Integrand size = 16, antiderivative size = 107 \[ \int \sin ^2\left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {(b c-a d) \operatorname {CosIntegral}\left (\frac {2 (b c-a d)}{d (c+d x)}\right ) \sin \left (\frac {2 b}{d}\right )}{d^2}+\frac {(c+d x) \sin ^2\left (\frac {a+b x}{c+d x}\right )}{d}-\frac {(b c-a d) \cos \left (\frac {2 b}{d}\right ) \text {Si}\left (\frac {2 (b c-a d)}{d (c+d x)}\right )}{d^2} \]
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Time = 0.24 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4659, 3394, 12, 3384, 3380, 3383} \[ \int \sin ^2\left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {\sin \left (\frac {2 b}{d}\right ) (b c-a d) \operatorname {CosIntegral}\left (\frac {2 (b c-a d)}{d (c+d x)}\right )}{d^2}-\frac {\cos \left (\frac {2 b}{d}\right ) (b c-a d) \text {Si}\left (\frac {2 (b c-a d)}{d (c+d x)}\right )}{d^2}+\frac {(c+d x) \sin ^2\left (\frac {a+b x}{c+d x}\right )}{d} \]
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Rule 12
Rule 3380
Rule 3383
Rule 3384
Rule 3394
Rule 4659
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\sin ^2\left (\frac {b}{d}-\frac {(b c-a d) x}{d}\right )}{x^2} \, dx,x,\frac {1}{c+d x}\right )}{d} \\ & = \frac {(c+d x) \sin ^2\left (\frac {a+b x}{c+d x}\right )}{d}+\frac {(2 (b c-a d)) \text {Subst}\left (\int \frac {\sin \left (\frac {2 b}{d}-\frac {2 (b c-a d) x}{d}\right )}{2 x} \, dx,x,\frac {1}{c+d x}\right )}{d^2} \\ & = \frac {(c+d x) \sin ^2\left (\frac {a+b x}{c+d x}\right )}{d}+\frac {(b c-a d) \text {Subst}\left (\int \frac {\sin \left (\frac {2 b}{d}-\frac {2 (b c-a d) x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{d^2} \\ & = \frac {(c+d x) \sin ^2\left (\frac {a+b x}{c+d x}\right )}{d}-\frac {\left ((b c-a d) \cos \left (\frac {2 b}{d}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 (b c-a d) x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{d^2}+\frac {\left ((b c-a d) \sin \left (\frac {2 b}{d}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 (b c-a d) x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{d^2} \\ & = \frac {(b c-a d) \operatorname {CosIntegral}\left (\frac {2 (b c-a d)}{d (c+d x)}\right ) \sin \left (\frac {2 b}{d}\right )}{d^2}+\frac {(c+d x) \sin ^2\left (\frac {a+b x}{c+d x}\right )}{d}-\frac {(b c-a d) \cos \left (\frac {2 b}{d}\right ) \text {Si}\left (\frac {2 (b c-a d)}{d (c+d x)}\right )}{d^2} \\ \end{align*}
Result contains complex when optimal does not.
Time = 3.92 (sec) , antiderivative size = 495, normalized size of antiderivative = 4.63 \[ \int \sin ^2\left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {-c d e^{-\frac {2 i (a+b x)}{c+d x}}-c d e^{\frac {2 i (a+b x)}{c+d x}}+2 d^2 x-2 d^2 x \cos \left (\frac {2 b}{d}\right ) \cos \left (\frac {2 (-b c+a d)}{d (c+d x)}\right )+2 (b c-a d) \operatorname {CosIntegral}\left (\frac {2 b c-2 a d}{c d+d^2 x}\right ) \left (-i \cos \left (\frac {2 b}{d}\right )+\sin \left (\frac {2 b}{d}\right )\right )+2 (b c-a d) \operatorname {CosIntegral}\left (\frac {2 (-b c+a d)}{d (c+d x)}\right ) \left (i \cos \left (\frac {2 b}{d}\right )+\sin \left (\frac {2 b}{d}\right )\right )+2 d^2 x \sin \left (\frac {2 b}{d}\right ) \sin \left (\frac {2 (-b c+a d)}{d (c+d x)}\right )+2 b c \cos \left (\frac {2 b}{d}\right ) \text {Si}\left (\frac {2 (-b c+a d)}{d (c+d x)}\right )-2 a d \cos \left (\frac {2 b}{d}\right ) \text {Si}\left (\frac {2 (-b c+a d)}{d (c+d x)}\right )+2 i b c \sin \left (\frac {2 b}{d}\right ) \text {Si}\left (\frac {2 (-b c+a d)}{d (c+d x)}\right )-2 i a d \sin \left (\frac {2 b}{d}\right ) \text {Si}\left (\frac {2 (-b c+a d)}{d (c+d x)}\right )-2 b c \cos \left (\frac {2 b}{d}\right ) \text {Si}\left (\frac {2 b c-2 a d}{c d+d^2 x}\right )+2 a d \cos \left (\frac {2 b}{d}\right ) \text {Si}\left (\frac {2 b c-2 a d}{c d+d^2 x}\right )+2 i b c \sin \left (\frac {2 b}{d}\right ) \text {Si}\left (\frac {2 b c-2 a d}{c d+d^2 x}\right )-2 i a d \sin \left (\frac {2 b}{d}\right ) \text {Si}\left (\frac {2 b c-2 a d}{c d+d^2 x}\right )}{4 d^2} \]
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Time = 1.49 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.82
method | result | size |
derivativedivides | \(-\frac {\left (a d -c b \right ) \left (-\frac {d}{2 \left (\left (\frac {b}{d}+\frac {a d -c b}{d \left (d x +c \right )}\right ) d -b \right )}-\frac {d^{2} \left (-\frac {2 \cos \left (\frac {2 a d -2 c b}{d \left (d x +c \right )}+\frac {2 b}{d}\right )}{\left (\left (\frac {b}{d}+\frac {a d -c b}{d \left (d x +c \right )}\right ) d -b \right ) d}-\frac {2 \left (\frac {2 \,\operatorname {Si}\left (\frac {2 a d -2 c b}{d \left (d x +c \right )}\right ) \cos \left (\frac {2 b}{d}\right )}{d}+\frac {2 \,\operatorname {Ci}\left (\frac {2 a d -2 c b}{d \left (d x +c \right )}\right ) \sin \left (\frac {2 b}{d}\right )}{d}\right )}{d}\right )}{4}\right )}{d^{2}}\) | \(195\) |
default | \(-\frac {\left (a d -c b \right ) \left (-\frac {d}{2 \left (\left (\frac {b}{d}+\frac {a d -c b}{d \left (d x +c \right )}\right ) d -b \right )}-\frac {d^{2} \left (-\frac {2 \cos \left (\frac {2 a d -2 c b}{d \left (d x +c \right )}+\frac {2 b}{d}\right )}{\left (\left (\frac {b}{d}+\frac {a d -c b}{d \left (d x +c \right )}\right ) d -b \right ) d}-\frac {2 \left (\frac {2 \,\operatorname {Si}\left (\frac {2 a d -2 c b}{d \left (d x +c \right )}\right ) \cos \left (\frac {2 b}{d}\right )}{d}+\frac {2 \,\operatorname {Ci}\left (\frac {2 a d -2 c b}{d \left (d x +c \right )}\right ) \sin \left (\frac {2 b}{d}\right )}{d}\right )}{d}\right )}{4}\right )}{d^{2}}\) | \(195\) |
risch | \(\frac {{\mathrm e}^{-\frac {2 i b}{d}} \operatorname {csgn}\left (\frac {a d -c b}{d \left (d x +c \right )}\right ) \pi a}{2 d}-\frac {{\mathrm e}^{-\frac {2 i b}{d}} \operatorname {csgn}\left (\frac {a d -c b}{d \left (d x +c \right )}\right ) \pi b c}{2 d^{2}}-\frac {{\mathrm e}^{-\frac {2 i b}{d}} \operatorname {Si}\left (\frac {2 a d -2 c b}{d \left (d x +c \right )}\right ) a}{d}+\frac {{\mathrm e}^{-\frac {2 i b}{d}} \operatorname {Si}\left (\frac {2 a d -2 c b}{d \left (d x +c \right )}\right ) b c}{d^{2}}+\frac {i {\mathrm e}^{-\frac {2 i b}{d}} \operatorname {Ei}_{1}\left (-\frac {2 i \left (a d -c b \right )}{d \left (d x +c \right )}\right ) a}{2 d}-\frac {i {\mathrm e}^{-\frac {2 i b}{d}} \operatorname {Ei}_{1}\left (-\frac {2 i \left (a d -c b \right )}{d \left (d x +c \right )}\right ) b c}{2 d^{2}}-\frac {i \operatorname {Ei}_{1}\left (-\frac {2 i \left (a d -c b \right )}{d \left (d x +c \right )}\right ) {\mathrm e}^{\frac {2 i b}{d}} a}{2 d}+\frac {i \operatorname {Ei}_{1}\left (-\frac {2 i \left (a d -c b \right )}{d \left (d x +c \right )}\right ) {\mathrm e}^{\frac {2 i b}{d}} c b}{2 d^{2}}+\frac {x}{2}-\frac {\cos \left (\frac {2 x b +2 a}{d x +c}\right ) x}{2}-\frac {\cos \left (\frac {2 x b +2 a}{d x +c}\right ) c}{2 d}\) | \(339\) |
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Time = 0.27 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.06 \[ \int \sin ^2\left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {d^{2} x - {\left (d^{2} x + c d\right )} \cos \left (\frac {b x + a}{d x + c}\right )^{2} + {\left (b c - a d\right )} \operatorname {Ci}\left (-\frac {2 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right ) \sin \left (\frac {2 \, b}{d}\right ) + {\left (b c - a d\right )} \cos \left (\frac {2 \, b}{d}\right ) \operatorname {Si}\left (-\frac {2 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right )}{d^{2}} \]
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\[ \int \sin ^2\left (\frac {a+b x}{c+d x}\right ) \, dx=\int \sin ^{2}{\left (\frac {a + b x}{c + d x} \right )}\, dx \]
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\[ \int \sin ^2\left (\frac {a+b x}{c+d x}\right ) \, dx=\int { \sin \left (\frac {b x + a}{d x + c}\right )^{2} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 681 vs. \(2 (107) = 214\).
Time = 25.07 (sec) , antiderivative size = 681, normalized size of antiderivative = 6.36 \[ \int \sin ^2\left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {{\left (2 \, b^{3} c^{2} \operatorname {Ci}\left (-\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) \sin \left (\frac {2 \, b}{d}\right ) - 4 \, a b^{2} c d \operatorname {Ci}\left (-\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) \sin \left (\frac {2 \, b}{d}\right ) - \frac {2 \, {\left (b x + a\right )} b^{2} c^{2} d \operatorname {Ci}\left (-\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) \sin \left (\frac {2 \, b}{d}\right )}{d x + c} + 2 \, a^{2} b d^{2} \operatorname {Ci}\left (-\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) \sin \left (\frac {2 \, b}{d}\right ) + \frac {4 \, {\left (b x + a\right )} a b c d^{2} \operatorname {Ci}\left (-\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) \sin \left (\frac {2 \, b}{d}\right )}{d x + c} - \frac {2 \, {\left (b x + a\right )} a^{2} d^{3} \operatorname {Ci}\left (-\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) \sin \left (\frac {2 \, b}{d}\right )}{d x + c} - 2 \, b^{3} c^{2} \cos \left (\frac {2 \, b}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) + 4 \, a b^{2} c d \cos \left (\frac {2 \, b}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) + \frac {2 \, {\left (b x + a\right )} b^{2} c^{2} d \cos \left (\frac {2 \, b}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right )}{d x + c} - 2 \, a^{2} b d^{2} \cos \left (\frac {2 \, b}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) - \frac {4 \, {\left (b x + a\right )} a b c d^{2} \cos \left (\frac {2 \, b}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right )}{d x + c} + \frac {2 \, {\left (b x + a\right )} a^{2} d^{3} \cos \left (\frac {2 \, b}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right )}{d x + c} - b^{2} c^{2} d \cos \left (\frac {2 \, {\left (b x + a\right )}}{d x + c}\right ) + 2 \, a b c d^{2} \cos \left (\frac {2 \, {\left (b x + a\right )}}{d x + c}\right ) - a^{2} d^{3} \cos \left (\frac {2 \, {\left (b x + a\right )}}{d x + c}\right ) + b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )}}{2 \, {\left (b d^{2} - \frac {{\left (b x + a\right )} d^{3}}{d x + c}\right )}} \]
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Timed out. \[ \int \sin ^2\left (\frac {a+b x}{c+d x}\right ) \, dx=\int {\sin \left (\frac {a+b\,x}{c+d\,x}\right )}^2 \,d x \]
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