Optimal. Leaf size=127 \[ -\frac {\cos (a+b x)}{3 d (c+d x)^3}+\frac {b^2 \cos (a+b x)}{6 d^3 (c+d x)}+\frac {b^3 \text {CosIntegral}\left (\frac {b c}{d}+b x\right ) \sin \left (a-\frac {b c}{d}\right )}{6 d^4}+\frac {b \sin (a+b x)}{6 d^2 (c+d x)^2}+\frac {b^3 \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{6 d^4} \]
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Rubi [A]
time = 0.11, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3378, 3384,
3380, 3383} \begin {gather*} \frac {b^3 \sin \left (a-\frac {b c}{d}\right ) \text {CosIntegral}\left (\frac {b c}{d}+b x\right )}{6 d^4}+\frac {b^3 \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{6 d^4}+\frac {b^2 \cos (a+b x)}{6 d^3 (c+d x)}+\frac {b \sin (a+b x)}{6 d^2 (c+d x)^2}-\frac {\cos (a+b x)}{3 d (c+d x)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 3378
Rule 3380
Rule 3383
Rule 3384
Rubi steps
\begin {align*} \int \frac {\cos (a+b x)}{(c+d x)^4} \, dx &=-\frac {\cos (a+b x)}{3 d (c+d x)^3}-\frac {b \int \frac {\sin (a+b x)}{(c+d x)^3} \, dx}{3 d}\\ &=-\frac {\cos (a+b x)}{3 d (c+d x)^3}+\frac {b \sin (a+b x)}{6 d^2 (c+d x)^2}-\frac {b^2 \int \frac {\cos (a+b x)}{(c+d x)^2} \, dx}{6 d^2}\\ &=-\frac {\cos (a+b x)}{3 d (c+d x)^3}+\frac {b^2 \cos (a+b x)}{6 d^3 (c+d x)}+\frac {b \sin (a+b x)}{6 d^2 (c+d x)^2}+\frac {b^3 \int \frac {\sin (a+b x)}{c+d x} \, dx}{6 d^3}\\ &=-\frac {\cos (a+b x)}{3 d (c+d x)^3}+\frac {b^2 \cos (a+b x)}{6 d^3 (c+d x)}+\frac {b \sin (a+b x)}{6 d^2 (c+d x)^2}+\frac {\left (b^3 \cos \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx}{6 d^3}+\frac {\left (b^3 \sin \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx}{6 d^3}\\ &=-\frac {\cos (a+b x)}{3 d (c+d x)^3}+\frac {b^2 \cos (a+b x)}{6 d^3 (c+d x)}+\frac {b^3 \text {Ci}\left (\frac {b c}{d}+b x\right ) \sin \left (a-\frac {b c}{d}\right )}{6 d^4}+\frac {b \sin (a+b x)}{6 d^2 (c+d x)^2}+\frac {b^3 \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{6 d^4}\\ \end {align*}
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Mathematica [A]
time = 0.39, size = 144, normalized size = 1.13 \begin {gather*} \frac {d \cos (b x) \left (\left (-2 d^2+b^2 (c+d x)^2\right ) \cos (a)+b d (c+d x) \sin (a)\right )+d \left (b d (c+d x) \cos (a)-\left (-2 d^2+b^2 (c+d x)^2\right ) \sin (a)\right ) \sin (b x)+b^3 (c+d x)^3 \left (\text {CosIntegral}\left (b \left (\frac {c}{d}+x\right )\right ) \sin \left (a-\frac {b c}{d}\right )+\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (b \left (\frac {c}{d}+x\right )\right )\right )}{6 d^4 (c+d x)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 184, normalized size = 1.45
method | result | size |
derivativedivides | \(b^{3} \left (-\frac {\cos \left (b x +a \right )}{3 \left (-d a +b c +d \left (b x +a \right )\right )^{3} d}-\frac {-\frac {\sin \left (b x +a \right )}{2 \left (-d a +b c +d \left (b x +a \right )\right )^{2} d}+\frac {-\frac {\cos \left (b x +a \right )}{\left (-d a +b c +d \left (b x +a \right )\right ) d}-\frac {-\frac {\sinIntegral \left (-b x -a -\frac {-d a +b c}{d}\right ) \cos \left (\frac {-d a +b c}{d}\right )}{d}-\frac {\cosineIntegral \left (b x +a +\frac {-d a +b c}{d}\right ) \sin \left (\frac {-d a +b c}{d}\right )}{d}}{d}}{2 d}}{3 d}\right )\) | \(184\) |
default | \(b^{3} \left (-\frac {\cos \left (b x +a \right )}{3 \left (-d a +b c +d \left (b x +a \right )\right )^{3} d}-\frac {-\frac {\sin \left (b x +a \right )}{2 \left (-d a +b c +d \left (b x +a \right )\right )^{2} d}+\frac {-\frac {\cos \left (b x +a \right )}{\left (-d a +b c +d \left (b x +a \right )\right ) d}-\frac {-\frac {\sinIntegral \left (-b x -a -\frac {-d a +b c}{d}\right ) \cos \left (\frac {-d a +b c}{d}\right )}{d}-\frac {\cosineIntegral \left (b x +a +\frac {-d a +b c}{d}\right ) \sin \left (\frac {-d a +b c}{d}\right )}{d}}{d}}{2 d}}{3 d}\right )\) | \(184\) |
risch | \(-\frac {i b^{3} {\mathrm e}^{-\frac {i \left (d a -b c \right )}{d}} \expIntegral \left (1, i b x +i a -\frac {i \left (d a -b c \right )}{d}\right )}{12 d^{4}}+\frac {i b^{3} {\mathrm e}^{\frac {i \left (d a -b c \right )}{d}} \expIntegral \left (1, -i b x -i a -\frac {-i a d +i b c}{d}\right )}{12 d^{4}}-\frac {\left (-2 b^{5} d^{5} x^{5}-10 b^{5} c \,d^{4} x^{4}-20 b^{5} c^{2} d^{3} x^{3}-20 b^{5} c^{3} d^{2} x^{2}-10 b^{5} c^{4} d x +4 b^{3} d^{5} x^{3}-2 b^{5} c^{5}+12 b^{3} c \,d^{4} x^{2}+12 b^{3} c^{2} d^{3} x +4 b^{3} c^{3} d^{2}\right ) \cos \left (b x +a \right )}{12 d^{3} \left (d x +c \right )^{3} \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +b^{3} c^{3}\right )}-\frac {i \left (2 i b^{4} d^{5} x^{4}+8 i b^{4} c \,d^{4} x^{3}+12 i b^{4} c^{2} d^{3} x^{2}+8 i b^{4} c^{3} d^{2} x +2 i b^{4} c^{4} d \right ) \sin \left (b x +a \right )}{12 d^{3} \left (d x +c \right )^{3} \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +b^{3} c^{3}\right )}\) | \(405\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.49, size = 249, normalized size = 1.96 \begin {gather*} -\frac {b^{4} {\left (E_{4}\left (\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right ) + E_{4}\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 ) + b^{4} {\left (-i \, E_{4}\left (\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right ) + i \, E_{4}\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 )}{2 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} + {\left (b x + a\right )}^{3} d^{4} - a^{3} d^{4} + 3 \, {\left (b c d^{3} - a d^{4}\right )} {\left (b x + a\right )}^{2} + 3 \, {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} {\left (b x + a\right )}\right )} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 295 vs.
\(2 (117) = 234\).
time = 0.38, size = 295, normalized size = 2.32 \begin {gather*} \frac {2 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {Si}\left (\frac {b d x + b c}{d}\right ) + 2 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d - 2 \, d^{3}\right )} \cos \left (b x + a\right ) + 2 \, {\left (b d^{3} x + b c d^{2}\right )} \sin \left (b x + a\right ) + {\left ({\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \operatorname {Ci}\left (\frac {b d x + b c}{d}\right ) + {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \operatorname {Ci}\left (-\frac {b d x + b c}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right )}{12 \, {\left (d^{7} x^{3} + 3 \, c d^{6} x^{2} + 3 \, c^{2} d^{5} x + c^{3} d^{4}\right )}} \end {gather*}
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 \begin {gather*} \int \frac {\cos {\left (a + b x \right )}}{\left (c + d x\right )^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.70, size = 8378, normalized size = 65.97 \begin {gather*} \text {Too large to display} \end {gather*}
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 \begin {gather*} \int \frac {\cos \left (a+b\,x\right )}{{\left (c+d\,x\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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