Integrand size = 20, antiderivative size = 162 \[ \int \frac {(a+a \sin (e+f x))^2}{(c+d x)^2} \, dx=\frac {2 a^2 f \cos \left (e-\frac {c f}{d}\right ) \operatorname {CosIntegral}\left (\frac {c f}{d}+f x\right )}{d^2}+\frac {a^2 f \operatorname {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{d^2}-\frac {4 a^2 \sin ^4\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{d (c+d x)}-\frac {2 a^2 f \sin \left (e-\frac {c f}{d}\right ) \text {Si}\left (\frac {c f}{d}+f x\right )}{d^2}+\frac {a^2 f \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{d^2} \]
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Time = 0.21 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3399, 3394, 3384, 3380, 3383} \[ \int \frac {(a+a \sin (e+f x))^2}{(c+d x)^2} \, dx=\frac {a^2 f \operatorname {CosIntegral}\left (2 x f+\frac {2 c f}{d}\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{d^2}+\frac {2 a^2 f \operatorname {CosIntegral}\left (x f+\frac {c f}{d}\right ) \cos \left (e-\frac {c f}{d}\right )}{d^2}-\frac {2 a^2 f \sin \left (e-\frac {c f}{d}\right ) \text {Si}\left (x f+\frac {c f}{d}\right )}{d^2}+\frac {a^2 f \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{d^2}-\frac {4 a^2 \sin ^4\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{d (c+d x)} \]
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Rule 3380
Rule 3383
Rule 3384
Rule 3394
Rule 3399
Rubi steps \begin{align*} \text {integral}& = \left (4 a^2\right ) \int \frac {\sin ^4\left (\frac {1}{2} \left (e+\frac {\pi }{2}\right )+\frac {f x}{2}\right )}{(c+d x)^2} \, dx \\ & = -\frac {4 a^2 \sin ^4\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{d (c+d x)}+\frac {\left (8 a^2 f\right ) \int \left (\frac {\cos (e+f x)}{4 (c+d x)}+\frac {\sin (2 e+2 f x)}{8 (c+d x)}\right ) \, dx}{d} \\ & = -\frac {4 a^2 \sin ^4\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{d (c+d x)}+\frac {\left (a^2 f\right ) \int \frac {\sin (2 e+2 f x)}{c+d x} \, dx}{d}+\frac {\left (2 a^2 f\right ) \int \frac {\cos (e+f x)}{c+d x} \, dx}{d} \\ & = -\frac {4 a^2 \sin ^4\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{d (c+d x)}+\frac {\left (a^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 (2 a^2 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 (a^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 {\left (2 a^2 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 {2 a^2 f \cos \left (e-\frac {c f}{d}\right ) \operatorname {CosIntegral}\left (\frac {c f}{d}+f x\right )}{d^2}+\frac {a^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 {4 a^2 \sin ^4\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{d (c+d x)}-\frac {2 a^2 f \sin \left (e-\frac {c f}{d}\right ) \text {Si}\left (\frac {c f}{d}+f x\right )}{d^2}+\frac {a^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 = 0.41 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.27 \[ \int \frac {(a+a \sin (e+f x))^2}{(c+d x)^2} \, dx=\frac {a^2 \left (-3 d+d \cos (2 (e+f x))+4 f (c+d x) \cos \left (e-\frac {c f}{d}\right ) \operatorname {CosIntegral}\left (f \left (\frac {c}{d}+x\right )\right )+2 f (c+d x) \operatorname {CosIntegral}\left (\frac {2 f (c+d x)}{d}\right ) \sin \left (2 e-\frac {2 c f}{d}\right )-4 d \sin (e+f x)-4 c f \sin \left (e-\frac {c f}{d}\right ) \text {Si}\left (f \left (\frac {c}{d}+x\right )\right )-4 d f x \sin \left (e-\frac {c f}{d}\right ) \text {Si}\left (f \left (\frac {c}{d}+x\right )\right )+2 c f \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 f (c+d x)}{d}\right )+2 d f x \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 f (c+d x)}{d}\right )\right )}{2 d^2 (c+d x)} \]
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Time = 0.31 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.69
method | result | size |
derivativedivides | \(\frac {-\frac {3 a^{2} f^{2}}{2 \left (c f -d e +d \left (f x +e \right )\right ) d}-\frac {a^{2} f^{2} \left (-\frac {2 \cos \left (2 f x +2 e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}-\frac {2 \left (\frac {2 \,\operatorname {Si}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \cos \left (\frac {2 c f -2 d e}{d}\right )}{d}-\frac {2 \,\operatorname {Ci}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \sin \left (\frac {2 c f -2 d e}{d}\right )}{d}\right )}{d}\right )}{4}+2 a^{2} f^{2} \left (-\frac {\sin \left (f x +e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}+\frac {\frac {\operatorname {Si}\left (f x +e +\frac {c f -d e}{d}\right ) \sin \left (\frac {c f -d e}{d}\right )}{d}+\frac {\operatorname {Ci}\left (f x +e +\frac {c f -d e}{d}\right ) \cos \left (\frac {c f -d e}{d}\right )}{d}}{d}\right )}{f}\) | \(274\) |
default | \(\frac {-\frac {3 a^{2} f^{2}}{2 \left (c f -d e +d \left (f x +e \right )\right ) d}-\frac {a^{2} f^{2} \left (-\frac {2 \cos \left (2 f x +2 e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}-\frac {2 \left (\frac {2 \,\operatorname {Si}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \cos \left (\frac {2 c f -2 d e}{d}\right )}{d}-\frac {2 \,\operatorname {Ci}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \sin \left (\frac {2 c f -2 d e}{d}\right )}{d}\right )}{d}\right )}{4}+2 a^{2} f^{2} \left (-\frac {\sin \left (f x +e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}+\frac {\frac {\operatorname {Si}\left (f x +e +\frac {c f -d e}{d}\right ) \sin \left (\frac {c f -d e}{d}\right )}{d}+\frac {\operatorname {Ci}\left (f x +e +\frac {c f -d e}{d}\right ) \cos \left (\frac {c f -d e}{d}\right )}{d}}{d}\right )}{f}\) | \(274\) |
parts | \(-\frac {a^{2}}{d \left (d x +c \right )}+\frac {a^{2} \left (-\frac {f^{2}}{2 \left (c f -d e +d \left (f x +e \right )\right ) d}-\frac {f^{2} \left (-\frac {2 \cos \left (2 f x +2 e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}-\frac {2 \left (\frac {2 \,\operatorname {Si}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \cos \left (\frac {2 c f -2 d e}{d}\right )}{d}-\frac {2 \,\operatorname {Ci}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \sin \left (\frac {2 c f -2 d e}{d}\right )}{d}\right )}{d}\right )}{4}\right )}{f}+2 a^{2} f \left (-\frac {\sin \left (f x +e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}+\frac {\frac {\operatorname {Si}\left (f x +e +\frac {c f -d e}{d}\right ) \sin \left (\frac {c f -d e}{d}\right )}{d}+\frac {\operatorname {Ci}\left (f x +e +\frac {c f -d e}{d}\right ) \cos \left (\frac {c f -d e}{d}\right )}{d}}{d}\right )\) | \(285\) |
risch | \(-\frac {f \,a^{2} {\mathrm e}^{\frac {i \left (c f -d e \right )}{d}} \operatorname {Ei}_{1}\left (i f x +i e +\frac {i \left (c f -d e \right )}{d}\right )}{d^{2}}-\frac {3 a^{2}}{2 d \left (d x +c \right )}-\frac {i a^{2} f \,{\mathrm e}^{\frac {2 i \left (c f -d e \right )}{d}} \operatorname {Ei}_{1}\left (2 i f x +2 i e +\frac {2 i \left (c f -d e \right )}{d}\right )}{2 d^{2}}+\frac {i f \,a^{2} {\mathrm e}^{-\frac {2 i \left (c f -d e \right )}{d}} \operatorname {Ei}_{1}\left (-2 i f x -2 i e -\frac {2 \left (i c f -i d e \right )}{d}\right )}{2 d^{2}}-\frac {a^{2} f \,{\mathrm e}^{-\frac {i \left (c f -d e \right )}{d}} \operatorname {Ei}_{1}\left (-i f x -i e -\frac {i c f -i d e}{d}\right )}{d^{2}}-\frac {a^{2} \left (-2 d x f -2 c f \right ) \sin \left (f x +e \right )}{d \left (d x +c \right ) \left (-d x f -c f \right )}+\frac {a^{2} \left (-2 d x f -2 c f \right ) \cos \left (2 f x +2 e \right )}{4 d \left (d x +c \right ) \left (-d x f -c f \right )}\) | \(312\) |
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Time = 0.30 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.35 \[ \int \frac {(a+a \sin (e+f x))^2}{(c+d x)^2} \, dx=\frac {a^{2} d \cos \left (f x + e\right )^{2} - 2 \, a^{2} d \sin \left (f x + e\right ) - 2 \, a^{2} d + 2 \, {\left (a^{2} d f x + a^{2} c f\right )} \cos \left (-\frac {d e - c f}{d}\right ) \operatorname {Ci}\left (\frac {d f x + c f}{d}\right ) - {\left (a^{2} d f x + a^{2} c f\right )} \operatorname {Ci}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sin \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) + {\left (a^{2} d f x + a^{2} c f\right )} \cos \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) + 2 \, {\left (a^{2} d f x + a^{2} c f\right )} \sin \left (-\frac {d e - c f}{d}\right ) \operatorname {Si}\left (\frac {d f x + c f}{d}\right )}{d^{3} x + c d^{2}} \]
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\[ \int \frac {(a+a \sin (e+f x))^2}{(c+d x)^2} \, dx=a^{2} \left (\int \frac {2 \sin {\left (e + f x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {\sin ^{2}{\left (e + f x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {1}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx\right ) \]
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Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 372, normalized size of antiderivative = 2.30 \[ \int \frac {(a+a \sin (e+f x))^2}{(c+d x)^2} \, dx=-\frac {\frac {4 \, a^{2} f^{2}}{{\left (f x + e\right )} d^{2} - d^{2} e + c d f} - \frac {4 \, {\left (f^{2} {\left (-i \, E_{2}\left (\frac {i \, {\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right ) + i \, E_{2}\left (-\frac {i \, {\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right )\right )} \cos \left (-\frac {d e - c f}{d}\right ) + f^{2} {\left (E_{2}\left (\frac {i \, {\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right ) + E_{2}\left (-\frac {i \, {\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right )\right )} \sin \left (-\frac {d e - c f}{d}\right )\right )} a^{2}}{{\left (f x + e\right )} d^{2} - d^{2} e + c d f} - \frac {{\left (f^{2} {\left (E_{2}\left (\frac {2 \, {\left (-i \, {\left (f x + e\right )} d + i \, d e - i \, c f\right )}}{d}\right ) + E_{2}\left (-\frac {2 \, {\left (-i \, {\left (f x + e\right )} d + i \, d e - i \, c f\right )}}{d}\right )\right )} \cos \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) - f^{2} {\left (i \, E_{2}\left (\frac {2 \, {\left (-i \, {\left (f x + e\right )} d + i \, d e - i \, c f\right )}}{d}\right ) - i \, E_{2}\left (-\frac {2 \, {\left (-i \, {\left (f x + e\right )} d + i \, d e - i \, c f\right )}}{d}\right )\right )} \sin \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) - 2 \, f^{2}\right )} a^{2}}{{\left (f x + e\right )} d^{2} - d^{2} e + c d f}}{4 \, f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1048 vs. \(2 (159) = 318\).
Time = 0.44 (sec) , antiderivative size = 1048, normalized size of antiderivative = 6.47 \[ \int \frac {(a+a \sin (e+f x))^2}{(c+d x)^2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(a+a \sin (e+f x))^2}{(c+d x)^2} \, dx=\int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^2}{{\left (c+d\,x\right )}^2} \,d x \]
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