Optimal. Leaf size=100 \[ \frac {\left (3 a^2+2 b^2\right ) \sec (c+d x)}{2 d}-\frac {\left (3 a^2+2 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {a^2 \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac {2 a b \tan (c+d x)}{d}-\frac {2 a b \cot (c+d x)}{d} \]
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Rubi [A] time = 0.25, antiderivative size = 124, normalized size of antiderivative = 1.24, number of steps used = 10, number of rules used = 9, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2911, 2620, 14, 3201, 446, 78, 51, 63, 206} \[ \frac {\left (3 a^2+2 b^2\right ) \sec (c+d x)}{2 d}-\frac {\left (3 a^2+2 b^2\right ) \sqrt {\cos ^2(c+d x)} \sec (c+d x) \tanh ^{-1}\left (\sqrt {\cos ^2(c+d x)}\right )}{2 d}-\frac {a^2 \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac {2 a b \tan (c+d x)}{d}-\frac {2 a b \cot (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 51
Rule 63
Rule 78
Rule 206
Rule 446
Rule 2620
Rule 2911
Rule 3201
Rubi steps
\begin {align*} \int \csc ^3(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^2 \, dx &=(2 a b) \int \csc ^2(c+d x) \sec ^2(c+d x) \, dx+\int \csc ^3(c+d x) \sec ^2(c+d x) \left (a^2+b^2 \sin ^2(c+d x)\right ) \, dx\\ &=\frac {(2 a b) \operatorname {Subst}\left (\int \frac {1+x^2}{x^2} \, dx,x,\tan (c+d x)\right )}{d}+\frac {\left (\sqrt {\cos ^2(c+d x)} \sec (c+d x)\right ) \operatorname {Subst}\left (\int \frac {a^2+b^2 x^2}{x^3 \left (1-x^2\right )^{3/2}} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac {(2 a b) \operatorname {Subst}\left (\int \left (1+\frac {1}{x^2}\right ) \, dx,x,\tan (c+d x)\right )}{d}+\frac {\left (\sqrt {\cos ^2(c+d x)} \sec (c+d x)\right ) \operatorname {Subst}\left (\int \frac {a^2+b^2 x}{(1-x)^{3/2} x^2} \, dx,x,\sin ^2(c+d x)\right )}{2 d}\\ &=-\frac {2 a b \cot (c+d x)}{d}-\frac {a^2 \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac {2 a b \tan (c+d x)}{d}-\frac {\left (\left (-3 a^2-2 b^2\right ) \sqrt {\cos ^2(c+d x)} \sec (c+d x)\right ) \operatorname {Subst}\left (\int \frac {1}{(1-x)^{3/2} x} \, dx,x,\sin ^2(c+d x)\right )}{4 d}\\ &=-\frac {2 a b \cot (c+d x)}{d}+\frac {\left (3 a^2+2 b^2\right ) \sec (c+d x)}{2 d}-\frac {a^2 \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac {2 a b \tan (c+d x)}{d}-\frac {\left (\left (-3 a^2-2 b^2\right ) \sqrt {\cos ^2(c+d x)} \sec (c+d x)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,\sin ^2(c+d x)\right )}{4 d}\\ &=-\frac {2 a b \cot (c+d x)}{d}+\frac {\left (3 a^2+2 b^2\right ) \sec (c+d x)}{2 d}-\frac {a^2 \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac {2 a b \tan (c+d x)}{d}+\frac {\left (\left (-3 a^2-2 b^2\right ) \sqrt {\cos ^2(c+d x)} \sec (c+d x)\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {\cos ^2(c+d x)}\right )}{2 d}\\ &=-\frac {2 a b \cot (c+d x)}{d}+\frac {\left (3 a^2+2 b^2\right ) \sec (c+d x)}{2 d}-\frac {\left (3 a^2+2 b^2\right ) \tanh ^{-1}\left (\sqrt {\cos ^2(c+d x)}\right ) \sqrt {\cos ^2(c+d x)} \sec (c+d x)}{2 d}-\frac {a^2 \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac {2 a b \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [B] time = 0.48, size = 238, normalized size = 2.38 \[ \frac {\csc ^4(c+d x) \left (-2 \left (3 a^2+2 b^2\right ) \cos (2 (c+d x))-\left (3 a^2+2 b^2\right ) \cos (c+d x) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+3 a^2 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-3 a^2 \cos (3 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+2 a^2+8 a b \sin (c+d x)-8 a b \sin (3 (c+d x))+2 b^2 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-2 b^2 \cos (3 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+4 b^2\right )}{2 d \left (\csc ^2\left (\frac {1}{2} (c+d x)\right )-\sec ^2\left (\frac {1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 186, normalized size = 1.86 \[ \frac {2 \, {\left (3 \, a^{2} + 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 4 \, a^{2} - 4 \, b^{2} - {\left ({\left (3 \, a^{2} + 2 \, b^{2}\right )} \cos \left (d x + c\right )^{3} - {\left (3 \, a^{2} + 2 \, b^{2}\right )} \cos \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left ({\left (3 \, a^{2} + 2 \, b^{2}\right )} \cos \left (d x + c\right )^{3} - {\left (3 \, a^{2} + 2 \, b^{2}\right )} \cos \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 8 \, {\left (2 \, a b \cos \left (d x + c\right )^{2} - a b\right )} \sin \left (d x + c\right )}{4 \, {\left (d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 157, normalized size = 1.57 \[ \frac {a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4 \, {\left (3 \, a^{2} + 2 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {16 \, {\left (2 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{2} + b^{2}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} - \frac {18 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.56, size = 140, normalized size = 1.40 \[ -\frac {a^{2}}{2 d \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {3 a^{2}}{2 d \cos \left (d x +c \right )}+\frac {3 a^{2} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2 d}+\frac {2 a b}{d \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {4 a b \cot \left (d x +c \right )}{d}+\frac {b^{2}}{d \cos \left (d x +c \right )}+\frac {b^{2} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 123, normalized size = 1.23 \[ \frac {a^{2} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 2\right )}}{\cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 2 \, b^{2} {\left (\frac {2}{\cos \left (d x + c\right )} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 8 \, a b {\left (\frac {1}{\tan \left (d x + c\right )} - \tan \left (d x + c\right )\right )}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.85, size = 148, normalized size = 1.48 \[ \frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {3\,a^2}{2}+b^2\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {17\,a^2}{2}+8\,b^2\right )-\frac {a^2}{2}+20\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-4\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}+\frac {a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d} \]
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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