Optimal. Leaf size=64 \[ \frac {a^4}{4 d (a-a \sin (c+d x))^2}-\frac {a^3}{4 d (a-a \sin (c+d x))}-\frac {a^2 \tanh ^{-1}(\sin (c+d x))}{4 d} \]
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Rubi [A] time = 0.09, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2836, 12, 77, 206} \[ \frac {a^4}{4 d (a-a \sin (c+d x))^2}-\frac {a^3}{4 d (a-a \sin (c+d x))}-\frac {a^2 \tanh ^{-1}(\sin (c+d x))}{4 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 77
Rule 206
Rule 2836
Rubi steps
\begin {align*} \int \sec ^4(c+d x) (a+a \sin (c+d x))^2 \tan (c+d x) \, dx &=\frac {a^5 \operatorname {Subst}\left (\int \frac {x}{a (a-x)^3 (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^4 \operatorname {Subst}\left (\int \frac {x}{(a-x)^3 (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^4 \operatorname {Subst}\left (\int \left (\frac {1}{2 (a-x)^3}-\frac {1}{4 a (a-x)^2}-\frac {1}{4 a \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^4}{4 d (a-a \sin (c+d x))^2}-\frac {a^3}{4 d (a-a \sin (c+d x))}-\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{4 d}\\ &=-\frac {a^2 \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac {a^4}{4 d (a-a \sin (c+d x))^2}-\frac {a^3}{4 d (a-a \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 36, normalized size = 0.56 \[ -\frac {a^2 \left (\tanh ^{-1}(\sin (c+d x))-\frac {\sin (c+d x)}{(\sin (c+d x)-1)^2}\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 120, normalized size = 1.88 \[ -\frac {2 \, a^{2} \sin \left (d x + c\right ) + {\left (a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} \sin \left (d x + c\right ) - 2 \, a^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} \sin \left (d x + c\right ) - 2 \, a^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right )}{8 \, {\left (d \cos \left (d x + c\right )^{2} + 2 \, d \sin \left (d x + c\right ) - 2 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 95, normalized size = 1.48 \[ -\frac {a^{2} \log \left ({\left | \frac {1}{\sin \left (d x + c\right )} + \sin \left (d x + c\right ) + 2 \right |}\right ) - a^{2} \log \left ({\left | \frac {1}{\sin \left (d x + c\right )} + \sin \left (d x + c\right ) - 2 \right |}\right ) + \frac {a^{2} {\left (\frac {1}{\sin \left (d x + c\right )} + \sin \left (d x + c\right )\right )} - 6 \, a^{2}}{\frac {1}{\sin \left (d x + c\right )} + \sin \left (d x + c\right ) - 2}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.26, size = 126, normalized size = 1.97 \[ \frac {a^{2} \left (\sin ^{4}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{4}}+\frac {a^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{4}}+\frac {a^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{2}}+\frac {a^{2} \sin \left (d x +c \right )}{4 d}-\frac {a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{4 d}+\frac {a^{2}}{4 d \cos \left (d x +c \right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 64, normalized size = 1.00 \[ -\frac {a^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - a^{2} \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac {2 \, a^{2} \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.13, size = 106, normalized size = 1.66 \[ \frac {\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}+\frac {a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}-\frac {a^2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,d} \]
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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