Optimal. Leaf size=58 \[ \frac {2 a^2 \tan (c+d x)}{d}-\frac {a^2 \cot (c+d x)}{d}+\frac {2 a^2 \sec (c+d x)}{d}-\frac {2 a^2 \tanh ^{-1}(\cos (c+d x))}{d} \]
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Rubi [A] time = 0.22, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2873, 3767, 8, 2622, 321, 207, 2620, 14} \[ \frac {2 a^2 \tan (c+d x)}{d}-\frac {a^2 \cot (c+d x)}{d}+\frac {2 a^2 \sec (c+d x)}{d}-\frac {2 a^2 \tanh ^{-1}(\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 14
Rule 207
Rule 321
Rule 2620
Rule 2622
Rule 2873
Rule 3767
Rubi steps
\begin {align*} \int \csc ^2(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^2 \, dx &=\int \left (a^2 \sec ^2(c+d x)+2 a^2 \csc (c+d x) \sec ^2(c+d x)+a^2 \csc ^2(c+d x) \sec ^2(c+d x)\right ) \, dx\\ &=a^2 \int \sec ^2(c+d x) \, dx+a^2 \int \csc ^2(c+d x) \sec ^2(c+d x) \, dx+\left (2 a^2\right ) \int \csc (c+d x) \sec ^2(c+d x) \, dx\\ &=-\frac {a^2 \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1+x^2}{x^2} \, dx,x,\tan (c+d x)\right )}{d}+\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac {2 a^2 \sec (c+d x)}{d}+\frac {a^2 \tan (c+d x)}{d}+\frac {a^2 \operatorname {Subst}\left (\int \left (1+\frac {1}{x^2}\right ) \, dx,x,\tan (c+d x)\right )}{d}+\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac {2 a^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac {a^2 \cot (c+d x)}{d}+\frac {2 a^2 \sec (c+d x)}{d}+\frac {2 a^2 \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.39, size = 96, normalized size = 1.66 \[ \frac {a^2 \left (\tan \left (\frac {1}{2} (c+d x)\right )-\cot \left (\frac {1}{2} (c+d x)\right )+4 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-4 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+\frac {8 \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 192, normalized size = 3.31 \[ -\frac {3 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2} \cos \left (d x + c\right ) - 2 \, a^{2} + {\left (a^{2} \cos \left (d x + c\right )^{2} - a^{2} + {\left (a^{2} \cos \left (d x + c\right ) + a^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - {\left (a^{2} \cos \left (d x + c\right )^{2} - a^{2} + {\left (a^{2} \cos \left (d x + c\right ) + a^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - {\left (3 \, a^{2} \cos \left (d x + c\right ) + 2 \, a^{2}\right )} \sin \left (d x + c\right )}{d \cos \left (d x + c\right )^{2} + {\left (d \cos \left (d x + c\right ) + d\right )} \sin \left (d x + c\right ) - d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 98, normalized size = 1.69 \[ \frac {4 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {2 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 7 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.60, size = 92, normalized size = 1.59 \[ \frac {a^{2} \tan \left (d x +c \right )}{d}+\frac {2 a^{2}}{d \cos \left (d x +c \right )}+\frac {2 a^{2} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{d}+\frac {a^{2}}{d \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {2 a^{2} \cot \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 72, normalized size = 1.24 \[ \frac {a^{2} {\left (\frac {2}{\cos \left (d x + c\right )} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - a^{2} {\left (\frac {1}{\tan \left (d x + c\right )} - \tan \left (d x + c\right )\right )} + a^{2} \tan \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.93, size = 86, normalized size = 1.48 \[ \frac {2\,a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {a^2-9\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}+\frac {a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d} \]
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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