Optimal. Leaf size=184 \[ -\frac {\cos ^3(a+b x)}{2 d (c+d x)^2}-\frac {3 b^2 \cos \left (a-\frac {b c}{d}\right ) \text {CosIntegral}\left (\frac {b c}{d}+b x\right )}{8 d^3}-\frac {9 b^2 \cos \left (3 a-\frac {3 b c}{d}\right ) \text {CosIntegral}\left (\frac {3 b c}{d}+3 b x\right )}{8 d^3}+\frac {3 b \cos ^2(a+b x) \sin (a+b x)}{2 d^2 (c+d x)}+\frac {3 b^2 \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{8 d^3}+\frac {9 b^2 \sin \left (3 a-\frac {3 b c}{d}\right ) \text {Si}\left (\frac {3 b c}{d}+3 b x\right )}{8 d^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.22, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {3395, 3384,
3380, 3383, 3393} \begin {gather*} -\frac {3 b^2 \cos \left (a-\frac {b c}{d}\right ) \text {CosIntegral}\left (\frac {b c}{d}+b x\right )}{8 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 )}{8 d^3}+\frac {3 b^2 \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{8 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 )}{8 d^3}+\frac {3 b \sin (a+b x) \cos ^2(a+b x)}{2 d^2 (c+d x)}-\frac {\cos ^3(a+b x)}{2 d (c+d x)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3380
Rule 3383
Rule 3384
Rule 3393
Rule 3395
Rubi steps
\begin {align*} \int \frac {\cos ^3(a+b x)}{(c+d x)^3} \, dx &=-\frac {\cos ^3(a+b x)}{2 d (c+d x)^2}+\frac {3 b \cos ^2(a+b x) \sin (a+b x)}{2 d^2 (c+d x)}+\frac {\left (3 b^2\right ) \int \frac {\cos (a+b x)}{c+d x} \, dx}{d^2}-\frac {\left (9 b^2\right ) \int \frac {\cos ^3(a+b x)}{c+d x} \, dx}{2 d^2}\\ &=-\frac {\cos ^3(a+b x)}{2 d (c+d x)^2}+\frac {3 b \cos ^2(a+b x) \sin (a+b x)}{2 d^2 (c+d x)}-\frac {\left (9 b^2\right ) \int \left (\frac {3 \cos (a+b x)}{4 (c+d x)}+\frac {\cos (3 a+3 b x)}{4 (c+d x)}\right ) \, dx}{2 d^2}+\frac {\left (3 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}{d^2}-\frac {\left (3 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}{d^2}\\ &=-\frac {\cos ^3(a+b x)}{2 d (c+d x)^2}+\frac {3 b^2 \cos \left (a-\frac {b c}{d}\right ) \text {Ci}\left (\frac {b c}{d}+b x\right )}{d^3}+\frac {3 b \cos ^2(a+b x) \sin (a+b x)}{2 d^2 (c+d x)}-\frac {3 b^2 \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{d^3}-\frac {\left (9 b^2\right ) \int \frac {\cos (3 a+3 b x)}{c+d x} \, dx}{8 d^2}-\frac {\left (27 b^2\right ) \int \frac {\cos (a+b x)}{c+d x} \, dx}{8 d^2}\\ &=-\frac {\cos ^3(a+b x)}{2 d (c+d x)^2}+\frac {3 b^2 \cos \left (a-\frac {b c}{d}\right ) \text {Ci}\left (\frac {b c}{d}+b x\right )}{d^3}+\frac {3 b \cos ^2(a+b x) \sin (a+b x)}{2 d^2 (c+d x)}-\frac {3 b^2 \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{d^3}-\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}{8 d^2}-\frac {\left (27 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}{8 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}{8 d^2}+\frac {\left (27 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}{8 d^2}\\ &=-\frac {\cos ^3(a+b x)}{2 d (c+d x)^2}-\frac {3 b^2 \cos \left (a-\frac {b c}{d}\right ) \text {Ci}\left (\frac {b c}{d}+b x\right )}{8 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 )}{8 d^3}+\frac {3 b \cos ^2(a+b x) \sin (a+b x)}{2 d^2 (c+d x)}+\frac {3 b^2 \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{8 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 )}{8 d^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.53, size = 221, normalized size = 1.20 \begin {gather*} \frac {6 d \cos (b x) (-d \cos (a)+b (c+d x) \sin (a))+2 d \cos (3 b x) (-d \cos (3 a)+3 b (c+d x) \sin (3 a))+6 d (b (c+d x) \cos (a)+d \sin (a)) \sin (b x)+2 d (3 b (c+d x) \cos (3 a)+d \sin (3 a)) \sin (3 b x)-6 b^2 (c+d x)^2 \left (\cos \left (a-\frac {b c}{d}\right ) \text {CosIntegral}\left (b \left (\frac {c}{d}+x\right )\right )+3 \cos \left (3 a-\frac {3 b c}{d}\right ) \text {CosIntegral}\left (\frac {3 b (c+d x)}{d}\right )-\sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (b \left (\frac {c}{d}+x\right )\right )-3 \sin \left (3 a-\frac {3 b c}{d}\right ) \text {Si}\left (\frac {3 b (c+d x)}{d}\right )\right )}{16 d^3 (c+d x)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.29, size = 316, normalized size = 1.72
method | result | size |
derivativedivides | \(\frac {\frac {b^{3} \left (-\frac {3 \cos \left (3 b x +3 a \right )}{2 \left (-d a +b c +d \left (b x +a \right )\right )^{2} d}-\frac {3 \left (-\frac {3 \sin \left (3 b x +3 a \right )}{\left (-d a +b c +d \left (b x +a \right )\right ) d}+\frac {-\frac {9 \sinIntegral \left (-3 b x -3 a -\frac {3 \left (-d a +b c \right )}{d}\right ) \sin \left (\frac {-3 d a +3 b c}{d}\right )}{d}+\frac {9 \cosineIntegral \left (3 b x +3 a +\frac {-3 d a +3 b c}{d}\right ) \cos \left (\frac {-3 d a +3 b c}{d}\right )}{d}}{d}\right )}{2 d}\right )}{12}+\frac {3 b^{3} \left (-\frac {\cos \left (b x +a \right )}{2 \left (-d a +b c +d \left (b x +a \right )\right )^{2} d}-\frac {-\frac {\sin \left (b x +a \right )}{\left (-d a +b c +d \left (b x +a \right )\right ) d}+\frac {-\frac {\sinIntegral \left (-b x -a -\frac {-d a +b c}{d}\right ) \sin \left (\frac {-d a +b c}{d}\right )}{d}+\frac {\cosineIntegral \left (b x +a +\frac {-d a +b c}{d}\right ) \cos \left (\frac {-d a +b c}{d}\right )}{d}}{d}}{2 d}\right )}{4}}{b}\) | \(316\) |
default | \(\frac {\frac {b^{3} \left (-\frac {3 \cos \left (3 b x +3 a \right )}{2 \left (-d a +b c +d \left (b x +a \right )\right )^{2} d}-\frac {3 \left (-\frac {3 \sin \left (3 b x +3 a \right )}{\left (-d a +b c +d \left (b x +a \right )\right ) d}+\frac {-\frac {9 \sinIntegral \left (-3 b x -3 a -\frac {3 \left (-d a +b c \right )}{d}\right ) \sin \left (\frac {-3 d a +3 b c}{d}\right )}{d}+\frac {9 \cosineIntegral \left (3 b x +3 a +\frac {-3 d a +3 b c}{d}\right ) \cos \left (\frac {-3 d a +3 b c}{d}\right )}{d}}{d}\right )}{2 d}\right )}{12}+\frac {3 b^{3} \left (-\frac {\cos \left (b x +a \right )}{2 \left (-d a +b c +d \left (b x +a \right )\right )^{2} d}-\frac {-\frac {\sin \left (b x +a \right )}{\left (-d a +b c +d \left (b x +a \right )\right ) d}+\frac {-\frac {\sinIntegral \left (-b x -a -\frac {-d a +b c}{d}\right ) \sin \left (\frac {-d a +b c}{d}\right )}{d}+\frac {\cosineIntegral \left (b x +a +\frac {-d a +b c}{d}\right ) \cos \left (\frac {-d a +b c}{d}\right )}{d}}{d}}{2 d}\right )}{4}}{b}\) | \(316\) |
risch | \(\frac {9 b^{2} {\mathrm e}^{-\frac {3 i \left (d a -b c \right )}{d}} \expIntegral \left (1, 3 i b x +3 i a -\frac {3 i \left (d a -b c \right )}{d}\right )}{16 d^{3}}+\frac {3 b^{2} {\mathrm e}^{-\frac {i \left (d a -b c \right )}{d}} \expIntegral \left (1, i b x +i a -\frac {i \left (d a -b c \right )}{d}\right )}{16 d^{3}}+\frac {3 b^{2} {\mathrm e}^{\frac {i \left (d a -b c \right )}{d}} \expIntegral \left (1, -i b x -i a -\frac {-i a d +i b c}{d}\right )}{16 d^{3}}+\frac {9 b^{2} {\mathrm e}^{\frac {3 i \left (d a -b c \right )}{d}} \expIntegral \left (1, -3 i b x -3 i a -\frac {3 \left (-i a d +i b c \right )}{d}\right )}{16 d^{3}}+\frac {3 \left (-2 b^{2} d^{3} x^{2}-4 b^{2} c \,d^{2} x -2 b^{2} c^{2} d \right ) \cos \left (b x +a \right )}{16 d^{2} \left (d x +c \right )^{2} \left (d^{2} x^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right )}-\frac {3 i \left (2 i b^{3} d^{3} x^{3}+6 i b^{3} c \,d^{2} x^{2}+6 i b^{3} c^{2} d x +2 i b^{3} c^{3}\right ) \sin \left (b x +a \right )}{16 d^{2} \left (d x +c \right )^{2} \left (d^{2} x^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right )}+\frac {\left (-2 b^{2} d^{3} x^{2}-4 b^{2} c \,d^{2} x -2 b^{2} c^{2} d \right ) \cos \left (3 b x +3 a \right )}{16 d^{2} \left (d x +c \right )^{2} \left (d^{2} x^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right )}-\frac {i \left (6 i b^{3} d^{3} x^{3}+18 i b^{3} c \,d^{2} x^{2}+18 i b^{3} c^{2} d x +6 i b^{3} c^{3}\right ) \sin \left (3 b x +3 a \right )}{16 d^{2} \left (d x +c \right )^{2} \left (d^{2} x^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right )}\) | \(548\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [C] Result contains complex when optimal does not.
time = 0.53, size = 339, normalized size = 1.84 \begin {gather*} -\frac {3 \, 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 ) + b^{3} {\left (E_{3}\left (\frac {3 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + E_{3}\left (-\frac {3 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + 3 \, b^{3} {\left (-i \, E_{3}\left (\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right ) + 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 (i \, E_{3}\left (\frac {3 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) - i \, E_{3}\left (-\frac {3 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )}{8 \, {\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 375 vs.
\(2 (172) = 344\).
time = 0.43, size = 375, normalized size = 2.04 \begin {gather*} -\frac {8 \, d^{2} \cos \left (b x + a\right )^{3} - 24 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )^{2} \sin \left (b x + a\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 ) - 6 \, {\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 ) + 3 \, {\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 )}{16 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos ^{3}{\left (a + b x \right )}}{\left (c + d x\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 2.71, size = 115446, normalized size = 627.42 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\cos \left (a+b\,x\right )}^3}{{\left (c+d\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________