Optimal. Leaf size=73 \[ \frac {a^2 (A-2 B) \tan (c+d x)}{3 d}+\frac {a^2 (A-2 B) \sec (c+d x)}{3 d}+\frac {(A+B) \sec ^3(c+d x) (a \sin (c+d x)+a)^2}{3 d} \]
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Rubi [A] time = 0.12, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {2855, 2669, 3767, 8} \[ \frac {a^2 (A-2 B) \tan (c+d x)}{3 d}+\frac {a^2 (A-2 B) \sec (c+d x)}{3 d}+\frac {(A+B) \sec ^3(c+d x) (a \sin (c+d x)+a)^2}{3 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2669
Rule 2855
Rule 3767
Rubi steps
\begin {align*} \int \sec ^4(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx &=\frac {(A+B) \sec ^3(c+d x) (a+a \sin (c+d x))^2}{3 d}+\frac {1}{3} (a (A-2 B)) \int \sec ^2(c+d x) (a+a \sin (c+d x)) \, dx\\ &=\frac {a^2 (A-2 B) \sec (c+d x)}{3 d}+\frac {(A+B) \sec ^3(c+d x) (a+a \sin (c+d x))^2}{3 d}+\frac {1}{3} \left (a^2 (A-2 B)\right ) \int \sec ^2(c+d x) \, dx\\ &=\frac {a^2 (A-2 B) \sec (c+d x)}{3 d}+\frac {(A+B) \sec ^3(c+d x) (a+a \sin (c+d x))^2}{3 d}-\frac {\left (a^2 (A-2 B)\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 d}\\ &=\frac {a^2 (A-2 B) \sec (c+d x)}{3 d}+\frac {(A+B) \sec ^3(c+d x) (a+a \sin (c+d x))^2}{3 d}+\frac {a^2 (A-2 B) \tan (c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 121, normalized size = 1.66 \[ -\frac {a^2 A \tan ^3(c+d x)}{3 d}+\frac {2 a^2 A \sec ^3(c+d x)}{3 d}+\frac {a^2 A \tan (c+d x) \sec ^2(c+d x)}{d}+\frac {2 a^2 B \tan ^3(c+d x)}{3 d}-\frac {a^2 B \sec ^3(c+d x)}{3 d}+\frac {a^2 B \tan ^2(c+d x) \sec (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 120, normalized size = 1.64 \[ -\frac {{\left (A - 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} + {\left (2 \, A - B\right )} a^{2} \cos \left (d x + c\right ) + {\left (A + B\right )} a^{2} - {\left ({\left (A - 2 \, B\right )} a^{2} \cos \left (d x + c\right ) - {\left (A + B\right )} a^{2}\right )} \sin \left (d x + c\right )}{3 \, {\left (d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + {\left (d \cos \left (d x + c\right ) + 2 \, d\right )} \sin \left (d x + c\right ) - 2 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 78, normalized size = 1.07 \[ -\frac {2 \, {\left (3 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, A a^{2} - B a^{2}\right )}}{3 \, d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.66, size = 162, normalized size = 2.22 \[ \frac {\frac {a^{2} A \left (\sin ^{3}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )^{3}}+B \,a^{2} \left (\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}\right )+\frac {2 a^{2} A}{3 \cos \left (d x +c \right )^{3}}+\frac {2 B \,a^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )^{3}}-a^{2} A \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+\frac {B \,a^{2}}{3 \cos \left (d x +c \right )^{3}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 108, normalized size = 1.48 \[ \frac {A a^{2} \tan \left (d x + c\right )^{3} + 2 \, B a^{2} \tan \left (d x + c\right )^{3} + {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{2} - \frac {{\left (3 \, \cos \left (d x + c\right )^{2} - 1\right )} B a^{2}}{\cos \left (d x + c\right )^{3}} + \frac {2 \, A a^{2}}{\cos \left (d x + c\right )^{3}} + \frac {B a^{2}}{\cos \left (d x + c\right )^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.15, size = 77, normalized size = 1.05 \[ -\frac {\sqrt {2}\,a^2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {B}{2}-\frac {5\,A}{2}+\frac {A\,\cos \left (c+d\,x\right )}{2}+\frac {B\,\cos \left (c+d\,x\right )}{2}+\frac {3\,A\,\sin \left (c+d\,x\right )}{2}-\frac {3\,B\,\sin \left (c+d\,x\right )}{2}\right )}{6\,d\,{\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d\,x}{2}\right )}^3} \]
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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