Optimal. Leaf size=138 \[ \frac {\left (2 a^2-b^2\right ) \csc ^6(c+d x)}{6 d}-\frac {\left (a^2-2 b^2\right ) \csc ^4(c+d x)}{4 d}-\frac {a^2 \csc ^8(c+d x)}{8 d}-\frac {2 a b \csc ^7(c+d x)}{7 d}+\frac {4 a b \csc ^5(c+d x)}{5 d}-\frac {2 a b \csc ^3(c+d x)}{3 d}-\frac {b^2 \csc ^2(c+d x)}{2 d} \]
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Rubi [A] time = 0.16, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2837, 12, 948} \[ \frac {\left (2 a^2-b^2\right ) \csc ^6(c+d x)}{6 d}-\frac {\left (a^2-2 b^2\right ) \csc ^4(c+d x)}{4 d}-\frac {a^2 \csc ^8(c+d x)}{8 d}-\frac {2 a b \csc ^7(c+d x)}{7 d}+\frac {4 a b \csc ^5(c+d x)}{5 d}-\frac {2 a b \csc ^3(c+d x)}{3 d}-\frac {b^2 \csc ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 948
Rule 2837
Rubi steps
\begin {align*} \int \cot ^5(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {b^9 (a+x)^2 \left (b^2-x^2\right )^2}{x^9} \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac {b^4 \operatorname {Subst}\left (\int \frac {(a+x)^2 \left (b^2-x^2\right )^2}{x^9} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {b^4 \operatorname {Subst}\left (\int \left (\frac {a^2 b^4}{x^9}+\frac {2 a b^4}{x^8}+\frac {-2 a^2 b^2+b^4}{x^7}-\frac {4 a b^2}{x^6}+\frac {a^2-2 b^2}{x^5}+\frac {2 a}{x^4}+\frac {1}{x^3}\right ) \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac {b^2 \csc ^2(c+d x)}{2 d}-\frac {2 a b \csc ^3(c+d x)}{3 d}-\frac {\left (a^2-2 b^2\right ) \csc ^4(c+d x)}{4 d}+\frac {4 a b \csc ^5(c+d x)}{5 d}+\frac {\left (2 a^2-b^2\right ) \csc ^6(c+d x)}{6 d}-\frac {2 a b \csc ^7(c+d x)}{7 d}-\frac {a^2 \csc ^8(c+d x)}{8 d}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 108, normalized size = 0.78 \[ -\frac {\csc ^2(c+d x) \left (-140 \left (2 a^2-b^2\right ) \csc ^4(c+d x)+210 \left (a^2-2 b^2\right ) \csc ^2(c+d x)+105 a^2 \csc ^6(c+d x)+240 a b \csc ^5(c+d x)-672 a b \csc ^3(c+d x)+560 a b \csc (c+d x)+420 b^2\right )}{840 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 148, normalized size = 1.07 \[ \frac {420 \, b^{2} \cos \left (d x + c\right )^{6} - 210 \, {\left (a^{2} + 4 \, b^{2}\right )} \cos \left (d x + c\right )^{4} + 140 \, {\left (a^{2} + 4 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 35 \, a^{2} - 140 \, b^{2} - 16 \, {\left (35 \, a b \cos \left (d x + c\right )^{4} - 28 \, a b \cos \left (d x + c\right )^{2} + 8 \, a b\right )} \sin \left (d x + c\right )}{840 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 118, normalized size = 0.86 \[ -\frac {420 \, b^{2} \sin \left (d x + c\right )^{6} + 560 \, a b \sin \left (d x + c\right )^{5} + 210 \, a^{2} \sin \left (d x + c\right )^{4} - 420 \, b^{2} \sin \left (d x + c\right )^{4} - 672 \, a b \sin \left (d x + c\right )^{3} - 280 \, a^{2} \sin \left (d x + c\right )^{2} + 140 \, b^{2} \sin \left (d x + c\right )^{2} + 240 \, a b \sin \left (d x + c\right ) + 105 \, a^{2}}{840 \, d \sin \left (d x + c\right )^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.56, size = 173, normalized size = 1.25 \[ \frac {a^{2} \left (-\frac {\cos ^{6}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{6}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{6}}\right )+2 a b \left (-\frac {\cos ^{6}\left (d x +c \right )}{7 \sin \left (d x +c \right )^{7}}-\frac {\cos ^{6}\left (d x +c \right )}{35 \sin \left (d x +c \right )^{5}}+\frac {\cos ^{6}\left (d x +c \right )}{105 \sin \left (d x +c \right )^{3}}-\frac {\cos ^{6}\left (d x +c \right )}{35 \sin \left (d x +c \right )}-\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{35}\right )-\frac {b^{2} \left (\cos ^{6}\left (d x +c \right )\right )}{6 \sin \left (d x +c \right )^{6}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 106, normalized size = 0.77 \[ -\frac {420 \, b^{2} \sin \left (d x + c\right )^{6} + 560 \, a b \sin \left (d x + c\right )^{5} - 672 \, a b \sin \left (d x + c\right )^{3} + 210 \, {\left (a^{2} - 2 \, b^{2}\right )} \sin \left (d x + c\right )^{4} + 240 \, a b \sin \left (d x + c\right ) - 140 \, {\left (2 \, a^{2} - b^{2}\right )} \sin \left (d x + c\right )^{2} + 105 \, a^{2}}{840 \, d \sin \left (d x + c\right )^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.78, size = 107, normalized size = 0.78 \[ -\frac {\frac {a^2}{8}+{\sin \left (c+d\,x\right )}^4\,\left (\frac {a^2}{4}-\frac {b^2}{2}\right )-{\sin \left (c+d\,x\right )}^2\,\left (\frac {a^2}{3}-\frac {b^2}{6}\right )+\frac {b^2\,{\sin \left (c+d\,x\right )}^6}{2}+\frac {2\,a\,b\,\sin \left (c+d\,x\right )}{7}-\frac {4\,a\,b\,{\sin \left (c+d\,x\right )}^3}{5}+\frac {2\,a\,b\,{\sin \left (c+d\,x\right )}^5}{3}}{d\,{\sin \left (c+d\,x\right )}^8} \]
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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