Optimal. Leaf size=155 \[ -\frac {2 b^2 \sqrt {a-b} \sqrt {a+b} \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 d}-\frac {b \tan (c+d x) \sec (c+d x)}{2 a^2 d}+\frac {b \left (a^2-2 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac {\left (a^2-3 b^2\right ) \tan (c+d x)}{3 a^3 d}+\frac {\tan (c+d x) \sec ^2(c+d x)}{3 a d} \]
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Rubi [A] time = 0.57, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3056, 3055, 3001, 3770, 2659, 205} \[ -\frac {2 b^2 \sqrt {a-b} \sqrt {a+b} \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 d}-\frac {\left (a^2-3 b^2\right ) \tan (c+d x)}{3 a^3 d}+\frac {b \left (a^2-2 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac {b \tan (c+d x) \sec (c+d x)}{2 a^2 d}+\frac {\tan (c+d x) \sec ^2(c+d x)}{3 a d} \]
Antiderivative was successfully verified.
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Rule 205
Rule 2659
Rule 3001
Rule 3055
Rule 3056
Rule 3770
Rubi steps
\begin {align*} \int \frac {\left (1-\cos ^2(c+d x)\right ) \sec ^4(c+d x)}{a+b \cos (c+d x)} \, dx &=\frac {\sec ^2(c+d x) \tan (c+d x)}{3 a d}+\frac {\int \frac {\left (-3 b-a \cos (c+d x)+2 b \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{a+b \cos (c+d x)} \, dx}{3 a}\\ &=-\frac {b \sec (c+d x) \tan (c+d x)}{2 a^2 d}+\frac {\sec ^2(c+d x) \tan (c+d x)}{3 a d}+\frac {\int \frac {\left (-2 \left (a^2-3 b^2\right )+a b \cos (c+d x)-3 b^2 \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{6 a^2}\\ &=-\frac {\left (a^2-3 b^2\right ) \tan (c+d x)}{3 a^3 d}-\frac {b \sec (c+d x) \tan (c+d x)}{2 a^2 d}+\frac {\sec ^2(c+d x) \tan (c+d x)}{3 a d}+\frac {\int \frac {\left (3 b \left (a^2-2 b^2\right )-3 a b^2 \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)} \, dx}{6 a^3}\\ &=-\frac {\left (a^2-3 b^2\right ) \tan (c+d x)}{3 a^3 d}-\frac {b \sec (c+d x) \tan (c+d x)}{2 a^2 d}+\frac {\sec ^2(c+d x) \tan (c+d x)}{3 a d}+\frac {\left (b \left (a^2-2 b^2\right )\right ) \int \sec (c+d x) \, dx}{2 a^4}-\frac {\left (b^2 \left (a^2-b^2\right )\right ) \int \frac {1}{a+b \cos (c+d x)} \, dx}{a^4}\\ &=\frac {b \left (a^2-2 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac {\left (a^2-3 b^2\right ) \tan (c+d x)}{3 a^3 d}-\frac {b \sec (c+d x) \tan (c+d x)}{2 a^2 d}+\frac {\sec ^2(c+d x) \tan (c+d x)}{3 a d}-\frac {\left (2 b^2 \left (a^2-b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^4 d}\\ &=-\frac {2 \sqrt {a-b} b^2 \sqrt {a+b} \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 d}+\frac {b \left (a^2-2 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac {\left (a^2-3 b^2\right ) \tan (c+d x)}{3 a^3 d}-\frac {b \sec (c+d x) \tan (c+d x)}{2 a^2 d}+\frac {\sec ^2(c+d x) \tan (c+d x)}{3 a d}\\ \end {align*}
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Mathematica [A] time = 2.56, size = 256, normalized size = 1.65 \[ -\frac {24 b^2 \sqrt {b^2-a^2} \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b^2-a^2}}\right )+\frac {1}{2} \sec ^3(c+d x) \left (4 a \sin (c+d x) \left (\left (a^2-3 b^2\right ) \cos (2 (c+d x))-a^2+3 a b \cos (c+d x)-3 b^2\right )+9 b \left (a^2-2 b^2\right ) \cos (c+d x) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )+3 b \left (a^2-2 b^2\right ) \cos (3 (c+d x)) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )\right )}{12 a^4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 431, normalized size = 2.78 \[ \left [\frac {6 \, \sqrt {-a^{2} + b^{2}} b^{2} \cos \left (d x + c\right )^{3} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) + {\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \left (d x + c\right ) + b\right )} \sin \left (d x + c\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}}\right ) + 3 \, {\left (a^{2} b - 2 \, b^{3}\right )} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (a^{2} b - 2 \, b^{3}\right )} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (3 \, a^{2} b \cos \left (d x + c\right ) - 2 \, a^{3} + 2 \, {\left (a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{12 \, a^{4} d \cos \left (d x + c\right )^{3}}, -\frac {12 \, \sqrt {a^{2} - b^{2}} b^{2} \arctan \left (-\frac {a \cos \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \sin \left (d x + c\right )}\right ) \cos \left (d x + c\right )^{3} - 3 \, {\left (a^{2} b - 2 \, b^{3}\right )} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, {\left (a^{2} b - 2 \, b^{3}\right )} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (3 \, a^{2} b \cos \left (d x + c\right ) - 2 \, a^{3} + 2 \, {\left (a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{12 \, a^{4} d \cos \left (d x + c\right )^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.90, size = 266, normalized size = 1.72 \[ \frac {\frac {3 \, {\left (a^{2} b - 2 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac {3 \, {\left (a^{2} b - 2 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{4}} + \frac {12 \, {\left (a^{2} b^{2} - b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} a^{4}} - \frac {2 \, {\left (3 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 8 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3} a^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.18, size = 407, normalized size = 2.63 \[ -\frac {2 \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right ) b^{2}}{d \,a^{2} \sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {2 b^{4} \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{d \,a^{4} \sqrt {\left (a -b \right ) \left (a +b \right )}}-\frac {1}{3 d a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 d a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {b}{2 d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {b}{2 d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {b^{2}}{d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d \,a^{2}}+\frac {b^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d \,a^{4}}-\frac {1}{3 d a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1}{2 d a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {b}{2 d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {b}{2 d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {b^{2}}{d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d \,a^{2}}-\frac {b^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d \,a^{4}} \]
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 = 1.82, size = 224, normalized size = 1.45 \[ \frac {\sin \left (c+d\,x\right )}{3\,a\,d\,{\cos \left (c+d\,x\right )}^3}-\frac {\sin \left (c+d\,x\right )}{3\,a\,d\,\cos \left (c+d\,x\right )}-\frac {2\,b^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{a^4\,d}+\frac {b\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{a^2\,d}+\frac {2\,b^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {b^2-a^2}}{a\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+b\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\sqrt {b^2-a^2}}{a^4\,d}-\frac {b\,\sin \left (c+d\,x\right )}{2\,a^2\,d\,{\cos \left (c+d\,x\right )}^2}+\frac {b^2\,\sin \left (c+d\,x\right )}{a^3\,d\,\cos \left (c+d\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \left (- \frac {\sec ^{4}{\left (c + d x \right )}}{a + b \cos {\left (c + d x \right )}}\right )\, dx - \int \frac {\cos ^{2}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}}{a + b \cos {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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