Integrand size = 20, antiderivative size = 183 \[ \int \frac {(a+b \sin (e+f x))^2}{(c+d x)^2} \, dx=-\frac {a^2}{d (c+d x)}+\frac {2 a b f \cos \left (e-\frac {c f}{d}\right ) \operatorname {CosIntegral}\left (\frac {c f}{d}+f x\right )}{d^2}+\frac {b^2 f \operatorname {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{d^2}-\frac {2 a b \sin (e+f x)}{d (c+d x)}-\frac {b^2 \sin ^2(e+f x)}{d (c+d x)}-\frac {2 a b f \sin \left (e-\frac {c f}{d}\right ) \text {Si}\left (\frac {c f}{d}+f x\right )}{d^2}+\frac {b^2 f \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{d^2} \]
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Time = 0.23 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {3398, 3378, 3384, 3380, 3383, 3394, 12} \[ \int \frac {(a+b \sin (e+f x))^2}{(c+d x)^2} \, dx=-\frac {a^2}{d (c+d x)}+\frac {2 a b f \operatorname {CosIntegral}\left (x f+\frac {c f}{d}\right ) \cos \left (e-\frac {c f}{d}\right )}{d^2}-\frac {2 a b f \sin \left (e-\frac {c f}{d}\right ) \text {Si}\left (x f+\frac {c f}{d}\right )}{d^2}-\frac {2 a b \sin (e+f x)}{d (c+d x)}+\frac {b^2 f \operatorname {CosIntegral}\left (2 x f+\frac {2 c f}{d}\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{d^2}+\frac {b^2 f \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{d^2}-\frac {b^2 \sin ^2(e+f x)}{d (c+d x)} \]
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Rule 12
Rule 3378
Rule 3380
Rule 3383
Rule 3384
Rule 3394
Rule 3398
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^2}{(c+d x)^2}+\frac {2 a b \sin (e+f x)}{(c+d x)^2}+\frac {b^2 \sin ^2(e+f x)}{(c+d x)^2}\right ) \, dx \\ & = -\frac {a^2}{d (c+d x)}+(2 a b) \int \frac {\sin (e+f x)}{(c+d x)^2} \, dx+b^2 \int \frac {\sin ^2(e+f x)}{(c+d x)^2} \, dx \\ & = -\frac {a^2}{d (c+d x)}-\frac {2 a b \sin (e+f x)}{d (c+d x)}-\frac {b^2 \sin ^2(e+f x)}{d (c+d x)}+\frac {(2 a b f) \int \frac {\cos (e+f x)}{c+d x} \, dx}{d}+\frac {\left (2 b^2 f\right ) \int \frac {\sin (2 e+2 f x)}{2 (c+d x)} \, dx}{d} \\ & = -\frac {a^2}{d (c+d x)}-\frac {2 a b \sin (e+f x)}{d (c+d x)}-\frac {b^2 \sin ^2(e+f x)}{d (c+d x)}+\frac {\left (b^2 f\right ) \int \frac {\sin (2 e+2 f x)}{c+d x} \, dx}{d}+\frac {\left (2 a b f \cos \left (e-\frac {c f}{d}\right )\right ) \int \frac {\cos \left (\frac {c f}{d}+f x\right )}{c+d x} \, dx}{d}-\frac {\left (2 a b f \sin \left (e-\frac {c f}{d}\right )\right ) \int \frac {\sin \left (\frac {c f}{d}+f x\right )}{c+d x} \, dx}{d} \\ & = -\frac {a^2}{d (c+d x)}+\frac {2 a b f \cos \left (e-\frac {c f}{d}\right ) \operatorname {CosIntegral}\left (\frac {c f}{d}+f x\right )}{d^2}-\frac {2 a b \sin (e+f x)}{d (c+d x)}-\frac {b^2 \sin ^2(e+f x)}{d (c+d x)}-\frac {2 a b f \sin \left (e-\frac {c f}{d}\right ) \text {Si}\left (\frac {c f}{d}+f x\right )}{d^2}+\frac {\left (b^2 f \cos \left (2 e-\frac {2 c f}{d}\right )\right ) \int \frac {\sin \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx}{d}+\frac {\left (b^2 f \sin \left (2 e-\frac {2 c f}{d}\right )\right ) \int \frac {\cos \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx}{d} \\ & = -\frac {a^2}{d (c+d x)}+\frac {2 a b f \cos \left (e-\frac {c f}{d}\right ) \operatorname {CosIntegral}\left (\frac {c f}{d}+f x\right )}{d^2}+\frac {b^2 f \operatorname {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{d^2}-\frac {2 a b \sin (e+f x)}{d (c+d x)}-\frac {b^2 \sin ^2(e+f x)}{d (c+d x)}-\frac {2 a b f \sin \left (e-\frac {c f}{d}\right ) \text {Si}\left (\frac {c f}{d}+f x\right )}{d^2}+\frac {b^2 f \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{d^2} \\ \end{align*}
Time = 0.43 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.27 \[ \int \frac {(a+b \sin (e+f x))^2}{(c+d x)^2} \, dx=\frac {-2 a^2 d-b^2 d+b^2 d \cos (2 (e+f x))+4 a b f (c+d x) \cos \left (e-\frac {c f}{d}\right ) \operatorname {CosIntegral}\left (f \left (\frac {c}{d}+x\right )\right )+2 b^2 f (c+d x) \operatorname {CosIntegral}\left (\frac {2 f (c+d x)}{d}\right ) \sin \left (2 e-\frac {2 c f}{d}\right )-4 a b d \sin (e+f x)-4 a b c f \sin \left (e-\frac {c f}{d}\right ) \text {Si}\left (f \left (\frac {c}{d}+x\right )\right )-4 a b d f x \sin \left (e-\frac {c f}{d}\right ) \text {Si}\left (f \left (\frac {c}{d}+x\right )\right )+2 b^2 c f \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 f (c+d x)}{d}\right )+2 b^2 d f x \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 f (c+d x)}{d}\right )}{2 d^2 (c+d x)} \]
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Time = 0.32 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.55
method | result | size |
parts | \(-\frac {a^{2}}{d \left (d x +c \right )}+\frac {b^{2} \left (-\frac {f^{2}}{2 \left (c f -d e +d \left (f x +e \right )\right ) d}-\frac {f^{2} \left (-\frac {2 \cos \left (2 f x +2 e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}-\frac {2 \left (\frac {2 \,\operatorname {Si}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \cos \left (\frac {2 c f -2 d e}{d}\right )}{d}-\frac {2 \,\operatorname {Ci}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \sin \left (\frac {2 c f -2 d e}{d}\right )}{d}\right )}{d}\right )}{4}\right )}{f}+2 a b f \left (-\frac {\sin \left (f x +e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}+\frac {\frac {\operatorname {Si}\left (f x +e +\frac {c f -d e}{d}\right ) \sin \left (\frac {c f -d e}{d}\right )}{d}+\frac {\operatorname {Ci}\left (f x +e +\frac {c f -d e}{d}\right ) \cos \left (\frac {c f -d e}{d}\right )}{d}}{d}\right )\) | \(284\) |
derivativedivides | \(\frac {-\frac {a^{2} f^{2}}{\left (c f -d e +d \left (f x +e \right )\right ) d}+2 f^{2} a b \left (-\frac {\sin \left (f x +e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}+\frac {\frac {\operatorname {Si}\left (f x +e +\frac {c f -d e}{d}\right ) \sin \left (\frac {c f -d e}{d}\right )}{d}+\frac {\operatorname {Ci}\left (f x +e +\frac {c f -d e}{d}\right ) \cos \left (\frac {c f -d e}{d}\right )}{d}}{d}\right )-\frac {f^{2} b^{2}}{2 \left (c f -d e +d \left (f x +e \right )\right ) d}-\frac {f^{2} b^{2} \left (-\frac {2 \cos \left (2 f x +2 e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}-\frac {2 \left (\frac {2 \,\operatorname {Si}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \cos \left (\frac {2 c f -2 d e}{d}\right )}{d}-\frac {2 \,\operatorname {Ci}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \sin \left (\frac {2 c f -2 d e}{d}\right )}{d}\right )}{d}\right )}{4}}{f}\) | \(301\) |
default | \(\frac {-\frac {a^{2} f^{2}}{\left (c f -d e +d \left (f x +e \right )\right ) d}+2 f^{2} a b \left (-\frac {\sin \left (f x +e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}+\frac {\frac {\operatorname {Si}\left (f x +e +\frac {c f -d e}{d}\right ) \sin \left (\frac {c f -d e}{d}\right )}{d}+\frac {\operatorname {Ci}\left (f x +e +\frac {c f -d e}{d}\right ) \cos \left (\frac {c f -d e}{d}\right )}{d}}{d}\right )-\frac {f^{2} b^{2}}{2 \left (c f -d e +d \left (f x +e \right )\right ) d}-\frac {f^{2} b^{2} \left (-\frac {2 \cos \left (2 f x +2 e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}-\frac {2 \left (\frac {2 \,\operatorname {Si}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \cos \left (\frac {2 c f -2 d e}{d}\right )}{d}-\frac {2 \,\operatorname {Ci}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \sin \left (\frac {2 c f -2 d e}{d}\right )}{d}\right )}{d}\right )}{4}}{f}\) | \(301\) |
risch | \(-\frac {f a b \,{\mathrm e}^{\frac {i \left (c f -d e \right )}{d}} \operatorname {Ei}_{1}\left (i f x +i e +\frac {i \left (c f -d e \right )}{d}\right )}{d^{2}}-\frac {a^{2}}{d \left (d x +c \right )}-\frac {b^{2}}{2 d \left (d x +c \right )}-\frac {i b^{2} f \,{\mathrm e}^{\frac {2 i \left (c f -d e \right )}{d}} \operatorname {Ei}_{1}\left (2 i f x +2 i e +\frac {2 i \left (c f -d e \right )}{d}\right )}{2 d^{2}}+\frac {i f \,b^{2} {\mathrm e}^{-\frac {2 i \left (c f -d e \right )}{d}} \operatorname {Ei}_{1}\left (-2 i f x -2 i e -\frac {2 \left (i c f -i d e \right )}{d}\right )}{2 d^{2}}-\frac {a b f \,{\mathrm e}^{-\frac {i \left (c f -d e \right )}{d}} \operatorname {Ei}_{1}\left (-i f x -i e -\frac {i c f -i d e}{d}\right )}{d^{2}}-\frac {a b \left (-2 d x f -2 c f \right ) \sin \left (f x +e \right )}{d \left (d x +c \right ) \left (-d x f -c f \right )}+\frac {b^{2} \left (-2 d x f -2 c f \right ) \cos \left (2 f x +2 e \right )}{4 d \left (d x +c \right ) \left (-d x f -c f \right )}\) | \(324\) |
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Time = 0.30 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.19 \[ \int \frac {(a+b \sin (e+f x))^2}{(c+d x)^2} \, dx=\frac {b^{2} d \cos \left (f x + e\right )^{2} - 2 \, a b d \sin \left (f x + e\right ) + 2 \, {\left (a b d f x + a b c f\right )} \cos \left (-\frac {d e - c f}{d}\right ) \operatorname {Ci}\left (\frac {d f x + c f}{d}\right ) - {\left (b^{2} d f x + b^{2} c f\right )} \operatorname {Ci}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sin \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) + {\left (b^{2} d f x + b^{2} c f\right )} \cos \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) + 2 \, {\left (a b d f x + a b c f\right )} \sin \left (-\frac {d e - c f}{d}\right ) \operatorname {Si}\left (\frac {d f x + c f}{d}\right ) - {\left (a^{2} + b^{2}\right )} d}{d^{3} x + c d^{2}} \]
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\[ \int \frac {(a+b \sin (e+f x))^2}{(c+d x)^2} \, dx=\int \frac {\left (a + b \sin {\left (e + f x \right )}\right )^{2}}{\left (c + d x\right )^{2}}\, dx \]
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Result contains complex when optimal does not.
Time = 0.43 (sec) , antiderivative size = 371, normalized size of antiderivative = 2.03 \[ \int \frac {(a+b \sin (e+f x))^2}{(c+d x)^2} \, dx=-\frac {\frac {4 \, a^{2} f^{2}}{{\left (f x + e\right )} d^{2} - d^{2} e + c d f} - \frac {4 \, {\left (f^{2} {\left (-i \, E_{2}\left (\frac {i \, {\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right ) + i \, E_{2}\left (-\frac {i \, {\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right )\right )} \cos \left (-\frac {d e - c f}{d}\right ) + f^{2} {\left (E_{2}\left (\frac {i \, {\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right ) + E_{2}\left (-\frac {i \, {\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right )\right )} \sin \left (-\frac {d e - c f}{d}\right )\right )} a b}{{\left (f x + e\right )} d^{2} - d^{2} e + c d f} - \frac {{\left (f^{2} {\left (E_{2}\left (\frac {2 \, {\left (-i \, {\left (f x + e\right )} d + i \, d e - i \, c f\right )}}{d}\right ) + E_{2}\left (-\frac {2 \, {\left (-i \, {\left (f x + e\right )} d + i \, d e - i \, c f\right )}}{d}\right )\right )} \cos \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) - f^{2} {\left (i \, E_{2}\left (\frac {2 \, {\left (-i \, {\left (f x + e\right )} d + i \, d e - i \, c f\right )}}{d}\right ) - i \, E_{2}\left (-\frac {2 \, {\left (-i \, {\left (f x + e\right )} d + i \, d e - i \, c f\right )}}{d}\right )\right )} \sin \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) - 2 \, f^{2}\right )} b^{2}}{{\left (f x + e\right )} d^{2} - d^{2} e + c d f}}{4 \, f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1050 vs. \(2 (186) = 372\).
Time = 0.45 (sec) , antiderivative size = 1050, normalized size of antiderivative = 5.74 \[ \int \frac {(a+b \sin (e+f x))^2}{(c+d x)^2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(a+b \sin (e+f x))^2}{(c+d x)^2} \, dx=\int \frac {{\left (a+b\,\sin \left (e+f\,x\right )\right )}^2}{{\left (c+d\,x\right )}^2} \,d x \]
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