Optimal. Leaf size=57 \[ \frac {a^2 \sin (c+d x)}{d}+\frac {(a-b)^2 \sin ^5(c+d x)}{5 d}-\frac {2 a (a-b) \sin ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.06, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3676, 194} \[ \frac {a^2 \sin (c+d x)}{d}+\frac {(a-b)^2 \sin ^5(c+d x)}{5 d}-\frac {2 a (a-b) \sin ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 194
Rule 3676
Rubi steps
\begin {align*} \int \cos ^5(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \left (a-(a-b) x^2\right )^2 \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^2-2 a (a-b) x^2+(a-b)^2 x^4\right ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac {a^2 \sin (c+d x)}{d}-\frac {2 a (a-b) \sin ^3(c+d x)}{3 d}+\frac {(a-b)^2 \sin ^5(c+d x)}{5 d}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 52, normalized size = 0.91 \[ \frac {15 a^2 \sin (c+d x)+3 (a-b)^2 \sin ^5(c+d x)-10 a (a-b) \sin ^3(c+d x)}{15 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 71, normalized size = 1.25 \[ \frac {{\left (3 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (2 \, a^{2} + a b - 3 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 8 \, a^{2} + 4 \, a b + 3 \, b^{2}\right )} \sin \left (d x + c\right )}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [A] time = 0.85, size = 89, normalized size = 1.56 \[ \frac {\frac {b^{2} \left (\sin ^{5}\left (d x +c \right )\right )}{5}+2 a b \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {\left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{15}\right )+\frac {a^{2} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 56, normalized size = 0.98 \[ \frac {3 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \sin \left (d x + c\right )^{5} - 10 \, {\left (a^{2} - a b\right )} \sin \left (d x + c\right )^{3} + 15 \, a^{2} \sin \left (d x + c\right )}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.20, size = 119, normalized size = 2.09 \[ \frac {\frac {5\,a^2\,\sin \left (c+d\,x\right )}{8}+\frac {b^2\,\sin \left (c+d\,x\right )}{8}+\frac {5\,a^2\,\sin \left (3\,c+3\,d\,x\right )}{48}+\frac {a^2\,\sin \left (5\,c+5\,d\,x\right )}{80}-\frac {b^2\,\sin \left (3\,c+3\,d\,x\right )}{16}+\frac {b^2\,\sin \left (5\,c+5\,d\,x\right )}{80}+\frac {a\,b\,\sin \left (c+d\,x\right )}{4}-\frac {a\,b\,\sin \left (3\,c+3\,d\,x\right )}{24}-\frac {a\,b\,\sin \left (5\,c+5\,d\,x\right )}{40}}{d} \]
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 \[ \int \left (a + b \tan ^{2}{\left (c + d x \right )}\right )^{2} \cos ^{5}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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