Optimal. Leaf size=247 \[ -\frac {16 b^2}{105 d^3 (c+d x)^{3/2}}-\frac {2 \cos ^2(a+b x)}{7 d (c+d x)^{7/2}}+\frac {32 b^2 \cos ^2(a+b x)}{105 d^3 (c+d x)^{3/2}}+\frac {128 b^{7/2} \sqrt {\pi } \cos \left (2 a-\frac {2 b c}{d}\right ) \text {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{105 d^{9/2}}-\frac {128 b^{7/2} \sqrt {\pi } S\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{105 d^{9/2}}+\frac {8 b \cos (a+b x) \sin (a+b x)}{35 d^2 (c+d x)^{5/2}}-\frac {128 b^3 \cos (a+b x) \sin (a+b x)}{105 d^4 \sqrt {c+d x}} \]
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Rubi [A]
time = 0.27, antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps
used = 11, 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 {128 \sqrt {\pi } b^{7/2} \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 )}{105 d^{9/2}}-\frac {128 \sqrt {\pi } b^{7/2} \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 )}{105 d^{9/2}}-\frac {128 b^3 \sin (a+b x) \cos (a+b x)}{105 d^4 \sqrt {c+d x}}+\frac {32 b^2 \cos ^2(a+b x)}{105 d^3 (c+d x)^{3/2}}+\frac {8 b \sin (a+b x) \cos (a+b x)}{35 d^2 (c+d x)^{5/2}}-\frac {2 \cos ^2(a+b x)}{7 d (c+d x)^{7/2}}-\frac {16 b^2}{105 d^3 (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)^{9/2}} \, dx &=-\frac {2 \cos ^2(a+b x)}{7 d (c+d x)^{7/2}}+\frac {8 b \cos (a+b x) \sin (a+b x)}{35 d^2 (c+d x)^{5/2}}+\frac {\left (8 b^2\right ) \int \frac {1}{(c+d x)^{5/2}} \, dx}{35 d^2}-\frac {\left (16 b^2\right ) \int \frac {\cos ^2(a+b x)}{(c+d x)^{5/2}} \, dx}{35 d^2}\\ &=-\frac {16 b^2}{105 d^3 (c+d x)^{3/2}}-\frac {2 \cos ^2(a+b x)}{7 d (c+d x)^{7/2}}+\frac {32 b^2 \cos ^2(a+b x)}{105 d^3 (c+d x)^{3/2}}+\frac {8 b \cos (a+b x) \sin (a+b x)}{35 d^2 (c+d x)^{5/2}}-\frac {128 b^3 \cos (a+b x) \sin (a+b x)}{105 d^4 \sqrt {c+d x}}-\frac {\left (128 b^4\right ) \int \frac {1}{\sqrt {c+d x}} \, dx}{105 d^4}+\frac {\left (256 b^4\right ) \int \frac {\cos ^2(a+b x)}{\sqrt {c+d x}} \, dx}{105 d^4}\\ &=-\frac {16 b^2}{105 d^3 (c+d x)^{3/2}}-\frac {256 b^4 \sqrt {c+d x}}{105 d^5}-\frac {2 \cos ^2(a+b x)}{7 d (c+d x)^{7/2}}+\frac {32 b^2 \cos ^2(a+b x)}{105 d^3 (c+d x)^{3/2}}+\frac {8 b \cos (a+b x) \sin (a+b x)}{35 d^2 (c+d x)^{5/2}}-\frac {128 b^3 \cos (a+b x) \sin (a+b x)}{105 d^4 \sqrt {c+d x}}+\frac {\left (256 b^4\right ) \int \left (\frac {1}{2 \sqrt {c+d x}}+\frac {\cos (2 a+2 b x)}{2 \sqrt {c+d x}}\right ) \, dx}{105 d^4}\\ &=-\frac {16 b^2}{105 d^3 (c+d x)^{3/2}}-\frac {2 \cos ^2(a+b x)}{7 d (c+d x)^{7/2}}+\frac {32 b^2 \cos ^2(a+b x)}{105 d^3 (c+d x)^{3/2}}+\frac {8 b \cos (a+b x) \sin (a+b x)}{35 d^2 (c+d x)^{5/2}}-\frac {128 b^3 \cos (a+b x) \sin (a+b x)}{105 d^4 \sqrt {c+d x}}+\frac {\left (128 b^4\right ) \int \frac {\cos (2 a+2 b x)}{\sqrt {c+d x}} \, dx}{105 d^4}\\ &=-\frac {16 b^2}{105 d^3 (c+d x)^{3/2}}-\frac {2 \cos ^2(a+b x)}{7 d (c+d x)^{7/2}}+\frac {32 b^2 \cos ^2(a+b x)}{105 d^3 (c+d x)^{3/2}}+\frac {8 b \cos (a+b x) \sin (a+b x)}{35 d^2 (c+d x)^{5/2}}-\frac {128 b^3 \cos (a+b x) \sin (a+b x)}{105 d^4 \sqrt {c+d x}}+\frac {\left (128 b^4 \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}{105 d^4}-\frac {\left (128 b^4 \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}{105 d^4}\\ &=-\frac {16 b^2}{105 d^3 (c+d x)^{3/2}}-\frac {2 \cos ^2(a+b x)}{7 d (c+d x)^{7/2}}+\frac {32 b^2 \cos ^2(a+b x)}{105 d^3 (c+d x)^{3/2}}+\frac {8 b \cos (a+b x) \sin (a+b x)}{35 d^2 (c+d x)^{5/2}}-\frac {128 b^3 \cos (a+b x) \sin (a+b x)}{105 d^4 \sqrt {c+d x}}+\frac {\left (256 b^4 \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 )}{105 d^5}-\frac {\left (256 b^4 \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 )}{105 d^5}\\ &=-\frac {16 b^2}{105 d^3 (c+d x)^{3/2}}-\frac {2 \cos ^2(a+b x)}{7 d (c+d x)^{7/2}}+\frac {32 b^2 \cos ^2(a+b x)}{105 d^3 (c+d x)^{3/2}}+\frac {128 b^{7/2} \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 )}{105 d^{9/2}}-\frac {128 b^{7/2} \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 )}{105 d^{9/2}}+\frac {8 b \cos (a+b x) \sin (a+b x)}{35 d^2 (c+d x)^{5/2}}-\frac {128 b^3 \cos (a+b x) \sin (a+b x)}{105 d^4 \sqrt {c+d x}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.54, size = 237, normalized size = 0.96 \begin {gather*} \frac {2 \left (-8 b^2 d (c+d x)^2-15 d^3 \cos ^2(a+b x)+16 b^2 d (c+d x)^2 \cos ^2(a+b x)+16 \sqrt {2} b^2 d e^{2 i \left (a-\frac {b c}{d}\right )} (c+d x)^2 \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 )+16 \sqrt {2} b^2 d e^{-2 i \left (a-\frac {b c}{d}\right )} (c+d x)^2 \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 )+6 b d^2 (c+d x) \sin (2 (a+b x))-32 b^3 (c+d x)^3 \sin (2 (a+b x))\right )}{105 d^4 (c+d x)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 273, normalized size = 1.11
method | result | size |
derivativedivides | \(\frac {-\frac {1}{7 \left (d x +c \right )^{\frac {7}{2}}}-\frac {\cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 d a -2 b c}{d}\right )}{7 \left (d x +c \right )^{\frac {7}{2}}}-\frac {4 b \left (-\frac {\sin \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 d a -2 b c}{d}\right )}{5 \left (d x +c \right )^{\frac {5}{2}}}+\frac {4 b \left (-\frac {\cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 d a -2 b c}{d}\right )}{3 \left (d x +c \right )^{\frac {3}{2}}}-\frac {4 b \left (-\frac {\sin \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 d a -2 b c}{d}\right )}{\sqrt {d x +c}}+\frac {2 b \sqrt {\pi }\, \left (\cos \left (\frac {2 d a -2 b c}{d}\right ) \FresnelC \left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {2 d a -2 b c}{d}\right ) \mathrm {S}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}\right )}{3 d}\right )}{5 d}\right )}{7 d}}{d}\) | \(273\) |
default | \(\frac {-\frac {1}{7 \left (d x +c \right )^{\frac {7}{2}}}-\frac {\cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 d a -2 b c}{d}\right )}{7 \left (d x +c \right )^{\frac {7}{2}}}-\frac {4 b \left (-\frac {\sin \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 d a -2 b c}{d}\right )}{5 \left (d x +c \right )^{\frac {5}{2}}}+\frac {4 b \left (-\frac {\cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 d a -2 b c}{d}\right )}{3 \left (d x +c \right )^{\frac {3}{2}}}-\frac {4 b \left (-\frac {\sin \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 d a -2 b c}{d}\right )}{\sqrt {d x +c}}+\frac {2 b \sqrt {\pi }\, \left (\cos \left (\frac {2 d a -2 b c}{d}\right ) \FresnelC \left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {2 d a -2 b c}{d}\right ) \mathrm {S}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}\right )}{3 d}\right )}{5 d}\right )}{7 d}}{d}\) | \(273\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.66, size = 136, normalized size = 0.55 \begin {gather*} -\frac {7 \, \sqrt {2} {\left ({\left (-\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {7}{2}, \frac {2 i \, {\left (d x + c\right )} b}{d}\right ) + \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {7}{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 {7}{2}, \frac {2 i \, {\left (d x + c\right )} b}{d}\right ) + \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {7}{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 {7}{2}} + 1}{7 \, {\left (d x + c\right )}^{\frac {7}{2}} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 417 vs.
\(2 (195) = 390\).
time = 0.45, size = 417, normalized size = 1.69 \begin {gather*} \frac {2 \, {\left (64 \, {\left (\pi b^{3} d^{4} x^{4} + 4 \, \pi b^{3} c d^{3} x^{3} + 6 \, \pi b^{3} c^{2} d^{2} x^{2} + 4 \, \pi b^{3} c^{3} d x + \pi b^{3} c^{4}\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 ) - 64 \, {\left (\pi b^{3} d^{4} x^{4} + 4 \, \pi b^{3} c d^{3} x^{3} + 6 \, \pi b^{3} c^{2} d^{2} x^{2} + 4 \, \pi b^{3} c^{3} d x + \pi b^{3} c^{4}\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 (8 \, b^{2} d^{3} x^{2} + 16 \, b^{2} c d^{2} x + 8 \, b^{2} c^{2} d - {\left (16 \, b^{2} d^{3} x^{2} + 32 \, b^{2} c d^{2} x + 16 \, b^{2} c^{2} d - 15 \, d^{3}\right )} \cos \left (b x + a\right )^{2} + 4 \, {\left (16 \, b^{3} d^{3} x^{3} + 48 \, b^{3} c d^{2} x^{2} + 16 \, b^{3} c^{3} - 3 \, b c d^{2} + 3 \, {\left (16 \, b^{3} c^{2} d - b d^{3}\right )} x\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right )\right )} \sqrt {d x + c}\right )}}{105 \, {\left (d^{8} x^{4} + 4 \, c d^{7} x^{3} + 6 \, c^{2} d^{6} x^{2} + 4 \, c^{3} d^{5} x + c^{4} d^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 = -0.00 \begin {gather*} \int \frac {{\cos \left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^{9/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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