Optimal. Leaf size=59 \[ \frac {\left (a^2+b^2\right ) \tan (c+d x)}{d}-\frac {a^2 \cot (c+d x)}{d}+\frac {2 a b \sec (c+d x)}{d}-\frac {2 a b \tanh ^{-1}(\cos (c+d x))}{d} \]
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Rubi [A] time = 0.28, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2911, 2622, 321, 207, 3200, 14} \[ \frac {\left (a^2+b^2\right ) \tan (c+d x)}{d}-\frac {a^2 \cot (c+d x)}{d}+\frac {2 a b \sec (c+d x)}{d}-\frac {2 a b \tanh ^{-1}(\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 207
Rule 321
Rule 2622
Rule 2911
Rule 3200
Rubi steps
\begin {align*} \int \csc ^2(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^2 \, dx &=(2 a b) \int \csc (c+d x) \sec ^2(c+d x) \, dx+\int \csc ^2(c+d x) \sec ^2(c+d x) \left (a^2+b^2 \sin ^2(c+d x)\right ) \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {a^2+\left (a^2+b^2\right ) x^2}{x^2} \, dx,x,\tan (c+d x)\right )}{d}+\frac {(2 a b) \operatorname {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac {2 a b \sec (c+d x)}{d}+\frac {\operatorname {Subst}\left (\int \left (a^2 \left (1+\frac {b^2}{a^2}\right )+\frac {a^2}{x^2}\right ) \, dx,x,\tan (c+d x)\right )}{d}+\frac {(2 a b) \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac {2 a b \tanh ^{-1}(\cos (c+d x))}{d}-\frac {a^2 \cot (c+d x)}{d}+\frac {2 a b \sec (c+d x)}{d}+\frac {\left (a^2+b^2\right ) \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 102, normalized size = 1.73 \[ -\frac {\csc \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {1}{2} (c+d x)\right ) \sec (c+d x) \left (\left (2 a^2+b^2\right ) \cos (2 (c+d x))-b \left (4 a \sin (c+d x)-2 a \sin (2 (c+d x)) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+b\right )\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 113, normalized size = 1.92 \[ -\frac {a b \cos \left (d x + c\right ) \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - a b \cos \left (d x + c\right ) \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + {\left (2 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}{d \cos \left (d x + c\right ) \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 128, normalized size = 2.17 \[ \frac {12 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {4 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 20 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.60, size = 90, normalized size = 1.53 \[ \frac {a^{2}}{d \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {2 a^{2} \cot \left (d x +c \right )}{d}+\frac {2 a b}{d \cos \left (d x +c \right )}+\frac {2 a b \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{d}+\frac {b^{2} \tan \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 71, normalized size = 1.20 \[ \frac {a b {\left (\frac {2}{\cos \left (d x + c\right )} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - a^{2} {\left (\frac {1}{\tan \left (d x + c\right )} - \tan \left (d x + c\right )\right )} + b^{2} \tan \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.87, size = 108, normalized size = 1.83 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (5\,a^2+4\,b^2\right )-a^2+8\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}+\frac {a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d}+\frac {2\,a\,b\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d} \]
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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