Optimal. Leaf size=69 \[ -\frac {\cos (c+d x)}{b d}-\frac {a \text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{b^2}-\frac {a \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.11, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6874, 2718,
3384, 3380, 3383} \begin {gather*} -\frac {a \sin \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{b^2}-\frac {a \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^2}-\frac {\cos (c+d x)}{b d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2718
Rule 3380
Rule 3383
Rule 3384
Rule 6874
Rubi steps
\begin {align*} \int \frac {x \sin (c+d x)}{a+b x} \, dx &=\int \left (\frac {\sin (c+d x)}{b}-\frac {a \sin (c+d x)}{b (a+b x)}\right ) \, dx\\ &=\frac {\int \sin (c+d x) \, dx}{b}-\frac {a \int \frac {\sin (c+d x)}{a+b x} \, dx}{b}\\ &=-\frac {\cos (c+d x)}{b d}-\frac {\left (a \cos \left (c-\frac {a d}{b}\right )\right ) \int \frac {\sin \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b}-\frac {\left (a \sin \left (c-\frac {a d}{b}\right )\right ) \int \frac {\cos \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b}\\ &=-\frac {\cos (c+d x)}{b d}-\frac {a \text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{b^2}-\frac {a \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.10, size = 63, normalized size = 0.91 \begin {gather*} -\frac {b \cos (c+d x)+a d \text {Ci}\left (d \left (\frac {a}{b}+x\right )\right ) \sin \left (c-\frac {a d}{b}\right )+a d \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )}{b^2 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(179\) vs.
\(2(70)=140\).
time = 0.06, size = 180, normalized size = 2.61
method | result | size |
risch | \(-\frac {\cos \left (d x +c \right )}{b d}+\frac {i a \cos \left (\frac {d a -c b}{b}\right ) \expIntegral \left (1, \frac {i d \left (b x +a \right )}{b}\right )}{2 b^{2}}-\frac {i a \cos \left (\frac {d a -c b}{b}\right ) \expIntegral \left (1, -\frac {i d \left (b x +a \right )}{b}\right )}{2 b^{2}}-\frac {a \sin \left (\frac {d a -c b}{b}\right ) \expIntegral \left (1, \frac {i d \left (b x +a \right )}{b}\right )}{2 b^{2}}-\frac {a \sin \left (\frac {d a -c b}{b}\right ) \expIntegral \left (1, -\frac {i d \left (b x +a \right )}{b}\right )}{2 b^{2}}\) | \(150\) |
derivativedivides | \(\frac {-\frac {\left (d a -c b \right ) d \left (\frac {\sinIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\cosineIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b}-\frac {d \cos \left (d x +c \right )}{b}-d c \left (\frac {\sinIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\cosineIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{d^{2}}\) | \(180\) |
default | \(\frac {-\frac {\left (d a -c b \right ) d \left (\frac {\sinIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\cosineIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b}-\frac {d \cos \left (d x +c \right )}{b}-d c \left (\frac {\sinIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\cosineIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{d^{2}}\) | \(180\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [C] Result contains complex when optimal does not.
time = 0.42, size = 776, normalized size = 11.25 \begin {gather*} -\frac {\frac {{\left (d {\left (-i \, E_{1}\left (\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right ) + i \, E_{1}\left (-\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right )\right )} \cos \left (-\frac {b c - a d}{b}\right ) + d {\left (E_{1}\left (\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right ) + E_{1}\left (-\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right )\right )} \sin \left (-\frac {b c - a d}{b}\right )\right )} c}{b} + \frac {{\left (d x + c\right )} b d \cos \left (d x + c\right )^{3} + {\left (d x + c\right )} b d \cos \left (d x + c\right ) - {\left ({\left (b c d {\left (E_{2}\left (\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right ) + E_{2}\left (-\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right )\right )} - a d^{2} {\left (E_{2}\left (\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right ) + E_{2}\left (-\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right )\right )}\right )} \cos \left (-\frac {b c - a d}{b}\right ) - {\left (a d^{2} {\left (i \, E_{2}\left (\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right ) - i \, E_{2}\left (-\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right )\right )} + b c d {\left (-i \, E_{2}\left (\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right ) + i \, E_{2}\left (-\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right )\right )}\right )} \sin \left (-\frac {b c - a d}{b}\right )\right )} \cos \left (d x + c\right )^{2} + {\left ({\left (d x + c\right )} b d \cos \left (d x + c\right ) - {\left (b c d {\left (E_{2}\left (\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right ) + E_{2}\left (-\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right )\right )} - a d^{2} {\left (E_{2}\left (\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right ) + E_{2}\left (-\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right )\right )}\right )} \cos \left (-\frac {b c - a d}{b}\right ) + {\left (a d^{2} {\left (i \, E_{2}\left (\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right ) - i \, E_{2}\left (-\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right )\right )} + b c d {\left (-i \, E_{2}\left (\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right ) + i \, E_{2}\left (-\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right )\right )}\right )} \sin \left (-\frac {b c - a d}{b}\right )\right )} \sin \left (d x + c\right )^{2}}{{\left ({\left (d x + c\right )} b^{2} - b^{2} c + a b d\right )} \cos \left (d x + c\right )^{2} + {\left ({\left (d x + c\right )} b^{2} - b^{2} c + a b d\right )} \sin \left (d x + c\right )^{2}}}{2 \, d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.35, size = 99, normalized size = 1.43 \begin {gather*} -\frac {2 \, a d \cos \left (-\frac {b c - a d}{b}\right ) \operatorname {Si}\left (\frac {b d x + a d}{b}\right ) + 2 \, b \cos \left (d x + c\right ) - {\left (a d \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + a d \operatorname {Ci}\left (-\frac {b d x + a d}{b}\right )\right )} \sin \left (-\frac {b c - a d}{b}\right )}{2 \, b^{2} d} \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 {x \sin {\left (c + d x \right )}}{a + b x}\, 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 = 4.39, size = 1647, normalized size = 23.87 \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 {x\,\sin \left (c+d\,x\right )}{a+b\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________