Optimal. Leaf size=59 \[ \frac {\left (a^2-b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {\sec ^2(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))}{2 d} \]
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Rubi [A] time = 0.06, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2668, 723, 206} \[ \frac {\left (a^2-b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {\sec ^2(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))}{2 d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 723
Rule 2668
Rubi steps
\begin {align*} \int \sec ^3(c+d x) (a+b \sin (c+d x))^2 \, dx &=\frac {b^3 \operatorname {Subst}\left (\int \frac {(a+x)^2}{\left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {\sec ^2(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))}{2 d}+\frac {\left (b \left (a^2-b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{2 d}\\ &=\frac {\left (a^2-b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {\sec ^2(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.92, size = 113, normalized size = 1.92 \[ \frac {-2 \left (a^4-b^4\right ) \tan (c+d x) \sec (c+d x)+\left (4 a b^3-6 a^3 b\right ) \tan ^2(c+d x)+2 a^3 b \sec ^2(c+d x)+\left (a^2-b^2\right )^2 (\log (1-\sin (c+d x))-\log (\sin (c+d x)+1))}{4 d \left (b^2-a^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 90, normalized size = 1.53 \[ \frac {{\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 4 \, a b + 2 \, {\left (a^{2} + b^{2}\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.15, size = 86, normalized size = 1.46 \[ \frac {{\left (a^{2} - b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - {\left (a^{2} - b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (a^{2} \sin \left (d x + c\right ) + b^{2} \sin \left (d x + c\right ) + 2 \, a b\right )}}{\sin \left (d x + c\right )^{2} - 1}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.25, size = 118, normalized size = 2.00 \[ \frac {a^{2} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {a b}{d \cos \left (d x +c \right )^{2}}+\frac {b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{2}}+\frac {b^{2} \sin \left (d x +c \right )}{2 d}-\frac {b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 78, normalized size = 1.32 \[ \frac {{\left (a^{2} - b^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (a^{2} - b^{2}\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left (2 \, a b + {\left (a^{2} + b^{2}\right )} \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{2} - 1}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 62, normalized size = 1.05 \[ \frac {\mathrm {atanh}\left (\sin \left (c+d\,x\right )\right )\,\left (\frac {a^2}{2}-\frac {b^2}{2}\right )}{d}-\frac {a\,b+\sin \left (c+d\,x\right )\,\left (\frac {a^2}{2}+\frac {b^2}{2}\right )}{d\,\left ({\sin \left (c+d\,x\right )}^2-1\right )} \]
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 \[ \int \left (a + b \sin {\left (c + d x \right )}\right )^{2} \sec ^{3}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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