Optimal. Leaf size=501 \[ \frac {d \cos (c) \text {Ci}(d x)}{a^2}+\frac {d \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a^2}+\frac {d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 a^2}+\frac {3 \sqrt {b} \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{5/2}}-\frac {3 \sqrt {b} \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{5/2}}-\frac {\sin (c+d x)}{a^2 x}+\frac {\sqrt {b} \sin (c+d x)}{4 a^2 \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {\sqrt {b} \sin (c+d x)}{4 a^2 \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {d \sin (c) \text {Si}(d x)}{a^2}+\frac {3 \sqrt {b} \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{5/2}}+\frac {d \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a^2}+\frac {3 \sqrt {b} \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 (-a)^{5/2}}-\frac {d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 a^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.00, antiderivative size = 501, normalized size of antiderivative = 1.00, number of steps
used = 32, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3426, 3378,
3384, 3380, 3383, 3414} \begin {gather*} \frac {d \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a^2}+\frac {d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 a^2}+\frac {d \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a^2}-\frac {d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 a^2}+\frac {\sqrt {b} \sin (c+d x)}{4 a^2 \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {\sqrt {b} \sin (c+d x)}{4 a^2 \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {d \cos (c) \text {CosIntegral}(d x)}{a^2}-\frac {d \sin (c) \text {Si}(d x)}{a^2}-\frac {\sin (c+d x)}{a^2 x}+\frac {3 \sqrt {b} \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 (-a)^{5/2}}-\frac {3 \sqrt {b} \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{5/2}}+\frac {3 \sqrt {b} \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{5/2}}+\frac {3 \sqrt {b} \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3378
Rule 3380
Rule 3383
Rule 3384
Rule 3414
Rule 3426
Rubi steps
\begin {align*} \int \frac {\sin (c+d x)}{x^2 \left (a+b x^2\right )^2} \, dx &=\int \left (\frac {\sin (c+d x)}{a^2 x^2}-\frac {b \sin (c+d x)}{a \left (a+b x^2\right )^2}-\frac {b \sin (c+d x)}{a^2 \left (a+b x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {\sin (c+d x)}{x^2} \, dx}{a^2}-\frac {b \int \frac {\sin (c+d x)}{a+b x^2} \, dx}{a^2}-\frac {b \int \frac {\sin (c+d x)}{\left (a+b x^2\right )^2} \, dx}{a}\\ &=-\frac {\sin (c+d x)}{a^2 x}-\frac {b \int \left (\frac {\sqrt {-a} \sin (c+d x)}{2 a \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {-a} \sin (c+d x)}{2 a \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{a^2}-\frac {b \int \left (-\frac {b \sin (c+d x)}{4 a \left (\sqrt {-a} \sqrt {b}-b x\right )^2}-\frac {b \sin (c+d x)}{4 a \left (\sqrt {-a} \sqrt {b}+b x\right )^2}-\frac {b \sin (c+d x)}{2 a \left (-a b-b^2 x^2\right )}\right ) \, dx}{a}+\frac {d \int \frac {\cos (c+d x)}{x} \, dx}{a^2}\\ &=-\frac {\sin (c+d x)}{a^2 x}+\frac {b \int \frac {\sin (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 (-a)^{5/2}}+\frac {b \int \frac {\sin (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 (-a)^{5/2}}+\frac {b^2 \int \frac {\sin (c+d x)}{\left (\sqrt {-a} \sqrt {b}-b x\right )^2} \, dx}{4 a^2}+\frac {b^2 \int \frac {\sin (c+d x)}{\left (\sqrt {-a} \sqrt {b}+b x\right )^2} \, dx}{4 a^2}+\frac {b^2 \int \frac {\sin (c+d x)}{-a b-b^2 x^2} \, dx}{2 a^2}+\frac {(d \cos (c)) \int \frac {\cos (d x)}{x} \, dx}{a^2}-\frac {(d \sin (c)) \int \frac {\sin (d x)}{x} \, dx}{a^2}\\ &=\frac {d \cos (c) \text {Ci}(d x)}{a^2}-\frac {\sin (c+d x)}{a^2 x}+\frac {\sqrt {b} \sin (c+d x)}{4 a^2 \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {\sqrt {b} \sin (c+d x)}{4 a^2 \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {d \sin (c) \text {Si}(d x)}{a^2}+\frac {b^2 \int \left (-\frac {\sqrt {-a} \sin (c+d x)}{2 a b \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {\sqrt {-a} \sin (c+d x)}{2 a b \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{2 a^2}-\frac {(b d) \int \frac {\cos (c+d x)}{\sqrt {-a} \sqrt {b}-b x} \, dx}{4 a^2}+\frac {(b d) \int \frac {\cos (c+d x)}{\sqrt {-a} \sqrt {b}+b x} \, dx}{4 a^2}+\frac {\left (b \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 (-a)^{5/2}}-\frac {\left (b \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 (-a)^{5/2}}+\frac {\left (b \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 (-a)^{5/2}}+\frac {\left (b \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 (-a)^{5/2}}\\ &=\frac {d \cos (c) \text {Ci}(d x)}{a^2}+\frac {\sqrt {b} \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 (-a)^{5/2}}-\frac {\sqrt {b} \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 (-a)^{5/2}}-\frac {\sin (c+d x)}{a^2 x}+\frac {\sqrt {b} \sin (c+d x)}{4 a^2 \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {\sqrt {b} \sin (c+d x)}{4 a^2 \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {d \sin (c) \text {Si}(d x)}{a^2}+\frac {\sqrt {b} \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 (-a)^{5/2}}+\frac {\sqrt {b} \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 (-a)^{5/2}}+\frac {b \int \frac {\sin (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{4 (-a)^{5/2}}+\frac {b \int \frac {\sin (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{4 (-a)^{5/2}}+\frac {\left (b d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a} \sqrt {b}+b x} \, dx}{4 a^2}-\frac {\left (b d \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a} \sqrt {b}-b x} \, dx}{4 a^2}-\frac {\left (b d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a} \sqrt {b}+b x} \, dx}{4 a^2}-\frac {\left (b d \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a} \sqrt {b}-b x} \, dx}{4 a^2}\\ &=\frac {d \cos (c) \text {Ci}(d x)}{a^2}+\frac {d \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a^2}+\frac {d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 a^2}+\frac {\sqrt {b} \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 (-a)^{5/2}}-\frac {\sqrt {b} \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 (-a)^{5/2}}-\frac {\sin (c+d x)}{a^2 x}+\frac {\sqrt {b} \sin (c+d x)}{4 a^2 \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {\sqrt {b} \sin (c+d x)}{4 a^2 \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {d \sin (c) \text {Si}(d x)}{a^2}+\frac {\sqrt {b} \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 (-a)^{5/2}}+\frac {d \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a^2}+\frac {\sqrt {b} \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 (-a)^{5/2}}-\frac {d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 a^2}+\frac {\left (b \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a}+\sqrt {b} x} \, dx}{4 (-a)^{5/2}}-\frac {\left (b \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a}-\sqrt {b} x} \, dx}{4 (-a)^{5/2}}+\frac {\left (b \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a}+\sqrt {b} x} \, dx}{4 (-a)^{5/2}}+\frac {\left (b \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a}-\sqrt {b} x} \, dx}{4 (-a)^{5/2}}\\ &=\frac {d \cos (c) \text {Ci}(d x)}{a^2}+\frac {d \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a^2}+\frac {d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 a^2}+\frac {3 \sqrt {b} \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{5/2}}-\frac {3 \sqrt {b} \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{5/2}}-\frac {\sin (c+d x)}{a^2 x}+\frac {\sqrt {b} \sin (c+d x)}{4 a^2 \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {\sqrt {b} \sin (c+d x)}{4 a^2 \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {d \sin (c) \text {Si}(d x)}{a^2}+\frac {3 \sqrt {b} \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{5/2}}+\frac {d \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a^2}+\frac {3 \sqrt {b} \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 (-a)^{5/2}}-\frac {d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 a^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 0.62, size = 768, normalized size = 1.53 \begin {gather*} \frac {4 \sqrt {a} d x \left (a+b x^2\right ) \cos (c) \text {Ci}(d x)+a^{3/2} d x \cos \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+\sqrt {a} b d x^3 \cos \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )-3 i a \sqrt {b} x \text {Ci}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right ) \sin \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right )-3 i b^{3/2} x^3 \text {Ci}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right ) \sin \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right )+x \left (a+b x^2\right ) \text {Ci}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right ) \left (\sqrt {a} d \cos \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right )+3 i \sqrt {b} \sin \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right )\right )-4 a^{3/2} \sin (c+d x)-6 \sqrt {a} b x^2 \sin (c+d x)-4 a^{3/2} d x \sin (c) \text {Si}(d x)-4 \sqrt {a} b d x^3 \sin (c) \text {Si}(d x)-3 i a \sqrt {b} x \cos \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )-3 i b^{3/2} x^3 \cos \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )-a^{3/2} d x \sin \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )-\sqrt {a} b d x^3 \sin \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )-3 i a \sqrt {b} x \cos \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )-3 i b^{3/2} x^3 \cos \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+a^{3/2} d x \sin \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+\sqrt {a} b d x^3 \sin \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )}{4 a^{5/2} x \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.19, size = 761, normalized size = 1.52 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] Result contains complex when optimal does not.
time = 0.39, size = 406, normalized size = 0.81 \begin {gather*} \frac {4 \, {\left (a b d^{2} x^{3} + a^{2} d^{2} x\right )} {\rm Ei}\left (i \, d x\right ) e^{\left (i \, c\right )} + 4 \, {\left (a b d^{2} x^{3} + a^{2} d^{2} x\right )} {\rm Ei}\left (-i \, d x\right ) e^{\left (-i \, c\right )} + {\left (a b d^{2} x^{3} + a^{2} d^{2} x - 3 \, {\left (b^{2} x^{3} + a b x\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (a b d^{2} x^{3} + a^{2} d^{2} x + 3 \, {\left (b^{2} x^{3} + a b x\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (a b d^{2} x^{3} + a^{2} d^{2} x - 3 \, {\left (b^{2} x^{3} + a b x\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (-i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (a b d^{2} x^{3} + a^{2} d^{2} x + 3 \, {\left (b^{2} x^{3} + a b x\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (-i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} - 4 \, {\left (3 \, a b d x^{2} + 2 \, a^{2} d\right )} \sin \left (d x + c\right )}{8 \, {\left (a^{3} b d x^{3} + a^{4} d x\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 {\sin {\left (c + d x \right )}}{x^{2} \left (a + b x^{2}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sin \left (c+d\,x\right )}{x^2\,{\left (b\,x^2+a\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________