\right )} d x - 3 \, {\left (2 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, {\left (b^{6} d \sin \left (d x + c\right ) + a b^{5} d\right )}}, -\frac {2 \, a b^{3} \cos \left (d x + c\right )^{3} + 3 \, {\left (4 \, a^{4} - 3 \, a^{2} b^{2}\right )} d x + 3 \, {\left (4 \, a^{3} - a b^{2} + {\left (4 \, a^{2} b - b^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) + 3 \, {\left (4 \, a^{3} b - 3 \, a b^{3}\right )} \cos \left (d x + c\right ) - {\left (b^{4} \cos \left (d x + c\right )^{3} - 3 \, {\left (4 \, a^{3} b - 3 \, a b^{3}\right )} d x - 3 \, {\left (2 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3 \, {\left (b^{6} d \sin \left (d x + c\right ) + a b^{5} d\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 300, normalized size = 1.84 \[ -\frac {\frac {3 \, {\left (4 \, a^{3} - 3 \, a b^{2}\right )} {\left (d x + c\right )}}{b^{5}} - \frac {6 \, {\left (4 \, a^{4} - 5 \, a^{2} b^{2} + b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (a) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} b^{5}} + \frac {6 \, {\left (a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3} - a b^{2}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )} b^{4}} + \frac {2 \, {\left (3 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 6 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 18 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 6 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, a^{2} - 4 \, b^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} b^{4}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.47, size = 627, normalized size = 3.85 \[ -\frac {2 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,b^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {6 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}}{d \,b^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {4 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,b^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {12 a^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,b^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,b^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,b^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {6 a^{2}}{d \,b^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {8}{3 d \,b^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {8 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{d \,b^{5}}+\frac {6 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{d \,b^{3}}-\frac {2 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,b^{3} \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right )}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d b \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right )}-\frac {2 a^{3}}{d \,b^{4} \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right )}+\frac {2 a}{d \,b^{2} \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right )}+\frac {8 a^{4} \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{d \,b^{5} \sqrt {a^{2}-b^{2}}}-\frac {10 a^{2} \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{d \,b^{3} \sqrt {a^{2}-b^{2}}}+\frac {2 \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{d b \sqrt {a^{2}-b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.53, size = 964, normalized size = 5.91 \[ \text {result too large to display} \]
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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