Optimal. Leaf size=84 \[ \frac {2 a \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^2 d \sqrt {a^2-b^2}}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))}-\frac {x}{b^2} \]
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Rubi [A] time = 0.11, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2693, 2735, 2660, 618, 204} \[ \frac {2 a \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^2 d \sqrt {a^2-b^2}}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))}-\frac {x}{b^2} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 2660
Rule 2693
Rule 2735
Rubi steps
\begin {align*} \int \frac {\cos ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx &=-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))}-\frac {\int \frac {\sin (c+d x)}{a+b \sin (c+d x)} \, dx}{b}\\ &=-\frac {x}{b^2}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))}+\frac {a \int \frac {1}{a+b \sin (c+d x)} \, dx}{b^2}\\ &=-\frac {x}{b^2}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))}+\frac {(2 a) \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^2 d}\\ &=-\frac {x}{b^2}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))}-\frac {(4 a) \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^2 d}\\ &=-\frac {x}{b^2}+\frac {2 a \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^2 \sqrt {a^2-b^2} d}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))}\\ \end {align*}
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Mathematica [B] time = 2.69, size = 414, normalized size = 4.93 \[ \frac {\cos (c+d x) \left (\sqrt {a+b} \left ((b-a) \sqrt {-\frac {b (\sin (c+d x)-1)}{a+b}} \left (\sqrt {a-b} (a+b) \sqrt {1-\sin (c+d x)} \sqrt {-\frac {b (\sin (c+d x)+1)}{a-b}}+2 \sqrt {b} (a+b \sin (c+d x)) \sinh ^{-1}\left (\frac {\sqrt {a-b} \sqrt {-\frac {b (\sin (c+d x)+1)}{a-b}}}{\sqrt {2} \sqrt {b}}\right )\right )+2 a \sqrt {a-b} \sqrt {1-\sin (c+d x)} (a+b \sin (c+d x)) \tanh ^{-1}\left (\frac {\sqrt {\frac {b (\sin (c+d x)+1)}{b-a}}}{\sqrt {-\frac {b (\sin (c+d x)-1)}{a+b}}}\right )\right )-2 a (a-b) \sqrt {1-\sin (c+d x)} (a+b \sin (c+d x)) \tanh ^{-1}\left (\frac {\sqrt {a-b} \sqrt {-\frac {b (\sin (c+d x)+1)}{a-b}}}{\sqrt {a+b} \sqrt {-\frac {b (\sin (c+d x)-1)}{a+b}}}\right )\right )}{b d (a-b)^{3/2} (a+b)^{3/2} \sqrt {1-\sin (c+d x)} \sqrt {-\frac {b (\sin (c+d x)-1)}{a+b}} \sqrt {-\frac {b (\sin (c+d x)+1)}{a-b}} (a+b \sin (c+d x))} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 388, normalized size = 4.62 \[ \left [-\frac {2 \, {\left (a^{2} b - b^{3}\right )} d x \sin \left (d x + c\right ) + 2 \, {\left (a^{3} - a b^{2}\right )} d x + {\left (a b \sin \left (d x + c\right ) + a^{2}\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} + 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) + 2 \, {\left (a^{2} b - b^{3}\right )} \cos \left (d x + c\right )}{2 \, {\left ({\left (a^{2} b^{3} - b^{5}\right )} d \sin \left (d x + c\right ) + {\left (a^{3} b^{2} - a b^{4}\right )} d\right )}}, -\frac {{\left (a^{2} b - b^{3}\right )} d x \sin \left (d x + c\right ) + {\left (a^{3} - a b^{2}\right )} d x + {\left (a b \sin \left (d x + c\right ) + a^{2}\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) + {\left (a^{2} b - b^{3}\right )} \cos \left (d x + c\right )}{{\left (a^{2} b^{3} - b^{5}\right )} d \sin \left (d x + c\right ) + {\left (a^{3} b^{2} - a b^{4}\right )} d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.38, size = 126, normalized size = 1.50 \[ \frac {\frac {2 \, {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (a) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )} a}{\sqrt {a^{2} - b^{2}} b^{2}} - \frac {d x + c}{b^{2}} - \frac {2 \, {\left (b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )} a b}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 153, normalized size = 1.82 \[ -\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,b^{2}}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right ) a}-\frac {2}{d b \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right )}+\frac {2 a \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{d \,b^{2} \sqrt {a^{2}-b^{2}}} \]
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 = 5.56, size = 329, normalized size = 3.92 \[ -\frac {b^2\,\sin \left (c+d\,x\right )+\frac {\left (2\,a^3\,\mathrm {atan}\left (\frac {\left (-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b+2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2\right )\,1{}\mathrm {i}}{\sqrt {b^2-a^2}\,\left (a\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+2\,b\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}\right )-a^2\,\sqrt {b^2-a^2}\,\left (c+d\,x\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{\sqrt {b^2-a^2}}-\frac {b\,\left (a\,\sqrt {b^2-a^2}\,1{}\mathrm {i}+a\,\cos \left (c+d\,x\right )\,\sqrt {b^2-a^2}\,1{}\mathrm {i}-2\,a^2\,\mathrm {atan}\left (\frac {\left (-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b+2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2\right )\,1{}\mathrm {i}}{\sqrt {b^2-a^2}\,\left (a\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+2\,b\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}\right )\,\sin \left (c+d\,x\right )+a\,\sin \left (c+d\,x\right )\,\sqrt {b^2-a^2}\,\left (c+d\,x\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{\sqrt {b^2-a^2}}}{a\,b^2\,d\,\left (a+b\,\sin \left (c+d\,x\right )\right )} \]
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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