Optimal. Leaf size=156 \[ -\frac {b^2 \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Ci}\left (\frac {2 c f}{d}+2 f x\right )}{2 d}+\frac {a^2 \log (c+d x)}{d}+\frac {b^2 \log (c+d x)}{2 d}+\frac {2 a b \text {Ci}\left (\frac {c f}{d}+f x\right ) \sin \left (e-\frac {c f}{d}\right )}{d}+\frac {2 a b \cos \left (e-\frac {c f}{d}\right ) \text {Si}\left (\frac {c f}{d}+f x\right )}{d}+\frac {b^2 \sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{2 d} \]
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Rubi [A]
time = 0.21, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3398, 3384,
3380, 3383, 3393} \begin {gather*} \frac {a^2 \log (c+d x)}{d}+\frac {2 a b \text {CosIntegral}\left (\frac {c f}{d}+f x\right ) \sin \left (e-\frac {c f}{d}\right )}{d}+\frac {2 a b \cos \left (e-\frac {c f}{d}\right ) \text {Si}\left (x f+\frac {c f}{d}\right )}{d}-\frac {b^2 \text {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right ) \cos \left (2 e-\frac {2 c f}{d}\right )}{2 d}+\frac {b^2 \sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{2 d}+\frac {b^2 \log (c+d x)}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3380
Rule 3383
Rule 3384
Rule 3393
Rule 3398
Rubi steps
\begin {align*} \int \frac {(a+b \sin (e+f x))^2}{c+d x} \, dx &=\int \left (\frac {a^2}{c+d x}+\frac {2 a b \sin (e+f x)}{c+d x}+\frac {b^2 \sin ^2(e+f x)}{c+d x}\right ) \, dx\\ &=\frac {a^2 \log (c+d x)}{d}+(2 a b) \int \frac {\sin (e+f x)}{c+d x} \, dx+b^2 \int \frac {\sin ^2(e+f x)}{c+d x} \, dx\\ &=\frac {a^2 \log (c+d x)}{d}+b^2 \int \left (\frac {1}{2 (c+d x)}-\frac {\cos (2 e+2 f x)}{2 (c+d x)}\right ) \, dx+\left (2 a b \cos \left (e-\frac {c f}{d}\right )\right ) \int \frac {\sin \left (\frac {c f}{d}+f x\right )}{c+d x} \, dx+\left (2 a b \sin \left (e-\frac {c f}{d}\right )\right ) \int \frac {\cos \left (\frac {c f}{d}+f x\right )}{c+d x} \, dx\\ &=\frac {a^2 \log (c+d x)}{d}+\frac {b^2 \log (c+d x)}{2 d}+\frac {2 a b \text {Ci}\left (\frac {c f}{d}+f x\right ) \sin \left (e-\frac {c f}{d}\right )}{d}+\frac {2 a b \cos \left (e-\frac {c f}{d}\right ) \text {Si}\left (\frac {c f}{d}+f x\right )}{d}-\frac {1}{2} b^2 \int \frac {\cos (2 e+2 f x)}{c+d x} \, dx\\ &=\frac {a^2 \log (c+d x)}{d}+\frac {b^2 \log (c+d x)}{2 d}+\frac {2 a b \text {Ci}\left (\frac {c f}{d}+f x\right ) \sin \left (e-\frac {c f}{d}\right )}{d}+\frac {2 a b \cos \left (e-\frac {c f}{d}\right ) \text {Si}\left (\frac {c f}{d}+f x\right )}{d}-\frac {1}{2} \left (b^2 \cos \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+\frac {1}{2} \left (b^2 \sin \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\\ &=-\frac {b^2 \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Ci}\left (\frac {2 c f}{d}+2 f x\right )}{2 d}+\frac {a^2 \log (c+d x)}{d}+\frac {b^2 \log (c+d x)}{2 d}+\frac {2 a b \text {Ci}\left (\frac {c f}{d}+f x\right ) \sin \left (e-\frac {c f}{d}\right )}{d}+\frac {2 a b \cos \left (e-\frac {c f}{d}\right ) \text {Si}\left (\frac {c f}{d}+f x\right )}{d}+\frac {b^2 \sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{2 d}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 134, normalized size = 0.86 \begin {gather*} \frac {-b^2 \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Ci}\left (\frac {2 f (c+d x)}{d}\right )+2 a^2 \log (c+d x)+b^2 \log (c+d x)+4 a b \text {Ci}\left (f \left (\frac {c}{d}+x\right )\right ) \sin \left (e-\frac {c f}{d}\right )+4 a b \cos \left (e-\frac {c f}{d}\right ) \text {Si}\left (f \left (\frac {c}{d}+x\right )\right )+b^2 \sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 f (c+d x)}{d}\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 221, normalized size = 1.42
method | result | size |
derivativedivides | \(\frac {\frac {a^{2} f \ln \left (c f -d e +d \left (f x +e \right )\right )}{d}+2 f a b \left (\frac {\sinIntegral \left (f x +e +\frac {c f -d e}{d}\right ) \cos \left (\frac {c f -d e}{d}\right )}{d}-\frac {\cosineIntegral \left (f x +e +\frac {c f -d e}{d}\right ) \sin \left (\frac {c f -d e}{d}\right )}{d}\right )+\frac {f \,b^{2} \ln \left (c f -d e +d \left (f x +e \right )\right )}{2 d}-\frac {f \,b^{2} \left (\frac {2 \sinIntegral \left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \sin \left (\frac {2 c f -2 d e}{d}\right )}{d}+\frac {2 \cosineIntegral \left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \cos \left (\frac {2 c f -2 d e}{d}\right )}{d}\right )}{4}}{f}\) | \(221\) |
default | \(\frac {\frac {a^{2} f \ln \left (c f -d e +d \left (f x +e \right )\right )}{d}+2 f a b \left (\frac {\sinIntegral \left (f x +e +\frac {c f -d e}{d}\right ) \cos \left (\frac {c f -d e}{d}\right )}{d}-\frac {\cosineIntegral \left (f x +e +\frac {c f -d e}{d}\right ) \sin \left (\frac {c f -d e}{d}\right )}{d}\right )+\frac {f \,b^{2} \ln \left (c f -d e +d \left (f x +e \right )\right )}{2 d}-\frac {f \,b^{2} \left (\frac {2 \sinIntegral \left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \sin \left (\frac {2 c f -2 d e}{d}\right )}{d}+\frac {2 \cosineIntegral \left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \cos \left (\frac {2 c f -2 d e}{d}\right )}{d}\right )}{4}}{f}\) | \(221\) |
risch | \(-\frac {i a b \,{\mathrm e}^{\frac {i \left (c f -d e \right )}{d}} \expIntegral \left (1, i f x +i e +\frac {i \left (c f -d e \right )}{d}\right )}{d}+\frac {a^{2} \ln \left (d x +c \right )}{d}+\frac {b^{2} \ln \left (d x +c \right )}{2 d}+\frac {b^{2} {\mathrm e}^{\frac {2 i \left (c f -d e \right )}{d}} \expIntegral \left (1, 2 i f x +2 i e +\frac {2 i \left (c f -d e \right )}{d}\right )}{4 d}+\frac {b^{2} {\mathrm e}^{-\frac {2 i \left (c f -d e \right )}{d}} \expIntegral \left (1, -2 i f x -2 i e -\frac {2 \left (i c f -i d e \right )}{d}\right )}{4 d}+\frac {i a b \,{\mathrm e}^{-\frac {i \left (c f -d e \right )}{d}} \expIntegral \left (1, -i f x -i e -\frac {i c f -i d e}{d}\right )}{d}\) | \(229\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.40, size = 358, normalized size = 2.29 \begin {gather*} \frac {\frac {4 \, a^{2} f \log \left (c + \frac {{\left (f x + e\right )} d}{f} - \frac {d e}{f}\right )}{d} + \frac {4 \, {\left (f {\left (-i \, E_{1}\left (\frac {i \, {\left (f x + e\right )} d + i \, c f - i \, d e}{d}\right ) + i \, E_{1}\left (-\frac {i \, {\left (f x + e\right )} d + i \, c f - i \, d e}{d}\right )\right )} \cos \left (\frac {c f - d e}{d}\right ) + f {\left (E_{1}\left (\frac {i \, {\left (f x + e\right )} d + i \, c f - i \, d e}{d}\right ) + E_{1}\left (-\frac {i \, {\left (f x + e\right )} d + i \, c f - i \, d e}{d}\right )\right )} \sin \left (\frac {c f - d e}{d}\right )\right )} a b}{d} + \frac {{\left (f {\left (E_{1}\left (\frac {2 \, {\left (-i \, {\left (f x + e\right )} d - i \, c f + i \, d e\right )}}{d}\right ) + E_{1}\left (-\frac {2 \, {\left (-i \, {\left (f x + e\right )} d - i \, c f + i \, d e\right )}}{d}\right )\right )} \cos \left (\frac {2 \, {\left (c f - d e\right )}}{d}\right ) + f {\left (-i \, E_{1}\left (\frac {2 \, {\left (-i \, {\left (f x + e\right )} d - i \, c f + i \, d e\right )}}{d}\right ) + i \, E_{1}\left (-\frac {2 \, {\left (-i \, {\left (f x + e\right )} d - i \, c f + i \, d e\right )}}{d}\right )\right )} \sin \left (\frac {2 \, {\left (c f - d e\right )}}{d}\right ) + 2 \, f \log \left ({\left (f x + e\right )} d + c f - d e\right )\right )} b^{2}}{d}}{4 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 194, normalized size = 1.24 \begin {gather*} \frac {2 \, b^{2} \sin \left (-\frac {2 \, {\left (c f - d e\right )}}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) + 8 \, a b \cos \left (-\frac {c f - d e}{d}\right ) \operatorname {Si}\left (\frac {d f x + c f}{d}\right ) - {\left (b^{2} \operatorname {Ci}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) + b^{2} \operatorname {Ci}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right )\right )} \cos \left (-\frac {2 \, {\left (c f - d e\right )}}{d}\right ) + 2 \, {\left (2 \, a^{2} + b^{2}\right )} \log \left (d x + c\right ) + 4 \, {\left (a b \operatorname {Ci}\left (\frac {d f x + c f}{d}\right ) + a b \operatorname {Ci}\left (-\frac {d f x + c f}{d}\right )\right )} \sin \left (-\frac {c f - d e}{d}\right )}{4 \, d} \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 \sin {\left (e + f x \right )}\right )^{2}}{c + d x}\, 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 = 3.90, size = 7397, normalized size = 47.42 \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 {{\left (a+b\,\sin \left (e+f\,x\right )\right )}^2}{c+d\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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