Optimal. Leaf size=164 \[ \frac {a^3 \tan (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {2 a^3 \cot (c+d x)}{d}+\frac {9 a^2 b \sec (c+d x)}{2 d}-\frac {9 a^2 b \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {3 a^2 b \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac {3 a b^2 \tan (c+d x)}{d}-\frac {3 a b^2 \cot (c+d x)}{d}+\frac {b^3 \sec (c+d x)}{d}-\frac {b^3 \tanh ^{-1}(\cos (c+d x))}{d} \]
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Rubi [A] time = 0.26, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2912, 2622, 321, 207, 2620, 14, 288, 270} \[ \frac {9 a^2 b \sec (c+d x)}{2 d}-\frac {9 a^2 b \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {3 a^2 b \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac {a^3 \tan (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {2 a^3 \cot (c+d x)}{d}+\frac {3 a b^2 \tan (c+d x)}{d}-\frac {3 a b^2 \cot (c+d x)}{d}+\frac {b^3 \sec (c+d x)}{d}-\frac {b^3 \tanh ^{-1}(\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 207
Rule 270
Rule 288
Rule 321
Rule 2620
Rule 2622
Rule 2912
Rubi steps
\begin {align*} \int \csc ^4(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^3 \, dx &=\int \left (b^3 \csc (c+d x) \sec ^2(c+d x)+3 a b^2 \csc ^2(c+d x) \sec ^2(c+d x)+3 a^2 b \csc ^3(c+d x) \sec ^2(c+d x)+a^3 \csc ^4(c+d x) \sec ^2(c+d x)\right ) \, dx\\ &=a^3 \int \csc ^4(c+d x) \sec ^2(c+d x) \, dx+\left (3 a^2 b\right ) \int \csc ^3(c+d x) \sec ^2(c+d x) \, dx+\left (3 a b^2\right ) \int \csc ^2(c+d x) \sec ^2(c+d x) \, dx+b^3 \int \csc (c+d x) \sec ^2(c+d x) \, dx\\ &=\frac {a^3 \operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x^4} \, dx,x,\tan (c+d x)\right )}{d}+\frac {\left (3 a^2 b\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (-1+x^2\right )^2} \, dx,x,\sec (c+d x)\right )}{d}+\frac {\left (3 a b^2\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{x^2} \, dx,x,\tan (c+d x)\right )}{d}+\frac {b^3 \operatorname {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac {b^3 \sec (c+d x)}{d}-\frac {3 a^2 b \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac {a^3 \operatorname {Subst}\left (\int \left (1+\frac {1}{x^4}+\frac {2}{x^2}\right ) \, dx,x,\tan (c+d x)\right )}{d}+\frac {\left (9 a^2 b\right ) \operatorname {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{2 d}+\frac {\left (3 a b^2\right ) \operatorname {Subst}\left (\int \left (1+\frac {1}{x^2}\right ) \, dx,x,\tan (c+d x)\right )}{d}+\frac {b^3 \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac {b^3 \tanh ^{-1}(\cos (c+d x))}{d}-\frac {2 a^3 \cot (c+d x)}{d}-\frac {3 a b^2 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{3 d}+\frac {9 a^2 b \sec (c+d x)}{2 d}+\frac {b^3 \sec (c+d x)}{d}-\frac {3 a^2 b \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac {a^3 \tan (c+d x)}{d}+\frac {3 a b^2 \tan (c+d x)}{d}+\frac {\left (9 a^2 b\right ) \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{2 d}\\ &=-\frac {9 a^2 b \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {b^3 \tanh ^{-1}(\cos (c+d x))}{d}-\frac {2 a^3 \cot (c+d x)}{d}-\frac {3 a b^2 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{3 d}+\frac {9 a^2 b \sec (c+d x)}{2 d}+\frac {b^3 \sec (c+d x)}{d}-\frac {3 a^2 b \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac {a^3 \tan (c+d x)}{d}+\frac {3 a b^2 \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 1.36, size = 287, normalized size = 1.75 \[ \frac {\csc ^5\left (\frac {1}{2} (c+d x)\right ) \sec ^3\left (\frac {1}{2} (c+d x)\right ) \left (-8 \left (4 a^3+9 a b^2\right ) \cos (2 (c+d x))+4 \left (4 a^3+9 a b^2\right ) \cos (4 (c+d x))+3 b \left (6 \left (5 a^2+2 b^2\right ) \sin (c+d x)-2 \left (9 a^2+2 b^2\right ) \sin (2 (c+d x)) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-18 a^2 \sin (3 (c+d x))-9 a^2 \sin (4 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+9 a^2 \sin (4 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+12 a b-4 b^2 \sin (3 (c+d x))-2 b^2 \sin (4 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+2 b^2 \sin (4 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )\right )}{384 d \left (\cot ^2\left (\frac {1}{2} (c+d x)\right )-1\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 252, normalized size = 1.54 \[ -\frac {8 \, {\left (4 \, a^{3} + 9 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} + 12 \, a^{3} + 36 \, a b^{2} - 12 \, {\left (4 \, a^{3} + 9 \, a b^{2}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left ({\left (9 \, a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{3} - {\left (9 \, a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 3 \, {\left ({\left (9 \, a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{3} - {\left (9 \, a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 6 \, {\left (6 \, a^{2} b + 2 \, b^{3} - {\left (9 \, a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{12 \, {\left (d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 245, normalized size = 1.49 \[ \frac {a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 21 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 36 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, {\left (9 \, a^{2} b + 2 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {48 \, {\left (a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a^{2} b + b^{3}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} - \frac {198 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 44 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 21 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 36 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 9 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.63, size = 209, normalized size = 1.27 \[ -\frac {a^{3}}{3 d \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {4 a^{3}}{3 d \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {8 a^{3} \cot \left (d x +c \right )}{3 d}-\frac {3 a^{2} b}{2 d \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {9 a^{2} b}{2 d \cos \left (d x +c \right )}+\frac {9 a^{2} b \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2 d}+\frac {3 a \,b^{2}}{d \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {6 a \,b^{2} \cot \left (d x +c \right )}{d}+\frac {b^{3}}{d \cos \left (d x +c \right )}+\frac {b^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 162, normalized size = 0.99 \[ \frac {9 \, a^{2} b {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 2\right )}}{\cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 6 \, b^{3} {\left (\frac {2}{\cos \left (d x + c\right )} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 36 \, a b^{2} {\left (\frac {1}{\tan \left (d x + c\right )} - \tan \left (d x + c\right )\right )} - 4 \, a^{3} {\left (\frac {6 \, \tan \left (d x + c\right )^{2} + 1}{\tan \left (d x + c\right )^{3}} - 3 \, \tan \left (d x + c\right )\right )}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.89, size = 218, normalized size = 1.33 \[ \frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {9\,a^2\,b}{2}+b^3\right )}{d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {20\,a^3}{3}+12\,a\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (23\,a^3+60\,a\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (51\,a^2\,b+16\,b^3\right )+\frac {a^3}{3}+3\,a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\right )}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {7\,a^3}{8}+\frac {3\,a\,b^2}{2}\right )}{d}+\frac {3\,a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d} \]
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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