Optimal. Leaf size=117 \[ -\frac {c^3 \cos \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a}{b}+\csc ^{-1}(c x)\right )}{4 b}+\frac {c^3 \cos \left (\frac {3 a}{b}\right ) \text {CosIntegral}\left (\frac {3 a}{b}+3 \csc ^{-1}(c x)\right )}{4 b}-\frac {c^3 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\csc ^{-1}(c x)\right )}{4 b}+\frac {c^3 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \csc ^{-1}(c x)\right )}{4 b} \]
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Rubi [A]
time = 0.19, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5331, 4491,
3384, 3380, 3383} \begin {gather*} -\frac {c^3 \cos \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a}{b}+\csc ^{-1}(c x)\right )}{4 b}+\frac {c^3 \cos \left (\frac {3 a}{b}\right ) \text {CosIntegral}\left (\frac {3 a}{b}+3 \csc ^{-1}(c x)\right )}{4 b}-\frac {c^3 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\csc ^{-1}(c x)\right )}{4 b}+\frac {c^3 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \csc ^{-1}(c x)\right )}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 3380
Rule 3383
Rule 3384
Rule 4491
Rule 5331
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (a+b \csc ^{-1}(c x)\right )} \, dx &=-\left (c^3 \text {Subst}\left (\int \frac {\cos (x) \sin ^2(x)}{a+b x} \, dx,x,\csc ^{-1}(c x)\right )\right )\\ &=-\left (c^3 \text {Subst}\left (\int \left (\frac {\cos (x)}{4 (a+b x)}-\frac {\cos (3 x)}{4 (a+b x)}\right ) \, dx,x,\csc ^{-1}(c x)\right )\right )\\ &=-\left (\frac {1}{4} c^3 \text {Subst}\left (\int \frac {\cos (x)}{a+b x} \, dx,x,\csc ^{-1}(c x)\right )\right )+\frac {1}{4} c^3 \text {Subst}\left (\int \frac {\cos (3 x)}{a+b x} \, dx,x,\csc ^{-1}(c x)\right )\\ &=-\left (\frac {1}{4} \left (c^3 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\csc ^{-1}(c x)\right )\right )+\frac {1}{4} \left (c^3 \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\csc ^{-1}(c x)\right )-\frac {1}{4} \left (c^3 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\csc ^{-1}(c x)\right )+\frac {1}{4} \left (c^3 \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\csc ^{-1}(c x)\right )\\ &=-\frac {c^3 \cos \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a}{b}+\csc ^{-1}(c x)\right )}{4 b}+\frac {c^3 \cos \left (\frac {3 a}{b}\right ) \text {Ci}\left (\frac {3 a}{b}+3 \csc ^{-1}(c x)\right )}{4 b}-\frac {c^3 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\csc ^{-1}(c x)\right )}{4 b}+\frac {c^3 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \csc ^{-1}(c x)\right )}{4 b}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 91, normalized size = 0.78 \begin {gather*} -\frac {c^3 \left (\cos \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a}{b}+\csc ^{-1}(c x)\right )-\cos \left (\frac {3 a}{b}\right ) \text {CosIntegral}\left (3 \left (\frac {a}{b}+\csc ^{-1}(c x)\right )\right )+\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\csc ^{-1}(c x)\right )-\sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\csc ^{-1}(c x)\right )\right )\right )}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.21, size = 102, normalized size = 0.87
method | result | size |
derivativedivides | \(c^{3} \left (-\frac {\sinIntegral \left (\frac {a}{b}+\mathrm {arccsc}\left (c x \right )\right ) \sin \left (\frac {a}{b}\right )}{4 b}-\frac {\cosineIntegral \left (\frac {a}{b}+\mathrm {arccsc}\left (c x \right )\right ) \cos \left (\frac {a}{b}\right )}{4 b}+\frac {\sinIntegral \left (\frac {3 a}{b}+3 \,\mathrm {arccsc}\left (c x \right )\right ) \sin \left (\frac {3 a}{b}\right )}{4 b}+\frac {\cosineIntegral \left (\frac {3 a}{b}+3 \,\mathrm {arccsc}\left (c x \right )\right ) \cos \left (\frac {3 a}{b}\right )}{4 b}\right )\) | \(102\) |
default | \(c^{3} \left (-\frac {\sinIntegral \left (\frac {a}{b}+\mathrm {arccsc}\left (c x \right )\right ) \sin \left (\frac {a}{b}\right )}{4 b}-\frac {\cosineIntegral \left (\frac {a}{b}+\mathrm {arccsc}\left (c x \right )\right ) \cos \left (\frac {a}{b}\right )}{4 b}+\frac {\sinIntegral \left (\frac {3 a}{b}+3 \,\mathrm {arccsc}\left (c x \right )\right ) \sin \left (\frac {3 a}{b}\right )}{4 b}+\frac {\cosineIntegral \left (\frac {3 a}{b}+3 \,\mathrm {arccsc}\left (c x \right )\right ) \cos \left (\frac {3 a}{b}\right )}{4 b}\right )\) | \(102\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {1}{x^{4} \left (a + b \operatorname {acsc}{\left (c x \right )}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 200, normalized size = 1.71 \begin {gather*} \frac {1}{4} \, {\left (\frac {4 \, c^{2} \cos \left (\frac {a}{b}\right )^{3} \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (\frac {1}{c x}\right )\right )}{b} + \frac {4 \, c^{2} \cos \left (\frac {a}{b}\right )^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (\frac {1}{c x}\right )\right )}{b} - \frac {3 \, c^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (\frac {1}{c x}\right )\right )}{b} - \frac {c^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (\frac {1}{c x}\right )\right )}{b} - \frac {c^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (\frac {1}{c x}\right )\right )}{b} - \frac {c^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (\frac {1}{c x}\right )\right )}{b}\right )} c \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 {1}{x^4\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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