Optimal. Leaf size=211 \[ \frac {3 d \sqrt {c+d x}}{16 b^2}+\frac {(c+d x)^{5/2}}{5 d}-\frac {3 d \sqrt {c+d x} \cosh ^2(a+b x)}{8 b^2}+\frac {3 d^{3/2} e^{-2 a+\frac {2 b c}{d}} \sqrt {\frac {\pi }{2}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{64 b^{5/2}}+\frac {3 d^{3/2} e^{2 a-\frac {2 b c}{d}} \sqrt {\frac {\pi }{2}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{64 b^{5/2}}+\frac {(c+d x)^{3/2} \cosh (a+b x) \sinh (a+b x)}{2 b} \]
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Rubi [A]
time = 0.23, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {3392, 32, 3393,
3388, 2211, 2235, 2236} \begin {gather*} \frac {3 \sqrt {\frac {\pi }{2}} d^{3/2} e^{\frac {2 b c}{d}-2 a} \text {Erf}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{64 b^{5/2}}+\frac {3 \sqrt {\frac {\pi }{2}} d^{3/2} e^{2 a-\frac {2 b c}{d}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{64 b^{5/2}}-\frac {3 d \sqrt {c+d x} \cosh ^2(a+b x)}{8 b^2}+\frac {(c+d x)^{3/2} \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {3 d \sqrt {c+d x}}{16 b^2}+\frac {(c+d x)^{5/2}}{5 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 3392
Rule 3393
Rubi steps
\begin {align*} \int (c+d x)^{3/2} \cosh ^2(a+b x) \, dx &=-\frac {3 d \sqrt {c+d x} \cosh ^2(a+b x)}{8 b^2}+\frac {(c+d x)^{3/2} \cosh (a+b x) \sinh (a+b x)}{2 b}+\frac {1}{2} \int (c+d x)^{3/2} \, dx+\frac {\left (3 d^2\right ) \int \frac {\cosh ^2(a+b x)}{\sqrt {c+d x}} \, dx}{16 b^2}\\ &=\frac {(c+d x)^{5/2}}{5 d}-\frac {3 d \sqrt {c+d x} \cosh ^2(a+b x)}{8 b^2}+\frac {(c+d x)^{3/2} \cosh (a+b x) \sinh (a+b x)}{2 b}+\frac {\left (3 d^2\right ) \int \left (\frac {1}{2 \sqrt {c+d x}}+\frac {\cosh (2 a+2 b x)}{2 \sqrt {c+d x}}\right ) \, dx}{16 b^2}\\ &=\frac {3 d \sqrt {c+d x}}{16 b^2}+\frac {(c+d x)^{5/2}}{5 d}-\frac {3 d \sqrt {c+d x} \cosh ^2(a+b x)}{8 b^2}+\frac {(c+d x)^{3/2} \cosh (a+b x) \sinh (a+b x)}{2 b}+\frac {\left (3 d^2\right ) \int \frac {\cosh (2 a+2 b x)}{\sqrt {c+d x}} \, dx}{32 b^2}\\ &=\frac {3 d \sqrt {c+d x}}{16 b^2}+\frac {(c+d x)^{5/2}}{5 d}-\frac {3 d \sqrt {c+d x} \cosh ^2(a+b x)}{8 b^2}+\frac {(c+d x)^{3/2} \cosh (a+b x) \sinh (a+b x)}{2 b}+\frac {\left (3 d^2\right ) \int \frac {e^{-i (2 i a+2 i b x)}}{\sqrt {c+d x}} \, dx}{64 b^2}+\frac {\left (3 d^2\right ) \int \frac {e^{i (2 i a+2 i b x)}}{\sqrt {c+d x}} \, dx}{64 b^2}\\ &=\frac {3 d \sqrt {c+d x}}{16 b^2}+\frac {(c+d x)^{5/2}}{5 d}-\frac {3 d \sqrt {c+d x} \cosh ^2(a+b x)}{8 b^2}+\frac {(c+d x)^{3/2} \cosh (a+b x) \sinh (a+b x)}{2 b}+\frac {(3 d) \text {Subst}\left (\int e^{i \left (2 i a-\frac {2 i b c}{d}\right )-\frac {2 b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{32 b^2}+\frac {(3 d) \text {Subst}\left (\int e^{-i \left (2 i a-\frac {2 i b c}{d}\right )+\frac {2 b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{32 b^2}\\ &=\frac {3 d \sqrt {c+d x}}{16 b^2}+\frac {(c+d x)^{5/2}}{5 d}-\frac {3 d \sqrt {c+d x} \cosh ^2(a+b x)}{8 b^2}+\frac {3 d^{3/2} e^{-2 a+\frac {2 b c}{d}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{64 b^{5/2}}+\frac {3 d^{3/2} e^{2 a-\frac {2 b c}{d}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{64 b^{5/2}}+\frac {(c+d x)^{3/2} \cosh (a+b x) \sinh (a+b x)}{2 b}\\ \end {align*}
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Mathematica [A]
time = 0.54, size = 163, normalized size = 0.77 \begin {gather*} \frac {32 b^3 (c+d x)^3+5 \sqrt {2} d^3 \sqrt {\frac {b (c+d x)}{d}} \Gamma \left (\frac {5}{2},\frac {2 b (c+d x)}{d}\right ) \left (-\cosh \left (2 a-\frac {2 b c}{d}\right )+\sinh \left (2 a-\frac {2 b c}{d}\right )\right )+5 \sqrt {2} d^3 \sqrt {-\frac {b (c+d x)}{d}} \Gamma \left (\frac {5}{2},-\frac {2 b (c+d x)}{d}\right ) \left (\cosh \left (2 a-\frac {2 b c}{d}\right )+\sinh \left (2 a-\frac {2 b c}{d}\right )\right )}{160 b^3 d \sqrt {c+d x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (d x +c \right )^{\frac {3}{2}} \left (\cosh ^{2}\left (b x +a \right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 239, normalized size = 1.13 \begin {gather*} \frac {128 \, {\left (d x + c\right )}^{\frac {5}{2}} + \frac {15 \, \sqrt {2} \sqrt {\pi } d^{2} \operatorname {erf}\left (\sqrt {2} \sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) e^{\left (2 \, a - \frac {2 \, b c}{d}\right )}}{b^{2} \sqrt {-\frac {b}{d}}} + \frac {15 \, \sqrt {2} \sqrt {\pi } d^{2} \operatorname {erf}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) e^{\left (-2 \, a + \frac {2 \, b c}{d}\right )}}{b^{2} \sqrt {\frac {b}{d}}} - \frac {20 \, {\left (4 \, {\left (d x + c\right )}^{\frac {3}{2}} b d e^{\left (\frac {2 \, b c}{d}\right )} + 3 \, \sqrt {d x + c} d^{2} e^{\left (\frac {2 \, b c}{d}\right )}\right )} e^{\left (-2 \, a - \frac {2 \, {\left (d x + c\right )} b}{d}\right )}}{b^{2}} + \frac {20 \, {\left (4 \, {\left (d x + c\right )}^{\frac {3}{2}} b d e^{\left (2 \, a\right )} - 3 \, \sqrt {d x + c} d^{2} e^{\left (2 \, a\right )}\right )} e^{\left (\frac {2 \, {\left (d x + c\right )} b}{d} - \frac {2 \, b c}{d}\right )}}{b^{2}}}{640 \, 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. 755 vs.
\(2 (159) = 318\).
time = 0.44, size = 755, normalized size = 3.58 \begin {gather*} \frac {15 \, \sqrt {2} \sqrt {\pi } {\left (d^{3} \cosh \left (b x + a\right )^{2} \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - d^{3} \cosh \left (b x + a\right )^{2} \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + {\left (d^{3} \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - d^{3} \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \sinh \left (b x + a\right )^{2} + 2 \, {\left (d^{3} \cosh \left (b x + a\right ) \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - d^{3} \cosh \left (b x + a\right ) \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \sinh \left (b x + a\right )\right )} \sqrt {\frac {b}{d}} \operatorname {erf}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) - 15 \, \sqrt {2} \sqrt {\pi } {\left (d^{3} \cosh \left (b x + a\right )^{2} \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + d^{3} \cosh \left (b x + a\right )^{2} \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + {\left (d^{3} \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + d^{3} \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \sinh \left (b x + a\right )^{2} + 2 \, {\left (d^{3} \cosh \left (b x + a\right ) \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + d^{3} \cosh \left (b x + a\right ) \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \sinh \left (b x + a\right )\right )} \sqrt {-\frac {b}{d}} \operatorname {erf}\left (\sqrt {2} \sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) - 4 \, {\left (20 \, b^{2} d^{2} x - 5 \, {\left (4 \, b^{2} d^{2} x + 4 \, b^{2} c d - 3 \, b d^{2}\right )} \cosh \left (b x + a\right )^{4} - 20 \, {\left (4 \, b^{2} d^{2} x + 4 \, b^{2} c d - 3 \, b d^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} - 5 \, {\left (4 \, b^{2} d^{2} x + 4 \, b^{2} c d - 3 \, b d^{2}\right )} \sinh \left (b x + a\right )^{4} + 20 \, b^{2} c d + 15 \, b d^{2} - 32 \, {\left (b^{3} d^{2} x^{2} + 2 \, b^{3} c d x + b^{3} c^{2}\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (16 \, b^{3} d^{2} x^{2} + 32 \, b^{3} c d x + 16 \, b^{3} c^{2} + 15 \, {\left (4 \, b^{2} d^{2} x + 4 \, b^{2} c d - 3 \, b d^{2}\right )} \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )^{2} - 4 \, {\left (5 \, {\left (4 \, b^{2} d^{2} x + 4 \, b^{2} c d - 3 \, b d^{2}\right )} \cosh \left (b x + a\right )^{3} + 16 \, {\left (b^{3} d^{2} x^{2} + 2 \, b^{3} c d x + b^{3} c^{2}\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\right )} \sqrt {d x + c}}{640 \, {\left (b^{3} d \cosh \left (b x + a\right )^{2} + 2 \, b^{3} d \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{3} d \sinh \left (b x + a\right )^{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 \left (c + d x\right )^{\frac {3}{2}} \cosh ^{2}{\left (a + b x \right )}\, 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 = -0.00 \begin {gather*} \int {\mathrm {cosh}\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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