Optimal. Leaf size=62 \[ \frac {(a-b)^2 \sin (c+d x)}{d}+\frac {b (4 a-3 b) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {b^2 \tan (c+d x) \sec (c+d x)}{2 d} \]
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Rubi [A] time = 0.09, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3676, 390, 385, 206} \[ \frac {(a-b)^2 \sin (c+d x)}{d}+\frac {b (4 a-3 b) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {b^2 \tan (c+d x) \sec (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 385
Rule 390
Rule 3676
Rubi steps
\begin {align*} \int \cos (c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a-(a-b) x^2\right )^2}{\left (1-x^2\right )^2} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left ((a-b)^2+\frac {(2 a-b) b-2 (a-b) b x^2}{\left (1-x^2\right )^2}\right ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac {(a-b)^2 \sin (c+d x)}{d}+\frac {\operatorname {Subst}\left (\int \frac {(2 a-b) b-2 (a-b) b x^2}{\left (1-x^2\right )^2} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac {(a-b)^2 \sin (c+d x)}{d}+\frac {b^2 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {((4 a-3 b) b) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (c+d x)\right )}{2 d}\\ &=\frac {(4 a-3 b) b \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {(a-b)^2 \sin (c+d x)}{d}+\frac {b^2 \sec (c+d x) \tan (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.46, size = 66, normalized size = 1.06 \[ \frac {\tan (c+d x) \sec (c+d x) \left (a^2+(a-b)^2 \cos (2 (c+d x))-2 a b+2 b^2\right )+b (4 a-3 b) \tanh ^{-1}(\sin (c+d x))}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 106, normalized size = 1.71 \[ \frac {{\left (4 \, a b - 3 \, b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (4 \, a b - 3 \, b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.60, size = 104, normalized size = 1.68 \[ \frac {4 \, a^{2} \sin \left (d x + c\right ) - 8 \, a b \sin \left (d x + c\right ) + 4 \, b^{2} \sin \left (d x + c\right ) + {\left (4 \, a b - 3 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - {\left (4 \, a b - 3 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac {2 \, b^{2} \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.40, size = 125, normalized size = 2.02 \[ \frac {b^{2} \left (\sin ^{5}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{2}}+\frac {b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{2 d}+\frac {3 b^{2} \sin \left (d x +c \right )}{2 d}-\frac {3 b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}-\frac {2 a b \sin \left (d x +c \right )}{d}+\frac {2 a b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {a^{2} \sin \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 105, normalized size = 1.69 \[ -\frac {b^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} + 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\sin \left (d x + c\right ) - 1\right ) - 4 \, \sin \left (d x + c\right )\right )} - 4 \, a b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right ) - 2 \, \sin \left (d x + c\right )\right )} - 4 \, a^{2} \sin \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 14.31, size = 148, normalized size = 2.39 \[ \frac {b\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (4\,a-3\,b\right )}{d}-\frac {\left (2\,a^2-4\,a\,b+3\,b^2\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-4\,a^2+8\,a\,b-2\,b^2\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,a^2-4\,a\,b+3\,b^2\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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 \[ \int \left (a + b \tan ^{2}{\left (c + d x \right )}\right )^{2} \cos {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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