Optimal. Leaf size=129 \[ \frac {\left (2 a^2-b^2\right ) \csc ^5(c+d x)}{5 d}-\frac {\left (a^2-2 b^2\right ) \csc ^3(c+d x)}{3 d}-\frac {a^2 \csc ^7(c+d x)}{7 d}-\frac {a b \csc ^6(c+d x)}{3 d}+\frac {a b \csc ^4(c+d x)}{d}-\frac {a b \csc ^2(c+d x)}{d}-\frac {b^2 \csc (c+d x)}{d} \]
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Rubi [A] time = 0.16, antiderivative size = 129, 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} \[ \frac {\left (2 a^2-b^2\right ) \csc ^5(c+d x)}{5 d}-\frac {\left (a^2-2 b^2\right ) \csc ^3(c+d x)}{3 d}-\frac {a^2 \csc ^7(c+d x)}{7 d}-\frac {a b \csc ^6(c+d x)}{3 d}+\frac {a b \csc ^4(c+d x)}{d}-\frac {a b \csc ^2(c+d x)}{d}-\frac {b^2 \csc (c+d x)}{d} \]
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 ^3(c+d x) (a+b \sin (c+d x))^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {b^8 (a+x)^2 \left (b^2-x^2\right )^2}{x^8} \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac {b^3 \operatorname {Subst}\left (\int \frac {(a+x)^2 \left (b^2-x^2\right )^2}{x^8} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {b^3 \operatorname {Subst}\left (\int \left (\frac {a^2 b^4}{x^8}+\frac {2 a b^4}{x^7}+\frac {-2 a^2 b^2+b^4}{x^6}-\frac {4 a b^2}{x^5}+\frac {a^2-2 b^2}{x^4}+\frac {2 a}{x^3}+\frac {1}{x^2}\right ) \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac {b^2 \csc (c+d x)}{d}-\frac {a b \csc ^2(c+d x)}{d}-\frac {\left (a^2-2 b^2\right ) \csc ^3(c+d x)}{3 d}+\frac {a b \csc ^4(c+d x)}{d}+\frac {\left (2 a^2-b^2\right ) \csc ^5(c+d x)}{5 d}-\frac {a b \csc ^6(c+d x)}{3 d}-\frac {a^2 \csc ^7(c+d x)}{7 d}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 104, normalized size = 0.81 \[ -\frac {\csc (c+d x) \left (21 \left (b^2-2 a^2\right ) \csc ^4(c+d x)+35 \left (a^2-2 b^2\right ) \csc ^2(c+d x)+15 a^2 \csc ^6(c+d x)+35 a b \csc ^5(c+d x)-105 a b \csc ^3(c+d x)+105 a b \csc (c+d x)+105 b^2\right )}{105 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 146, normalized size = 1.13 \[ -\frac {105 \, b^{2} \cos \left (d x + c\right )^{6} - 35 \, {\left (a^{2} + 7 \, b^{2}\right )} \cos \left (d x + c\right )^{4} + 28 \, {\left (a^{2} + 7 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 8 \, a^{2} - 56 \, b^{2} - 35 \, {\left (3 \, a b \cos \left (d x + c\right )^{4} - 3 \, a b \cos \left (d x + c\right )^{2} + a b\right )} \sin \left (d x + c\right )}{105 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 118, normalized size = 0.91 \[ -\frac {105 \, b^{2} \sin \left (d x + c\right )^{6} + 105 \, a b \sin \left (d x + c\right )^{5} + 35 \, a^{2} \sin \left (d x + c\right )^{4} - 70 \, b^{2} \sin \left (d x + c\right )^{4} - 105 \, a b \sin \left (d x + c\right )^{3} - 42 \, a^{2} \sin \left (d x + c\right )^{2} + 21 \, b^{2} \sin \left (d x + c\right )^{2} + 35 \, a b \sin \left (d x + c\right ) + 15 \, a^{2}}{105 \, d \sin \left (d x + c\right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.56, size = 218, normalized size = 1.69 \[ \frac {a^{2} \left (-\frac {\cos ^{6}\left (d x +c \right )}{7 \sin \left (d x +c \right )^{7}}-\frac {\cos ^{6}\left (d x +c \right )}{35 \sin \left (d x +c \right )^{5}}+\frac {\cos ^{6}\left (d x +c \right )}{105 \sin \left (d x +c \right )^{3}}-\frac {\cos ^{6}\left (d x +c \right )}{35 \sin \left (d x +c \right )}-\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{35}\right )-\frac {a b \left (\cos ^{6}\left (d x +c \right )\right )}{3 \sin \left (d x +c \right )^{6}}+b^{2} \left (-\frac {\cos ^{6}\left (d x +c \right )}{5 \sin \left (d x +c \right )^{5}}+\frac {\cos ^{6}\left (d x +c \right )}{15 \sin \left (d x +c \right )^{3}}-\frac {\cos ^{6}\left (d x +c \right )}{5 \sin \left (d x +c \right )}-\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 106, normalized size = 0.82 \[ -\frac {105 \, b^{2} \sin \left (d x + c\right )^{6} + 105 \, a b \sin \left (d x + c\right )^{5} - 105 \, a b \sin \left (d x + c\right )^{3} + 35 \, {\left (a^{2} - 2 \, b^{2}\right )} \sin \left (d x + c\right )^{4} + 35 \, a b \sin \left (d x + c\right ) - 21 \, {\left (2 \, a^{2} - b^{2}\right )} \sin \left (d x + c\right )^{2} + 15 \, a^{2}}{105 \, d \sin \left (d x + c\right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.79, size = 105, normalized size = 0.81 \[ -\frac {\frac {a^2}{7}+{\sin \left (c+d\,x\right )}^4\,\left (\frac {a^2}{3}-\frac {2\,b^2}{3}\right )-{\sin \left (c+d\,x\right )}^2\,\left (\frac {2\,a^2}{5}-\frac {b^2}{5}\right )+b^2\,{\sin \left (c+d\,x\right )}^6+\frac {a\,b\,\sin \left (c+d\,x\right )}{3}-a\,b\,{\sin \left (c+d\,x\right )}^3+a\,b\,{\sin \left (c+d\,x\right )}^5}{d\,{\sin \left (c+d\,x\right )}^7} \]
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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