Optimal. Leaf size=247 \[ \frac{128 \sqrt{\pi } b^{7/2} \cos \left (2 a-\frac{2 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{\pi } \sqrt{d}}\right )}{105 d^{9/2}}-\frac{128 \sqrt{\pi } b^{7/2} \sin \left (2 a-\frac{2 b c}{d}\right ) S\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right )}{105 d^{9/2}}+\frac{32 b^2 \cos ^2(a+b x)}{105 d^3 (c+d x)^{3/2}}-\frac{128 b^3 \sin (a+b x) \cos (a+b x)}{105 d^4 \sqrt{c+d x}}+\frac{8 b \sin (a+b x) \cos (a+b x)}{35 d^2 (c+d x)^{5/2}}-\frac{2 \cos ^2(a+b x)}{7 d (c+d x)^{7/2}}-\frac{16 b^2}{105 d^3 (c+d x)^{3/2}} \]
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Rubi [A] time = 0.407135, antiderivative size = 247, normalized size of antiderivative = 1., 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 = {3314, 32, 3312, 3306, 3305, 3351, 3304, 3352} \[ \frac{128 \sqrt{\pi } b^{7/2} \cos \left (2 a-\frac{2 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{\pi } \sqrt{d}}\right )}{105 d^{9/2}}-\frac{128 \sqrt{\pi } b^{7/2} \sin \left (2 a-\frac{2 b c}{d}\right ) S\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right )}{105 d^{9/2}}+\frac{32 b^2 \cos ^2(a+b x)}{105 d^3 (c+d x)^{3/2}}-\frac{128 b^3 \sin (a+b x) \cos (a+b x)}{105 d^4 \sqrt{c+d x}}+\frac{8 b \sin (a+b x) \cos (a+b x)}{35 d^2 (c+d x)^{5/2}}-\frac{2 \cos ^2(a+b x)}{7 d (c+d x)^{7/2}}-\frac{16 b^2}{105 d^3 (c+d x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3314
Rule 32
Rule 3312
Rule 3306
Rule 3305
Rule 3351
Rule 3304
Rule 3352
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 ) \operatorname{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 ) \operatorname{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*}
Mathematica [C] time = 0.857093, size = 237, normalized size = 0.96 \[ \frac{2 \left (16 \sqrt{2} b^2 d (c+d x)^2 e^{2 i \left (a-\frac{b c}{d}\right )} \left (-\frac{i b (c+d x)}{d}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-\frac{2 i b (c+d x)}{d}\right )+16 \sqrt{2} b^2 d (c+d x)^2 e^{-2 i \left (a-\frac{b c}{d}\right )} \left (\frac{i b (c+d x)}{d}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},\frac{2 i b (c+d x)}{d}\right )-32 b^3 (c+d x)^3 \sin (2 (a+b x))+16 b^2 d (c+d x)^2 \cos ^2(a+b x)+6 b d^2 (c+d x) \sin (2 (a+b x))-15 d^3 \cos ^2(a+b x)-8 b^2 d (c+d x)^2\right )}{105 d^4 (c+d x)^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 273, normalized size = 1.1 \begin{align*} 2\,{\frac{1}{d} \left ( -1/14\, \left ( dx+c \right ) ^{-7/2}-1/14\,{\frac{1}{ \left ( dx+c \right ) ^{7/2}}\cos \left ( 2\,{\frac{ \left ( dx+c \right ) b}{d}}+2\,{\frac{da-cb}{d}} \right ) }-2/7\,{\frac{b}{d} \left ( -1/5\,{\frac{1}{ \left ( dx+c \right ) ^{5/2}}\sin \left ( 2\,{\frac{ \left ( dx+c \right ) b}{d}}+2\,{\frac{da-cb}{d}} \right ) }+4/5\,{\frac{b}{d} \left ( -1/3\,{\frac{1}{ \left ( dx+c \right ) ^{3/2}}\cos \left ( 2\,{\frac{ \left ( dx+c \right ) b}{d}}+2\,{\frac{da-cb}{d}} \right ) }-4/3\,{\frac{b}{d} \left ( -{\frac{1}{\sqrt{dx+c}}\sin \left ( 2\,{\frac{ \left ( dx+c \right ) b}{d}}+2\,{\frac{da-cb}{d}} \right ) }+2\,{\frac{b\sqrt{\pi }}{d} \left ( \cos \left ( 2\,{\frac{da-cb}{d}} \right ){\it FresnelC} \left ( 2\,{\frac{b\sqrt{dx+c}}{\sqrt{\pi }d}{\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) -\sin \left ( 2\,{\frac{da-cb}{d}} \right ){\it FresnelS} \left ( 2\,{\frac{b\sqrt{dx+c}}{\sqrt{\pi }d}{\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) \right ){\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) } \right ) } \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.71768, size = 644, normalized size = 2.61 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.23948, size = 927, normalized size = 3.75 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (b x + a\right )^{2}}{{\left (d x + c\right )}^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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