Optimal. Leaf size=170 \[ -\frac {2 \cos ^2(a+b x)}{3 d (c+d x)^{3/2}}-\frac {8 b^{3/2} \sqrt {\pi } \cos \left (2 a-\frac {2 b c}{d}\right ) \text {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{3 d^{5/2}}+\frac {8 b^{3/2} \sqrt {\pi } S\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{3 d^{5/2}}+\frac {8 b \cos (a+b x) \sin (a+b x)}{3 d^2 \sqrt {c+d x}} \]
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Rubi [A]
time = 0.21, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3395, 32, 3393,
3387, 3386, 3432, 3385, 3433} \begin {gather*} -\frac {8 \sqrt {\pi } b^{3/2} \cos \left (2 a-\frac {2 b c}{d}\right ) \text {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {\pi } \sqrt {d}}\right )}{3 d^{5/2}}+\frac {8 \sqrt {\pi } b^{3/2} \sin \left (2 a-\frac {2 b c}{d}\right ) S\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{3 d^{5/2}}+\frac {8 b \sin (a+b x) \cos (a+b x)}{3 d^2 \sqrt {c+d x}}-\frac {2 \cos ^2(a+b x)}{3 d (c+d x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 3385
Rule 3386
Rule 3387
Rule 3393
Rule 3395
Rule 3432
Rule 3433
Rubi steps
\begin {align*} \int \frac {\cos ^2(a+b x)}{(c+d x)^{5/2}} \, dx &=-\frac {2 \cos ^2(a+b x)}{3 d (c+d x)^{3/2}}+\frac {8 b \cos (a+b x) \sin (a+b x)}{3 d^2 \sqrt {c+d x}}+\frac {\left (8 b^2\right ) \int \frac {1}{\sqrt {c+d x}} \, dx}{3 d^2}-\frac {\left (16 b^2\right ) \int \frac {\cos ^2(a+b x)}{\sqrt {c+d x}} \, dx}{3 d^2}\\ &=\frac {16 b^2 \sqrt {c+d x}}{3 d^3}-\frac {2 \cos ^2(a+b x)}{3 d (c+d x)^{3/2}}+\frac {8 b \cos (a+b x) \sin (a+b x)}{3 d^2 \sqrt {c+d x}}-\frac {\left (16 b^2\right ) \int \left (\frac {1}{2 \sqrt {c+d x}}+\frac {\cos (2 a+2 b x)}{2 \sqrt {c+d x}}\right ) \, dx}{3 d^2}\\ &=-\frac {2 \cos ^2(a+b x)}{3 d (c+d x)^{3/2}}+\frac {8 b \cos (a+b x) \sin (a+b x)}{3 d^2 \sqrt {c+d x}}-\frac {\left (8 b^2\right ) \int \frac {\cos (2 a+2 b x)}{\sqrt {c+d x}} \, dx}{3 d^2}\\ &=-\frac {2 \cos ^2(a+b x)}{3 d (c+d x)^{3/2}}+\frac {8 b \cos (a+b x) \sin (a+b x)}{3 d^2 \sqrt {c+d x}}-\frac {\left (8 b^2 \cos \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {2 b c}{d}+2 b x\right )}{\sqrt {c+d x}} \, dx}{3 d^2}+\frac {\left (8 b^2 \sin \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {2 b c}{d}+2 b x\right )}{\sqrt {c+d x}} \, dx}{3 d^2}\\ &=-\frac {2 \cos ^2(a+b x)}{3 d (c+d x)^{3/2}}+\frac {8 b \cos (a+b x) \sin (a+b x)}{3 d^2 \sqrt {c+d x}}-\frac {\left (16 b^2 \cos \left (2 a-\frac {2 b c}{d}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {2 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{3 d^3}+\frac {\left (16 b^2 \sin \left (2 a-\frac {2 b c}{d}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {2 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{3 d^3}\\ &=-\frac {2 \cos ^2(a+b x)}{3 d (c+d x)^{3/2}}-\frac {8 b^{3/2} \sqrt {\pi } \cos \left (2 a-\frac {2 b c}{d}\right ) C\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{3 d^{5/2}}+\frac {8 b^{3/2} \sqrt {\pi } S\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{3 d^{5/2}}+\frac {8 b \cos (a+b x) \sin (a+b x)}{3 d^2 \sqrt {c+d x}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.93, size = 181, normalized size = 1.06 \begin {gather*} \frac {e^{-\frac {2 i (b c+a d)}{d}} \left (-2 \sqrt {2} d e^{4 i a} \left (-\frac {i b (c+d x)}{d}\right )^{3/2} \text {Gamma}\left (\frac {1}{2},-\frac {2 i b (c+d x)}{d}\right )-2 \sqrt {2} d e^{\frac {4 i b c}{d}} \left (\frac {i b (c+d x)}{d}\right )^{3/2} \text {Gamma}\left (\frac {1}{2},\frac {2 i b (c+d x)}{d}\right )+2 e^{\frac {2 i (b c+a d)}{d}} \left (-d \cos ^2(a+b x)+2 b (c+d x) \sin (2 (a+b x))\right )\right )}{3 d^2 (c+d x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 189, normalized size = 1.11
method | result | size |
derivativedivides | \(\frac {-\frac {1}{3 \left (d x +c \right )^{\frac {3}{2}}}-\frac {\cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 d a -2 b c}{d}\right )}{3 \left (d x +c \right )^{\frac {3}{2}}}-\frac {4 b \left (-\frac {\sin \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 d a -2 b c}{d}\right )}{\sqrt {d x +c}}+\frac {2 b \sqrt {\pi }\, \left (\cos \left (\frac {2 d a -2 b c}{d}\right ) \FresnelC \left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {2 d a -2 b c}{d}\right ) \mathrm {S}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}\right )}{3 d}}{d}\) | \(189\) |
default | \(\frac {-\frac {1}{3 \left (d x +c \right )^{\frac {3}{2}}}-\frac {\cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 d a -2 b c}{d}\right )}{3 \left (d x +c \right )^{\frac {3}{2}}}-\frac {4 b \left (-\frac {\sin \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 d a -2 b c}{d}\right )}{\sqrt {d x +c}}+\frac {2 b \sqrt {\pi }\, \left (\cos \left (\frac {2 d a -2 b c}{d}\right ) \FresnelC \left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {2 d a -2 b c}{d}\right ) \mathrm {S}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}\right )}{3 d}}{d}\) | \(189\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.64, size = 136, normalized size = 0.80 \begin {gather*} \frac {3 \, \sqrt {2} {\left ({\left (-\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, \frac {2 i \, {\left (d x + c\right )} b}{d}\right ) + \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -\frac {2 i \, {\left (d x + c\right )} b}{d}\right )\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + {\left (-\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, \frac {2 i \, {\left (d x + c\right )} b}{d}\right ) + \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -\frac {2 i \, {\left (d x + c\right )} b}{d}\right )\right )} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \left (\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {3}{2}} - 4}{12 \, {\left (d x + c\right )}^{\frac {3}{2}} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 206, normalized size = 1.21 \begin {gather*} -\frac {2 \, {\left (4 \, {\left (\pi b d^{2} x^{2} + 2 \, \pi b c d x + \pi b c^{2}\right )} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {C}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 4 \, {\left (\pi b d^{2} x^{2} + 2 \, \pi b c d x + \pi b c^{2}\right )} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + {\left (d \cos \left (b x + a\right )^{2} - 4 \, {\left (b d x + b c\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right )\right )} \sqrt {d x + c}\right )}}{3 \, {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}} \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 {\cos ^{2}{\left (a + b x \right )}}{\left (c + d x\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\cos \left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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