Optimal. Leaf size=104 \[ \frac {\left (a^2+b^2\right ) \tan (c+d x)}{d}-\frac {\left (2 a^2+b^2\right ) \cot (c+d x)}{d}-\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {3 a b \sec (c+d x)}{d}-\frac {3 a b \tanh ^{-1}(\cos (c+d x))}{d}-\frac {a b \csc ^2(c+d x) \sec (c+d x)}{d} \]
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Rubi [A] time = 0.23, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2911, 2622, 288, 321, 207, 3200, 448} \[ \frac {\left (a^2+b^2\right ) \tan (c+d x)}{d}-\frac {\left (2 a^2+b^2\right ) \cot (c+d x)}{d}-\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {3 a b \sec (c+d x)}{d}-\frac {3 a b \tanh ^{-1}(\cos (c+d x))}{d}-\frac {a b \csc ^2(c+d x) \sec (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 207
Rule 288
Rule 321
Rule 448
Rule 2622
Rule 2911
Rule 3200
Rubi steps
\begin {align*} \int \csc ^4(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^2 \, dx &=(2 a b) \int \csc ^3(c+d x) \sec ^2(c+d x) \, dx+\int \csc ^4(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 {\left (1+x^2\right ) \left (a^2+\left (a^2+b^2\right ) x^2\right )}{x^4} \, dx,x,\tan (c+d x)\right )}{d}+\frac {(2 a b) \operatorname {Subst}\left (\int \frac {x^4}{\left (-1+x^2\right )^2} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac {a b \csc ^2(c+d x) \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^4}+\frac {2 a^2+b^2}{x^2}\right ) \, dx,x,\tan (c+d x)\right )}{d}+\frac {(3 a b) \operatorname {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac {\left (2 a^2+b^2\right ) \cot (c+d x)}{d}-\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {3 a b \sec (c+d x)}{d}-\frac {a b \csc ^2(c+d x) \sec (c+d x)}{d}+\frac {\left (a^2+b^2\right ) \tan (c+d x)}{d}+\frac {(3 a b) \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac {3 a b \tanh ^{-1}(\cos (c+d x))}{d}-\frac {\left (2 a^2+b^2\right ) \cot (c+d x)}{d}-\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {3 a b \sec (c+d x)}{d}-\frac {a b \csc ^2(c+d x) \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 = 0.99, size = 196, normalized size = 1.88 \[ \frac {\csc ^5\left (\frac {1}{2} (c+d x)\right ) \sec ^3\left (\frac {1}{2} (c+d x)\right ) \left (-4 \left (4 a^2+3 b^2\right ) \cos (2 (c+d x))+\left (8 a^2+6 b^2\right ) \cos (4 (c+d x))+3 b \left (10 a \sin (c+d x)-6 a \sin (3 (c+d x))-3 a \sin (4 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-6 a \sin (2 (c+d x)) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+3 a \sin (4 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+2 b\right )\right )}{192 d \left (\cot ^2\left (\frac {1}{2} (c+d x)\right )-1\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 192, normalized size = 1.85 \[ -\frac {4 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )} \cos \left (d x + c\right )^{4} - 6 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 9 \, {\left (a b \cos \left (d x + c\right )^{3} - a b \cos \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 9 \, {\left (a b \cos \left (d x + c\right )^{3} - a b \cos \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 6 \, a^{2} + 6 \, b^{2} - 6 \, {\left (3 \, a b \cos \left (d x + c\right )^{2} - 2 \, a b\right )} \sin \left (d x + c\right )}{6 \, {\left (d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 204, normalized size = 1.96 \[ \frac {a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 72 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 21 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {48 \, {\left (a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, a b\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} - \frac {132 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 21 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 6 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.57, size = 162, normalized size = 1.56 \[ -\frac {a^{2}}{3 d \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {4 a^{2}}{3 d \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {8 a^{2} \cot \left (d x +c \right )}{3 d}-\frac {a b}{d \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {3 a b}{d \cos \left (d x +c \right )}+\frac {3 a b \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{d}+\frac {b^{2}}{d \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {2 b^{2} \cot \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 123, normalized size = 1.18 \[ \frac {3 \, a b {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 2\right )}}{\cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 6 \, b^{2} {\left (\frac {1}{\tan \left (d x + c\right )} - \tan \left (d x + c\right )\right )} - 2 \, a^{2} {\left (\frac {6 \, \tan \left (d x + c\right )^{2} + 1}{\tan \left (d x + c\right )^{3}} - 3 \, \tan \left (d x + c\right )\right )}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.86, size = 194, normalized size = 1.87 \[ \frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {20\,a^2}{3}+4\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (23\,a^2+20\,b^2\right )+\frac {a^2}{3}-34\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+2\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\right )}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {7\,a^2}{8}+\frac {b^2}{2}\right )}{d}+\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4\,d}+\frac {3\,a\,b\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d} \]
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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