Optimal. Leaf size=163 \[ -\frac {(5 A+12 C) \sin ^3(c+d x)}{3 a^2 d}+\frac {(5 A+12 C) \sin (c+d x)}{a^2 d}-\frac {2 (2 A+5 C) \sin (c+d x) \cos ^3(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac {(2 A+5 C) \sin (c+d x) \cos (c+d x)}{a^2 d}-\frac {x (2 A+5 C)}{a^2}-\frac {(A+C) \sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2} \]
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Rubi [A] time = 0.33, antiderivative size = 163, 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 = {3042, 2977, 2748, 2635, 8, 2633} \[ -\frac {(5 A+12 C) \sin ^3(c+d x)}{3 a^2 d}+\frac {(5 A+12 C) \sin (c+d x)}{a^2 d}-\frac {2 (2 A+5 C) \sin (c+d x) \cos ^3(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac {(2 A+5 C) \sin (c+d x) \cos (c+d x)}{a^2 d}-\frac {x (2 A+5 C)}{a^2}-\frac {(A+C) \sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 2748
Rule 2977
Rule 3042
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx &=-\frac {(A+C) \cos ^4(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac {\int \frac {\cos ^3(c+d x) (-a (A+4 C)+3 a (A+2 C) \cos (c+d x))}{a+a \cos (c+d x)} \, dx}{3 a^2}\\ &=-\frac {2 (2 A+5 C) \cos ^3(c+d x) \sin (c+d x)}{3 a^2 d (1+\cos (c+d x))}-\frac {(A+C) \cos ^4(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac {\int \cos ^2(c+d x) \left (-6 a^2 (2 A+5 C)+3 a^2 (5 A+12 C) \cos (c+d x)\right ) \, dx}{3 a^4}\\ &=-\frac {2 (2 A+5 C) \cos ^3(c+d x) \sin (c+d x)}{3 a^2 d (1+\cos (c+d x))}-\frac {(A+C) \cos ^4(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}-\frac {(2 (2 A+5 C)) \int \cos ^2(c+d x) \, dx}{a^2}+\frac {(5 A+12 C) \int \cos ^3(c+d x) \, dx}{a^2}\\ &=-\frac {(2 A+5 C) \cos (c+d x) \sin (c+d x)}{a^2 d}-\frac {2 (2 A+5 C) \cos ^3(c+d x) \sin (c+d x)}{3 a^2 d (1+\cos (c+d x))}-\frac {(A+C) \cos ^4(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}-\frac {(2 A+5 C) \int 1 \, dx}{a^2}-\frac {(5 A+12 C) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{a^2 d}\\ &=-\frac {(2 A+5 C) x}{a^2}+\frac {(5 A+12 C) \sin (c+d x)}{a^2 d}-\frac {(2 A+5 C) \cos (c+d x) \sin (c+d x)}{a^2 d}-\frac {2 (2 A+5 C) \cos ^3(c+d x) \sin (c+d x)}{3 a^2 d (1+\cos (c+d x))}-\frac {(A+C) \cos ^4(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}-\frac {(5 A+12 C) \sin ^3(c+d x)}{3 a^2 d}\\ \end {align*}
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Mathematica [B] time = 0.68, size = 341, normalized size = 2.09 \[ \frac {\sec \left (\frac {c}{2}\right ) \cos \left (\frac {1}{2} (c+d x)\right ) \left (-72 d x (2 A+5 C) \cos \left (c+\frac {d x}{2}\right )-120 A \sin \left (c+\frac {d x}{2}\right )+164 A \sin \left (c+\frac {3 d x}{2}\right )+36 A \sin \left (2 c+\frac {3 d x}{2}\right )+12 A \sin \left (2 c+\frac {5 d x}{2}\right )+12 A \sin \left (3 c+\frac {5 d x}{2}\right )-48 A d x \cos \left (c+\frac {3 d x}{2}\right )-48 A d x \cos \left (2 c+\frac {3 d x}{2}\right )-72 d x (2 A+5 C) \cos \left (\frac {d x}{2}\right )+264 A \sin \left (\frac {d x}{2}\right )-156 C \sin \left (c+\frac {d x}{2}\right )+342 C \sin \left (c+\frac {3 d x}{2}\right )+118 C \sin \left (2 c+\frac {3 d x}{2}\right )+30 C \sin \left (2 c+\frac {5 d x}{2}\right )+30 C \sin \left (3 c+\frac {5 d x}{2}\right )-3 C \sin \left (3 c+\frac {7 d x}{2}\right )-3 C \sin \left (4 c+\frac {7 d x}{2}\right )+C \sin \left (4 c+\frac {9 d x}{2}\right )+C \sin \left (5 c+\frac {9 d x}{2}\right )-120 C d x \cos \left (c+\frac {3 d x}{2}\right )-120 C d x \cos \left (2 c+\frac {3 d x}{2}\right )+516 C \sin \left (\frac {d x}{2}\right )\right )}{48 a^2 d (\cos (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 148, normalized size = 0.91 \[ -\frac {3 \, {\left (2 \, A + 5 \, C\right )} d x \cos \left (d x + c\right )^{2} + 6 \, {\left (2 \, A + 5 \, C\right )} d x \cos \left (d x + c\right ) + 3 \, {\left (2 \, A + 5 \, C\right )} d x - {\left (C \cos \left (d x + c\right )^{4} - C \cos \left (d x + c\right )^{3} + 3 \, {\left (A + 2 \, C\right )} \cos \left (d x + c\right )^{2} + {\left (14 \, A + 33 \, C\right )} \cos \left (d x + c\right ) + 10 \, A + 24 \, C\right )} \sin \left (d x + c\right )}{3 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 191, normalized size = 1.17 \[ -\frac {\frac {6 \, {\left (d x + c\right )} {\left (2 \, A + 5 \, C\right )}}{a^{2}} - \frac {4 \, {\left (3 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 20 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} a^{2}} + \frac {A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 27 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.12, size = 322, normalized size = 1.98 \[ -\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{6 d \,a^{2}}-\frac {C \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d \,a^{2}}+\frac {5 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d \,a^{2}}+\frac {9 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d \,a^{2}}+\frac {2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {10 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {4 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {40 C \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {2 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {6 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {4 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{d \,a^{2}}-\frac {10 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 325, normalized size = 1.99 \[ \frac {C {\left (\frac {4 \, {\left (\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {20 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{2} + \frac {3 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {\frac {27 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {60 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}\right )} + A {\left (\frac {\frac {15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {24 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} + \frac {12 \, \sin \left (d x + c\right )}{{\left (a^{2} + \frac {a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}}\right )}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.01, size = 181, normalized size = 1.11 \[ \frac {\left (2\,A+10\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (4\,A+\frac {40\,C}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,A+6\,C\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^2\right )}-\frac {x\,\left (2\,A+5\,C\right )}{a^2}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {2\,\left (A+C\right )}{a^2}+\frac {A+5\,C}{2\,a^2}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (A+C\right )}{6\,a^2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.01, size = 1426, normalized size = 8.75 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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