Optimal. Leaf size=76 \[ \frac {1}{2} x \left (a^2-3 b^2\right )-\frac {2 a b \log (\cos (c+d x))}{d}-\frac {\sin (c+d x) \cos (c+d x) (a+b \tan (c+d x))^2}{2 d}+\frac {3 b^2 \tan (c+d x)}{2 d} \]
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Rubi [A] time = 0.11, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3516, 1645, 774, 635, 203, 260} \[ \frac {1}{2} x \left (a^2-3 b^2\right )-\frac {2 a b \log (\cos (c+d x))}{d}-\frac {\sin (c+d x) \cos (c+d x) (a+b \tan (c+d x))^2}{2 d}+\frac {3 b^2 \tan (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 635
Rule 774
Rule 1645
Rule 3516
Rubi steps
\begin {align*} \int \sin ^2(c+d x) (a+b \tan (c+d x))^2 \, dx &=\frac {b \operatorname {Subst}\left (\int \frac {x^2 (a+x)^2}{\left (b^2+x^2\right )^2} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=-\frac {\cos (c+d x) \sin (c+d x) (a+b \tan (c+d x))^2}{2 d}-\frac {\operatorname {Subst}\left (\int \frac {(a+x) \left (-a b^2-3 b^2 x\right )}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{2 b d}\\ &=\frac {3 b^2 \tan (c+d x)}{2 d}-\frac {\cos (c+d x) \sin (c+d x) (a+b \tan (c+d x))^2}{2 d}-\frac {\operatorname {Subst}\left (\int \frac {-a^2 b^2+3 b^4-4 a b^2 x}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{2 b d}\\ &=\frac {3 b^2 \tan (c+d x)}{2 d}-\frac {\cos (c+d x) \sin (c+d x) (a+b \tan (c+d x))^2}{2 d}+\frac {(2 a b) \operatorname {Subst}\left (\int \frac {x}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{d}+\frac {\left (b \left (a^2-3 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{2 d}\\ &=\frac {1}{2} \left (a^2-3 b^2\right ) x-\frac {2 a b \log (\cos (c+d x))}{d}+\frac {3 b^2 \tan (c+d x)}{2 d}-\frac {\cos (c+d x) \sin (c+d x) (a+b \tan (c+d x))^2}{2 d}\\ \end {align*}
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Mathematica [B] time = 2.50, size = 162, normalized size = 2.13 \[ \frac {b \left (\frac {\left (b^2-a^2\right ) \sin (2 (c+d x))}{2 b}+\frac {\left (b^2-a^2\right ) \tan ^{-1}(\tan (c+d x))}{b}+\left (\frac {a^2-2 b^2}{\sqrt {-b^2}}+2 a\right ) \log \left (\sqrt {-b^2}-b \tan (c+d x)\right )+\left (\frac {2 b^2-a^2}{\sqrt {-b^2}}+2 a\right ) \log \left (\sqrt {-b^2}+b \tan (c+d x)\right )+2 a \cos ^2(c+d x)+2 b \tan (c+d x)\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 101, normalized size = 1.33 \[ \frac {2 \, a b \cos \left (d x + c\right )^{3} - 4 \, a b \cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right ) + {\left ({\left (a^{2} - 3 \, b^{2}\right )} d x - a b\right )} \cos \left (d x + c\right ) - {\left ({\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, b^{2}\right )} \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.17, size = 1061, normalized size = 13.96 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.34, size = 145, normalized size = 1.91 \[ -\frac {a^{2} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}+\frac {a^{2} x}{2}+\frac {a^{2} c}{2 d}-\frac {a b \left (\sin ^{2}\left (d x +c \right )\right )}{d}-\frac {2 a b \ln \left (\cos \left (d x +c \right )\right )}{d}+\frac {b^{2} \left (\sin ^{5}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )}+\frac {b^{2} \cos \left (d x +c \right ) \left (\sin ^{3}\left (d x +c \right )\right )}{d}+\frac {3 b^{2} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}-\frac {3 b^{2} x}{2}-\frac {3 c \,b^{2}}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 82, normalized size = 1.08 \[ \frac {2 \, a b \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 2 \, b^{2} \tan \left (d x + c\right ) + {\left (a^{2} - 3 \, b^{2}\right )} {\left (d x + c\right )} + \frac {2 \, a b - {\left (a^{2} - b^{2}\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.69, size = 75, normalized size = 0.99 \[ \frac {{\cos \left (c+d\,x\right )}^2\,\left (a\,b-\mathrm {tan}\left (c+d\,x\right )\,\left (\frac {a^2}{2}-\frac {b^2}{2}\right )\right )+b^2\,\mathrm {tan}\left (c+d\,x\right )+a\,b\,\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )+d\,x\,\left (\frac {a^2}{2}-\frac {3\,b^2}{2}\right )}{d} \]
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 \[ \int \left (a + b \tan {\left (c + d x \right )}\right )^{2} \sin ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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