Optimal. Leaf size=185 \[ -\frac {\left (12 a^2+15 a b+4 b^2\right ) \log (1-\sin (c+d x))}{8 d}+\frac {\left (3 a^2+b^2\right ) \log (\sin (c+d x))}{d}-\frac {\left (12 a^2-15 a b+4 b^2\right ) \log (\sin (c+d x)+1)}{8 d}+\frac {\sec ^4(c+d x) \left (a^2+2 a b \sin (c+d x)+b^2\right )}{4 d}+\frac {\sec ^2(c+d x) \left (2 \left (2 a^2+b^2\right )+7 a b \sin (c+d x)\right )}{4 d}-\frac {a^2 \csc ^2(c+d x)}{2 d}-\frac {2 a b \csc (c+d x)}{d} \]
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Rubi [A] time = 0.38, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2837, 12, 1805, 1802} \[ -\frac {\left (12 a^2+15 a b+4 b^2\right ) \log (1-\sin (c+d x))}{8 d}+\frac {\left (3 a^2+b^2\right ) \log (\sin (c+d x))}{d}-\frac {\left (12 a^2-15 a b+4 b^2\right ) \log (\sin (c+d x)+1)}{8 d}+\frac {\sec ^4(c+d x) \left (a^2+2 a b \sin (c+d x)+b^2\right )}{4 d}+\frac {\sec ^2(c+d x) \left (2 \left (2 a^2+b^2\right )+7 a b \sin (c+d x)\right )}{4 d}-\frac {a^2 \csc ^2(c+d x)}{2 d}-\frac {2 a b \csc (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 1802
Rule 1805
Rule 2837
Rubi steps
\begin {align*} \int \csc ^3(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^2 \, dx &=\frac {b^5 \operatorname {Subst}\left (\int \frac {b^3 (a+x)^2}{x^3 \left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {b^8 \operatorname {Subst}\left (\int \frac {(a+x)^2}{x^3 \left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {\sec ^4(c+d x) \left (a^2+b^2+2 a b \sin (c+d x)\right )}{4 d}-\frac {b^6 \operatorname {Subst}\left (\int \frac {-4 a^2-8 a x-4 \left (1+\frac {a^2}{b^2}\right ) x^2-\frac {6 a x^3}{b^2}}{x^3 \left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{4 d}\\ &=\frac {\sec ^4(c+d x) \left (a^2+b^2+2 a b \sin (c+d x)\right )}{4 d}+\frac {\sec ^2(c+d x) \left (2 \left (2 a^2+b^2\right )+7 a b \sin (c+d x)\right )}{4 d}+\frac {b^4 \operatorname {Subst}\left (\int \frac {8 a^2+16 a x+8 \left (1+\frac {2 a^2}{b^2}\right ) x^2+\frac {14 a x^3}{b^2}}{x^3 \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{8 d}\\ &=\frac {\sec ^4(c+d x) \left (a^2+b^2+2 a b \sin (c+d x)\right )}{4 d}+\frac {\sec ^2(c+d x) \left (2 \left (2 a^2+b^2\right )+7 a b \sin (c+d x)\right )}{4 d}+\frac {b^4 \operatorname {Subst}\left (\int \left (\frac {12 a^2+15 a b+4 b^2}{b^4 (b-x)}+\frac {8 a^2}{b^2 x^3}+\frac {16 a}{b^2 x^2}+\frac {8 \left (3 a^2+b^2\right )}{b^4 x}+\frac {-12 a^2+15 a b-4 b^2}{b^4 (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{8 d}\\ &=-\frac {2 a b \csc (c+d x)}{d}-\frac {a^2 \csc ^2(c+d x)}{2 d}-\frac {\left (12 a^2+15 a b+4 b^2\right ) \log (1-\sin (c+d x))}{8 d}+\frac {\left (3 a^2+b^2\right ) \log (\sin (c+d x))}{d}-\frac {\left (12 a^2-15 a b+4 b^2\right ) \log (1+\sin (c+d x))}{8 d}+\frac {\sec ^4(c+d x) \left (a^2+b^2+2 a b \sin (c+d x)\right )}{4 d}+\frac {\sec ^2(c+d x) \left (2 \left (2 a^2+b^2\right )+7 a b \sin (c+d x)\right )}{4 d}\\ \end {align*}
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Mathematica [A] time = 3.66, size = 182, normalized size = 0.98 \[ \frac {-2 \left (12 a^2+15 a b+4 b^2\right ) \log (1-\sin (c+d x))+16 \left (3 a^2+b^2\right ) \log (\sin (c+d x))-2 \left (12 a^2-15 a b+4 b^2\right ) \log (\sin (c+d x)+1)-8 a^2 \csc ^2(c+d x)+\frac {(a-b)^2}{(\sin (c+d x)+1)^2}+\frac {(9 a-5 b) (a-b)}{\sin (c+d x)+1}-\frac {(a+b) (9 a+5 b)}{\sin (c+d x)-1}+\frac {(a+b)^2}{(\sin (c+d x)-1)^2}-32 a b \csc (c+d x)}{16 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 285, normalized size = 1.54 \[ \frac {4 \, {\left (3 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (3 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - 2 \, b^{2} + 8 \, {\left ({\left (3 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{6} - {\left (3 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - {\left ({\left (12 \, a^{2} - 15 \, a b + 4 \, b^{2}\right )} \cos \left (d x + c\right )^{6} - {\left (12 \, a^{2} - 15 \, a b + 4 \, b^{2}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left ({\left (12 \, a^{2} + 15 \, a b + 4 \, b^{2}\right )} \cos \left (d x + c\right )^{6} - {\left (12 \, a^{2} + 15 \, a b + 4 \, b^{2}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (15 \, a b \cos \left (d x + c\right )^{4} - 5 \, a b \cos \left (d x + c\right )^{2} - 2 \, a b\right )} \sin \left (d x + c\right )}{8 \, {\left (d \cos \left (d x + c\right )^{6} - d \cos \left (d x + c\right )^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 190, normalized size = 1.03 \[ -\frac {{\left (12 \, a^{2} - 15 \, a b + 4 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) + {\left (12 \, a^{2} + 15 \, a b + 4 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - 8 \, {\left (3 \, a^{2} + b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + \frac {2 \, {\left (15 \, a b \sin \left (d x + c\right )^{5} + 6 \, a^{2} \sin \left (d x + c\right )^{4} + 2 \, b^{2} \sin \left (d x + c\right )^{4} - 25 \, a b \sin \left (d x + c\right )^{3} - 9 \, a^{2} \sin \left (d x + c\right )^{2} - 3 \, b^{2} \sin \left (d x + c\right )^{2} + 8 \, a b \sin \left (d x + c\right ) + 2 \, a^{2}\right )}}{{\left (\sin \left (d x + c\right )^{3} - \sin \left (d x + c\right )\right )}^{2}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.54, size = 209, normalized size = 1.13 \[ \frac {a^{2}}{4 d \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{4}}+\frac {3 a^{2}}{4 d \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}}-\frac {3 a^{2}}{2 d \sin \left (d x +c \right )^{2}}+\frac {3 a^{2} \ln \left (\tan \left (d x +c \right )\right )}{d}+\frac {a b}{2 d \sin \left (d x +c \right ) \cos \left (d x +c \right )^{4}}+\frac {5 a b}{4 d \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {15 a b}{4 d \sin \left (d x +c \right )}+\frac {15 a b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{4 d}+\frac {b^{2}}{4 d \cos \left (d x +c \right )^{4}}+\frac {b^{2}}{2 d \cos \left (d x +c \right )^{2}}+\frac {b^{2} \ln \left (\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 183, normalized size = 0.99 \[ -\frac {{\left (12 \, a^{2} - 15 \, a b + 4 \, b^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left (12 \, a^{2} + 15 \, a b + 4 \, b^{2}\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) - 8 \, {\left (3 \, a^{2} + b^{2}\right )} \log \left (\sin \left (d x + c\right )\right ) + \frac {2 \, {\left (15 \, a b \sin \left (d x + c\right )^{5} - 25 \, a b \sin \left (d x + c\right )^{3} + 2 \, {\left (3 \, a^{2} + b^{2}\right )} \sin \left (d x + c\right )^{4} + 8 \, a b \sin \left (d x + c\right ) - 3 \, {\left (3 \, a^{2} + b^{2}\right )} \sin \left (d x + c\right )^{2} + 2 \, a^{2}\right )}}{\sin \left (d x + c\right )^{6} - 2 \, \sin \left (d x + c\right )^{4} + \sin \left (d x + c\right )^{2}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 194, normalized size = 1.05 \[ \frac {\ln \left (\sin \left (c+d\,x\right )\right )\,\left (3\,a^2+b^2\right )}{d}-\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,\left (\frac {3\,a^2}{2}-\frac {15\,a\,b}{8}+\frac {b^2}{2}\right )}{d}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (\frac {3\,a^2}{2}+\frac {15\,a\,b}{8}+\frac {b^2}{2}\right )}{d}-\frac {\frac {a^2}{2}+{\sin \left (c+d\,x\right )}^4\,\left (\frac {3\,a^2}{2}+\frac {b^2}{2}\right )-{\sin \left (c+d\,x\right )}^2\,\left (\frac {9\,a^2}{4}+\frac {3\,b^2}{4}\right )+2\,a\,b\,\sin \left (c+d\,x\right )-\frac {25\,a\,b\,{\sin \left (c+d\,x\right )}^3}{4}+\frac {15\,a\,b\,{\sin \left (c+d\,x\right )}^5}{4}}{d\,\left ({\sin \left (c+d\,x\right )}^6-2\,{\sin \left (c+d\,x\right )}^4+{\sin \left (c+d\,x\right )}^2\right )} \]
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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