Optimal. Leaf size=99 \[ -\frac{4 \cos \left (\frac{2 c}{d}\right ) \text{CosIntegral}\left (\frac{2 c}{d}+2 x\right )}{d^3}-\frac{4 \sin \left (\frac{2 c}{d}\right ) \text{Si}\left (\frac{2 c}{d}+2 x\right )}{d^3}+\frac{4 \sin (x) \cos (x)}{d^2 (c+d x)}+\frac{\sin ^2(x)}{2 d (c+d x)^2}-\frac{3 \cos ^2(x)}{2 d (c+d x)^2} \]
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Rubi [A] time = 0.328565, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, 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 4431
Rule 3314
Rule 31
Rule 3312
Rule 3303
Rule 3299
Rule 3302
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*}
Mathematica [A] time = 0.233797, size = 77, normalized size = 0.78 \[ \frac{-8 \cos \left (\frac{2 c}{d}\right ) \text{CosIntegral}\left (2 \left (\frac{c}{d}+x\right )\right )-8 \sin \left (\frac{2 c}{d}\right ) \text{Si}\left (2 \left (\frac{c}{d}+x\right )\right )+\frac{d (4 \sin (2 x) (c+d x)-2 d \cos (2 x)-d)}{(c+d x)^2}}{2 d^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 104, normalized size = 1.1 \begin{align*} -{\frac{\cos \left ( 2\,x \right ) }{ \left ( dx+c \right ) ^{2}d}}-{\frac{1}{d} \left ( -2\,{\frac{\sin \left ( 2\,x \right ) }{ \left ( dx+c \right ) d}}+2\,{\frac{1}{d} \left ( 2\,{\frac{1}{d}{\it Si} \left ( 2\,{\frac{c}{d}}+2\,x \right ) \sin \left ( 2\,{\frac{c}{d}} \right ) }+2\,{\frac{1}{d}{\it Ci} \left ( 2\,{\frac{c}{d}}+2\,x \right ) \cos \left ( 2\,{\frac{c}{d}} \right ) } \right ) } \right ) }-{\frac{1}{2\, \left ( dx+c \right ) ^{2}d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.37194, size = 489, normalized size = 4.94 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.529877, size = 392, normalized size = 3.96 \begin{align*} -\frac{4 \, d^{2} \cos \left (x\right )^{2} - 8 \,{\left (d^{2} x + c d\right )} \cos \left (x\right ) \sin \left (x\right ) + 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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.11449, size = 271, normalized size = 2.74 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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