Optimal. Leaf size=241 \[ -\frac {a^2 d^2 \sin \left (c-\frac {a d}{b}\right ) \text {Ci}\left (x d+\frac {a d}{b}\right )}{2 b^5}-\frac {a^2 d^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{2 b^5}-\frac {a^2 d \cos (c+d x)}{2 b^4 (a+b x)}-\frac {a^2 \sin (c+d x)}{2 b^3 (a+b x)^2}-\frac {2 a d \cos \left (c-\frac {a d}{b}\right ) \text {Ci}\left (x d+\frac {a d}{b}\right )}{b^4}+\frac {2 a d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^4}+\frac {\sin \left (c-\frac {a d}{b}\right ) \text {Ci}\left (x d+\frac {a d}{b}\right )}{b^3}+\frac {\cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^3}+\frac {2 a \sin (c+d x)}{b^3 (a+b x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.54, antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {6742, 3297, 3303, 3299, 3302} \[ -\frac {a^2 d^2 \sin \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{2 b^5}-\frac {a^2 d^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{2 b^5}-\frac {a^2 \sin (c+d x)}{2 b^3 (a+b x)^2}-\frac {a^2 d \cos (c+d x)}{2 b^4 (a+b x)}+\frac {\sin \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{b^3}-\frac {2 a d \cos \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{b^4}+\frac {2 a d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^4}+\frac {\cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^3}+\frac {2 a \sin (c+d x)}{b^3 (a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rule 6742
Rubi steps
\begin {align*} \int \frac {x^2 \sin (c+d x)}{(a+b x)^3} \, dx &=\int \left (\frac {a^2 \sin (c+d x)}{b^2 (a+b x)^3}-\frac {2 a \sin (c+d x)}{b^2 (a+b x)^2}+\frac {\sin (c+d x)}{b^2 (a+b x)}\right ) \, dx\\ &=\frac {\int \frac {\sin (c+d x)}{a+b x} \, dx}{b^2}-\frac {(2 a) \int \frac {\sin (c+d x)}{(a+b x)^2} \, dx}{b^2}+\frac {a^2 \int \frac {\sin (c+d x)}{(a+b x)^3} \, dx}{b^2}\\ &=-\frac {a^2 \sin (c+d x)}{2 b^3 (a+b x)^2}+\frac {2 a \sin (c+d x)}{b^3 (a+b x)}-\frac {(2 a d) \int \frac {\cos (c+d x)}{a+b x} \, dx}{b^3}+\frac {\left (a^2 d\right ) \int \frac {\cos (c+d x)}{(a+b x)^2} \, dx}{2 b^3}+\frac {\cos \left (c-\frac {a d}{b}\right ) \int \frac {\sin \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b^2}+\frac {\sin \left (c-\frac {a d}{b}\right ) \int \frac {\cos \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b^2}\\ &=-\frac {a^2 d \cos (c+d x)}{2 b^4 (a+b x)}+\frac {\text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{b^3}-\frac {a^2 \sin (c+d x)}{2 b^3 (a+b x)^2}+\frac {2 a \sin (c+d x)}{b^3 (a+b x)}+\frac {\cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^3}-\frac {\left (a^2 d^2\right ) \int \frac {\sin (c+d x)}{a+b x} \, dx}{2 b^4}-\frac {\left (2 a d \cos \left (c-\frac {a d}{b}\right )\right ) \int \frac {\cos \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b^3}+\frac {\left (2 a d \sin \left (c-\frac {a d}{b}\right )\right ) \int \frac {\sin \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b^3}\\ &=-\frac {a^2 d \cos (c+d x)}{2 b^4 (a+b x)}-\frac {2 a d \cos \left (c-\frac {a d}{b}\right ) \text {Ci}\left (\frac {a d}{b}+d x\right )}{b^4}+\frac {\text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{b^3}-\frac {a^2 \sin (c+d x)}{2 b^3 (a+b x)^2}+\frac {2 a \sin (c+d x)}{b^3 (a+b x)}+\frac {\cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^3}+\frac {2 a d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^4}-\frac {\left (a^2 d^2 \cos \left (c-\frac {a d}{b}\right )\right ) \int \frac {\sin \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{2 b^4}-\frac {\left (a^2 d^2 \sin \left (c-\frac {a d}{b}\right )\right ) \int \frac {\cos \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{2 b^4}\\ &=-\frac {a^2 d \cos (c+d x)}{2 b^4 (a+b x)}-\frac {2 a d \cos \left (c-\frac {a d}{b}\right ) \text {Ci}\left (\frac {a d}{b}+d x\right )}{b^4}+\frac {\text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{b^3}-\frac {a^2 d^2 \text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{2 b^5}-\frac {a^2 \sin (c+d x)}{2 b^3 (a+b x)^2}+\frac {2 a \sin (c+d x)}{b^3 (a+b x)}+\frac {\cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^3}-\frac {a^2 d^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{2 b^5}+\frac {2 a d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.25, size = 154, normalized size = 0.64 \[ -\frac {-\text {Ci}\left (d \left (\frac {a}{b}+x\right )\right ) \left (\left (2 b^2-a^2 d^2\right ) \sin \left (c-\frac {a d}{b}\right )-4 a b d \cos \left (c-\frac {a d}{b}\right )\right )+\text {Si}\left (d \left (\frac {a}{b}+x\right )\right ) \left (\left (a^2 d^2-2 b^2\right ) \cos \left (c-\frac {a d}{b}\right )-4 a b d \sin \left (c-\frac {a d}{b}\right )\right )+\frac {a b (a d (a+b x) \cos (c+d x)-b (3 a+4 b x) \sin (c+d x))}{(a+b x)^2}}{2 b^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.88, size = 438, normalized size = 1.82 \[ -\frac {2 \, {\left (a^{2} b^{2} d x + a^{3} b d\right )} \cos \left (d x + c\right ) + 2 \, {\left (2 \, {\left (a b^{3} d x^{2} + 2 \, a^{2} b^{2} d x + a^{3} b d\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + 2 \, {\left (a b^{3} d x^{2} + 2 \, a^{2} b^{2} d x + a^{3} b d\right )} \operatorname {Ci}\left (-\frac {b d x + a d}{b}\right ) + {\left (a^{4} d^{2} - 2 \, a^{2} b^{2} + {\left (a^{2} b^{2} d^{2} - 2 \, b^{4}\right )} x^{2} + 2 \, {\left (a^{3} b d^{2} - 2 \, a b^{3}\right )} x\right )} \operatorname {Si}\left (\frac {b d x + a d}{b}\right )\right )} \cos \left (-\frac {b c - a d}{b}\right ) - 2 \, {\left (4 \, a b^{3} x + 3 \, a^{2} b^{2}\right )} \sin \left (d x + c\right ) - {\left ({\left (a^{4} d^{2} - 2 \, a^{2} b^{2} + {\left (a^{2} b^{2} d^{2} - 2 \, b^{4}\right )} x^{2} + 2 \, {\left (a^{3} b d^{2} - 2 \, a b^{3}\right )} x\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + {\left (a^{4} d^{2} - 2 \, a^{2} b^{2} + {\left (a^{2} b^{2} d^{2} - 2 \, b^{4}\right )} x^{2} + 2 \, {\left (a^{3} b d^{2} - 2 \, a b^{3}\right )} x\right )} \operatorname {Ci}\left (-\frac {b d x + a d}{b}\right ) - 8 \, {\left (a b^{3} d x^{2} + 2 \, a^{2} b^{2} d x + a^{3} b d\right )} \operatorname {Si}\left (\frac {b d x + a d}{b}\right )\right )} \sin \left (-\frac {b c - a d}{b}\right )}{4 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.03, size = 779, normalized size = 3.23 \[ \frac {\frac {d^{3} \left (d a -c b \right )^{2} \left (-\frac {\sin \left (d x +c \right )}{2 \left (\left (d x +c \right ) b +d a -c b \right )^{2} b}+\frac {-\frac {\cos \left (d x +c \right )}{\left (\left (d x +c \right ) b +d a -c b \right ) b}-\frac {\frac {\Si \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\Ci \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}}{b}}{2 b}\right )}{b^{2}}+\frac {d^{3} \left (\frac {\Si \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\Ci \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b^{2}}-\frac {2 d^{3} \left (d a -c b \right ) \left (-\frac {\sin \left (d x +c \right )}{\left (\left (d x +c \right ) b +d a -c b \right ) b}+\frac {\frac {\Si \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}+\frac {\Ci \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}}{b}\right )}{b^{2}}+\frac {2 d^{3} \left (d a -c b \right ) c \left (-\frac {\sin \left (d x +c \right )}{2 \left (\left (d x +c \right ) b +d a -c b \right )^{2} b}+\frac {-\frac {\cos \left (d x +c \right )}{\left (\left (d x +c \right ) b +d a -c b \right ) b}-\frac {\frac {\Si \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\Ci \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}}{b}}{2 b}\right )}{b}-\frac {2 d^{3} c \left (-\frac {\sin \left (d x +c \right )}{\left (\left (d x +c \right ) b +d a -c b \right ) b}+\frac {\frac {\Si \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}+\frac {\Ci \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}}{b}\right )}{b}+d^{3} c^{2} \left (-\frac {\sin \left (d x +c \right )}{2 \left (\left (d x +c \right ) b +d a -c b \right )^{2} b}+\frac {-\frac {\cos \left (d x +c \right )}{\left (\left (d x +c \right ) b +d a -c b \right ) b}-\frac {\frac {\Si \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\Ci \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}}{b}}{2 b}\right )}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2\,\sin \left (c+d\,x\right )}{{\left (a+b\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \sin {\left (c + d x \right )}}{\left (a + b x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________