Optimal. Leaf size=134 \[ -\frac {b^2 \cos (c+d x)}{d}-\frac {a^2 d \cos (c+d x)}{6 x^2}+2 a b d \cos (c) \text {Ci}(d x)-\frac {1}{6} a^2 d^3 \cos (c) \text {Ci}(d x)-\frac {a^2 \sin (c+d x)}{3 x^3}-\frac {2 a b \sin (c+d x)}{x}+\frac {a^2 d^2 \sin (c+d x)}{6 x}-2 a b d \sin (c) \text {Si}(d x)+\frac {1}{6} a^2 d^3 \sin (c) \text {Si}(d x) \]
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Rubi [A]
time = 0.16, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3420, 2718,
3378, 3384, 3380, 3383} \begin {gather*} -\frac {1}{6} a^2 d^3 \cos (c) \text {CosIntegral}(d x)+\frac {1}{6} a^2 d^3 \sin (c) \text {Si}(d x)+\frac {a^2 d^2 \sin (c+d x)}{6 x}-\frac {a^2 \sin (c+d x)}{3 x^3}-\frac {a^2 d \cos (c+d x)}{6 x^2}+2 a b d \cos (c) \text {CosIntegral}(d x)-2 a b d \sin (c) \text {Si}(d x)-\frac {2 a b \sin (c+d x)}{x}-\frac {b^2 \cos (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rule 3378
Rule 3380
Rule 3383
Rule 3384
Rule 3420
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2 \sin (c+d x)}{x^4} \, dx &=\int \left (b^2 \sin (c+d x)+\frac {a^2 \sin (c+d x)}{x^4}+\frac {2 a b \sin (c+d x)}{x^2}\right ) \, dx\\ &=a^2 \int \frac {\sin (c+d x)}{x^4} \, dx+(2 a b) \int \frac {\sin (c+d x)}{x^2} \, dx+b^2 \int \sin (c+d x) \, dx\\ &=-\frac {b^2 \cos (c+d x)}{d}-\frac {a^2 \sin (c+d x)}{3 x^3}-\frac {2 a b \sin (c+d x)}{x}+\frac {1}{3} \left (a^2 d\right ) \int \frac {\cos (c+d x)}{x^3} \, dx+(2 a b d) \int \frac {\cos (c+d x)}{x} \, dx\\ &=-\frac {b^2 \cos (c+d x)}{d}-\frac {a^2 d \cos (c+d x)}{6 x^2}-\frac {a^2 \sin (c+d x)}{3 x^3}-\frac {2 a b \sin (c+d x)}{x}-\frac {1}{6} \left (a^2 d^2\right ) \int \frac {\sin (c+d x)}{x^2} \, dx+(2 a b d \cos (c)) \int \frac {\cos (d x)}{x} \, dx-(2 a b d \sin (c)) \int \frac {\sin (d x)}{x} \, dx\\ &=-\frac {b^2 \cos (c+d x)}{d}-\frac {a^2 d \cos (c+d x)}{6 x^2}+2 a b d \cos (c) \text {Ci}(d x)-\frac {a^2 \sin (c+d x)}{3 x^3}-\frac {2 a b \sin (c+d x)}{x}+\frac {a^2 d^2 \sin (c+d x)}{6 x}-2 a b d \sin (c) \text {Si}(d x)-\frac {1}{6} \left (a^2 d^3\right ) \int \frac {\cos (c+d x)}{x} \, dx\\ &=-\frac {b^2 \cos (c+d x)}{d}-\frac {a^2 d \cos (c+d x)}{6 x^2}+2 a b d \cos (c) \text {Ci}(d x)-\frac {a^2 \sin (c+d x)}{3 x^3}-\frac {2 a b \sin (c+d x)}{x}+\frac {a^2 d^2 \sin (c+d x)}{6 x}-2 a b d \sin (c) \text {Si}(d x)-\frac {1}{6} \left (a^2 d^3 \cos (c)\right ) \int \frac {\cos (d x)}{x} \, dx+\frac {1}{6} \left (a^2 d^3 \sin (c)\right ) \int \frac {\sin (d x)}{x} \, dx\\ &=-\frac {b^2 \cos (c+d x)}{d}-\frac {a^2 d \cos (c+d x)}{6 x^2}+2 a b d \cos (c) \text {Ci}(d x)-\frac {1}{6} a^2 d^3 \cos (c) \text {Ci}(d x)-\frac {a^2 \sin (c+d x)}{3 x^3}-\frac {2 a b \sin (c+d x)}{x}+\frac {a^2 d^2 \sin (c+d x)}{6 x}-2 a b d \sin (c) \text {Si}(d x)+\frac {1}{6} a^2 d^3 \sin (c) \text {Si}(d x)\\ \end {align*}
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Mathematica [A]
time = 0.27, size = 114, normalized size = 0.85 \begin {gather*} \frac {1}{6} \left (-\frac {6 b^2 \cos (c+d x)}{d}-\frac {a^2 d \cos (c+d x)}{x^2}-a d \left (-12 b+a d^2\right ) \cos (c) \text {Ci}(d x)-\frac {2 a^2 \sin (c+d x)}{x^3}-\frac {12 a b \sin (c+d x)}{x}+\frac {a^2 d^2 \sin (c+d x)}{x}+a d \left (-12 b+a d^2\right ) \sin (c) \text {Si}(d x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 120, normalized size = 0.90
method | result | size |
derivativedivides | \(d^{3} \left (a^{2} \left (-\frac {\sin \left (d x +c \right )}{3 d^{3} x^{3}}-\frac {\cos \left (d x +c \right )}{6 d^{2} x^{2}}+\frac {\sin \left (d x +c \right )}{6 d x}+\frac {\sinIntegral \left (d x \right ) \sin \left (c \right )}{6}-\frac {\cosineIntegral \left (d x \right ) \cos \left (c \right )}{6}\right )+\frac {2 a b \left (-\frac {\sin \left (d x +c \right )}{d x}-\sinIntegral \left (d x \right ) \sin \left (c \right )+\cosineIntegral \left (d x \right ) \cos \left (c \right )\right )}{d^{2}}-\frac {b^{2} \cos \left (d x +c \right )}{d^{4}}\right )\) | \(120\) |
default | \(d^{3} \left (a^{2} \left (-\frac {\sin \left (d x +c \right )}{3 d^{3} x^{3}}-\frac {\cos \left (d x +c \right )}{6 d^{2} x^{2}}+\frac {\sin \left (d x +c \right )}{6 d x}+\frac {\sinIntegral \left (d x \right ) \sin \left (c \right )}{6}-\frac {\cosineIntegral \left (d x \right ) \cos \left (c \right )}{6}\right )+\frac {2 a b \left (-\frac {\sin \left (d x +c \right )}{d x}-\sinIntegral \left (d x \right ) \sin \left (c \right )+\cosineIntegral \left (d x \right ) \cos \left (c \right )\right )}{d^{2}}-\frac {b^{2} \cos \left (d x +c \right )}{d^{4}}\right )\) | \(120\) |
risch | \(\frac {\expIntegral \left (1, -i d x \right ) \cos \left (c \right ) a^{2} d^{3}}{12}+\frac {\expIntegral \left (1, i d x \right ) \cos \left (c \right ) a^{2} d^{3}}{12}-\cos \left (c \right ) \expIntegral \left (1, -i d x \right ) a b d -\cos \left (c \right ) \expIntegral \left (1, i d x \right ) a b d +\frac {i \expIntegral \left (1, -i d x \right ) \sin \left (c \right ) a^{2} d^{3}}{12}-\frac {i \expIntegral \left (1, i d x \right ) \sin \left (c \right ) a^{2} d^{3}}{12}-i \sin \left (c \right ) \expIntegral \left (1, -i d x \right ) a b d +i \sin \left (c \right ) \expIntegral \left (1, i d x \right ) a b d -\frac {\left (2 a^{2} d^{8} x^{4}+12 b^{2} x^{6} d^{6}\right ) \cos \left (d x +c \right )}{12 d^{7} x^{6}}+\frac {i \left (-2 i a^{2} d^{9} x^{5}+24 i a b \,d^{7} x^{5}+4 i a^{2} d^{7} x^{3}\right ) \sin \left (d x +c \right )}{12 d^{7} x^{6}}\) | \(218\) |
meijerg | \(\frac {b^{2} \sin \left (c \right ) \sin \left (d x \right )}{d}+\frac {b^{2} \sqrt {\pi }\, \cos \left (c \right ) \left (\frac {1}{\sqrt {\pi }}-\frac {\cos \left (d x \right )}{\sqrt {\pi }}\right )}{d}+\frac {d^{2} a b \sqrt {\pi }\, \sin \left (c \right ) \left (-\frac {4 d^{2} \cos \left (x \sqrt {d^{2}}\right )}{x \left (d^{2}\right )^{\frac {3}{2}} \sqrt {\pi }}-\frac {4 \sinIntegral \left (x \sqrt {d^{2}}\right )}{\sqrt {\pi }}\right )}{2 \sqrt {d^{2}}}+\frac {d a b \sqrt {\pi }\, \cos \left (c \right ) \left (\frac {4 \gamma -4+4 \ln \left (x \right )+4 \ln \left (d \right )}{\sqrt {\pi }}+\frac {4}{\sqrt {\pi }}-\frac {4 \gamma }{\sqrt {\pi }}-\frac {4 \ln \left (2\right )}{\sqrt {\pi }}-\frac {4 \ln \left (\frac {d x}{2}\right )}{\sqrt {\pi }}-\frac {4 \sin \left (d x \right )}{\sqrt {\pi }\, d x}+\frac {4 \cosineIntegral \left (d x \right )}{\sqrt {\pi }}\right )}{2}+\frac {a^{2} \sqrt {\pi }\, \sin \left (c \right ) d^{4} \left (-\frac {8 \left (-d^{2} x^{2}+2\right ) d^{2} \cos \left (x \sqrt {d^{2}}\right )}{3 x^{3} \left (d^{2}\right )^{\frac {5}{2}} \sqrt {\pi }}+\frac {8 \sin \left (x \sqrt {d^{2}}\right )}{3 d^{2} x^{2} \sqrt {\pi }}+\frac {8 \sinIntegral \left (x \sqrt {d^{2}}\right )}{3 \sqrt {\pi }}\right )}{16 \sqrt {d^{2}}}+\frac {a^{2} \sqrt {\pi }\, \cos \left (c \right ) d^{3} \left (-\frac {8}{\sqrt {\pi }\, x^{2} d^{2}}-\frac {4 \left (2 \gamma -\frac {11}{3}+2 \ln \left (x \right )+2 \ln \left (d \right )\right )}{3 \sqrt {\pi }}+\frac {-\frac {44 d^{2} x^{2}}{9}+8}{\sqrt {\pi }\, x^{2} d^{2}}+\frac {8 \gamma }{3 \sqrt {\pi }}+\frac {8 \ln \left (2\right )}{3 \sqrt {\pi }}+\frac {8 \ln \left (\frac {d x}{2}\right )}{3 \sqrt {\pi }}-\frac {8 \cos \left (d x \right )}{3 \sqrt {\pi }\, d^{2} x^{2}}-\frac {16 \left (-\frac {5 d^{2} x^{2}}{2}+5\right ) \sin \left (d x \right )}{15 \sqrt {\pi }\, d^{3} x^{3}}-\frac {8 \cosineIntegral \left (d x \right )}{3 \sqrt {\pi }}\right )}{16}\) | \(397\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 1.82, size = 140, normalized size = 1.04 \begin {gather*} -\frac {{\left ({\left (a^{2} {\left (\Gamma \left (-3, i \, d x\right ) + \Gamma \left (-3, -i \, d x\right )\right )} \cos \left (c\right ) + a^{2} {\left (-i \, \Gamma \left (-3, i \, d x\right ) + i \, \Gamma \left (-3, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{5} - 12 \, {\left (a b {\left (\Gamma \left (-3, i \, d x\right ) + \Gamma \left (-3, -i \, d x\right )\right )} \cos \left (c\right ) + a b {\left (-i \, \Gamma \left (-3, i \, d x\right ) + i \, \Gamma \left (-3, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{3}\right )} x^{3} + 8 \, a b \sin \left (d x + c\right ) + 2 \, {\left (b^{2} d x^{3} + 2 \, a b d x\right )} \cos \left (d x + c\right )}{2 \, d^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 145, normalized size = 1.08 \begin {gather*} \frac {2 \, {\left (a^{2} d^{4} - 12 \, a b d^{2}\right )} x^{3} \sin \left (c\right ) \operatorname {Si}\left (d x\right ) - 2 \, {\left (a^{2} d^{2} x + 6 \, b^{2} x^{3}\right )} \cos \left (d x + c\right ) - {\left ({\left (a^{2} d^{4} - 12 \, a b d^{2}\right )} x^{3} \operatorname {Ci}\left (d x\right ) + {\left (a^{2} d^{4} - 12 \, a b d^{2}\right )} x^{3} \operatorname {Ci}\left (-d x\right )\right )} \cos \left (c\right ) - 2 \, {\left (2 \, a^{2} d - {\left (a^{2} d^{3} - 12 \, a b d\right )} x^{2}\right )} \sin \left (d x + c\right )}{12 \, d x^{3}} \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 {\left (a + b x^{2}\right )^{2} \sin {\left (c + d x \right )}}{x^{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 = 3.82, size = 1032, normalized size = 7.70 \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 {\sin \left (c+d\,x\right )\,{\left (b\,x^2+a\right )}^2}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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