Optimal. Leaf size=173 \[ \frac {(9 A-5 B+C) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a \cos (c+d x)+a}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {(3 A-2 B) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{a^{3/2} d}+\frac {(3 A-B+C) \tan (c+d x)}{2 a d \sqrt {a \cos (c+d x)+a}}-\frac {(A-B+C) \tan (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}} \]
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Rubi [A] time = 0.57, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {3041, 2984, 2985, 2649, 206, 2773} \[ \frac {(9 A-5 B+C) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a \cos (c+d x)+a}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {(3 A-2 B) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{a^{3/2} d}+\frac {(3 A-B+C) \tan (c+d x)}{2 a d \sqrt {a \cos (c+d x)+a}}-\frac {(A-B+C) \tan (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2649
Rule 2773
Rule 2984
Rule 2985
Rule 3041
Rubi steps
\begin {align*} \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx &=-\frac {(A-B+C) \tan (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {\int \frac {\left (a (3 A-B+C)-\frac {1}{2} a (3 A-3 B-C) \cos (c+d x)\right ) \sec ^2(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx}{2 a^2}\\ &=-\frac {(A-B+C) \tan (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(3 A-B+C) \tan (c+d x)}{2 a d \sqrt {a+a \cos (c+d x)}}+\frac {\int \frac {\left (-a^2 (3 A-2 B)+\frac {1}{2} a^2 (3 A-B+C) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx}{2 a^3}\\ &=-\frac {(A-B+C) \tan (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(3 A-B+C) \tan (c+d x)}{2 a d \sqrt {a+a \cos (c+d x)}}-\frac {(3 A-2 B) \int \sqrt {a+a \cos (c+d x)} \sec (c+d x) \, dx}{2 a^2}+\frac {(9 A-5 B+C) \int \frac {1}{\sqrt {a+a \cos (c+d x)}} \, dx}{4 a}\\ &=-\frac {(A-B+C) \tan (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(3 A-B+C) \tan (c+d x)}{2 a d \sqrt {a+a \cos (c+d x)}}+\frac {(3 A-2 B) \operatorname {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,-\frac {a \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{a d}-\frac {(9 A-5 B+C) \operatorname {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,-\frac {a \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{2 a d}\\ &=-\frac {(3 A-2 B) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{a^{3/2} d}+\frac {(9 A-5 B+C) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a+a \cos (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {(A-B+C) \tan (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(3 A-B+C) \tan (c+d x)}{2 a d \sqrt {a+a \cos (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 1.62, size = 196, normalized size = 1.13 \[ \frac {\cos ^3\left (\frac {1}{2} (c+d x)\right ) \cos ^2(c+d x) \left (A \sec ^2(c+d x)+B \sec (c+d x)+C\right ) \left (2 (9 A-5 B+C) \tanh ^{-1}\left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {4 \sqrt {2} (3 A-2 B) \cos ^2\left (\frac {1}{2} (c+d x)\right ) \tanh ^{-1}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right )-2 \sin \left (\frac {1}{2} (c+d x)\right ) (2 A \sec (c+d x)+3 A-B+C)}{\sin ^2\left (\frac {1}{2} (c+d x)\right )-1}\right )}{d (a (\cos (c+d x)+1))^{3/2} (2 A+2 B \cos (c+d x)+C \cos (2 (c+d x))+C)} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.51, size = 343, normalized size = 1.98 \[ \frac {\sqrt {2} {\left ({\left (9 \, A - 5 \, B + C\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (9 \, A - 5 \, B + C\right )} \cos \left (d x + c\right )^{2} + {\left (9 \, A - 5 \, B + C\right )} \cos \left (d x + c\right )\right )} \sqrt {a} \log \left (-\frac {a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {a} \sin \left (d x + c\right ) - 2 \, a \cos \left (d x + c\right ) - 3 \, a}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) - 2 \, {\left ({\left (3 \, A - 2 \, B\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (3 \, A - 2 \, B\right )} \cos \left (d x + c\right )^{2} + {\left (3 \, A - 2 \, B\right )} \cos \left (d x + c\right )\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {a} {\left (\cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) + 8 \, a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}\right ) + 4 \, {\left ({\left (3 \, A - B + C\right )} \cos \left (d x + c\right ) + 2 \, A\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{8 \, {\left (a^{2} d \cos \left (d x + c\right )^{3} + 2 \, a^{2} d \cos \left (d x + c\right )^{2} + a^{2} d \cos \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.14, size = 384, normalized size = 2.22 \[ -\frac {\frac {\sqrt {2} {\left (9 \, A \sqrt {a} - 5 \, B \sqrt {a} + C \sqrt {a}\right )} \log \left ({\left (\sqrt {a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2}\right )}{a^{2}} + \frac {4 \, {\left (3 \, A \sqrt {a} - 2 \, B \sqrt {a}\right )} \log \left ({\left | {\left (\sqrt {a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} - a {\left (2 \, \sqrt {2} + 3\right )} \right |}\right )}{a^{2}} - \frac {4 \, {\left (3 \, A \sqrt {a} - 2 \, B \sqrt {a}\right )} \log \left ({\left | {\left (\sqrt {a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} + a {\left (2 \, \sqrt {2} - 3\right )} \right |}\right )}{a^{2}} - \frac {16 \, \sqrt {2} {\left (3 \, {\left (\sqrt {a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} A \sqrt {a} - A a^{\frac {3}{2}}\right )}}{{\left ({\left (\sqrt {a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{4} - 6 \, {\left (\sqrt {a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} a + a^{2}\right )} a} - \frac {2 \, \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} {\left (\sqrt {2} A a - \sqrt {2} B a + \sqrt {2} C a\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{3}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.78, size = 1222, normalized size = 7.06 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A}{{\cos \left (c+d\,x\right )}^2\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (A + B \cos {\left (c + d x \right )} + C \cos ^{2}{\left (c + d x \right )}\right ) \sec ^{2}{\left (c + d x \right )}}{\left (a \left (\cos {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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