Optimal. Leaf size=99 \[ -\frac {4 \cos \left (\frac {2 c}{d}\right ) \text {Ci}\left (\frac {2 c}{d}+2 x\right )}{d^3}-\frac {4 \sin \left (\frac {2 c}{d}\right ) \text {Si}\left (\frac {2 c}{d}+2 x\right )}{d^3}+\frac {4 \sin (x) \cos (x)}{d^2 (c+d x)}+\frac {\sin ^2(x)}{2 d (c+d x)^2}-\frac {3 \cos ^2(x)}{2 d (c+d x)^2} \]
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Rubi [A] time = 0.33, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4431, 3314, 31, 3312, 3303, 3299, 3302} \[ -\frac {4 \cos \left (\frac {2 c}{d}\right ) \text {CosIntegral}\left (\frac {2 c}{d}+2 x\right )}{d^3}-\frac {4 \sin \left (\frac {2 c}{d}\right ) \text {Si}\left (\frac {2 c}{d}+2 x\right )}{d^3}+\frac {4 \sin (x) \cos (x)}{d^2 (c+d x)}+\frac {\sin ^2(x)}{2 d (c+d x)^2}-\frac {3 \cos ^2(x)}{2 d (c+d x)^2} \]
Antiderivative was successfully verified.
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Rule 31
Rule 3299
Rule 3302
Rule 3303
Rule 3312
Rule 3314
Rule 4431
Rubi steps
\begin {align*} \int \frac {\csc (x) \sin (3 x)}{(c+d x)^3} \, dx &=\int \left (\frac {3 \cos ^2(x)}{(c+d x)^3}-\frac {\sin ^2(x)}{(c+d x)^3}\right ) \, dx\\ &=3 \int \frac {\cos ^2(x)}{(c+d x)^3} \, dx-\int \frac {\sin ^2(x)}{(c+d x)^3} \, dx\\ &=-\frac {3 \cos ^2(x)}{2 d (c+d x)^2}+\frac {4 \cos (x) \sin (x)}{d^2 (c+d x)}+\frac {\sin ^2(x)}{2 d (c+d x)^2}-\frac {\int \frac {1}{c+d x} \, dx}{d^2}+\frac {2 \int \frac {\sin ^2(x)}{c+d x} \, dx}{d^2}+\frac {3 \int \frac {1}{c+d x} \, dx}{d^2}-\frac {6 \int \frac {\cos ^2(x)}{c+d x} \, dx}{d^2}\\ &=-\frac {3 \cos ^2(x)}{2 d (c+d x)^2}+\frac {2 \log (c+d x)}{d^3}+\frac {4 \cos (x) \sin (x)}{d^2 (c+d x)}+\frac {\sin ^2(x)}{2 d (c+d x)^2}+\frac {2 \int \left (\frac {1}{2 (c+d x)}-\frac {\cos (2 x)}{2 (c+d x)}\right ) \, dx}{d^2}-\frac {6 \int \left (\frac {1}{2 (c+d x)}+\frac {\cos (2 x)}{2 (c+d x)}\right ) \, dx}{d^2}\\ &=-\frac {3 \cos ^2(x)}{2 d (c+d x)^2}+\frac {4 \cos (x) \sin (x)}{d^2 (c+d x)}+\frac {\sin ^2(x)}{2 d (c+d x)^2}-\frac {\int \frac {\cos (2 x)}{c+d x} \, dx}{d^2}-\frac {3 \int \frac {\cos (2 x)}{c+d x} \, dx}{d^2}\\ &=-\frac {3 \cos ^2(x)}{2 d (c+d x)^2}+\frac {4 \cos (x) \sin (x)}{d^2 (c+d x)}+\frac {\sin ^2(x)}{2 d (c+d x)^2}-\frac {\cos \left (\frac {2 c}{d}\right ) \int \frac {\cos \left (\frac {2 c}{d}+2 x\right )}{c+d x} \, dx}{d^2}-\frac {\left (3 \cos \left (\frac {2 c}{d}\right )\right ) \int \frac {\cos \left (\frac {2 c}{d}+2 x\right )}{c+d x} \, dx}{d^2}-\frac {\sin \left (\frac {2 c}{d}\right ) \int \frac {\sin \left (\frac {2 c}{d}+2 x\right )}{c+d x} \, dx}{d^2}-\frac {\left (3 \sin \left (\frac {2 c}{d}\right )\right ) \int \frac {\sin \left (\frac {2 c}{d}+2 x\right )}{c+d x} \, dx}{d^2}\\ &=-\frac {3 \cos ^2(x)}{2 d (c+d x)^2}-\frac {4 \cos \left (\frac {2 c}{d}\right ) \text {Ci}\left (\frac {2 c}{d}+2 x\right )}{d^3}+\frac {4 \cos (x) \sin (x)}{d^2 (c+d x)}+\frac {\sin ^2(x)}{2 d (c+d x)^2}-\frac {4 \sin \left (\frac {2 c}{d}\right ) \text {Si}\left (\frac {2 c}{d}+2 x\right )}{d^3}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 77, normalized size = 0.78 \[ \frac {-8 \cos \left (\frac {2 c}{d}\right ) \text {Ci}\left (2 \left (\frac {c}{d}+x\right )\right )-8 \sin \left (\frac {2 c}{d}\right ) \text {Si}\left (2 \left (\frac {c}{d}+x\right )\right )+\frac {d (4 \sin (2 x) (c+d x)-2 d \cos (2 x)-d)}{(c+d x)^2}}{2 d^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 158, normalized size = 1.60 \[ -\frac {4 \, d^{2} \cos \relax (x)^{2} - 8 \, {\left (d^{2} x + c d\right )} \cos \relax (x) \sin \relax (x) + 8 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \sin \left (\frac {2 \, c}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (d x + c\right )}}{d}\right ) - d^{2} + 4 \, {\left ({\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \operatorname {Ci}\left (\frac {2 \, {\left (d x + c\right )}}{d}\right ) + {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \operatorname {Ci}\left (-\frac {2 \, {\left (d x + c\right )}}{d}\right )\right )} \cos \left (\frac {2 \, c}{d}\right )}{2 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.73, size = 201, normalized size = 2.03 \[ -\frac {8 \, d^{2} x^{2} \cos \left (\frac {2 \, c}{d}\right ) \operatorname {Ci}\left (\frac {2 \, {\left (d x + c\right )}}{d}\right ) + 8 \, d^{2} x^{2} \sin \left (\frac {2 \, c}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (d x + c\right )}}{d}\right ) + 16 \, c d x \cos \left (\frac {2 \, c}{d}\right ) \operatorname {Ci}\left (\frac {2 \, {\left (d x + c\right )}}{d}\right ) + 16 \, c d x \sin \left (\frac {2 \, c}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (d x + c\right )}}{d}\right ) + 8 \, c^{2} \cos \left (\frac {2 \, c}{d}\right ) \operatorname {Ci}\left (\frac {2 \, {\left (d x + c\right )}}{d}\right ) - 4 \, d^{2} x \sin \left (2 \, x\right ) + 8 \, c^{2} \sin \left (\frac {2 \, c}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (d x + c\right )}}{d}\right ) + 2 \, d^{2} \cos \left (2 \, x\right ) - 4 \, c d \sin \left (2 \, x\right ) + d^{2}}{2 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 104, normalized size = 1.05 \[ -\frac {\cos \left (2 x \right )}{\left (d x +c \right )^{2} d}-\frac {-\frac {2 \sin \left (2 x \right )}{\left (d x +c \right ) d}+\frac {\frac {4 \Si \left (\frac {2 c}{d}+2 x \right ) \sin \left (\frac {2 c}{d}\right )}{d}+\frac {4 \Ci \left (\frac {2 c}{d}+2 x \right ) \cos \left (\frac {2 c}{d}\right )}{d}}{d}}{d}-\frac {1}{2 d \left (d x +c \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.42, size = 362, normalized size = 3.66 \[ -\frac {2 \, {\left (E_{3}\left (\frac {2 i \, d x + 2 i \, c}{d}\right ) + E_{3}\left (-\frac {2 i \, d x + 2 i \, c}{d}\right )\right )} \cos \left (\frac {2 \, c}{d}\right )^{3} + {\left (2 i \, E_{3}\left (\frac {2 i \, d x + 2 i \, c}{d}\right ) - 2 i \, E_{3}\left (-\frac {2 i \, d x + 2 i \, c}{d}\right )\right )} \sin \left (\frac {2 \, c}{d}\right )^{3} + 2 \, {\left ({\left (E_{3}\left (\frac {2 i \, d x + 2 i \, c}{d}\right ) + E_{3}\left (-\frac {2 i \, d x + 2 i \, c}{d}\right )\right )} \cos \left (\frac {2 \, c}{d}\right ) + 1\right )} \sin \left (\frac {2 \, c}{d}\right )^{2} + 2 \, {\left (E_{3}\left (\frac {2 i \, d x + 2 i \, c}{d}\right ) + E_{3}\left (-\frac {2 i \, d x + 2 i \, c}{d}\right )\right )} \cos \left (\frac {2 \, c}{d}\right ) + 2 \, \cos \left (\frac {2 \, c}{d}\right )^{2} + {\left ({\left (2 i \, E_{3}\left (\frac {2 i \, d x + 2 i \, c}{d}\right ) - 2 i \, E_{3}\left (-\frac {2 i \, d x + 2 i \, c}{d}\right )\right )} \cos \left (\frac {2 \, c}{d}\right )^{2} + 2 i \, E_{3}\left (\frac {2 i \, d x + 2 i \, c}{d}\right ) - 2 i \, E_{3}\left (-\frac {2 i \, d x + 2 i \, c}{d}\right )\right )} \sin \left (\frac {2 \, c}{d}\right )}{4 \, {\left ({\left (\cos \left (\frac {2 \, c}{d}\right )^{2} + \sin \left (\frac {2 \, c}{d}\right )^{2}\right )} d^{3} x^{2} + 2 \, {\left (c \cos \left (\frac {2 \, c}{d}\right )^{2} + c \sin \left (\frac {2 \, c}{d}\right )^{2}\right )} d^{2} x + {\left (c^{2} \cos \left (\frac {2 \, c}{d}\right )^{2} + c^{2} \sin \left (\frac {2 \, c}{d}\right )^{2}\right )} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sin \left (3\,x\right )}{\sin \relax (x)\,{\left (c+d\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin {\left (3 x \right )} \csc {\relax (x )}}{\left (c + d x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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