Optimal. Leaf size=338 \[ -\frac {b^2 \cos \left (a-\frac {b c}{d}\right ) \text {Ci}\left (\frac {b c}{d}+b x\right )}{16 d^3}+\frac {9 b^2 \cos \left (3 a-\frac {3 b c}{d}\right ) \text {Ci}\left (\frac {3 b c}{d}+3 b x\right )}{32 d^3}+\frac {25 b^2 \cos \left (5 a-\frac {5 b c}{d}\right ) \text {Ci}\left (\frac {5 b c}{d}+5 b x\right )}{32 d^3}+\frac {b^2 \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{16 d^3}-\frac {9 b^2 \sin \left (3 a-\frac {3 b c}{d}\right ) \text {Si}\left (\frac {3 b c}{d}+3 b x\right )}{32 d^3}-\frac {25 b^2 \sin \left (5 a-\frac {5 b c}{d}\right ) \text {Si}\left (\frac {5 b c}{d}+5 b x\right )}{32 d^3}+\frac {b \sin (a+b x)}{16 d^2 (c+d x)}-\frac {3 b \sin (3 a+3 b x)}{32 d^2 (c+d x)}-\frac {5 b \sin (5 a+5 b x)}{32 d^2 (c+d x)}-\frac {\cos (a+b x)}{16 d (c+d x)^2}+\frac {\cos (3 a+3 b x)}{32 d (c+d x)^2}+\frac {\cos (5 a+5 b x)}{32 d (c+d x)^2} \]
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Rubi [A] time = 0.44, antiderivative size = 338, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {4406, 3297, 3303, 3299, 3302} \[ -\frac {b^2 \cos \left (a-\frac {b c}{d}\right ) \text {CosIntegral}\left (\frac {b c}{d}+b x\right )}{16 d^3}+\frac {9 b^2 \cos \left (3 a-\frac {3 b c}{d}\right ) \text {CosIntegral}\left (\frac {3 b c}{d}+3 b x\right )}{32 d^3}+\frac {25 b^2 \cos \left (5 a-\frac {5 b c}{d}\right ) \text {CosIntegral}\left (\frac {5 b c}{d}+5 b x\right )}{32 d^3}+\frac {b^2 \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{16 d^3}-\frac {9 b^2 \sin \left (3 a-\frac {3 b c}{d}\right ) \text {Si}\left (\frac {3 b c}{d}+3 b x\right )}{32 d^3}-\frac {25 b^2 \sin \left (5 a-\frac {5 b c}{d}\right ) \text {Si}\left (\frac {5 b c}{d}+5 b x\right )}{32 d^3}+\frac {b \sin (a+b x)}{16 d^2 (c+d x)}-\frac {3 b \sin (3 a+3 b x)}{32 d^2 (c+d x)}-\frac {5 b \sin (5 a+5 b x)}{32 d^2 (c+d x)}-\frac {\cos (a+b x)}{16 d (c+d x)^2}+\frac {\cos (3 a+3 b x)}{32 d (c+d x)^2}+\frac {\cos (5 a+5 b x)}{32 d (c+d x)^2} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rule 4406
Rubi steps
\begin {align*} \int \frac {\cos ^3(a+b x) \sin ^2(a+b x)}{(c+d x)^3} \, dx &=\int \left (\frac {\cos (a+b x)}{8 (c+d x)^3}-\frac {\cos (3 a+3 b x)}{16 (c+d x)^3}-\frac {\cos (5 a+5 b x)}{16 (c+d x)^3}\right ) \, dx\\ &=-\left (\frac {1}{16} \int \frac {\cos (3 a+3 b x)}{(c+d x)^3} \, dx\right )-\frac {1}{16} \int \frac {\cos (5 a+5 b x)}{(c+d x)^3} \, dx+\frac {1}{8} \int \frac {\cos (a+b x)}{(c+d x)^3} \, dx\\ &=-\frac {\cos (a+b x)}{16 d (c+d x)^2}+\frac {\cos (3 a+3 b x)}{32 d (c+d x)^2}+\frac {\cos (5 a+5 b x)}{32 d (c+d x)^2}-\frac {b \int \frac {\sin (a+b x)}{(c+d x)^2} \, dx}{16 d}+\frac {(3 b) \int \frac {\sin (3 a+3 b x)}{(c+d x)^2} \, dx}{32 d}+\frac {(5 b) \int \frac {\sin (5 a+5 b x)}{(c+d x)^2} \, dx}{32 d}\\ &=-\frac {\cos (a+b x)}{16 d (c+d x)^2}+\frac {\cos (3 a+3 b x)}{32 d (c+d x)^2}+\frac {\cos (5 a+5 b x)}{32 d (c+d x)^2}+\frac {b \sin (a+b x)}{16 d^2 (c+d x)}-\frac {3 b \sin (3 a+3 b x)}{32 d^2 (c+d x)}-\frac {5 b \sin (5 a+5 b x)}{32 d^2 (c+d x)}-\frac {b^2 \int \frac {\cos (a+b x)}{c+d x} \, dx}{16 d^2}+\frac {\left (9 b^2\right ) \int \frac {\cos (3 a+3 b x)}{c+d x} \, dx}{32 d^2}+\frac {\left (25 b^2\right ) \int \frac {\cos (5 a+5 b x)}{c+d x} \, dx}{32 d^2}\\ &=-\frac {\cos (a+b x)}{16 d (c+d x)^2}+\frac {\cos (3 a+3 b x)}{32 d (c+d x)^2}+\frac {\cos (5 a+5 b x)}{32 d (c+d x)^2}+\frac {b \sin (a+b x)}{16 d^2 (c+d x)}-\frac {3 b \sin (3 a+3 b x)}{32 d^2 (c+d x)}-\frac {5 b \sin (5 a+5 b x)}{32 d^2 (c+d x)}+\frac {\left (25 b^2 \cos \left (5 a-\frac {5 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {5 b c}{d}+5 b x\right )}{c+d x} \, dx}{32 d^2}+\frac {\left (9 b^2 \cos \left (3 a-\frac {3 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {3 b c}{d}+3 b x\right )}{c+d x} \, dx}{32 d^2}-\frac {\left (b^2 \cos \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx}{16 d^2}-\frac {\left (25 b^2 \sin \left (5 a-\frac {5 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {5 b c}{d}+5 b x\right )}{c+d x} \, dx}{32 d^2}-\frac {\left (9 b^2 \sin \left (3 a-\frac {3 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {3 b c}{d}+3 b x\right )}{c+d x} \, dx}{32 d^2}+\frac {\left (b^2 \sin \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx}{16 d^2}\\ &=-\frac {\cos (a+b x)}{16 d (c+d x)^2}+\frac {\cos (3 a+3 b x)}{32 d (c+d x)^2}+\frac {\cos (5 a+5 b x)}{32 d (c+d x)^2}-\frac {b^2 \cos \left (a-\frac {b c}{d}\right ) \text {Ci}\left (\frac {b c}{d}+b x\right )}{16 d^3}+\frac {9 b^2 \cos \left (3 a-\frac {3 b c}{d}\right ) \text {Ci}\left (\frac {3 b c}{d}+3 b x\right )}{32 d^3}+\frac {25 b^2 \cos \left (5 a-\frac {5 b c}{d}\right ) \text {Ci}\left (\frac {5 b c}{d}+5 b x\right )}{32 d^3}+\frac {b \sin (a+b x)}{16 d^2 (c+d x)}-\frac {3 b \sin (3 a+3 b x)}{32 d^2 (c+d x)}-\frac {5 b \sin (5 a+5 b x)}{32 d^2 (c+d x)}+\frac {b^2 \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{16 d^3}-\frac {9 b^2 \sin \left (3 a-\frac {3 b c}{d}\right ) \text {Si}\left (\frac {3 b c}{d}+3 b x\right )}{32 d^3}-\frac {25 b^2 \sin \left (5 a-\frac {5 b c}{d}\right ) \text {Si}\left (\frac {5 b c}{d}+5 b x\right )}{32 d^3}\\ \end {align*}
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Mathematica [A] time = 3.37, size = 283, normalized size = 0.84 \[ \frac {-2 b^2 \cos \left (a-\frac {b c}{d}\right ) \text {Ci}\left (b \left (\frac {c}{d}+x\right )\right )+9 b^2 \cos \left (3 a-\frac {3 b c}{d}\right ) \text {Ci}\left (\frac {3 b (c+d x)}{d}\right )+25 b^2 \cos \left (5 a-\frac {5 b c}{d}\right ) \text {Ci}\left (\frac {5 b (c+d x)}{d}\right )+2 b^2 \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (b \left (\frac {c}{d}+x\right )\right )-9 b^2 \sin \left (3 a-\frac {3 b c}{d}\right ) \text {Si}\left (\frac {3 b (c+d x)}{d}\right )-25 b^2 \sin \left (5 a-\frac {5 b c}{d}\right ) \text {Si}\left (\frac {5 b (c+d x)}{d}\right )+\frac {d^2 \cos (3 (a+b x))}{(c+d x)^2}+\frac {d^2 \cos (5 (a+b x))}{(c+d x)^2}-\frac {3 b d \sin (3 (a+b x))}{c+d x}-\frac {5 b d \sin (5 (a+b x))}{c+d x}+\frac {2 d (b (c+d x) \sin (a+b x)-d \cos (a+b x))}{(c+d x)^2}}{32 d^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 567, normalized size = 1.68 \[ \frac {32 \, d^{2} \cos \left (b x + a\right )^{5} - 32 \, d^{2} \cos \left (b x + a\right )^{3} - 50 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \sin \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {5 \, {\left (b d x + b c\right )}}{d}\right ) - 18 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) + 4 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \sin \left (-\frac {b c - a d}{d}\right ) \operatorname {Si}\left (\frac {b d x + b c}{d}\right ) - 2 \, {\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \operatorname {Ci}\left (\frac {b d x + b c}{d}\right ) + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \operatorname {Ci}\left (-\frac {b d x + b c}{d}\right )\right )} \cos \left (-\frac {b c - a d}{d}\right ) + 9 \, {\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \operatorname {Ci}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \operatorname {Ci}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right )\right )} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + 25 \, {\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \operatorname {Ci}\left (\frac {5 \, {\left (b d x + b c\right )}}{d}\right ) + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \operatorname {Ci}\left (-\frac {5 \, {\left (b d x + b c\right )}}{d}\right )\right )} \cos \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right ) - 32 \, {\left (5 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )^{4} - 3 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )^{2}\right )} \sin \left (b x + a\right )}{64 \, {\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 [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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maple [A] time = 0.02, size = 473, normalized size = 1.40 \[ \frac {\frac {b^{3} \left (-\frac {\cos \left (b x +a \right )}{2 \left (\left (b x +a \right ) d -d a +c b \right )^{2} d}-\frac {-\frac {\sin \left (b x +a \right )}{\left (\left (b x +a \right ) d -d a +c b \right ) d}+\frac {\frac {\Si \left (b x +a +\frac {-d a +c b}{d}\right ) \sin \left (\frac {-d a +c b}{d}\right )}{d}+\frac {\Ci \left (b x +a +\frac {-d a +c b}{d}\right ) \cos \left (\frac {-d a +c b}{d}\right )}{d}}{d}}{2 d}\right )}{8}-\frac {b^{3} \left (-\frac {5 \cos \left (5 b x +5 a \right )}{2 \left (\left (b x +a \right ) d -d a +c b \right )^{2} d}-\frac {5 \left (-\frac {5 \sin \left (5 b x +5 a \right )}{\left (\left (b x +a \right ) d -d a +c b \right ) d}+\frac {\frac {25 \Si \left (5 b x +5 a +\frac {-5 d a +5 c b}{d}\right ) \sin \left (\frac {-5 d a +5 c b}{d}\right )}{d}+\frac {25 \Ci \left (5 b x +5 a +\frac {-5 d a +5 c b}{d}\right ) \cos \left (\frac {-5 d a +5 c b}{d}\right )}{d}}{d}\right )}{2 d}\right )}{80}-\frac {b^{3} \left (-\frac {3 \cos \left (3 b x +3 a \right )}{2 \left (\left (b x +a \right ) d -d a +c b \right )^{2} d}-\frac {3 \left (-\frac {3 \sin \left (3 b x +3 a \right )}{\left (\left (b x +a \right ) d -d a +c b \right ) d}+\frac {\frac {9 \Si \left (3 b x +3 a +\frac {-3 d a +3 c b}{d}\right ) \sin \left (\frac {-3 d a +3 c b}{d}\right )}{d}+\frac {9 \Ci \left (3 b x +3 a +\frac {-3 d a +3 c b}{d}\right ) \cos \left (\frac {-3 d a +3 c b}{d}\right )}{d}}{d}\right )}{2 d}\right )}{48}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.37, size = 474, normalized size = 1.40 \[ -\frac {1073741824 \, b^{3} {\left (E_{3}\left (\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right ) + E_{3}\left (-\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right )\right )} \cos \left (-\frac {b c - a d}{d}\right ) - 536870912 \, b^{3} {\left (E_{3}\left (\frac {3 i \, b c + 3 i \, {\left (b x + a\right )} d - 3 i \, a d}{d}\right ) + E_{3}\left (-\frac {3 i \, b c + 3 i \, {\left (b x + a\right )} d - 3 i \, a d}{d}\right )\right )} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) - 536870912 \, b^{3} {\left (E_{3}\left (\frac {5 i \, b c + 5 i \, {\left (b x + a\right )} d - 5 i \, a d}{d}\right ) + E_{3}\left (-\frac {5 i \, b c + 5 i \, {\left (b x + a\right )} d - 5 i \, a d}{d}\right )\right )} \cos \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right ) + b^{3} {\left (-1073741824 i \, E_{3}\left (\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right ) + 1073741824 i \, E_{3}\left (-\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right ) + b^{3} {\left (536870912 i \, E_{3}\left (\frac {3 i \, b c + 3 i \, {\left (b x + a\right )} d - 3 i \, a d}{d}\right ) - 536870912 i \, E_{3}\left (-\frac {3 i \, b c + 3 i \, {\left (b x + a\right )} d - 3 i \, a d}{d}\right )\right )} \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + b^{3} {\left (536870912 i \, E_{3}\left (\frac {5 i \, b c + 5 i \, {\left (b x + a\right )} d - 5 i \, a d}{d}\right ) - 536870912 i \, E_{3}\left (-\frac {5 i \, b c + 5 i \, {\left (b x + a\right )} d - 5 i \, a d}{d}\right )\right )} \sin \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right )}{17179869184 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + {\left (b x + a\right )}^{2} d^{3} + a^{2} d^{3} + 2 \, {\left (b c d^{2} - a d^{3}\right )} {\left (b x + a\right )}\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\cos \left (a+b\,x\right )}^3\,{\sin \left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^3} \,d x \]
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 \[ \int \frac {\sin ^{2}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{\left (c + d x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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